Enhanced Damping Method for Suppressing Sub-Synchronous Oscillations of Grid-Forming Permanent Magnet Synchronous Generator

Abstract

With the increase in wind power penetration, the stable operation of wind turbines under the new power system is facing severe challenges. The grid-forming wind power technology operates in a self-synchronous mode, which can provide voltage and frequency support for the system without being affected by the phase-locked loop, and is also suitable for operation under weak power grids. However, the current research for the grid-forming (GFM) permanent magnet synchronous generator (PMSG) ignores the DC-link dynamics generated by the wind turbine, which makes the sub-synchronous oscillation (SSO) phenomenon under different grid conditions and lacks a physical explanation. In this paper, the SSO problem in the grid-forming PMSG is studied, and the study reveals that the reduction in the DC-link voltage control bandwidth of the machine-side converter (MSC) is the main cause. To this end, an improved damping method is proposed, which introduces a low-pass filter branch in the reactive power control loop and takes the DC-link voltage tracking error as a compensation term. The small-signal analysis and simulation results show that the proposed method has significant effectiveness.

1. Introduction

With the rapid development of wind power generation technology and the transformation of the global energy structure to low-carbon and clean, the proportion of wind power in the power system continues to rise [1]. However, higher wind penetration introduces intermittency and fluctuation challenges to grid frequency regulation and stability. To address these challenges, grid-forming wind power technology provides critical virtual inertia and voltage support, significantly enhancing weak grid stability [2,3].

Unlike traditional grid-following control in wind power systems [4,5], grid-forming control maintains stable operation during grid disturbances and weak-grid conditions by independently establishing voltage/frequency references, providing inertial support, and enhancing renewable energy’s active support capability [6,7]. Moreover, GFM wind power systems differ from standalone GFM converters [8,9], as wind turbines necessitate active DC-link voltage regulation—implemented via either the GSC or the MSC—to ensure operational stability. In grid-forming wind turbines, the prevailing control approach employs the GSC to regulate the DC-link voltage while simultaneously providing grid support. For example, the study in [10] demonstrates a configuration where the GSC not only maintains DC-link voltage but also accomplishes grid synchronization through virtual synchronous machine control. Concurrently, the MSC integrates a frequency control loop into its maximum power point tracking structure and utilizes feedforward control to mitigate DC-link voltage variations. Ref. [11] further pointed out that when the system is disturbed or operated under weak grid conditions, the control burden of GSC will increase sharply, and its voltage regulation rate and network construction performance will deteriorate.

In practice, requiring the GSC to simultaneously achieve both DC-link voltage control and grid support increases the coupling between control objectives and raises complexity. To address the challenges of the GSC handling both tasks, a subsequent strategy proposes assigning DC-link voltage control to the MSC, enabling the GSC to focus exclusively on grid-forming capabilities. For example, in [12], the MSC regulates the DC-link voltage via vector control. This involves generating reference currents based on outer loops for reactive power and DC-link voltage and producing modulation voltages through current control loops. Simultaneously, the GSC implements grid-forming control based on droop control to provide grid support. In [13], a small-signal model incorporating DC-link voltage dynamics is established. However, the analysis primarily focuses on the GFM performance on the AC-side without considering the influence of the DC-link voltage dynamics. Ref. [14] investigates the stability mechanism of shaft torsional vibration in a VSG-based direct-drive fan system by establishing a corresponding transfer function model. Nevertheless, the model complexity is high, and a quantitative evaluation of the shaft damping effect is not fully discussed. In [15], simulation and comparative analysis under normal operation, fault response, and black-start conditions demonstrate that utilizing the MSC for DC-link voltage control effectively decouples the system from the AC power grid, leading to improved control outcomes. In [16], a small-signal model integrating both DC-side and AC-side controllers is developed. However, the DC-side devices are modeled as constant current sources rather than controlled current sources, lacking a dynamic representation of the DC-link voltage.

