A Double-PLL-Based Impedance Reshaping Strategy for DFIG System Under Grid Frequency Deviation

Abstract

The stable operation of doubly fed induction generator (DFIG) systems under weak-grid and off-nominal-frequency conditions is important for reliable wind-power integration. However, the phase-locked loop (PLL) dynamics can degrade DFIG impedance and damping, while conventional single-PLL reshaping may suffer from compensation drift under grid-frequency deviations and often relies on a high-pass filter. Therefore, this paper proposes an integrated Double-PLL-Based impedance-reshaping strategy for DFIG systems. Firstly, a complete DFIG admittance model incorporating the rotor-side converter, grid-side converter, DC link, and PLL dynamics is established to identify the critical coupling channel responsible for the adverse impedance characteristics. Secondly, a supplementary rotor-current compensation path is constructed to directly reshape the adverse impedance characteristics and improve system damping. Thirdly, the relative phase angle between the main and auxiliary PLLs is used to generate a frequency-adaptive compensation signal, thereby avoiding continuous compensation drift under persistent frequency deviations and reducing reliance on a dedicated high-pass filter. In this complete strategy, the compensation path directly performs impedance reshaping, while the Double-PLL-Based implementation provides frequency adaptation. Finally, generalized Nyquist analysis and MATLAB/Simulink simulations demonstrate improved impedance matching and oscillation suppression under the considered weak-grid and continuous-frequency-deviation conditions, while hardware-in-the-loop (HIL) experiments corroborate the robustness under PLL-parameter variations.

1. Introduction

With the continuous growth in the installed capacity of renewable energy generation, renewable energy sources—represented primarily by wind power—are progressively becoming a vital component of modern power systems. Due to the advantages of a smaller converter capacity rating, high operational efficiency, and flexible variable-speed constant-frequency (VSCF) control, the doubly fed induction generator (DFIG) maintains a broad application basis in onshore wind farms [1]. However, large-scale wind power is typically integrated into relatively weak grid areas. The increased impedance from long-distance transmission and collection lines reduces the system’s short-circuit ratio (SCR), thereby significantly intensifying the dynamic interactions between the wind turbines and the grid impedance [2]. Under weak grid conditions, wind power grid-connected systems are highly susceptible to stability issues, including subsynchronous, supersynchronous, and broadband oscillations [3]. At the broader smart-grid level, reliable real-time monitoring and communication schemes have also been investigated to improve the detection of critical grid events [4], further highlighting the need for robust converter operation under dynamically changing grid conditions. For grid-following DFIGs, the phase-locked loop (PLL) not only fulfills the function of synchronous phase detection but also alters the system’s equivalent impedance characteristics via coordinate transformations and current control loops, which subsequently deteriorates the system’s stability margin under weak grids [5].

Currently, extensive research has been conducted by scholars worldwide on the stability issues of renewable energy grid-connected systems. Common analytical methods primarily include eigenvalue analysis based on state-space models, frequency-domain analysis based on transfer functions, and stability analysis based on impedance models. The state-space method can comprehensively describe the internal dynamic processes of the system, making it suitable for modal analysis and parameter sensitivity analysis; however, in scenarios involving multiple control loops and multi-converter coupling, it suffers from issues such as high model order and strong parameter dependency [6]. The transfer function method facilitates the analysis of the impact of local control loops on stability, but its applicability is limited in multi-input multi-output (MIMO) coupled systems and scenarios with significant frequency coupling [7].

In contrast, the impedance analysis method can characterize the dynamic interactions between the wind turbine and the grid from the perspective of terminal characteristics, and it can be combined with the Nyquist criterion to evaluate system stability [8]. The introduction of the impedance stability criterion provided a foundational framework for the stability analysis of grid-connected converters [9]. Subsequently, dq-frame impedance modeling methods for grid-connected voltage source converters (VSCs) have continuously evolved, providing an essential tool for analyzing the interactive effects among control loops, the PLL, and grid impedance [10]. Further investigations into the dq-frame small-signal impedance characteristics of three-phase grid-connected converters have offered a valuable reference for impedance modeling and stability analysis under weak grid conditions [11].

Regarding DFIG grid-connected systems under weak grid conditions, existing studies have demonstrated that the dynamics of the PLL are crucial factors affecting the system’s impedance characteristics and stability margin. The utilization of complex transfer functions and transfer matrices to describe three-phase dynamic systems has provided a theoretical foundation for impedance modeling under frequency coupling conditions [12]. Systematic reviews of the structures and dynamic characteristics of three-phase PLLs have pointed out that PLL parameters significantly influence the dynamic response of grid-connected control systems [13]. Investigations into the modeling and control of DFIGs under unbalanced grid conditions have demonstrated that grid disturbances can affect the rotor-side dynamic characteristics through the control loops [14]. Studies on the subsynchronous interactions between wind farms and the grid in weak AC systems have revealed that the dynamic coupling between renewable energy grid-connected systems and weak grid impedance may induce oscillation issues [15]. Furthermore, a unified impedance model for grid-connected VSCs was proposed, noting that the PLL and outer-loop control alter the equivalent impedance matrix and consequently affect system stability [16]. To address the stability analysis of grid-connected converters with coupling terms, a generalized impedance stability criterion has also been proposed [17].

Regarding stability enhancement, existing research has primarily focused on PLL parameter tuning, virtual impedance compensation, voltage-disturbance compensation, and supplementary impedance reshaping. A symmetrical PLL structure was developed to improve impedance-modeling accuracy and stability analysis for grid-connected converters in weak grids [18]. An improved virtual-inductance controller considering PLL dynamics was proposed to enhance DFIG stability [16], while voltage-disturbance compensation was introduced to reshape the terminal characteristics through an additional voltage-related control path [19]. More closely related to the present work, supplementary compensation methods have been developed to mitigate PLL-induced frequency coupling and adverse impedance characteristics by acting on the rotor-current dynamics or converter control path [20,21]. These studies demonstrate the effectiveness of targeted impedance compensation without fundamentally changing the primary power- and current-control objectives.

A concise comparison of representative impedance-reshaping approaches is provided in

Table 1. Comparison of representative impedance-reshaping approaches for DFIG systems under grid-frequency deviations.

Method	Control Structure and Required Signals	Additional Filter	Frequency-Deviation Capability	Comp. Tuning	Major Limitation
Virtual-impedance control [16]	Virtual impedance generated from voltage/current feedback	Method-dependent	Not specifically designed for persistent frequency deviations	Medium	Parameter selection depends on operating conditions
Voltage-disturbance compensation [19]	PCC-voltage disturbance injected into the converter control path	Method-dependent	Not specifically designed for compensation drift	Medium	The frequency-reference issue is not directly addressed
Single-PLL compensation-based reshaping [20,21]	Main-PLL angle and rotor-current reference form the compensation signal	Dedicated HPF typically required	May suffer from drift under persistent frequency deviations	Medium; PLL/HPF	HPF tuning and fixed nominal-frequency reference
Proposed Double-PLL-Based impedance reshaping	Rotor-current compensation driven by the relative angle of the main and auxiliary PLLs	No dedicated HPF for drift suppression	Designed for persistent frequency deviations	Medium–high; auxiliary PLL	Adds one PLL and related tuning parameters

.

