Impedance-Based Criterion Design for Grid-Following/Grid-Forming Switching Control of Wind Generation System

Abstract

As wind farms connect with power grids via long-distance transmission lines and multiple transformers, the resulting networks exhibit inherently weak and fluctuating grid strength—subject to dynamic variations from external disturbances like generation tripping; thus, ensuring stable operation presents a critical challenge. To address this issue, grid-following/grid-forming mode switching has emerged as a strategic approach, where wind turbine generators strategically switch between grid-following control in strong grids and grid-forming control in weak grids. This paper proposes an impedance-based switching criteria design methodology, ensuring system stability under dynamic grid strength variations. Leveraging sequence impedance analysis in frequency domain, we establish a stability boundary for mode transitions. Simulation results demonstrate that the proposed criteria maintain stable operation, enhancing robustness against grid strength fluctuations.

1. Introduction

The widespread integration of renewable energy, especially wind and solar power [1,2], through power electronic converters has significantly transformed modern power systems [3]. Wind generation systems (as depicted in Figure 1), predominantly connected via long-distance transmission lines and multiple transformers [4], result in weak grid conditions in networks [5]. Moreover, increasing renewable penetration increases the complexity of power systems, where grid connection or disconnection of large-scale sources or loads can induce significant grid strength fluctuations [6]. Grid strength, quantified by the short circuit ratio (SCR), is categorized into strong (SCR > 3), weak (2 < SCR < 3), and very weak (SCR ≤ 2) [7,8].

Wind turbine stability is critically dependent on SCR variations across these regimes. Within wind farms, internal line impedances are negligible due to short electrical distances between turbines, ensuring stable inter-turbine connections [9]. The primary stability challenge arises from wind farm–grid interaction instabilities under weak grid conditions. Ensuring stable operation across wide SCR ranges therefore constitutes an urgent research priority for modern power systems.

Most wind turbines employ grid-following (GFL) control, utilizing phase-locked loop (PLL) for grid synchronization and vector current control for output power regulation [10]. Consequently, GFL-controlled wind turbines require strong grid conditions. Under weak grids, the PLL introduces negative damping that may trigger system instability. As demonstrated in [11,12], excessive PLL bandwidth may induce harmonic resonance in converters operating under weak grid conditions. Therefore, researchers propose reducing PLL bandwidth to enhance stability [13]; however, this approach compromises the dynamic performance.

In contrast to GFL control, grid-forming (GFM) control has been proposed from multiple perspectives [14,15]. GFM control eliminates PLL and emulates synchronous generator behavior, enabling wind turbines to operate as voltage-controlled sources [16]. This capability enables autonomous voltage and frequency support, independent of strong grid conditions [17,18]. Consequently, GFM-controlled wind turbines maintain robust small-signal stability even under weak to very weak grids. However, as noted in [19], excessively strong grids with small grid impedance may induce synchronous oscillations in GFM-controlled units.

Facing this instability challenge of GFL vulnerability in weak grids and GFM oscillations in strong grids, single-mode control exposes wind turbines to instability risks under wide-range grid strength variations [19,20]; thus, dual-mode control becomes essential. To realize this, ref. [21] proposes a dual-characteristic control integrating GFL and GFM controls, enhancing system damping and grid-connection stability. Ref. [22] develops an adaptive switching control strategy, where converters operate in GFL mode under strong grids (SCR > 2) and switch to GFM mode in weak grids (SCR ≤ 2).

Since both GFL and GFM control, as well as their switching strategy, are implemented within the converter controller, no additional physical switching devices are required for the transition. By embedding preprocessing functions in the control loop, a smooth and disturbance-free transition between the two modes can be achieved. For example, ref. [23] designed a self closing loop regulated current switch to ensure reference current balance during the mode transition of permanent magnet synchronous generators (PMSG) under VSG control, which guarantees stable operation. Similarly, ref. [24] proposed a seamless GFL/GFM switching method for renewable energy conversion systems, effectively mitigating voltage and current distortion during the transition and thereby ensuring reliable converter operation. These studies preliminarily verify the feasibility of applying such switching strategies in practical wind farms.

These methods ensure continuous stability during grid strength transitions, though the fixed threshold, e.g., SCR = 2 in [22], lacks adaptability to actual situations. This paper derives the switching criterion through sequential impedance modeling and stability analysis under both GFL and GFM controls, enabling real-time mode adaptation to grid impedance variations. The major contributions of this paper are as follows:

• A switching criterion design methodology ensuring small-signal stability across wide SCR variations, based on impedance stability analysis;
• Dynamic triggered switching mechanism via Extended Kalman Filter (EKF)-based impedance tracking, enabling real-time response to grid condition changes.

The remainder of this paper is arranged as follows. Section 2 presents the GFL/GFM switching control framework of wind turbines. Section 3 establishes sequence impedance models for both control modes. Section 4 investigates small-signal stability and proposes the switching criterion derivation process. Section 5 describes the method of identifying the grid impedance using EKF method. Section 6 provides simulation validation. Section 7 introduces the experimental verification plan. Section 8 concludes the article.