Furthermore, in [17], the GSC achieves grid-forming control through virtual synchronous control, while the MSC obtains reference currents from the DC-link voltage outer loop and generates modulation voltages via current loops. This demonstrates that wind turbines employing GFM exhibit enhanced adaptability to weak grids. In [18,19], the MSC maintains DC-link voltage stability by converting the voltage control objective into torque and power control references and then employing either droop control or virtual synchronous control strategies. Although these studies improve the stability of DC-link voltage control, they fail to address the issue of SSO caused by DC-link voltage dynamics, nor do they provide effective suppression strategies. As shown in Ref. [20], variations in the inertia constant can readily cause the DC-link voltage to diverge at an oscillation frequency of 62.5 Hz. Similarly, Ref. [21] reports that changes in grid impedance may induce DC-link voltage oscillations around 55 Hz. These phenomena indicate that the DC-link voltage is susceptible to SSO behavior. To mitigate such issues, GFM control often employs DC-link voltage regulation strategies to enhance system stability. For example, Ref. [22] proposes a q-axis voltage feedforward-based damping method that avoids synchronous resonance but suffers from degraded stability under grid voltage fluctuations. Zhao et al. [23] introduce a parallel combination of power DC-link voltage control and active power feedforward to improve transient stability, albeit with a large number of parameters and high design complexity. A comparison in [24] further reveals that constant power loads significantly elevate instability risks under DC-link voltage synchronization control.

On the modeling side, Ref. [16] develops a dual-port admittance model using harmonic linearization to assess AC–DC converter coupling characteristics. In [25], a small-signal model is established that includes controllers and components on both AC and DC sides; however, the representation of DC-side components as constant current sources limits its ability to fully capture DC-link dynamics. Ref. [26] constructs a sequence impedance model that accounts for DC-side dynamics and AC–DC frequency coupling, yet it focuses mainly on converter output behavior and offers limited systematic insight into the intrinsic instability mechanisms of GFM converters.

To fill this research gap, this paper investigates the control strategy of the MSC for regulating the DC-link voltage, examines the dynamics of the DC-link voltage in grid-forming PMSG and its optimization, identifies SSO related to the DC-link voltage, and proposes a novel damping control method to address these oscillation issues.

The rest of this paper is organized as follows. In Section 2, a simplified model of PMSG is proposed, a small-signal model is developed, and the transfer function of DC-link voltage is established. Section 3 analyzes the mechanism of SSO problems and reveals the shortcomings of traditional damping strategies. In Section 4, a damping optimization strategy is proposed, and the changes in the corresponding parameters are theoretically analyzed through the bode diagram and the eigenvalue. In Section 5, the effectiveness of the damping strategy is verified by simulation, and the adaptability of different short circuit ratios (SCRs) and the C_dc is verified. Finally, the conclusions are drawn in Section 6.

2. Model of Grid-Forming PMSG

Figure 1 shows the Grid-forming PMSG. The DC-link capacitor is denoted as C_dc. The output current of the AC–DC rectifier is denoted as i_dc. The DC-link voltage is denoted as v_dc. The output current of the DC–AC inverter is denoted as i_gabc. The output voltage phasor and the grid voltage phasor are represented as V_δo and V_g0, where δ_o is the power angle. The grid impedance is denoted as Z_g, where Z_g = R_g + jX_g. The equivalent circuits of MSC and GSC are also depicted in Figure 1. The input and output power of the DC-link are denoted as p_dc and p_ac, respectively. The output active power and reactive power on the AC-side are denoted as p and q, respectively.

2.1. Model of PMSG

The PMSG is controlled in d–q rotating coordinates, aligning the d-axis with the magnetic flux linkage of the rotor ψ_f. The stator voltage is expressed as follows:

$$\left\{\begin{array}{l}{u}_{sd}=-R_s{i}_{sd}-{\displaystyle \frac{L_d}{{\omega }_{eb}}}{\displaystyle \frac{d{i}_{sd}}{dt}}+{\omega }_e{L}_q{i}_{sq}\\ u_{sq}=-R_s{i}_{sq}-{\displaystyle \frac{L_d}{{\omega }_{eb}}}{\displaystyle \frac{d{i}_{sd}}{dt}}-{\omega }_e{L}_d{i}_{sd}+{\omega }_e{\psi }_f\end{array}\right.$$

(1)

where u_sd and u_sq are the d- and q-axis stator terminal voltages, respectively; i_sd and i_sq are the d- and q-axis stator currents, respectively; R_s is the resistance of PMSG stator; ω_e is the base value of stator angular frequency; ω_e is the angular velocity of generator rotor; and L_d and L_q are the d- and q-axis self-inductances of PMSG stator, respectively.