To address these limitations, this paper proposes an integrated Double-PLL-Based impedance-reshaping strategy for DFIGs under grid-frequency deviations. In this terminology, the proposed strategy consists of two coordinated parts: a supplementary rotor-current compensation path and a Double-PLL-Based relative-phase-angle generator. The supplementary path directly reshapes the adverse impedance characteristics by compensating the coupling from the point of common coupling (PCC) disturbance to the rotor-side control output, whereas the main and auxiliary PLLs generate a frequency-adaptive relative-angle signal for this compensation path. Therefore, the additional PLL alone is not regarded as an independent impedance-reshaping mechanism. Unlike the closest single-PLL implementation, the proposed strategy constructs the compensation signal from the relative dynamics of two PLL estimates rather than from the deviation between one PLL and the fixed nominal-frequency reference. This integrated architecture avoids continuous compensation drift under persistent frequency deviations and reduces reliance on a dedicated high-pass filter. Its effectiveness is evaluated through generalized Nyquist analysis, MATLAB/Simulink (2020b) time-domain simulations, and hardware-in-the-loop (HIL) experiments. The main contributions of this paper are summarized as follows:

1. Admittance Modeling and Mechanism Analysis: A complete admittance model of the DFIG grid-connected system considering the RSC, GSC, DC link, and PLL dynamics is established, and the critical PLL-related coupling channel that degrades the impedance characteristics and stability margin under weak-grid conditions is identified.

2. Impedance Reshaping via Supplementary Compensation: A rotor-current compensation path is constructed to act on the identified adverse coupling channel and reshape the DFIG impedance characteristics within the target frequency range without changing the primary power- and current-control objectives.

3. Double-PLL-Based Frequency-Adaptive Implementation: The relative phase angle between the main and auxiliary PLLs is integrated into the rotor-current compensation path to form the complete Double-PLL-Based impedance-reshaping strategy. This implementation replaces the compensation signal referenced to the fixed nominal frequency, prevents continuous drift under persistent frequency deviations, and reduces reliance on a dedicated high-pass filter.

2. Admittance Model of DFIG Grid-Connected System Under Weak Grid Conditions

Figure 1 illustrates the topology and control strategy of a typical grid-following DFIG system. To accurately characterize the dynamics of the DFIG system, this paper models the rotor-side converter (RSC), grid-side converter (GSC), and DC-link, respectively.

2.1. Modeling of the RSC

To obtain a tractable small-signal model around a balanced operating point, magnetic saturation is neglected and the stator and rotor windings are assumed to be symmetrically distributed. Accordingly, the inductances are treated as constants and spatial-harmonic effects are excluded; the model is therefore intended for small perturbations and may lose accuracy during deep saturation, severe unbalance, or large disturbances. Under these assumptions, the mathematical model of the DFIG in the natural $abc$ reference frame can be expressed as:

$$\left\{\begin{array}{c}{V}_{sabc}=R_s{i}_{sabc}+{\displaystyle \frac{d{\psi }_{sabc}}{dt}}\hfill \\ V_{rabc}=R_r{i}_{rabc}+{\displaystyle \frac{d{\psi }_{rabc}}{dt}}\hfill \end{array}\right.$$

(1)

where $V_s$, $V_r$, $i_s$, and $i_r$ represent the stator and rotor voltages and currents, respectively; $R_s$ and $R_r$ represent the stator and rotor resistances; ${\psi }_s$ and ${\psi }_r$ represent the stator and rotor flux linkages; and the subscript $abc$ denotes that the physical quantities are components in the $abc$ reference frame.

The control strategy illustrated in Figure 1 is implemented in the dq reference frame, where (1) can be expressed as (2) and (3).

$$\left\{\begin{array}{c}{V}_{sd}=D{\psi }_{sd}-\omega {\psi }_{sq}+R_s{i}_{sd}\hfill \\ V_{sq}=D{\psi }_{sq}+\omega {\psi }_{sd}+R_s{i}_{sq}\hfill \\ {\psi }_{sd}=L_s{i}_{sd}+L_m{i}_{rd}\hfill \\ {\psi }_{sq}=L_s{i}_{sq}+L_m{i}_{rq}\hfill \end{array}\right.$$

(2)

$$\left\{\begin{array}{c}{V}_{rd}=D{\psi }_{rd}-{\omega }_r{\psi }_{rq}+R_r{i}_{rd}\hfill \\ V_{rq}=D{\psi }_{rq}+{\omega }_r{\psi }_{rd}+R_r{i}_{rq}\hfill \\ {\psi }_{rd}=L_r{i}_{rd}+L_m{i}_{sd}\hfill \\ {\psi }_{rq}=L_r{i}_{rq}+L_m{i}_{sq}\hfill \end{array}\right.$$

(3)

where D is the differential operator; $\omega$ and ${\omega }_r$ represent the grid angular frequency and the rotor current angular frequency, respectively; $L_s=L_{ls}+L_m$ and $L_r=L_{lr}+L_m$ are the stator and rotor self-inductances corresponding to the leakage and magnetizing inductances listed in Table 2; and the subscript $dq$ indicates that the physical quantities are components in the $dq$ reference frame.

By introducing small perturbations into the system and neglecting higher-order terms, the small-signal linearization of (2) and (3) yields:

$$\left\{\begin{array}{c}\Delta V_s=G_1\Delta i_s+G_2\Delta i_r\hfill \\ \Delta V_r=G_3\Delta i_r+G_4\Delta i_s\hfill \end{array}\right.$$

(4)

where:

$$\begin{array}{cc}\hfill G_1& =\left[\begin{array}{cc}{R}_s+s{L}_s& -\omega L_s\\ \omega L_s& R_s+s{L}_s\end{array}\right],G_2=\left[\begin{array}{cc}s{L}_m& -\omega L_m\\ \omega L_m& s{L}_m\end{array}\right],\hfill \\ \hfill G_3& =\left[\begin{array}{cc}{R}_r+s{L}_r& -{\omega }_r{L}_r\\ {\omega }_r{L}_r& R_r+s{L}_r\end{array}\right],G_4=\left[\begin{array}{cc}s{L}_m& -{\omega }_r{L}_m\\ {\omega }_r{L}_m& s{L}_m\end{array}\right].\hfill \end{array}$$

(5)

Building upon the small-signal admittance modeling of the DFIG electrical side, a small-signal model of the RSC is further established to characterize the impact of the control loops on the system’s impedance characteristics. As illustrated in Figure 1, the RSC adopts a dual-closed-loop control structure consisting of an outer power loop and an inner current loop. The outer power loop generates the rotor current reference values based on the deviations in stator active and reactive power, while the inner current loop derives the rotor-side voltage commands through proportional–integral (PI) regulation and feedforward decoupling compensation. The corresponding control relationships are expressed in (6) and (7). Since the PLL dynamics introduce a phase angle deviation between the control reference frame and the actual electrical quantities, the superscript c is employed in this paper to denote the variables within the control frame.

$$\left\{\begin{array}{cc}\hfill V_{rd}& =\left(k_{pir}+{\displaystyle \frac{k_{iir}}{s}}\right)(i_{rdref}-i_{rd})-{\omega }_r{L}_r{i}_{rq}-{\omega }_r{L}_m{i}_{sq}\hfill \\ \hfill V_{rq}& =\left(k_{pir}+{\displaystyle \frac{k_{iir}}{s}}\right)(i_{rqref}-i_{rq})+{\omega }_r{L}_r{i}_{rd}+{\omega }_r{L}_m{i}_{sd}\hfill \end{array}\right.$$

(6)

$$\left\{\begin{array}{cc}\hfill i_{rdref}& =\left(k_{ppq}+{\displaystyle \frac{k_{ipq}}{s}}\right)(P_{sref}-P_s)\hfill \\ \hfill i_{rqref}& =\left(k_{ppq}+{\displaystyle \frac{k_{ipq}}{s}}\right)(Q_{sref}-Q_s)-{\displaystyle \frac{V_{sd}^c}{{\omega }_r{L}_m}}\hfill \end{array}\right.$$

(7)

Applying small-signal linearization to (6) and (7), respectively, yields:

$$\left\{\begin{array}{cc}\hfill \Delta V_r& =G_5\left(\Delta i_{r\_ref}-\Delta i_r\right)+G_6\Delta i_r+G_7\Delta i_s\hfill \\ \hfill G_5& =\left[\begin{array}{cc}{k}_{pir}+{\displaystyle \frac{k_{iir}}{s}}& 0\\ 0& k_{pir}+{\displaystyle \frac{k_{iir}}{s}}\end{array}\right]\hfill \\ \hfill G_6& =\left[\begin{array}{cc}0& -{\omega }_r{L}_r\\ {\omega }_r{L}_r& 0\end{array}\right]\hfill \\ \hfill G_7& =\left[\begin{array}{cc}0& -{\omega }_r{L}_m\\ {\omega }_r{L}_m& 0\end{array}\right]\hfill \end{array}\right.$$