2. Wind Turbine System Under GFL/GFM Switching Scheme

2.1. Circuit Topology

In contrast to doubly-fed induction generators, the simpler control structure of PMSGs makes them more amenable to GFM control implementation [25]. Hence, this study concentrates on PMSG-based turbines. For these systems, the focus is on grid-connection characteristics, while turbine-side dynamics are omitted for simplicity by assuming a constant DC voltage $u_{\mathrm{dc}}$ [26].

The circuit topology of the PMSG-based wind turbine generator is illustrated in Figure 2, connected to the grid through the grid-connected converter. Figure 3 depicts the converter structure viewed from the grid end. $e_{\mathrm{a}}$, $e_{\mathrm{b}}$, $e_{\mathrm{c}}$ are internal electric potentials; $i_{\mathrm{a}}$, $i_{\mathrm{b}}$, $i_{\mathrm{c}}$, and $u_{\mathrm{a}}$, $u_{\mathrm{b}}$, $u_{\mathrm{c}}$ are output currents and output voltages. The LC filter is composed of $L_{\mathrm{f}}$, $C_{\mathrm{f}}$, and $R_{\mathrm{f}}$, which represent the filter inductance, filter capacitor, and damping resistance. The equivalent line inductance and resistance of the grid are represented by $L_{\mathrm{g}}$ and $R_{\mathrm{g}}$. In weak girds, $L_{\mathrm{g}}$ dominates the impedance characteristics, rendering $R_{\mathrm{g}}$ negligible for analysis. Grid voltages $u_{\mathrm{ga}}$, $u_{\mathrm{gb}}$, $u_{\mathrm{gc}}$ and grid-connected currents $i_{\mathrm{ga}}$, $i_{\mathrm{gb}}$, $i_{\mathrm{gc}}$ are measured at the point of common coupling.

2.2. Switching Control Scheme

Figure 2 illustrates the architecture of the GFL-GFM switching controller. The detailed structure of GFL and GFM control is shown in Figure 4. The GFL control comprises the power loop, the current loop, and the PLL. Active/reactive power references are $P^{\mathrm{ref}}$ and $Q^{\mathrm{ref}}$. $u_{\mathrm{abc}}$ denotes the three-phase output voltage, while $V_1$ represents its amplitude. The output currents in the dq-axis coordinate system are denoted by $i_{\mathrm{d}}$ and $i_{\mathrm{q}}$. Under GFL control, $u_{\mathrm{GFL},\mathrm{d}}$ and $u_{\mathrm{GFL},\mathrm{q}}$ correspond to the dq-axis output voltages, with $I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}$ and $I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}}$ representing the dq-axis current references. The phase angle generated by GFL synchronization is ${\theta }_{\mathrm{GFL}}$, and $c_{\mathrm{GFL},\mathrm{dq}}$ signifies the dq-frame modulation signal for GFL control, which is converted to switching signal $S_{\mathrm{abc}}$.

The GFM control implements the droop-controlled power loop, the voltage loop and the current loop. Output angular frequency ${\omega }_{\mathrm{GFM}}$, rated angular frequency ${\omega }^{\mathrm{ref}}$ relate through phase-drooping coefficient $K_{\mathrm{P}}$, and output voltage $U_{\mathrm{GFM},\mathrm{d}}$, rated voltage $U^{\mathrm{ref}}$ relate through voltage-drooping coefficient $K_{\mathrm{Q}}$, with instantaneous active/reactive power represented as P and Q. $\mathsf{\Delta }\omega$ and $\mathsf{\Delta }U$ indicating angular frequency and voltage respective deviations. $I_{\mathrm{GFM},\mathrm{q}}^{\mathrm{ref}}$ and $I_{\mathrm{GFM},\mathrm{q}}^{\mathrm{ref}}$ specify the dq-axis current references. The GFM controller generates phase angle ${\theta }_{\mathrm{GFM}}$, producing dq-frame modulation signal $c_{\mathrm{GFM},\mathrm{dq}}$.

Mode switching is triggered by real-time EKF-based grid impedance identification: transition to GFM occurs when grid strength falls below threshold ${\lambda }_{\mathrm{b}}$, while reversion to GFL initiates when strength exceeds ${\lambda }_{\mathrm{b}}$.

To achieve disturbance-free mode transitions, a pre-synchronization control strategy is implemented [19]. Leveraging the identical current loop structure of both control modes, the current reference should be maintained constant at the switching moment. For GFL-to-GFM transitions, pre-synchronization activates prior to switching, where phase and amplitude tracking loop synchronizes the GFM parameters to the grid conditions via

$$\left\{\begin{array}{cc}\hfill \omega & ={\omega }^{\mathrm{ref}}+\mathsf{\Delta }{\omega }_{\mathrm{syn}}={\omega }^{\mathrm{ref}}+{\displaystyle \frac{K_{\omega }}{s}}\left({\theta }_{\mathrm{GFL}}-{\theta }_{\mathrm{GFM}}\right)\hfill \\ \hfill U_{\mathrm{GFM},d}& =U^{\mathrm{ref}}+\mathsf{\Delta }{U}_{\mathrm{syn}}=U^{\mathrm{ref}}+{\displaystyle \frac{K_U}{s}}\left(V_1-U_{\mathrm{GFM},d}\right)\hfill \end{array}\right.$$

(1)

where integral gains for voltage phase and amplitude synchronization are denoted by $K_{\omega }$ and $K_U$, respectively. Note that GFM-to-GFL transitions require no pre-synchronization since grid voltage directly locks the phase in GFL mode.