The megawatt PMSGs have relatively low speeds and are mostly mounted with non-salient surfaces (L_d = L_q). Setting L_d = L_q is a common simplification for PMSG modeling, primarily aimed at reducing model complexity during theoretical analysis and controller design, thereby allowing better focus on the core DC-link voltage control issue. Therefore, the electromagnetic torque of PMSG T_e can be expressed as:

$$T_e=i_{sq}{\psi }_f$$

(2)

2.2. Controller Model

In order to simplify the analysis, the models of MSC and GSC are simplified accordingly, and the corresponding equivalent circuit models are shown in Figure 1. Therein, the MSC is equivalent to a controlled current source i_dc, which is regulated by the DC-link voltage control loop [27]. K_pdc and K_idc are proportional and integral coefficients. The GSC is represented by a voltage source, which is determined by the virtual synchronous generator control. K_f and K_V are droop coefficients, J is the inertial coefficient, and K is the integral coefficient. i_dc0, ω₀, and E₀ are nominal values. Small-signal expressions of these control loops are given as follows:

$$\left\{\begin{array}{l}{\widehat{i}}_{\mathrm{dc}}=G_{\mathrm{DVC}}\cdot \left({\widehat{V}}_{\mathrm{dcref}}-{\widehat{v}}_{\mathrm{dc}}\right),G_{\mathrm{DVC}}=\left(s{K}_{\mathrm{pd}}+K_{\mathrm{id}}\right)/s\\ \widehat{\omega }=G_{\mathrm{APC}}\cdot \left({\widehat{P}}_{\mathrm{ref}}-\widehat{p}\right),G_{\mathrm{APC}}=1/\left(Js+K_f\right)\\ \widehat{E}=G_{\mathrm{RPC}}\cdot \left({\widehat{Q}}_{\mathrm{ref}}-\widehat{q}\right),G_{\mathrm{RPC}}=1/\left(Ks+K_V\right)\end{array}\right.$$

(3)

where the symbol ‘^’ represents the small-signal variable, ‘s’ represents the Laplace operator.

2.3. LCL Filter and Grid-Connected Line Model

As shown in Figure 2a, the model of AC-side circuit dynamics can be built in different reference frames. The control frame dq^c is defined by the internal voltage E and the rotational angular frequency is denoted by ω. The grid frame dq^g is defined by the grid voltage and the rotational angular frequency is denoted by ω_g. δ is the angle difference between the dq^c and dq^g frame.

If the dq^c frame is used as the reference frame, there are v_sdq = [v_sd; v_sq] = [E; 0] and v_gdq = [v_gd; v_gq] = [V_g·cosδ; −V_g·sinδ]. If the dq^g frame is used as the reference frame, there are v_sdq = [v_sd; v_sq] = [E·cosδ; E·sinδ] and v_gdq = [v_gd; v_gq] = [V_g; 0]. They are equivalent after the coordinate transformation.

This article selects the first set of frames. Thus, the models of AC-side circuit dynamics are given by the following:

$$v_{sdq}-v_{dq}=Z_{Lf}{i}_{dq}$$

(4)

$$i_{dq}-i_{gdq}=Z_{Cf}{v}_{dq}$$

(5)

$$v_{dq}-v_{gdq}=Z_{Lg}{i}_{gdq}$$

(6)

Among them, v_sdq is the inverter output voltage, v_dq is the point of common coupling (PCC) voltage, and i_dq is the inverter output current. Z_Lf, Z_Cf, and Z_Lg are the following:

$$Z_{Lf}=\left[s{L}_{\mathrm{f}}+R_{\mathrm{f}},-\omega L_{\mathrm{f}};\omega L_{\mathrm{f}},s{L}_{\mathrm{f}}+R_{\mathrm{f}}\right]$$

(7)

$$Z_{Cf}=\left[s{C}_{\mathrm{f}},-\omega C_{\mathrm{f}};\omega C_{\mathrm{f}},s{C}_{\mathrm{f}}\right]$$

(8)

$$Z_{Lg}=\left[s{L}_{\mathrm{g}}+R_{\mathrm{g}},-\omega L_{\mathrm{g}};\omega L_{\mathrm{g}},s{L}_{\mathrm{g}}+R_{\mathrm{g}}\right]$$

(9)