(8)

$$\left\{\begin{array}{cc}\hfill \Delta i_{rref}& =G_8\left[\begin{array}{c}-\Delta P_s\\ -\Delta Q_s\end{array}\right]+G_9\Delta V_s\hfill \\ \hfill G_8& =\left[\begin{array}{cc}{k}_{ppq}+{\displaystyle \frac{k_{ipq}}{s}}& 0\\ 0& k_{ppq}+{\displaystyle \frac{k_{ipq}}{s}}\end{array}\right]\hfill \\ \hfill G_9& =\left[\begin{array}{cc}0& 0\\ -{\displaystyle \frac{1}{{\omega }_r{L}_m}}& 0\end{array}\right]\hfill \end{array}\right.$$

(9)

In the $dq$ reference frame, the stator power of the DFIG can be expressed in terms of voltage and current as follows:

$$\left\{\begin{array}{cc}\hfill P_s& ={\displaystyle \frac{3}{2}}(V_{sd}{i}_{sd}+V_{sq}{i}_{sq})\hfill \\ \hfill Q_s& ={\displaystyle \frac{3}{2}}(V_{sq}{i}_{sd}-V_{sd}{i}_{sq})\hfill \end{array}\right.$$

(10)

Substituting (10) into the previous equation and linearizing yields:

$$\left\{\begin{array}{cc}\hfill & \left[\begin{array}{c}\Delta P_s\\ \Delta Q_s\end{array}\right]=G_{10}\Delta V_s+G_{11}\Delta i_s\hfill \\ \hfill & G_{10}={\displaystyle \frac{3}{2}}\left[\begin{array}{cc}{I}_{sd}^0& I_{sq}^0\\ I_{sq}^0& -I_{sd}^0\end{array}\right],G_{11}={\displaystyle \frac{3}{2}}\left[\begin{array}{cc}{V}_{sd}^0& V_{sq}^0\\ -V_{sq}^0& V_{sd}^0\end{array}\right]\hfill \end{array}\right.$$

(11)

At this stage, the small-signal modeling of the power and current loops is complete. As observed from (11), the voltage and current variables required for the subsequent admittance formulation have been incorporated into the model. Since the PLL introduces a deviation between the physical quantities in the control system and the actual electrical reference frame, its impact must be taken into account to ensure modeling accuracy. Figure 2 illustrates the typical structure of a synchronous reference frame PLL (SRF-PLL), while Figure 3 depicts the relationship between the control reference frame and the grid $dq$ reference frame. The relationship between a given physical quantity and its counterpart obtained after coordinate transformation is as follows:

$$x^c=\left[\begin{array}{cc}\cos ({\theta }_g-{\theta }_{pll})& \sin ({\theta }_g-{\theta }_{pll})\\ -\sin ({\theta }_g-{\theta }_{pll})& \cos ({\theta }_g-{\theta }_{pll})\end{array}\right]x$$

(12)

After small-signal linearization, let $\Delta \theta =\Delta {\theta }_g-\Delta {\theta }_{pll}$ and assume $|\Delta \theta |\ll 1$, so that $\sin (\Delta \theta )\approx \Delta \theta$ and $\cos (\Delta \theta )\approx 1$. Equation (13) is then obtained:

$$\Delta x^c=\left[\begin{array}{cc}1& \Delta \theta \\ -\Delta \theta & 1\end{array}\right]\Delta x$$

(13)

Based on Figure 2, it can be obtained that:

$$\Delta \theta =\left(k_{ppll,1}+{\displaystyle \frac{k_{ipll,1}}{s}}\right)\Delta V_{sq}^s{\displaystyle \frac{1}{s}}$$

(14)

Substituting (14) into (13) and letting the variable x denote the stator and rotor voltages and currents, respectively, yields (15).

$$\left\{\begin{array}{cc}\hfill \Delta i_s^c& =\Delta i_s+G_{12}\Delta V_s\hfill \\ \hfill \Delta i_r^c& =\Delta i_r+G_{13}\Delta V_s\hfill \\ \hfill \Delta V_s^c& =(E+G_{14})\Delta V_s\hfill \\ \hfill \Delta V_r^c& =\Delta V_r+G_{15}\Delta V_s\hfill \end{array}\right.$$

(15)

where:

$$\left\{\begin{array}{cc}\hfill G_{12}& =G_{pll}\left[\begin{array}{cc}0& I_{sq}^0\\ 0& -I_{sd}^0\end{array}\right]\hfill \\ \hfill G_{13}& =G_{pll}\left[\begin{array}{cc}0& I_{rq}^0\\ 0& -I_{rd}^0\end{array}\right]\hfill \\ \hfill G_{14}& =G_{pll}\left[\begin{array}{cc}0& V_{sq}^0\\ 0& -V_{sd}^0\end{array}\right]\hfill \\ \hfill G_{15}& =G_{pll}\left[\begin{array}{cc}0& V_{rq}^0\\ 0& -V_{rd}^0\end{array}\right]\hfill \end{array}\right.$$

(16)

where $G_{pll}$ denotes the closed-loop transfer function of the PLL.

$$G_{pll}={\displaystyle \frac{k_{ppll,1}s+k_{ipll,1}}{s^2+V_{sd}^0(k_{ppll,1}s+k_{ipll,1})}}$$

(17)

By combining the above equations, the rotor-side small-signal model can be obtained as:

$$\begin{array}{cc}\hfill \Delta V_r& =(G_6-G_5)\Delta i_r+(G_7-G_5{G}_8{G}_{11})\Delta i_s+G_{V_{pcc}\to V_r}\Delta V_s\hfill \end{array}$$

(18)

$$G_{V_{pcc}\to V_r}=G_5\left[G_9(E+G_{14})-G_8{G}_{10}(E+G_{14})+G_8{G}_{11}{G}_{12}-G_{13}\right]+G_6{G}_{13}-G_7{G}_{12}-G_{15}$$

(19)

2.2. Modeling of the GSC

The GSC of the DFIG primarily functions to process slip power and maintain a constant DC-link voltage. Consequently, compared to the rotor side, its impact on the stability of the DFIG is relatively small. However, to achieve an accurate system stability analysis, the modeling of the GSC cannot be neglected. Based on the structure illustrated in Figure 1, the following can be obtained:

$$\left\{\begin{array}{cc}\hfill V_{ld}& =-(R_f+s{L}_f)i_{ld}+\omega L_f{i}_{lq}+V_{sd}\hfill \\ \hfill V_{lq}& =-(R_f+s{L}_f)i_{lq}-\omega L_f{i}_{ld}+V_{sq}\hfill \end{array}\right.$$

(20)

where $V_{ld}$, $V_{lq}$ and $i_{ld}$, $i_{lq}$ represent the voltage and current components of the GSC in the $dq$ reference frame, respectively. Applying small-signal linearization yields:

$$\left\{\begin{array}{cc}\hfill \Delta V_l& =\Delta V_s-G_{16}\Delta i_l\hfill \\ \hfill G_{16}& =\left[\begin{array}{cc}s{L}_f+R_f& -\omega L_f\\ \omega L_f& s{L}_f+R_f\end{array}\right]\hfill \end{array}\right.$$

(21)

The control section of the grid-side converter shown in Figure 1 can be described by (22), where the superscript c indicates that the variable is expressed in the control-system reference frame:

$$\left\{\begin{array}{cc}\hfill V_{ld}^c& =-\left(k_{pii}+{\displaystyle \frac{k_{iii}}{s}}\right)(i_{ldref}^c-i_{ld}^c)+\omega L_f{i}_{lq}^c\hfill \\ \hfill V_{lq}^c& =-\left(k_{pii}+{\displaystyle \frac{k_{iii}}{s}}\right)(i_{lqref}^c-i_{lq}^c)-\omega L_f{i}_{ld}^c\hfill \end{array}\right.$$