3. Derivation of the Wind Turbine’s Sequence Impedance Model

The harmonic linearization method [27,28,29] is employed to derive the sequential impedance model for GFL/GFM-controlled wind turbines. Second and higher-order nonlinear terms are disregarded in this approach.

3.1. Sequential Impedance Modeling for GFL-Controlled Turbines

Under the injection of a small-signal perturbation into the system, the resulting time-domain expressions for grid-side converter output voltages and currents are

$$u_{\mathrm{a}}\left(t\right)=V_{\mathrm{l}}cos\left(2\pi f_{\mathrm{l}}t\right)+V_{\mathrm{p}}cos(2\pi f_{\mathrm{p}}t+{\phi }_{\mathrm{v}\mathrm{p}})+V_{\mathrm{n}}cos(2\pi f_{\mathrm{n}}t+{\phi }_{\mathrm{v}\mathrm{n}}),$$

(2)

$$i_{\mathrm{a}}\left(t\right)=I_{\mathrm{l}}cos(2\pi f_{\mathrm{l}}t+{\phi }_{\mathrm{i}1})+I_{\mathrm{p}}cos(2\pi f_{\mathrm{p}}t+{\phi }_{\mathrm{i}\mathrm{p}})+I_{\mathrm{n}}cos(2\pi f_{\mathrm{n}}t+{\phi }_{\mathrm{i}\mathrm{n}}),$$

(3)

where $V_1$, $V_{\mathrm{p}}$, $V_{\mathrm{n}}$ denote amplitudes of fundamental voltage, positive-sequence voltage perturbation, and negative-sequence voltage perturbation; $I_1$, $I_{\mathrm{p}}$, $I_{\mathrm{n}}$ represent amplitudes of fundamental current, positive-sequence current response, and negative-sequence current response; $f_1$, $f_{\mathrm{p}}$, $f_{\mathrm{n}}$ indicate fundamental frequency, positive-sequence perturbation frequency, and negative-sequence perturbation frequency; ${\phi }_{\mathrm{vp}}$, ${\phi }_{\mathrm{vn}}$ denote the initial phase angles for the positive- and negative-sequence voltage perturbations, respectively; ${\phi }_{\mathrm{i}1}$, ${\phi }_{\mathrm{ip}}$, ${\phi }_{\mathrm{in}}$ are initial phase angles of fundamental, positive-sequence, negative-sequence current responses.

The frequency-domain expressions can be described as follows:

$${\mathbf{U}}_{\mathrm{a}}\left[f\right]=\left\{\begin{array}{cc}{\mathbf{V}}_{\mathrm{l}},\hfill & f=\pm f_{\mathrm{l}}\hfill \\ {\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,{\mathbf{I}}_{\mathrm{a}}\left[f\right]=\left\{\begin{array}{cc}{\mathbf{I}}_{\mathrm{l}},\hfill & f=\pm f_{\mathrm{l}}\hfill \\ {\mathbf{I}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\mathbf{I}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,$$

(4)

where ${\mathbf{V}}_{\mathrm{l}}$, ${\mathbf{V}}_{\mathrm{p}}$, ${\mathbf{V}}_{\mathrm{n}}$, ${\mathbf{I}}_{\mathrm{l}}$, ${\mathbf{I}}_{\mathrm{p}}$, ${\mathbf{I}}_{\mathrm{n}}$ represent the phasors of fundamental and disturbed voltages and currents. ${\mathbf{V}}_1=V_1/2$; ${\mathbf{V}}_{\mathrm{p}}=V_{\mathrm{p}}/2·e^{\pm j{\phi }_{\mathrm{vp}}}$; ${\mathbf{V}}_{\mathrm{n}}=V_{\mathrm{p}}/2·e^{\pm j{\phi }_{\mathrm{vn}}}$; ${\mathbf{I}}_1=I_1/2·e^{\pm j{\phi }_{\mathrm{i}1}}$; ${\mathbf{I}}_{\mathrm{p}}=I_{\mathrm{p}}/2·e^{\pm j{\phi }_{\mathrm{ip}}}$; ${\mathbf{I}}_{\mathrm{n}}=I_{\mathrm{n}}/2·e^{\pm j{\phi }_{\mathrm{in}}}$.