By conducting the small-signal variations at the steady-state operating point (I_d0, I_q0, V_d0, V_q0, I_gd0, I_gq0), the small-signal model of (7)–(9) can be expressed as follows:

$${\widehat{v}}_{sdq}-{\widehat{v}}_{\mathrm{dc}}=Z_{Lf0}{\widehat{i}}_{dq}$$

(10)

$${\widehat{i}}_{dq}-{\widehat{i}}_{gdq}=Z_{Cf0}{\widehat{v}}_{dq}$$

(11)

$${\widehat{v}}_{dq}-{\widehat{v}}_{gdq}=Z_{Lg0}{\widehat{i}}_{gdq}$$

(12)

Among them, Z_Lf0, Z_Cf0 and Z_Lg0 are the following:

$$Z_{Lf0}=\left[s{L}_{\mathrm{f}}+R_{\mathrm{f}},-{\omega }_0{L}_{\mathrm{f}};{\omega }_0{L}_{\mathrm{f}},s{L}_{\mathrm{f}}+R_{\mathrm{f}}\right]$$

(13)

$$Z_{Cf0}=\left[s{C}_{\mathrm{f}},-{\omega }_0{C}_{\mathrm{f}};{\omega }_0{C}_{\mathrm{f}},s{C}_{\mathrm{f}}\right]$$

(14)

$$Z_{Lg0}=\left[s{L}_{\mathrm{g}}+R_{\mathrm{g}},-{\omega }_0{L}_{\mathrm{g}};{\omega }_0{L}_{\mathrm{g}},s{L}_{\mathrm{g}}+R_{\mathrm{g}}\right]$$

(15)

By conducting the small-signal variations at the steady-state operating point (δ₀, V_g0), v_gdq in (11) can be expressed as follows:

$${\widehat{v}}_{gdq}=[\cos {\delta }_0;\sin {\delta }_0]{\widehat{V}}_g+[-\sin {\delta }_0;\cos {\delta }_0]V_{g0}\widehat{\delta }$$

(16)

Furthermore, according to the relationship between the dq^c-frame and dq^g-frame shown in Figure 2a, the angle dynamics in (6) can be represented as follows:

$$\widehat{\delta }=(\widehat{\omega }-{\widehat{\omega }}_g)/s$$

(17)

The output active power and reactive power are calculated by the following:

$$p=1.5\cdot [v_d,v_q]\cdot {[i_d,i_q]}^{\mathrm{T}},q=1.5\cdot [v_d,v_q]\cdot {[-i_q,i_d]}^{\mathrm{T}}$$

(18)

The small-signal models of (18) can be represented as follows:

$$\widehat{p}=1.5\cdot ([I_{d0},I_{q0}]\cdot {[{\widehat{v}}_d,{\widehat{v}}_q]}^{\mathrm{T}}+[V_{d0},V_{q0}]\cdot {[{\widehat{i}}_d,{\widehat{i}}_q]}^{\mathrm{T}})$$

(19)

$$\widehat{q}=1.5\cdot ([-I_{q0},I_{q0}]\cdot {[{\widehat{v}}_d,{\widehat{v}}_q]}^{\mathrm{T}}+[V_{q0},-V_{d0}]\cdot {[{\widehat{i}}_d,{\widehat{i}}_q]}^{\mathrm{T}})$$

(20)

The PCC voltage magnitude V is calculated by the following:

$$V=\sqrt{v_d^2+v_q^2}$$

(21)

Then, the small-signal variation in V is expressed as follows:

$$\widehat{V}=[{\displaystyle \frac{V_{d0}}{V_0}},{\displaystyle \frac{V_{q0}}{V_0}}]\cdot {[{\widehat{v}}_d,{\widehat{v}}_q]}^{\mathrm{T}}$$

(22)

where V₀ is the steady-state value of V.

2.4. Model of DC-Side Dynamics

Figure 2b shows the instantaneous power balance of the DC-link, where p_dc and p_ac denote the input and output active power. The DC-link dynamics can be modeled as follows:

$$p_{\mathrm{dc}}-p_{\mathrm{ac}}=v_{\mathrm{dc}}{i}_{\mathrm{Cdc}}$$

(23)

where i_Cdc is the current that passes through the DC capacitor. In the steady-state operating point (V_dc0, I_dc0) for a small-signal change in Equation (23), the power balance equation for the DC capacitor can be expressed as follows:

$$I_{\mathrm{dc}0}{\widehat{v}}_{\mathrm{dc}}+V_{\mathrm{dc}0}{\widehat{i}}_{\mathrm{dc}}-{\widehat{p}}_{\mathrm{ac}}=s{C}_{\mathrm{dc}}{V}_{\mathrm{dc}0}{\widehat{v}}_{\mathrm{dc}}$$