(22)

where $i_{lqref}^c=0$, and $i_{ldref}^c$ is obtained by passing the error between the DC-link voltage reference and its actual value through a PI controller, which can be expressed as (23):

$$i_{ldref}^c=\left(k_{pdc}+{\displaystyle \frac{k_{idc}}{s}}\right)(V_{dcref}-V_{dc})$$

(23)

Linearizing (22) and (23), respectively, yields:

$$\left\{\begin{array}{cc}\hfill \Delta V_l^c& =G_{17}(\Delta i_{lref}^c-\Delta i_l^c)+G_{18}\Delta i_l^c\hfill \\ \hfill \Delta i_{lref}^c& =-G_{19}\Delta V_{dc}\hfill \end{array}\right.$$

(24)

The specific expressions for each matrix can be represented by (25), as shown below:

$$\left\{\begin{array}{cc}\hfill G_{17}& =-\left[\begin{array}{cc}{k}_{pii}+{\displaystyle \frac{k_{iii}}{s}}& 0\\ 0& k_{pii}+{\displaystyle \frac{k_{iii}}{s}}\end{array}\right]\hfill \\ \hfill G_{18}& =\left[\begin{array}{cc}0& \omega L_f\\ -\omega L_f& 0\end{array}\right]\hfill \\ \hfill G_{19}& =\left[\begin{array}{cc}{k}_{pdc}+{\displaystyle \frac{k_{idc}}{s}}& 0\\ 0& 0\end{array}\right]\hfill \end{array}\right.$$

(25)

The grid side also requires the angle reference provided by the PLL. Similar to the rotor side, the relationship between the voltage and current variables within the control system and those in the actual circuit structure is as follows:

$$\left\{\begin{array}{cc}\hfill \Delta V_l^c& =\Delta V_l+G_{20}\Delta V_s\hfill \\ \hfill \Delta i_l^c& =\Delta i_l+G_{21}\Delta V_s\hfill \\ \hfill G_{20}& =\left[\begin{array}{cc}0& V_{lq}^0{G}_{pll}\\ 0& -V_{ld}^0{G}_{pll}\end{array}\right]\hfill \\ \hfill G_{21}& =\left[\begin{array}{cc}0& I_{lq}^0{G}_{pll}\\ 0& -I_{ld}^0{G}_{pll}\end{array}\right]\hfill \end{array}\right.$$

(26)

By combining (21) and (24)–(26), the grid-side small-signal model accounting for the circuit structure, control loops, and PLL dynamics can be obtained as:

$$\left\{\begin{array}{cc}\hfill \Delta V_l& =(G_{18}-G_{17})\Delta i_l-G_{17}{G}_{19}\Delta V_{dc}+G_{22}\Delta V_s\hfill \\ \hfill G_{22}& =(G_{18}-G_{17})G_{21}-G_{20}\hfill \end{array}\right.$$

(27)

2.3. Modeling of the DC-Link

Similar to the grid side, the DC-link dynamics are also frequently omitted from DFIG admittance modeling. However, maintaining a constant DC-link voltage is essential for normal DFIG operation; therefore, the influence of the DC link should be retained in the overall admittance model. The power relationship between the rotor side and the grid side is given by (28):

$$P_l-P_r=V_{dc}{C}_{dc}{\displaystyle \frac{d{V}_{dc}}{dt}}$$

(28)

where $C_{dc}$ is the DC-link capacitance, and the grid-side power $P_l$ can be expressed as:

$$P_l={\displaystyle \frac{3}{2}}(V_{ld}{i}_{ld}+V_{lq}{i}_{lq})$$

(29)

Applying small-signal linearization to the power expressions of the rotor- and grid-side converters yields:

$$\left\{\begin{array}{cc}\hfill \Delta P_r& =G_{23}\Delta V_r+G_{24}\Delta i_r\hfill \\ \hfill \Delta P_l& =G_{25}\Delta V_l+G_{26}\Delta i_l\hfill \end{array}\right.$$

(30)

where:

$$\left\{\begin{array}{cccc}\hfill G_{23}& ={\displaystyle \frac{3}{2}}\left[\begin{array}{cc}{I}_{rd}^0& I_{rq}^0\\ 0& 0\end{array}\right]\hfill & \hfill G_{24}& ={\displaystyle \frac{3}{2}}\left[\begin{array}{cc}{V}_{rd}^0& V_{rq}^0\\ 0& 0\end{array}\right]\hfill \\ \hfill G_{25}& ={\displaystyle \frac{3}{2}}\left[\begin{array}{cc}{I}_{ld}^0& I_{lq}^0\\ 0& 0\end{array}\right]\hfill & \hfill G_{26}& ={\displaystyle \frac{3}{2}}\left[\begin{array}{cc}{V}_{ld}^0& V_{lq}^0\\ 0& 0\end{array}\right]\hfill \end{array}\right.$$

(31)

Furthermore, applying small-signal linearization to (28) yields:

$$G_{25}\Delta V_l+G_{26}\Delta i_l-G_{23}\Delta V_r-G_{24}\Delta i_r=G_{27}\Delta V_{dc}$$

(32)

where:

$$G_{27}=\left[\begin{array}{cc}s{C}_{dc}{V}_{dc}^0& 0\\ 0& 0\end{array}\right]$$

(33)

Considering the impact of small perturbations in the DC link on various parts of the system, (19) and (27) should be rewritten as:

$$\left\{\begin{array}{cc}\hfill \Delta V_r& =(G_6-G_5)\Delta i_r+(G_7-G_5{G}_8{G}_{11})\Delta i_s\hfill \\ \hfill & +G_{V_{pcc}\to V_r}\Delta V_s+G_{28}\Delta V_{dc}\hfill \\ \hfill \Delta V_l& =(G_{18}-G_{17})\Delta i_l-G_{17}{G}_{19}\Delta V_{dc}+G_{22}\Delta V_s\hfill \\ \hfill & +G_{29}\Delta V_{dc}\hfill \end{array}\right.$$

(34)

where:

$$\left\{\begin{array}{cc}\hfill G_{28}& =\left[\begin{array}{cc}{\displaystyle \frac{V_{rd}^0}{V_{dc}^0}}& {\displaystyle \frac{V_{rq}^0}{V_{dc}^0}}\\ 0& 0\end{array}\right]\hfill \\ \hfill G_{29}& =\left[\begin{array}{cc}{\displaystyle \frac{V_{ld}^0}{V_{dc}^0}}& {\displaystyle \frac{V_{lq}^0}{V_{dc}^0}}\\ 0& 0\end{array}\right]\hfill \end{array}\right.$$

(35)

2.4. Overall Admittance Model of the DFIG System

By combining the small-signal model equations of the various parts of the DFIG, the overall admittance model of the DFIG system can be obtained:

$$AX=B\Delta V_s$$

(36)

$$A=\left[\begin{array}{cccccc}0& G_1& G_2& 0& 0& 0\\ E& G_3& G_4& 0& 0& 0\\ E& G_5+G_6& G_5{G}_8{G}_{11}-G_7& 0& 0& -G_{28}\\ 0& 0& 0& E& G_{16}& 0\\ 0& 0& 0& E& G_{17}-G_{18}& G_{17}{G}_{19}+G_{18}\\ G_{24}& G_{23}& 0& -G_{25}& -G_{26}& G_{27}\end{array}\right]$$

(37)

$$B={\left[\begin{array}{cccccc}E& 0& G_{vsr}& E& G_{vsl}& 0\end{array}\right]}^T$$

(38)

$$X={\left[\begin{array}{cccccc}\Delta V_r& \Delta i_r& \Delta i_s& \Delta V_l& \Delta i_l& \Delta V_{dc}\end{array}\right]}^T$$

(39)

Based on the above derivation, the following can be obtained:

$$\left\{\begin{array}{c}\Delta i_r=Y_{rsc}\Delta V_s\hfill \\ \Delta i_s=Y_{gsc}\Delta V_s\hfill \end{array}\right.$$

(40)

where $Y_{rsc}$ and $Y_{gsc}$ represent the admittance models of the machine-side converter (rotor side) and the grid-side converter (stator side), respectively, then the overall admittance model of the doubly fed induction generator is:

$$Y_{dfig\_total}=Y_{rsc}+Y_{gsc}$$

(41)

where $A^{-1}B$ is a $12\times 2$ matrix, $Y_{rsc}$ corresponds to the fifth and sixth rows of the $A^{-1}B$ matrix, and $Y_{gsc}$ corresponds to the ninth and tenth rows of the matrix. If an impedance model is required, it can be obtained simply by taking the inverse of the admittance model.