According to the topology in Figure 3, the following relationship between voltages can be obtained:

$$s{L}_{\mathrm{f}}\left[\begin{array}{c}{i}_{\mathrm{a}}\hfill \\ i_{\mathrm{b}}\hfill \\ i_{\mathrm{c}}\hfill \end{array}\right]=\left[\begin{array}{c}{e}_{\mathrm{a}}\hfill \\ e_{\mathrm{b}}\hfill \\ e_{\mathrm{c}}\hfill \end{array}\right]-\left[\begin{array}{c}{u}_{\mathrm{a}}\hfill \\ u_{\mathrm{b}}\hfill \\ u_{\mathrm{c}}\hfill \end{array}\right].$$

(5)

Taking a-phase as example, $e_{\mathrm{a}}=K_{\mathrm{PWM}}{u}_{\mathrm{dc}}{c}_{\mathrm{a}}+K_{\mathrm{f}}{v}_{\mathrm{a}}$, where $K_{\mathrm{PWM}}$ represents the PWM coefficient and $K_{\mathrm{f}}$ denotes the feedforward coefficient.

The phase angle ${\theta }_{\mathrm{PLL}}$ (${\theta }_{\mathrm{GFL}}$) combines the positive-sequence angle ${\theta }_1$ from the fundamental voltage, and the perturbation component $\mathsf{\Delta }\theta$ associated with voltage disturbances, resulting in ${\theta }_{\mathrm{PLL}}={\theta }_1+\mathsf{\Delta }\theta$. The coordinate transformation matrix $T\left({\theta }_{\mathrm{PLL}}\right)$, accounting for perturbations, is

$$\begin{array}{cc}\hfill T\left({\theta }_{\mathrm{PLL}}\right)& ={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}cos{\theta }_{\mathrm{PLL}}& cos({\theta }_{\mathrm{PLL}}-2\pi /3)& cos({\theta }_{\mathrm{PLL}}+2\pi /3)\\ -sin{\theta }_{\mathrm{PLL}}& -sin({\theta }_{\mathrm{PLL}}-2\pi /3)& -sin({\theta }_{\mathrm{PLL}}+2\pi /3)\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\end{array}\right]\hfill \\ \hfill & \approx \left[\begin{array}{ccc}1& \mathsf{\Delta }\theta & 0\\ -\mathsf{\Delta }\theta & 1& 0\\ 0& 0& 1\end{array}\right]·{\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}cos{\theta }_1& cos({\theta }_1-2\pi /3)& cos({\theta }_1+2\pi /3)\\ -sin{\theta }_1& -sin({\theta }_1-2\pi /3)& -sin({\theta }_1+2\pi /3)\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\end{array}\right]\hfill \end{array}.$$

(6)

In the frequency-domain, PLL generates perturbation variable $\mathsf{\Delta }\theta \left[f\right]$ as follows [20]:

$$\mathsf{\Delta }\theta \left[f\right]=\left\{\begin{array}{cc}\mp j{H}_{\mathrm{PLL}}\left(s\right)/\left[1+V_1{H}_{\mathrm{PLL}}\left(s\right)\right]G_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_{\mathrm{l}})\hfill \\ \pm j{H}_{\mathrm{PLL}}\left(s\right)/\left[1+V_1{H}_{\mathrm{PLL}}\left(s\right)\right]G_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_{\mathrm{l}})\hfill \end{array}\right.,$$

(7)

where $H_{\mathrm{P}\mathrm{L}\mathrm{L}}\left(s\right)=(K_{\mathrm{p}\mathrm{P}\mathrm{L}\mathrm{L}}+K_{\mathrm{i}\mathrm{P}\mathrm{L}\mathrm{L}}/s)/s$. $K_{\mathrm{p}\mathrm{P}\mathrm{L}\mathrm{L}}$ and $K_{\mathrm{i}\mathrm{P}\mathrm{L}\mathrm{L}}$ denote the proportional and integral gains of the PLL. The transfer function $G_{\mathrm{v}}\left(s\right)=e^{-T_{\mathrm{s}}s}(1-e^{-T_{\mathrm{s}}s})/\left[\left(T_{\mathrm{s}}s\right)(1+s/{\omega }_{\mathrm{v}})\right]$ models the combined effect of sampling delay, PWM delay, and a sampling low-pass filter, where $T_{\mathrm{s}}$ is the sampling period and ${\omega }_{\mathrm{v}}$ is the filter’s cutoff angular frequency.

Given that $\mathsf{\Delta }\theta$ constitutes a small perturbation, it can be approximated that

$$\begin{array}{cc}\hfill cos{\theta }_{\mathrm{PLL}}& =cos({\theta }_1+\mathsf{\Delta }\theta )=cos{\theta }_{1}cos\mathsf{\Delta }\theta -sin{\theta }_{1}sin\mathsf{\Delta }\theta \approx cos{\theta }_1-\mathsf{\Delta }\theta sin{\theta }_1\hfill \\ \hfill sin{\theta }_{\mathrm{PLL}}& =sin({\theta }_1+\mathsf{\Delta }\theta )=sin{\theta }_{1}cos\mathsf{\Delta }\theta +cos{\theta }_{1}sin\mathsf{\Delta }\theta \approx sin{\theta }_1+\mathsf{\Delta }\theta cos{\theta }_1\hfill \end{array}.$$

(8)