(24)

Eventually, it can be converted to the following:

$${\widehat{v}}_{\mathrm{dc}}={\displaystyle \frac{1}{s{C}_{\mathrm{dc}}{V}_{\mathrm{dc}0}-I_{\mathrm{dc}0}}}(V_{\mathrm{dc}0}{\widehat{i}}_{\mathrm{dc}}-{\widehat{p}}_{\mathrm{ac}})$$

(25)

where the small-signal equation of p_ac can be calculated as follows:

$${\widehat{p}}_{\mathrm{ac}}=1.5\cdot (I_{sd0}\widehat{E}+E_0{\widehat{i}}_{sd})$$

(26)

Finally, the block diagram of the DC-link voltage control loop is shown in Figure 3. Here, e^-τs represents the time-delay element (τ = 0.0001 s). The impact of this element on both the system and subsequent methods is minimal and can be disregarded.

2.5. Complete Model

Finally, the open-loop transfer function from ${\widehat{V}}_{\mathrm{d}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{f}}$ to ${\widehat{v}}_{\mathrm{d}\mathrm{c}}$ can be obtained through the following derivation:

$$T_{open}=({\displaystyle \frac{s{K}_{\mathrm{pdc}}+K_{\mathrm{idc}}}{s}}){\displaystyle \frac{V_{\mathrm{dc}0}}{s{C}_{\mathrm{dc}}{V}_{\mathrm{dc}0}-I_{\mathrm{dc}0}}}$$

(27)

The closed-loop transfer function is presented as follows:

$$T_{closed}={\displaystyle \frac{T_{open}}{1+T_{open}}}$$

(28)

This section develops a complete small-signal model for a grid-forming PMSG. The model incorporates both DC-link voltage control and grid-forming control of the MSC. The analysis focuses on deriving the closed-loop transfer function for the DC-link voltage control, which determines the control system’s bandwidth characteristics. Subsequently, the study examines the impact of reduced DC-link voltage control bandwidth on AC-side system stability.

3. SSO Issue Caused by DC-Link Voltage Dynamics and Effect of Conventional Damping Method

Based on the model established in Section 2, this section studies the effect of DC voltage control bandwidth reduction by analyzing the bode plot and characteristic root trajectory of the DC-link voltage transfer function.

Figure 4a illustrates the bode plot of the DC-link voltage open-loop transfer function. Attributable to the non-minimum phase behavior inherent in the dynamics of DC-link voltage [28], the phase–frequency curve demonstrates phase-lead characteristics. Notably, when the control bandwidth (f_DVC) of DC-link voltage is set to 19 Hz, the magnitude–frequency curve intersects 0 dB at 11.9 Hz, exhibiting a phase margin (PM) of 0.3°.

Figure 4b depicts the locus of poles dominated by DC-link voltage. As f_DVC decreases, λ₁ and λ₂ gradually move rightward, approaching the imaginary axis, and ultimately entering the right half of the s-plane. This shift indicates a reduction in the stability margin. At an f_DVC value of 19 Hz, the poles are positioned at −2.13 ± 74.9 with a damping ratio of 0.03 and a resonant frequency of 11.9 Hz.

The PM of 0.3° shown in Figure 4a and the damping ratio of 0.03 derived from Figure 4b together indicate that the system will operate under a condition of weak damping. Further, the crossover frequency of 11.9 Hz presented in Figure 4a and the imaginary component of 74.9 rad/s depicted in Figure 4b suggest that the DC-link voltage may be susceptible to encountering low-frequency oscillations. Considering the power interactions between DC- and AC-sides, sub-synchronous oscillations may occur on the AC-side of the converter.

Note that it is difficult to redesign the DC-link voltage bandwidth when the controls of the AC–DC rectifier and DC–AC inverter are realized in separate chips. Hence, this oscillation issue must be suppressed by adding a damping control to the grid-forming converter. A conventional damping method is to introduce DC-link voltage dynamics into the active power control loop via a proportional gain K_DCP [27], the corresponding control structure is shown in Figure 5.