To validate the accuracy of the derived admittance model, a frequency-sweep test is performed for all four $dq$-domain channels. As shown in Figure 4, the swept frequency-response points closely agree with the analytical magnitude and phase curves of $Y_{dd}$, $Y_{dq}$, $Y_{qd}$, and $Y_{qq}$, including the resonance region. This agreement verifies the accuracy of the established small-signal admittance model over the considered frequency range.

3. Admittance-Based Stability Analysis Method for Grid-Connected DFIGs

3.1. Generalized Nyquist Stability Criterion

In the context of the integration of renewable energy grid, for the grid-connected converter system shown in Figure 5, the grid-connected circuit of renewable energy inverters can be divided into two parts: a renewable energy inverter subsystem represented by a Norton equivalent and an AC grid subsystem represented by a Thevenin equivalent. Herein, the grid side can be regarded as a voltage source, providing voltage support at the PCC; meanwhile, the inverter side can be treated as a current source, injecting current into the grid that corresponds to the given power reference.

Generally, the equivalent impedance of a weak grid exhibits resistive-inductive characteristics. Let $R_g$ and $L_g$ denote the equivalent resistance and equivalent inductance of the weak grid, respectively, and ${\omega }_0=2\pi f_0$, where $f_0$ represents the fundamental grid frequency. The weak-grid-side impedance model can be expressed as:

$$Z_g\left(s\right)=\left[\begin{array}{cc}{R}_g+s{L}_g& -{\omega }_0{L}_g\\ {\omega }_0{L}_g& R_g+s{L}_g\end{array}\right]$$

(42)

Define the minor-loop gain matrix of the interconnected DFIG-grid system as $L\left(s\right)=Y_{dfig\_total}\left(s\right)Z_g\left(s\right)$. The closed-loop characteristic equation is $\det [I+L(s\left)\right]=0$, which can be equivalently expressed as:

$${\lambda }_i\left[L\left(s\right)\right]=-1,i=1,2$$

(43)

where ${\lambda }_i[·]$ denotes the ith eigenvalue. Let P denote the total number of open-loop right-half-plane poles and let N denote the net counterclockwise encirclement number of $-1$ by the two eigenloci ${\lambda }_i\left[L\left(j\omega \right)\right]$. According to the generalized Nyquist criterion, the number of closed-loop right-half-plane poles is $Z=P+N$. For the parameter sets considered in this paper, pole evaluation confirms $P=0$. Therefore, closed-loop stability is established when the eigenloci do not encircle $-1$, i.e., when $N=0$ and $Z=0$.

3.2. Identifying Negative Impedance Components in Weak Power Networks

The overall admittance of the grid-connected DFIG system is decomposed in (44), in which the GSC mainly assumes the functions of DC-bus voltage support and bidirectional transmission of slip power. Compared with the RSC, which directly participates in electromagnetic-transient regulation and torque control, the influence of the GSC on the overall external characteristics of the system is relatively limited. As shown in Figure 6, the overall admittance models with and without the GSC exhibit only minor differences within the primary frequency band of interest, indicating that the degradation of system stability under weak-grid conditions is more attributable to changes in the admittance characteristics caused by the RSC control loops and their synchronization mechanism. The reduction in DFIG grid-connected stability results from the adverse interaction between the DFIG output admittance and the grid impedance within specific frequency bands. Therefore, the effects of the different control loops are identified separately below.

$$Y_{dfig\_total}^{dq}=Y_{str}^{dq}+Y_{ctrl}^{dq}+Y_{PLL}^{dq}$$

(44)

In the $dq$ reference frame, the overall admittance model of the wind turbine, denoted as $Y_{dfig\_total}^{dq}$, can be expressed as the sum of the DFIG circuit-topology admittance $Y_{str}^{dq}$, the control-loop admittance contribution excluding the PLL $Y_{ctrl}^{dq}$, and the PLL-related admittance contribution $Y_{PLL}^{dq}$. To independently analyze the impact of each component on the system characteristics, this section hierarchically constructs and compares three categories of frequency-domain models based on the previously established admittance model.

Figure 7 compares the admittance magnitude–frequency characteristics of the four channels before and after the introduction of different control loops. When only the circuit topology is considered, the four channels exhibit relatively large admittance magnitudes and distinct peaks in specific frequency bands, indicating that the basic admittance characteristics are determined by the electromagnetic parameters of the physical system. After the RSC and GSC controllers are included, the magnitude–frequency curves change significantly, demonstrating the dominant influence of the conventional control loops on the admittance distribution. A further comparison of the models without and with the PLL shows that the PLL mainly modifies the local resonance peaks and coupling strength within the critical frequency bands, rather than reconstructing the magnitude characteristics over the entire frequency range.

Both terms in (45) are associated with the q-axis voltage component, and the SRF-PLL utilizes the q-axis voltage component as the synchronization error signal. Therefore, upon the introduction of the PLL into the system, the q-axis voltage disturbance will directly enter the dynamic equations of the PLL, subsequently causing variations in the estimated phase angle $\theta$. In turn, the phase angle disturbance is further coupled into the current control and admittance expressions via coordinate transformations, thus affecting the channels related to the q-axis voltage.

$$\left\{\begin{array}{c}{Y}_{dq}={\displaystyle \frac{\Delta I_d}{\Delta V_q}}\hfill \\ Y_{qq}={\displaystyle \frac{\Delta I_q}{\Delta V_q}}\hfill \end{array}\right.$$

(45)

Figure 7, together with (45), indicates that the PLL-related admittance variations are concentrated in the channels associated with the q-axis voltage at the PCC. This conclusion is further examined by comparing the generalized Nyquist eigenloci of the system before and after the PLL is included.

Figure 8 presents the generalized Nyquist eigenloci before and after the inclusion of the PLL. Since $P=0$ and the eigenloci do not encircle the critical point $-1$ in either case, $N=0$ and $Z=0$, confirming closed-loop stability. Following the introduction of the PLL, the eigenloci move closer to $-1$, indicating a reduced stability margin and a greater susceptibility to weak-grid-induced oscillations. As clarified in Section 2, the PLL primarily affects the rotor-side voltage, which subsequently influences the system’s impedance characteristics. Therefore, it is necessary to compensate for this specific control loop.

4. Impedance Reshaping Method for DFIGs

4.1. DFIG Impedance Reshaping Method Based on Feedforward Compensation

The preceding analysis demonstrates that the negative impedance characteristics introduced by the PLL significantly diminish the stability margin of the grid-connected DFIG system under weak grid conditions, constituting one of the primary causes of system instability. To address this issue, a common approach is to attenuate the influence of PLL dynamics on the system’s impedance characteristics by decreasing the PLL bandwidth or reducing the relevant control parameters, thereby improving the grid-connected stability. However, this method inherently sacrifices synchronization tracking speed and dynamic response capability in exchange for an increased stability margin. Consequently, it often leads to a sluggish system response during grid disturbances, operating condition variations, and fault recovery processes, making it difficult to simultaneously satisfy the requirements for both dynamic performance and stable operation.