Consequently, the frequency-domain expressions for $sin{\theta }_{\mathrm{PLL}}$ and $cos{\theta }_{\mathrm{PLL}}$ become

$$\begin{array}{cc}\hfill sin{\theta }_{\mathrm{PLL}}\left[f\right]& =\left\{\begin{array}{cc}\mp {\displaystyle \frac{j}{2}},\hfill & f=\pm f_1\hfill \\ \mp {\displaystyle \frac{j}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\mp j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\mp j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ \pm {\displaystyle \frac{j}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\pm j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\pm j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,\hfill \\ \hfill cos{\theta }_{\mathrm{PLL}}\left[f\right]& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & f=\pm f_1\hfill \\ {\displaystyle \frac{1}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\mp j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\mp j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\displaystyle \frac{1}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\pm j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\pm j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right..\hfill \end{array}$$

(9)

Inverter currents $i_{\mathrm{a}}$, $i_{\mathrm{b}}$, $i_{\mathrm{c}}$ are transformed into $i_{\mathrm{d}}$, $i_{\mathrm{q}}$ using the transformation matrix $T\left({\theta }_{\mathrm{PLL}}\right)$. Their frequency-domain representations are

$$\begin{array}{cc}\hfill I_{\mathrm{d}}\left[f\right]& =\left\{\begin{array}{cc}{I}_{1}cos{\theta }_1,\hfill & \mathrm{dc}\hfill \\ \mp j{I}_{1}sin{\theta }_1{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{p}}+G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \pm j{I}_{1}sin{\theta }_1{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{n}}+G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.,\hfill \\ \hfill I_{\mathrm{q}}\left[f\right]& =\left\{\begin{array}{cc}{I}_{1}sin{\theta }_1,\hfill & \mathrm{dc}\hfill \\ \pm j{I}_{1}cos{\theta }_1{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{p}}\mp j{G}_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \mp j{I}_{1}cos{\theta }_1{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}}\pm j{G}_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.,\hfill \end{array}$$

(10)

where $G_{\mathrm{i}}\left(s\right)=e^{-T_{\mathrm{s}}s}(1-e^{-T_{\mathrm{s}}s})/\left[\left(T_{\mathrm{s}}s\right)(1+s/{\omega }_{\mathrm{i}})\right]$, which simulates sampling delay, PWM delay, and sampling low-pass filter. ${\omega }_{\mathrm{i}}$ is the cut-off angular frequency of the low-pass filter of current signal.

With current loop $c_{\mathrm{GFL},\mathrm{d}}=(I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}})H_{\mathrm{i}}\left(s\right)-K_{\mathrm{d}}{i}_{\mathrm{q}}$, $c_{\mathrm{GFL},\mathrm{q}}=(I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}}-i_{\mathrm{q}})H_{\mathrm{i}}\left(s\right)+K_{\mathrm{d}}{i}_{\mathrm{d}}$, and inverse coordinate transformation matrix $T^{-1}\left({\theta }_{\mathrm{PLL}}\right)$, the a-phase modulation wave $c_{\mathrm{GFL},\mathrm{a}}$ in the frequency domain are obtained as follows:

$$C_{\mathrm{a}}\left[f\right]=\left\{\begin{array}{c}\mp j{\displaystyle \frac{1}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\mp j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\mp j2\pi f_1)+1}}{H}_{\mathrm{i}}(s\mp j2\pi f_1)\left(-I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}}\pm j{I}_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}\right)G_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}}\hfill \\ +\left[-H_{\mathrm{i}}(s\mp j2\pi f_1)\pm j{K}_{\mathrm{d}}\right]G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{p}},f=\pm f_{\mathrm{p}}\hfill \\ \pm j{\displaystyle \frac{1}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\pm j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\pm j2\pi f_1)+1}}{H}_{\mathrm{i}}(s\pm j2\pi f_1)\left(-I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}}\mp j{I}_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}\right)G_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{n}}\hfill \\ +\left[-H_{\mathrm{i}}(s\pm j2\pi f_1)\mp j{K}_{\mathrm{d}}\right]G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{n}},f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,$$

(11)

where $K_{\mathrm{d}}$ is the decoupling gain. $H_{\mathrm{i}}\left(s\right)=(K_{\mathrm{p}\mathrm{i}}+K_{\mathrm{i}\mathrm{i}}/s)/s$. $K_{\mathrm{p}\mathrm{i}}$ and $K_{\mathrm{i}\mathrm{i}}$ are the proportional and integral coefficients of the current loop.

By substituting Equation (11) into Equation (5), the positive- and negative-sequence impedance for GFL-controlled turbines can be derived as follows:

$$\begin{array}{cc}\hfill Z_{\mathrm{p}}\left(s\right)& =-{\displaystyle \frac{{\mathbf{V}}_{\mathrm{p}}\left(s\right)}{{\mathbf{I}}_{\mathrm{p}}\left(s\right)}}={\displaystyle \frac{s{L}_{\mathrm{f}}+K_{\mathrm{PWM}}{u}_{\mathrm{dc}}{G}_{\mathrm{i}}\left(s\right)\left(H_{\mathrm{i}}(s-j2\pi f_1)-j{K}_{\mathrm{d}}\right)}{\begin{array}{c}\hfill 1-K_{\mathrm{PWM}}{u}_{\mathrm{dc}}{G}_{\mathrm{v}}\left(s\right)\left(K_{\mathrm{f}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s-j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s-j2\pi f_1)+1}}{H}_{\mathrm{i}}(s-j2\pi f_1)(I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}+j{I}_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\right)\end{array}}},\hfill \\ \hfill Z_{\mathrm{n}}\left(s\right)& =-{\displaystyle \frac{{\mathbf{V}}_{\mathrm{n}}\left(s\right)}{{\mathbf{I}}_{\mathrm{n}}\left(s\right)}}={\displaystyle \frac{s{L}_{\mathrm{f}}+K_{\mathrm{PWM}}{u}_{\mathrm{dc}}{G}_{\mathrm{i}}\left(s\right)\left(H_{\mathrm{i}}(s+j2\pi f_1)+j{K}_{\mathrm{d}}\right)}{\begin{array}{c}\hfill 1-K_{\mathrm{PWM}}{u}_{\mathrm{dc}}{G}_{\mathrm{v}}\left(s\right)\left(K_{\mathrm{f}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s+j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s+j2\pi f_1)+1}}{H}_{\mathrm{i}}(s+j2\pi f_1)(I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}-j{I}_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\right)\end{array}}}.\hfill \end{array}$$

(12)

3.2. Sequential Impedance Modeling for GFM-Controlled Turbines

Using Equation (4), a conversion from three-phase coordinates to two-phase coordinates is performed via the Clarke transform:

$$\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right]\left[\begin{array}{c}{u}_{\mathrm{a}}\\ u_{\mathrm{b}}\\ u_{\mathrm{c}}\end{array}\right],\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\end{array}\right]\left[\begin{array}{c}{i}_{\mathrm{a}}\\ i_{\mathrm{b}}\\ i_{\mathrm{c}}\end{array}\right].$$

(13)

The frequency-domain expressions of $u_{\alpha }$, $u_{\beta }$, $i_{\alpha }$, and $i_{\beta }$ can be obtained as follows:

$$\begin{array}{cc}\hfill {\mathbf{V}}_{\alpha }\left[f\right]& =\left\{\begin{array}{cc}{\mathbf{V}}_1,\hfill & f=\pm f_1\hfill \\ {\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,{\mathbf{V}}_{\beta }\left[f\right]=\left\{\begin{array}{cc}\mp j{\mathbf{V}}_1,\hfill & f=\pm f_1\hfill \\ \mp j{\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ \pm j{\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,\hfill \\ \hfill {\mathbf{I}}_{\alpha }\left[f\right]& =\left\{\begin{array}{cc}{\mathbf{I}}_1,\hfill & f=\pm f_1\hfill \\ {\mathbf{I}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\mathbf{I}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,{\mathbf{I}}_{\beta }\left[f\right]=\left\{\begin{array}{cc}\mp j{\mathbf{I}}_1,\hfill & f=\pm f_1\hfill \\ \mp j{\mathbf{I}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ \pm j{\mathbf{I}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right..\hfill \end{array}$$

(14)

The instantaneous active power P and reactive power Q are expressed in the frequency domain by applying the power formulation $P=1.5\left(v_{\alpha }{i}_{\alpha }+v_{\beta }{i}_{\beta }\right)$, $Q=1.5\left(v_{\beta }{i}_{\alpha }-v_{\alpha }{i}_{\beta }\right)$, and frequency-domain convolution theorem:

$$P\left[f\right]=\left\{\begin{array}{cc}{\displaystyle \frac{3}{2}}{I}_1{V}_{1}cos{\phi }_{\mathrm{i}1},\hfill & \mathrm{dc}\hfill \\ \\ {\displaystyle \frac{3}{2}}\left[G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{p}}{V}_1+G_{\mathrm{v}}(s\pm j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{p}}{e}^{\mp j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \\ {\displaystyle \frac{3}{2}}\left[G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}}{V}_1+G_{\mathrm{v}}(s\mp j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{n}}{e}^{\pm j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.,$$

(15)

$$Q\left[f\right]=\left\{\begin{array}{cc}-{\displaystyle \frac{3}{2}}{I}_1{V}_{1}sin{\phi }_{\mathrm{i}1},\hfill & \mathrm{dc}\hfill \\ \\ {\displaystyle \frac{3j}{2}}\left[\pm G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{p}}{V}_1\mp G_{\mathrm{v}}(s\pm j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{p}}{e}^{\mp j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \\ {\displaystyle \frac{3j}{2}}\left[\mp G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}}{V}_1\pm G_{\mathrm{v}}(s\mp j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{n}}{e}^{\pm j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right..$$

(16)

For GFM-controlled turbines, ${\theta }_{\mathrm{GFM}}$ is calculated by:

$${\theta }_{\mathrm{GFM}}=1/s·({\omega }^{\mathrm{ref}}+K_{\mathrm{P}}(P^{\mathrm{ref}}-P)).$$

(17)