Figure 6a shows the bode plots of the active power open-loop transfer function. For K_DCP values of 0, 5, 10, and 15 p.u., the phase margins are measured as 12.5°, 29.2°, 44.2°, and 53.1°, respectively. Notably, when K_DCP is set at 15 p.u., the magnitude–frequency curve crosses 0 dB again at 11.9 Hz with a phase margin of −45.5°. Figure 6b shows the bode plots of the DC-link voltage open-loop transfer function. For K_DCP values of 0, 5, 10, and 15 p.u., the phase margins are measured as 4.86°, 3.06°, 1.23°, and −0.63°, respectively. The negative phase margin indicates a 11.9 Hz oscillation in the DC-link voltage.

These phenomena indicate that the K_DCP is excessively large. While it improves stability around 1 Hz, it causes instability around 11.9 Hz. The traditional damping method not only cannot effectively solve the above mentioned sub-synchronous oscillation problem, but may aggravate the instability of the DC-link voltage.

4. Proposed Damping Method

4.1. Theoretical Analysis

As demonstrated in Section 3, conventional damping methods remain ineffective in suppressing SSO issues even with properly tuned parameters. Moreover, they may exacerbate DC-link voltage fluctuations and increase the risk of system instability. Therefore, to enhance oscillation suppression capability, this paper proposes an oscillation mitigation strategy inspired by the active-reactive power coupling principle of power system stabilizers. The strategy introduces the DC-link voltage error signal into the reactive power control loop, thereby providing additional damping to the system, as shown in Figure 7. Under the proposed control strategy, when DC-link voltage fluctuates, the tracking error is introduced into the reactive power loop through a proportional gain and a phase compensation element, as expressed in Equation (27). Through this mechanism, the DC voltage error is incorporated into the reactive power loop, which then couples into the DC-link voltage dynamics via the transfer path from p_ac to v_dc in Figure 3, thereby influencing the dynamic response of the DC-link voltage. This approach effectively enhances the damping characteristics of the system during fluctuations, suppressing oscillatory behavior.

The proposed method incorporates the DC-link voltage tracking error into the reactive power control loop via a low-pass filter. The low-pass filter is denoted as follows:

$$\widehat{E}=G_{\mathrm{RPC}}\cdot \left({\widehat{Q}}_{\mathrm{ref}}-\widehat{q}\right)+G_{\mathrm{DCQ}}\cdot \left({\widehat{v}}_{\mathrm{dc}}-{\widehat{V}}_{\mathrm{dcref}}\right)$$

(29)

where G_DCQ is presented as follows:

$$G_{\mathrm{DCQ}}=K_{\mathrm{DCQ}}\cdot {\omega }_{\mathrm{DCQ}}/\left(s+{\omega }_{\mathrm{DCQ}}\right)$$

(30)

The incorporation of the low-pass filter branch serves to reshape the non-minimum phase characteristics inherent in the DC-link voltage dynamics [16]. In the proposed damping method, the introduced low-pass filter provides phase lag, which effectively compensates for the phase anomalies induced by the non-minimum phase nature. This reshapes the system transfer function characteristics and thereby enhances the system damping effect. Through the overall control structure shown in Figure 8, the bode diagram of the DC-link voltage open-loop transfer function is plotted, as shown in Figure 9. In Figure 9a, f_DVC is set at 19 Hz and ω_DCQ is set at 30 rad/s. For K_DCQ values of 0, 0.5 and 1.0 p.u., the phase margins are measured as 4.86°, 8.34°, and 10.61°, respectively. Notably, an increase in K_DCQ results in a higher PM value. In Figure 9b, f_DVC is set at 19 Hz and K_DCQ is set at 1.0 p.u. For ω_DCQ values of 30, 60, and 90 rad/s, the phase margins are measured as 10.6°, 17.9°, and 24.2°, respectively. This indicates that increasing ω_DCQ value can also contribute to enhancing the stability margin.

Figure 10 depicts the locus of poles dominated by DC-link voltage. In Figure 10a, as K_DCQ increases from 0 to 1.0 p.u., the poles gradually move leftward, indicating an improved stability margin. Figure 10b illustrates that as ω_DCQ increases from 1 to 250 rad/s, λ₁ and λ₂ exhibit a significant leftward shift followed by a gradual rightward movement. These observations indicate that the proposed damping method can lead to a substantial leftward shift in the poles dominated by DC-link voltage and enhance the stability margin effectively.

4.2. Parameter Design Method

In the proposed damping strategy, the parameters for the active and reactive power loops are designed primarily based on the voltage/frequency regulation requirements and inertia response characteristics of the grid-forming converter. The key focus of the parameter design lies in determining the proportional gain K_DCQ and the low-pass filter cutoff frequency ω_DCQ after introducing the DC-link voltage error term. The detailed parameter design procedure is illustrated in Figure 11.