By introducing a feedforward compensation loop, targeted adjustments are made to the magnitude and phase distribution of the DFIG output impedance, driving the system’s impedance characteristics within the frequency bands of interest toward a direction favorable for stable operation. Rather than directly modifying the main structure and fundamental control objectives of the original controller, this method mitigates the adverse effects of the PLL’s negative impedance effect on grid-connected stability by reconstructing the impedance shaping path via supplementary compensation signals.

The term $G_{13}$ in the DFIG admittance model, which represents the perturbation effect of voltage fluctuations at the PCC on the rotor-side voltage of the DFIG, is a primary factor contributing to the system’s instability under weak-grid conditions. Therefore, it is imperative to redesign the control strategy to counteract the adverse impact on system stability caused by the negative-impedance loop introduced by this component. This matrix evidently exhibits an asymmetrical structure and is entirely associated with the q-axis voltage perturbation. Furthermore, following the incorporation of the PLL, the expression for $G_{13}$ resembles a second-order integral term; at lower frequencies, the low-frequency rotor current acquires an excessively large gain, leading to a deviation of the rotor current from its expected value. Consequently, a second-order high-pass filter (HPF) is incorporated into the feedforward control path to suppress the DC and low-frequency bias components of the phase-angle compensation signal.

In Figure 9, firstly, the phase angle deviation signal $\delta$ is obtained from the output of the PLL. Next, a high-pass filter is utilized to extract its AC component associated with oscillations. Finally, this dynamic deviation is combined with the reference current channel to generate an additional compensation term, which is subsequently injected into the rotor-side voltage control loop. Since this compensation quantity inherently originates from the synchronization phase deviation, it can more directly reflect the influence path of the PLL dynamics on the system’s impedance characteristics.

The scheme illustrated in Figure 9 can essentially be regarded as a small-signal feedforward correction based on phase angle deviation. Its physical significance lies in the following: when weak grid disturbances induce a dynamic shift in the output phase angle of the PLL, this shift will react on the rotor-side control output via coordinate transformations and controller coupling paths, thereby deteriorating the impedance characteristics of the DFIG. Conversely, the proposed compensation branch partially counteracts this original adverse effect by introducing an additional control variable related to this phase angle deviation.

4.2. DFIG Impedance Reshaping Method Considering Grid Frequency Deviation

However, the aforementioned method exhibits limitations under grid-frequency deviations. A high-pass filter is required to suppress the DC and low-frequency bias components of the phase-angle compensation signal. If a lower cutoff frequency ${\omega }_{\mathrm{HPF}}$ is used, $\Delta \omega \ne 0$ can cause continuous error accumulation and ultimately destabilize the system. Conversely, increasing ${\omega }_{\mathrm{HPF}}$ can intensify frequency coupling.

To address this issue, this paper proposes a Double-PLL-Based impedance-reshaping strategy for the DFIG, as shown in Figure 10.

Compared with the conventional single-PLL implementation, the proposed Double-PLL-Based impedance-reshaping strategy introduces an auxiliary PLL in addition to the original main PLL. The main PLL maintains the synchronous reference frame for the DFIG vector controller, whereas the auxiliary PLL is used only to generate the relative phase-angle signal for the impedance-reshaping branch; both PLLs are driven by the PCC voltage.

The PLL error signal $e_{pll1}$ can be expressed as:

$$e_{pll1}=v_{sq}$$

(46)

After passing through the PI controller of the PLL:

$$\Delta {\omega }_1=\left(k_{ppll,1}+{\displaystyle \frac{k_{ipll,1}}{s}}\right)e_{pll1}$$

(47)

Following integration, the deviation $\Delta {\theta }_1$ can be expressed as:

$$\Delta {\theta }_1={\displaystyle \frac{1}{s}}\Delta {\omega }_1$$

(48)

Let the angle deviation $\delta$ be the difference between the two angles; after small-signal linearization, $\Delta \delta$ can be expressed as:

$$\left\{\begin{array}{c}\delta ={\theta }_2-{\theta }_1\hfill \\ \Delta \delta =\Delta {\theta }_2-\Delta {\theta }_1={\displaystyle \frac{1}{s}}(\Delta {\omega }_2-\Delta {\omega }_1)\hfill \end{array}\right.$$

(49)

Under the condition of a single PLL, utilizing the rated power frequency ${\omega }_N$ as the reference frequency, the compensation angle can be expressed as:

$${\delta }_0=\int ({\omega }_g-{\omega }_N)dt$$

(50)

When a deviation exists in the actual grid frequency, i.e., $\Delta {\omega }_g={\omega }_g-{\omega }_N\ne 0$:

$${\delta }_0\left(t\right)=\int \Delta {\omega }_{g}dt$$

(51)

If the frequency deviation persists, ${\delta }_0$ will continuously accumulate, resulting in a drift in the compensation amount.

However, if the output ${\omega }_2$ of the auxiliary PLL is adopted as the reference frequency, the compensation angle can be rewritten as:

$$\delta ={\theta }_2-{\theta }_1=\int ({\omega }_2-{\omega }_1)dt$$

(52)

For the ith stable PLL, define $D_i\left(s\right)=s^2+2{\zeta }_i{\omega }_{n,i}s+{\omega }_{n,i}^2$. Its closed-loop frequency-tracking transfer function is $T_i\left(s\right)=\Delta {\omega }_i\left(s\right)/\Delta {\omega }_g\left(s\right)=(2{\zeta }_i{\omega }_{n,i}s+{\omega }_{n,i}^2)/D_i\left(s\right)$, which satisfies $T_i\left(0\right)=1$. Therefore,

$${\displaystyle \frac{\Delta \delta \left(s\right)}{\Delta {\omega }_g\left(s\right)}}={\displaystyle \frac{T_2\left(s\right)-T_1\left(s\right)}{s}}={\displaystyle \frac{s[D_2\left(s\right)-D_1\left(s\right)]}{D_1\left(s\right)D_2\left(s\right)}}.$$

(53)

For ${\zeta }_i>0$ and ${\omega }_{n,i}>0$, this transfer function has no pole at the origin and all its poles are in the left-half plane; hence, $\Delta \delta$ is bounded for any bounded grid-frequency deviation. In particular, a constant frequency deviation gives $\Delta \delta (\infty )=0$, while a grid-frequency ramp with rate r gives the finite value $\Delta \delta (\infty )=r(1/{\omega }_{n,1}^2-1/{\omega }_{n,2}^2)$. Thus, the relative-angle signal does not continuously accumulate, provided that both PLLs remain locked within the small-signal operating range.

For systematic tuning, the closed-loop denominator of the ith PLL ($i=1$ for the main PLL and $i=2$ for the auxiliary PLL) is matched to the standard second-order form $s^2+2{\zeta }_i{\omega }_{n,i}s+{\omega }_{n,i}^2$. The corresponding PI gains are selected as

$$k_{ppll,i}={\displaystyle \frac{2{\zeta }_i{\omega }_{n,i}}{V_{sd}^0}},k_{ipll,i}={\displaystyle \frac{{\omega }_{n,i}^2}{V_{sd}^0}},i=1,2,$$

(54)

where $V_{sd}^0$ is the steady-state d-axis PCC voltage. The main-PLL parameters retain the original synchronization performance, while the auxiliary-PLL parameters are selected to provide adequate relative-angle dynamics within the target impedance-reshaping range. The bandwidth separation should not be excessive, because identical PLL dynamics yield a weak compensation signal, whereas an overly fast auxiliary PLL increases sensitivity to measurement noise and harmonics.