Substituting Equation (15) into Equation (17) within the frequency domain provides the perturbation variable $\mathsf{\Delta }\theta$ for the droop-controlled power loop:

$$\mathsf{\Delta }\theta \left[f\right]=\left\{\begin{array}{cc}-{\displaystyle \frac{3K_{\mathrm{P}}}{2s}}\left[G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{p}}{V}_1+G_{\mathrm{v}}(s\pm j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{p}}{e}^{\mp j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \\ -{\displaystyle \frac{3K_{\mathrm{P}}}{2s}}\left[G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}}{V}_1+G_{\mathrm{v}}(s\mp j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{n}}{e}^{\pm j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right..$$

(18)

Therefore, the frequency-domain expressions for $sin{\theta }_{\mathrm{GFM}}$ and $cos{\theta }_{\mathrm{GFM}}$ are obtained:

$$\begin{array}{cc}\hfill sin{\theta }_{\mathrm{GFM}}\left[f\right]& =\left\{\begin{array}{cc}\mp {\displaystyle \frac{j}{2}},\hfill & f=\pm f_1\hfill \\ -{\displaystyle \frac{3K_{\mathrm{P}}}{4s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{p}}{V}_1+G_{\mathrm{v}}\left(s\right)I_1{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right),\hfill & f=\pm f_{\mathrm{p}}\hfill \\ -{\displaystyle \frac{3K_{\mathrm{P}}}{4s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{n}}{V}_1+G_{\mathrm{v}}\left(s\right)I_1{e}^{\pm j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right),\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,\hfill \\ \hfill cos{\theta }_{\mathrm{GFM}}\left[f\right]& =\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & f=\pm f_1\hfill \\ \mp {\displaystyle \frac{3K_{\mathrm{P}}}{4s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{p}}{V}_1+j{G}_{\mathrm{v}}\left(s\right)I_1{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right),\hfill & f=\pm f_{\mathrm{p}}\hfill \\ \pm {\displaystyle \frac{3K_{\mathrm{P}}}{4s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{n}}{V}_1+j{G}_{\mathrm{v}}\left(s\right)I_1{e}^{\pm j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right),\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right..\hfill \end{array}$$

(19)

The output voltage $U_{\mathrm{GFM},\mathrm{d}}^{\mathrm{ref}}$ is calculated by

$$U_{\mathrm{GFM},\mathrm{d}}^{\mathrm{ref}}=U^{\mathrm{ref}}+K_{\mathrm{Q}}(Q^{\mathrm{ref}}-Q).$$

(20)

Substituting Equation (16) into Equation (20), yields

$$U_{\mathrm{GFM},\mathrm{d}}^{\mathrm{ref}}\left[f\right]=\left\{\begin{array}{cc}{V}_1,\hfill & \mathrm{dc}\hfill \\ \\ \mp K_{\mathrm{Q}}·{\displaystyle \frac{3j}{2}}\left[G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{p}}{V}_1-G_{\mathrm{v}}(s\pm j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{p}}{e}^{\mp j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \\ \pm K_{\mathrm{Q}}·{\displaystyle \frac{3j}{2}}\left[G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}}{V}_1-G_{\mathrm{v}}(s\mp j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{n}}{e}^{\pm j{\phi }_{\mathrm{i}1}}\right],\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right..$$

(21)

Similar with GFL control, outcurrents $i_{\mathrm{a}}$, $i_{\mathrm{b}}$, $i_{\mathrm{c}}$ transform to $i_{\mathrm{d}}$, $i_{\mathrm{q}}$ via $T\left({\theta }_{\mathrm{GFM}}\right)$, and out voltages $u_{\mathrm{a}}$, $u_{\mathrm{b}}$, $u_{\mathrm{c}}$ transform to $u_{\mathrm{d}}$, $u_{\mathrm{q}}$ via $T\left({\theta }_{\mathrm{GFM}}\right)$.

With the following voltage loop and current loop, the dq-frame modulation signal is derived.

$$\left\{\begin{array}{cc}\hfill I_{\mathrm{GFM},\mathrm{d}}& =(U_{\mathrm{GFM},\mathrm{d}}^{\mathrm{ref}}-u_{\mathrm{d}})H_{\mathrm{v}}\left(s\right)\hfill \\ \hfill I_{\mathrm{GFM},\mathrm{q}}& =(U_{\mathrm{GFM},\mathrm{q}}^{\mathrm{ref}}-u_{\mathrm{q}})H_{\mathrm{v}}\left(s\right)\hfill \end{array}\right.,$$

(22)

$$\left\{\begin{array}{cc}\hfill c_{\mathrm{GFM},\mathrm{d}}& =(I_{\mathrm{GFM},\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}})H_{\mathrm{i}}\left(s\right)-K_{\mathrm{d}}{i}_{\mathrm{q}}\hfill \\ \hfill c_{\mathrm{GFM},\mathrm{q}}& =(I_{\mathrm{GFM},\mathrm{q}}^{\mathrm{ref}}-i_{\mathrm{q}})H_{\mathrm{i}}\left(s\right)+K_{\mathrm{d}}{i}_{\mathrm{d}}\hfill \end{array}\right.,$$

(23)

where $H_{\mathrm{v}}\left(s\right)=(K_{\mathrm{p}\mathrm{v}}+K_{\mathrm{i}\mathrm{v}}/s)/s$. $K_{\mathrm{p}\mathrm{v}}$ and $K_{\mathrm{i}\mathrm{v}}$ represent the proportional and integral gains for the voltage control loop within the GFM strategy.