Step 1: Determine the system’s state matrix A.

Step 2: Set the traversal matrix for K_DCQ and ω_DCQ.

Step 3: Select a set of K_DCQ and ω_DCQ values.

Step 4: Calculate the eigenvalues.

Step 5: Assess system stability based on the location of eigenvalues in the s-plane. If stable, proceed to the next step; otherwise, return to Step 3.

Step 6: Evaluate whether the system achieves the optimal stability margin based on the eigenvalue distribution in the s-plane. If yes, proceed; otherwise, return to Step 3.

Step 7: Determine the optimal K_DCQ and ω_DCQ.

Step 8: Conclude the parameter design process.

For a stable system, eigenvalues in the left-half plane are categorized into two regions: the overdamped region with a damping ratio ξ > 0.707, and the underdamped region with 0 < ξ < 0.707. When the damping ratio satisfies 0 < ξ < 0.05, poles below 2.5 Hz represent low-frequency oscillations, while poles between 2.5 Hz and 50 Hz represent sub-synchronous oscillations. In the parameter design process described above, achieving the optimal stability margin involves adjusting parameters to shift poles as much as possible toward the overdamped region, thereby enhancing system stability. In controller design, there is an inherent trade-off between damping strength and response speed. To ensure global system stability, it is often necessary to sacrifice some response speed in exchange for sufficient damping to suppress oscillations.

4.3. Comparison of Damping Methods

For reader convenience,

Table 1. Abbreviations.

Abbreviation	Significance
PMSG	permanent magnet synchronous generator
SSO	sub-synchronous oscillation
MSC	machine-side converter
GSC	grid-side converter
GFM	grid-forming
SCR	short circuit ratio
PCC	point of common coupling
VSG	virtual synchronous generator
PM	phase margin

lists the abbreviations used in this paper, while

Table 2. Comparisons with several typical articles about DC-link dynamics.

Refs	Synchronization	SCR Adaptability	Controller	Innovation
[12]	Active power	Weak grid	1-order	Vector control regulates the DC-link voltage.
[14]	Active power	——	VSG	VSG-Based Torsional Vibration Control for Shaft Systems.
[18]	Active power	——	1-order	Torque-based droop control regulates the DC-link voltage.
[19]	Active power	Weak grid	1-order	VSG controls the DC-link voltage by converting voltage to power.
[22]	DC-link voltage	Stiff grid	3-order	Enhancing Transient Stability Using Parallel Synchronization Control.
[23]	DC-link voltage	Weak grid	1-order	The system exhibits lower stability under a constant power source.
[24]	DC-link voltage	Weak grid	1-order	Hybrid synchronization control causes synchronous oscillations.
This	DC-link voltage	Yes (SCR:1.5~3.5)	1-order	Solve SSO issues caused by DC-link dynamics.

provides a comparative analysis of the relevant literature. Specifically, Table 2 details the similarities and differences between the proposed method and existing studies on grid-connected converters that account for DC-link voltage dynamics. Unlike [12], which focuses on achieving rapid frequency regulation via MSC-controlled DC-link voltage, this work places emphasis on analyzing the impact of DC-link voltage dynamics, thereby contributing to improved bandwidth and stability. When contrasted with [14], the established transfer function in this study is of lower order, substantially simplifying the modeling process and facilitating a more straightforward analytical approach. Compared to [23,24], this paper specifically addresses the SSO issue induced by DC-link voltage dynamics, highlighting the limitations of conventional damping strategies and proposing an effective mitigation approach that reduces system coupling and enhances overall stability. In comparison with [22,23,24], the damping method proposed in this paper features a significantly simpler structure, does not increase controller order, and offers greater ease in modeling and analysis.

5. Simulation Verification

In order to verify the effectiveness of the proposed damping method, a grid-forming PMSG model based on MATLAB (Version R2022b)/Simulink (Version 10.6) is constructed. The power rating in this model is 6 MVA, the DC-link voltage is 1800 V, and the AC-side voltage and frequency are rated at 1140 V and 50 Hz. The AC-side parameters are: 43,680 μF DC-link capacitor, 0.05 mH filter inductor, 900 μF filter capacitor, 0.04 mH gate-side inductor, and 0.06 Ω gate-side resistor.