For the normalized PLL input, $V_{sd}^0=1$ p.u. Under this condition, the auxiliary-PLL gains listed in Table 2, namely $k_{ppll,2}=5$ and $k_{ipll,2}=50$, correspond to ${\zeta }_2\approx 0.354$ and ${\omega }_{n,2}\approx 7.071\mathrm{rad}/\mathrm{s}$. By comparison, the main-PLL parameters give ${\zeta }_1=0.5$ and ${\omega }_{n,1}=10\mathrm{rad}/\mathrm{s}$. Therefore, the auxiliary PLL has a lower characteristic frequency and provides slower and smoother tracking dynamics than the main PLL. The final parameter selection is verified using the generalized Nyquist criterion for the complete system.

From an implementation perspective, the proposed strategy introduces one additional PI-PLL, including an integrator, a PI controller, and a relative-angle calculation. Therefore, its computational burden and number of dynamic states are slightly higher than those of the single-PLL implementation. However, the main-PLL parameters remain unchanged, and the auxiliary PLL can be tuned directly through the desired damping ratio ${\zeta }_2$ and natural frequency ${\omega }_{n,2}$ using (54). In contrast, the conventional method additionally requires coordinated selection of the HPF cutoff frequency and filter parameters to balance drift suppression and frequency-coupling performance. Hence, the proposed method does not claim a lower total implementation cost; rather, it replaces HPF-related tuning trade-offs with a structured auxiliary-PLL tuning procedure.

The two PLLs interact through the relative angle $\delta$ and the supplementary compensation path. The auxiliary PLL does not alter the main synchronization loop directly; instead, $\delta$ is injected into the rotor-current compensation branch, which subsequently affects the PCC dynamics. Therefore, the final parameter selection should also preserve adequate stability margin for the complete Double-PLL-Based impedance-reshaping system.

Figure 11 compares the generalized Nyquist eigenloci of the original system and the system using the proposed Double-PLL-Based method. The enlarged view shows that the proposed method shifts the critical eigenloci away from the critical point $-1$. After compensation, the eigenloci do not encircle $-1$; since the open-loop system has $P=0$, the proposed system has $N=0$ and hence $Z=0$, confirming closed-loop stability according to the generalized Nyquist criterion. This result demonstrates that the supplementary compensation branch reshapes the DFIG output impedance, mitigates the adverse influence of PLL dynamics, and improves the system stability margin.

5. Simulation and HIL Experimental Results

The Double-PLL-Based implementation replaces the original method of formulating the compensation amount relative to a fixed nominal-frequency reference with an approach based on the relative synchronous angle difference, thereby enhancing the adaptability of the control strategy to grid-frequency deviations. The parameters in Table 2 are selected for simulations of a representative 1.5 MW, 690 V DFIG system and are not manufacturer data. The rated and electrical parameters define the operating point, whereas the controller gains are selected for the considered simulations; the PLL gains follow the tuning principle in Section 4.2. Under weak-grid conditions (SCR=2), the wind turbine utilizes the parameters provided in Table 2.

At 2.5 s, the system SCR is altered to 1.95, and the resulting waveforms are illustrated in Figure 12. Upon changing the system short-circuit ratio at 2.5 s, the harmonics of the system waveforms increase significantly, and the active and reactive powers exhibit apparent oscillations, indicating that the system has become unstable due to oscillations. The FFT analysis of the system voltage reveals oscillation components at 140 Hz and 250 Hz. This indicates that, under weak-grid conditions, the system is highly susceptible to oscillations induced by q-axis voltage disturbances in the PLL. Therefore, the proposed method targets PLL-induced oscillations within the 10–300 Hz range, including the dominant supersynchronous components at 140 Hz and 250 Hz observed in this case.

Table 2. Simulation parameters of the representative DFIG system.

Parameter	Symbol	Value
Rated capacity	$S_N$	1.5 MW
Rated voltage	$U_N$	690 V
Rated active power	$P_N$	1.2 MW
Rated reactive power	$Q_N$	0 Mvar
Rated frequency	$f_N$	50 Hz
DC-link capacitance	$C_{dc}$	1 × 10−2 F
DC-link voltage	$V_{dc}$	1150 V
Filter resistance	$R_f$	3.03 × 10−5 Ω
Filter inductance	$L_f$	2.02 × 10−5 H
Stator resistance	$R_s$	9.52 × 10−3 Ω
Rotor resistance	$R_r$	5.08 × 10−3 Ω
Magnetizing inductance	$L_m$	2.93 × 10−3 H
Stator leakage inductance	$L_{ls}$	1.82 × 10−4 H
Rotor leakage inductance	$L_{lr}$	1.62 × 10−4 H
Power loop proportional gain	$k_{ppq}$	0.02
Power loop integral gain	$k_{ipq}$	0.002
Rotor current loop proportional gain	$k_{pir}$	1.2
Rotor current loop integral gain	$k_{iir}$	10
Main PLL proportional gain	$k_{ppll,1}$	10
Main PLL integral gain	$k_{ipll,1}$	100
Auxiliary PLL proportional gain	$k_{ppll,2}$	5
Auxiliary PLL integral gain	$k_{ipll,2}$	50
DC voltage loop proportional gain	$k_{pdc}$	6
DC voltage loop integral gain	$k_{idc}$	400
Grid-side current loop proportional gain	$k_{pii}$	10
Grid-side current loop integral gain	$k_{iii}$	100

Table 2. Simulation parameters of the representative DFIG system.

Parameter	Symbol	Value
Rated capacity	$S_N$	1.5 MW
Rated voltage	$U_N$	690 V
Rated active power	$P_N$	1.2 MW
Rated reactive power	$Q_N$	0 Mvar
Rated frequency	$f_N$	50 Hz
DC-link capacitance	$C_{dc}$	1 × 10−2 F
DC-link voltage	$V_{dc}$	1150 V
Filter resistance	$R_f$	3.03 × 10−5 Ω
Filter inductance	$L_f$	2.02 × 10−5 H
Stator resistance	$R_s$	9.52 × 10−3 Ω
Rotor resistance	$R_r$	5.08 × 10−3 Ω
Magnetizing inductance	$L_m$	2.93 × 10−3 H
Stator leakage inductance	$L_{ls}$	1.82 × 10−4 H
Rotor leakage inductance	$L_{lr}$	1.62 × 10−4 H
Power loop proportional gain	$k_{ppq}$	0.02
Power loop integral gain	$k_{ipq}$	0.002
Rotor current loop proportional gain	$k_{pir}$	1.2
Rotor current loop integral gain	$k_{iir}$	10
Main PLL proportional gain	$k_{ppll,1}$	10
Main PLL integral gain	$k_{ipll,1}$	100
Auxiliary PLL proportional gain	$k_{ppll,2}$	5
Auxiliary PLL integral gain	$k_{ipll,2}$	50
DC voltage loop proportional gain	$k_{pdc}$	6
DC voltage loop integral gain	$k_{idc}$	400
Grid-side current loop proportional gain	$k_{pii}$	10
Grid-side current loop integral gain	$k_{iii}$	100

Under the proposed control strategy, when the SCR drops from 2 to 1.9 at 1.5 s, the simulation results in Figure 13 show no obvious harmonic amplification. The voltage harmonic content remains below 5%. When the SCR drops to 1.8 at 1.5 s, Figure 14 shows that the system returns to stable operation after a brief oscillation of approximately 0.1 s, and the steady-state voltage harmonic content remains below 5%. The quantitative comparison in

Table 3. Comparison between the proposed control strategy and the conventional strategy under different SCR conditions.

Method	SCR	Stability	Oscillation Frequency	Harmonic Content
Single PLL	1.95	Oscillation	140 Hz, 250 Hz	18.56%
Proposed method	2	Stable	-	<5%
	1.9	Stable	-	<5%
	1.8	Stable after 0.1 s oscillation	-	<5%

therefore indicates that the proposed strategy retains adaptability under weak-grid conditions.

To demonstrate the adaptability of the proposed control strategy to grid-frequency deviations, simulations were conducted under three frequency-variation scenarios: continuous frequency drop, continuous frequency rise, and step frequency change. The results were compared with those obtained using the rotor-current feedforward compensation method [21], as detailed in

Table 4. Comparison between the proposed control strategy and rotor-current feedforward compensation under different frequency-variation scenarios.

Method	Variation	Oscillation Amplitude (Voltage)	Damping Sign	Settling Time	Voltage Overshoot	Stability
Method [21]	Drop 0.5 Hz/s	Divergent	−	-	System divergence	Unstable
Method [21] with increased ${\omega }_{\mathrm{HPF}}$	Drop 0.5 Hz/s	250 V	+	0.07 s	41%	Stable
	Drop 1 Hz/s	250 V	+	0.07 s	41%	Stable
	Rise 0.5 Hz/s	250 V	+	0.07 s	41%	Stable
	Rise 1 Hz/s	260 V	+	0.07 s	43%	Stable
	Step 2 Hz	250 V	+	0.08 s	41%	Stable
	Step −2 Hz	220 V	+	0.08 s	36%	Stable
Proposed strategy	Drop 0.5 Hz/s	1.2 V	+	0 s	0.2%	Stable
	Drop 1 Hz/s	1.3 V	+	0 s	0.2%	Stable
	Rise 0.5 Hz/s	1.2 V	+	0 s	0.2%	Stable
	Rise 1 Hz/s	1.2 V	+	0 s	0.2%	Stable
	Step 2 Hz	1.3 V	+	0 s	0.2%	Stable
	Step −2 Hz	1.2 V	+	0 s	0.2%	Stable

. Figure 15 shows the response of the system using rotor-current feedforward compensation under a continuous grid-frequency drop. The grid frequency decreases linearly from 50 Hz at $t=1\mathrm{s}$ to 49.5 Hz at $t=2\mathrm{s}$, corresponding to a drop rate of 0.5 Hz/s. With ${\omega }_{\mathrm{HPF}}=2\pi \mathrm{rad}/\mathrm{s}$, the gain coefficient $A(\infty )=1$, and the quality factor $Q=1$, the nonzero frequency deviation causes the compensation error to accumulate continuously, ultimately resulting in instability and oscillation.

Figure 16, Figure 17 and Figure 18 show the results obtained using rotor-current feedforward compensation under continuous frequency drop, continuous frequency rise, and step frequency change after increasing ${\omega }_{\mathrm{HPF}}$. In all cases, the disturbance applied at $t=1\mathrm{s}$ causes brief but pronounced transients in the stator-voltage, stator-current, and rotor-current waveforms. The active and reactive powers also exhibit short spikes before rapidly returning to steady values without sustained oscillation.

This indicates that the filtering stage can more effectively mitigate the low-frequency bias components induced by frequency deviations, allowing the compensation branch to act primarily on dynamic disturbance components and thus preventing the continuous accumulation of system errors. Consequently, the impedance reshaping strategy can function properly, and the system waveforms recover to a stable state relatively quickly. This demonstrates that, after appropriately increasing ${\omega }_{\mathrm{HPF}}$, the single-PLL-based impedance reshaping strategy can still achieve a certain degree of stability enhancement under frequency deviation conditions.

Figure 19, Figure 20 and Figure 21 present the simulation results of the proposed method under scenarios of continuous grid frequency drop, continuous frequency rise, and step frequency changes (step-up or step-down). The simulation results reveal that when the frequency disturbance occurs at $t=1\mathrm{s}$, the three-phase stator voltage, stator current, and rotor current waveforms of the system maintain a highly smooth, symmetrical sinusoidal profile throughout the entire simulation period under both continuous frequency variation and step disturbance scenarios. Unlike the rotor-current feedforward compensation method, the voltage and current waveforms under the proposed method exhibit no visible transient distortion, amplitude fluctuation, or harmonic amplification throughout the process. The active and reactive powers rapidly converge to their steady-state values.

These results demonstrate that the proposed control strategy maintains stable operation and suppresses pronounced transients under the grid-frequency variations considered in this study, thereby improving its adaptability to off-nominal-frequency conditions.

It is worth noting that, after the Double-PLL-Based impedance-reshaping strategy is introduced, the compensation branch no longer constructs the control variable from the deviation of the main PLL relative to a fixed nominal-frequency reference. Instead, it uses the relative deviation between the main and auxiliary PLLs for adaptive correction. Because both PLLs track the actual grid frequency in steady state, their steady-state frequency difference is zero, and the relative-angle signal does not accumulate continuously under grid-frequency deviations. Consequently, the compensation signal primarily reflects system dynamics without introducing a steady-state bias into the control channel. This feature eliminates the need for a dedicated high-pass filter for compensation-drift suppression and avoids the associated filter-tuning trade-off.

The aforementioned simulation results demonstrate that the proposed method exhibits strong robustness under scenarios of weak grids and frequency variations. To further investigate its performance when the internal control parameters change, the following simulation tests were conducted.

Figure 22 shows the system waveforms before and after changing the current-loop parameters. At $t=1\mathrm{s}$ and $t=2\mathrm{s}$, these parameters are stepped to 1.2 and 0.8 times their nominal values, respectively. No obvious deterioration is observed in the waveforms, and the system remains stable throughout the test, demonstrating the robustness of the proposed method to current-loop parameter variations.

Figure 23 presents the simulated system waveforms before and after changing the PLL parameters. At $t=1\mathrm{s}$ and $t=2\mathrm{s}$, the PLL parameters are stepped to 1.2 and 0.8 times their nominal values, respectively. The waveforms exhibit no obvious deterioration, sustained oscillation, or pronounced power overshoot, and the system remains stable throughout the test. These results demonstrate the robustness of the proposed control strategy under the considered PLL-parameter variations.

Under weak grid conditions, multiple operating conditions that cause system instability may occur simultaneously. Therefore, it is necessary to evaluate the robustness of the proposed method under complex operating conditions. This paper considers the following complex scenario: At $t=1\mathrm{s}$, the frequency steps to 52 Hz, and the SCR decreases from 2 to 1.8; at $t=3\mathrm{s}$, the frequency undergoes a step decrease of 2 Hz, and the current loop parameters abruptly step to 1.2 times their nominal values; at $t=4\mathrm{s}$, the PLL parameters drop to 80% of their nominal values. Under these conditions, the simulation results of the system are illustrated in Figure 24.

To further validate the simulation results in Figure 24, an HIL experiment was performed using the same PLL-parameter variation sequence. The measured three-phase stator-voltage and stator-current waveforms are shown in Figure 25. Following the PLL-parameter variations, both waveforms remain balanced and exhibit no sustained oscillatory growth, which is consistent with the simulation results and confirms the robustness of the proposed strategy under the considered PLL-parameter perturbations.

6. Conclusions

To address the susceptibility of DFIGs to PLL-induced oscillations under weak-grid conditions, alongside the inadequate adaptability of conventional single-PLL impedance-reshaping methods during frequency deviations, this paper has proposed a Double-PLL-Based impedance-reshaping control strategy. The analysis has revealed that the additional phase lag introduced by the PLL within specific frequency bands is a critical factor inducing the observed oscillations. Accordingly, a supplementary compensation channel has been constructed using the relative dynamic deviation between the main and auxiliary PLLs, effectively reshaping the impedance characteristics within the target frequency band. This method not only avoids the steady-state bias caused by frequency deviations but also eliminates the need for a dedicated high-pass filter for compensation-drift suppression, thereby avoiding the associated filter-tuning trade-off. Theoretical analysis and simulation results have demonstrated that the proposed strategy improves the stability margin of the system under the weak-grid conditions considered. The extended simulations further demonstrate stable operation at an SCR of 1.8, effective oscillation suppression during the tested frequency variations, and robustness under the considered internal-parameter changes and combined operating conditions. The HIL experimental results corroborate the stable three-phase stator-voltage and stator-current responses under PLL-parameter variations. Overall, the results indicate improved transient performance within the investigated operating range. Under extremely weak-grid conditions, however, instability may still occur; future work will therefore investigate control strategies for stable operation during frequency variations in such scenarios.