Figure 12 shows the simulation waveforms. The parameters of active and reactive power loops are set as K_f is 100 p.u., K_V is 20 p.u., J is 58 p.u., and K is 0.9 p.u. Before t₁, f_DVC is set as 19 Hz, and the converter can operate stably. However, after adjusting f_DVC from 19 Hz to 18.7 Hz at t₁, the GFM converter encounters sub-synchronous oscillations. Figure 12b shows the zoomed view of sub-synchronous oscillations, indicating that fluctuations in the DC-link voltage can indeed induce instability issues.

Figure 12c displays the FFT analysis results of v_dc and i_oa. Notably, v_dc includes a DC component and a significant 11.9 Hz component. The output current i_oa contains an obvious 50 Hz component, a sub-synchronous frequency component of 38.1 Hz, and a super-synchronous frequency component of 61.9 Hz. The 11.9 Hz frequency component closely aligns with the resonant frequency 11.9 Hz observed in Figure 4, indicating the accuracy of the small-signal analysis. Moreover, when the proposed damping method is utilized at t₂, with K_DCQ = 1.0 p.u. and ω_DCQ = 90 rad/s, the sub-synchronous oscillations can be effectively damped.

In addition, in order to test the SCR adaptability of the proposed damping method, simulation tests were carried out under two different grid strengths. As can be seen from Figure 13a,b, the response of the proposed damping method is almost identical when the bandwidth is reduced, and the method is also applicable at smaller SCRs.

Figure 14 illustrates the adaptability of the proposed damping method to observe when the bandwidth is reduced at different C_dc sizes. It can be found that the SSO caused by the DC-link voltage dynamics can be effectively suppressed, and the steady state can be quickly reached under different C_dc.

To further validate the theoretical conclusions from the damping strategy analysis in Section 4 regarding stability effects when increasing K_DCQ and ω_DCQ, simulation tests were conducted, and the simulation waveforms are shown in Figure 15 and Figure 16. The simulation results align with theoretical conclusions: when ω_DCQ is held constant, increasing K_DCQ enhances the stability margin, improves damping performance, and accelerates the system’s return to steady state. As shown in Figure 16, when K_DCQ is maintained constant, increasing ω_DCQ also contributes to an improved stability margin.

In order to further verify the effectiveness of the proposed damping method, the following simulation tests are carried out. The parameters used in the regular virtual synchronous generator are the same as before. The parameters K_DCQ and ω_DCQ in the proposed damping method are set to 1.0 p.u. and 150 rad/s, respectively. Figure 17 shows the simulated waveform when the DC voltage reference value jumps from 1 p.u. order to 1.05 p.u. It can be found that the proposed damping method can reduce oscillations in the case of voltage response and quickly reach a steady state, as well as the power response. The proposed damping method can effectively improve the response performance when the reference value changes, provide a damping effect, and improve the stability of the system.

For the simulation test waveforms shown in Figure 17, we have defined quantitative evaluation metrics and calculated the key dynamic performance parameters listed in

Table 3. Key performance indicator comparison.

DC-Link Voltage	Traditional VSG Control	the Proposed Damping Strategy
Damping ratio	0.03	0.27
Overshoot	0.096 p.u. (9.6%)	0.062 p.u. (6.2%)
Steady-state time	3.2 s	0.38 s
Frequency deviation	0.9%	0.1%

. The data clearly demonstrate that after applying the proposed damping strategy, system oscillations are significantly suppressed, with notable improvements in all key indicators. Specifically, the damping ratio increases substantially from 0.03 to 0.27, indicating rapid oscillation suppression and enhanced stability. Both the overshoot and settling time are significantly reduced, resulting in a smoother and faster dynamic response. Furthermore, the decreased frequency deviation confirms that the proposed method not only stabilizes the DC voltage but also improves frequency stability.

6. Conclusions

This paper focuses on grid-forming PMSG, establishing a small-signal model that considers the dynamics of DC-link voltage, and reveals the mechanism by which DC-link voltage dynamics induce SSO in grid-forming PMSG. These instabilities arise from: (1) reduced bandwidth in DC-link voltage control, and (2) the implementation of conventional damping control in the converter’s later stages. To mitigate these SSO risks, we propose a novel damping method, which demonstrates significant damping effects by feeding the DC-link voltage tracking error processed through a low-pass filter into the reactive power control loop. Simulation results validate the effectiveness of the proposed approach in suppressing oscillations and its satisfactory adaptability under SCR conditions.