The result from Equation (23) is then multiplied by the inverse coordinate transformation matrix $T^{-1}\left({\theta }_{\mathrm{GFM}}\right)$, yielding the frequency-domain representation of the a-phase modulation wave $c_{\mathrm{GFM},\mathrm{a}}$.

Combining with Equation (5), the positive-sequence and negative-sequence impedance are given by

$$Z_p\left(s\right)=-{\displaystyle \frac{{\mathbf{V}}_{\mathrm{p}}\left(s\right)}{{\mathbf{I}}_{\mathrm{p}}\left(s\right)}}={\displaystyle \frac{\begin{array}{c}-j{e}^{-j{\phi }_{\mathrm{i}1}}·\left[\begin{array}{c}4js(s-j2\pi f_1)L_{\mathrm{f}}+4(s-j2\pi f_1)G_{\mathrm{i}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}(j{H}_{\mathrm{i}}+K_{\mathrm{d}})\hfill \\ -3G_{\mathrm{i}}{H}_{\mathrm{i}}{H}_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}{V}_1\left[K_{\mathrm{P}}{V}_1+(s-j2\pi f_1)K_{\mathrm{Q}}\right]\hfill \end{array}\right]\times \\ \left\{\begin{array}{c}4e^{j{\phi }_{\mathrm{i}1}}(s-j2\pi f_1)\left[1-G_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}(K_{\mathrm{f}}-H_{\mathrm{i}}{H}_{\mathrm{v}})\right]\hfill \\ -3j{G}_{\mathrm{v}}{H}_{\mathrm{i}}{H}_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}{I}_1\left[K_{\mathrm{P}}{V}_1+K_{\mathrm{Q}}(s-j2\pi f_1)\right]\hfill \end{array}\right\}\end{array}}{\begin{array}{cc}\hfill & 16{(s-j2\pi f_1)}^2{\left[1-G_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}(K_{\mathrm{f}}-H_{\mathrm{i}}{H}_{\mathrm{v}})\right]}^2\hfill \\ \hfill & +9e^{-2j{\phi }_{\mathrm{i}1}}{G}_{\mathrm{v}}^2{H}_{\mathrm{i}}^2{H}_{\mathrm{v}}^2{K}_{\mathrm{PWM}}^2{u}_{\mathrm{dc}}^2{I}_1^2{\left[K_{\mathrm{P}}{V}_1+K_{\mathrm{Q}}(s-j2\pi f_1)\right]}^2\hfill \end{array}}},$$

(24)

$$Z_n\left(s\right)=-{\displaystyle \frac{{\mathbf{V}}_{\mathrm{n}}\left(s\right)}{{\mathbf{I}}_{\mathrm{n}}\left(s\right)}}={\displaystyle \frac{\begin{array}{c}j{e}^{j{\phi }_{\mathrm{i}1}}·\left[\begin{array}{c}-4js(s+j2\pi f_1)L_{\mathrm{f}}+4(s+j2\pi f_1)G_{\mathrm{i}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}(-j{H}_{\mathrm{i}}+K_{\mathrm{d}})\hfill \\ -3G_{\mathrm{i}}{H}_{\mathrm{i}}{H}_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}{V}_1\left[K_{\mathrm{P}}{V}_1+(s+j2\pi f_1)K_{\mathrm{Q}}\right]\hfill \end{array}\right]\times \\ \left\{\begin{array}{c}4e^{-j{\phi }_{\mathrm{i}1}}(s+j2\pi f_1)\left[1-G_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}(K_{\mathrm{f}}-H_{\mathrm{i}}{H}_{\mathrm{v}})\right]\hfill \\ +3j{G}_{\mathrm{v}}{H}_{\mathrm{i}}{H}_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}{I}_1\left[K_{\mathrm{P}}{V}_1+K_{\mathrm{Q}}(s+j2\pi f_1)\right]\hfill \end{array}\right\}\end{array}}{\begin{array}{cc}\hfill & 16{(s+j2\pi f_1)}^2{\left[1-G_{\mathrm{v}}{K}_{\mathrm{PWM}}{u}_{\mathrm{dc}}(K_{\mathrm{f}}-H_{\mathrm{i}}{H}_{\mathrm{v}})\right]}^2\hfill \\ \hfill & +9e^{2j{\phi }_{\mathrm{i}1}}{G}_{\mathrm{v}}^2{H}_{\mathrm{i}}^2{H}_{\mathrm{v}}^2{K}_{\mathrm{PWM}}^2{u}_{\mathrm{dc}}^2{I}_1^2{\left[K_{\mathrm{P}}{V}_1+K_{\mathrm{Q}}(s+j2\pi f_1)\right]}^2\hfill \end{array}}},$$

(25)

where simplified notations are employed; the complete expressions can be stated as $G_{\mathrm{i}}=G_{\mathrm{i}}\left(s\right)$, $G_{\mathrm{v}}=G_{\mathrm{v}}\left(s\right)$, $H_{\mathrm{i}}=H_{\mathrm{i}}(s\mp j2\pi f_1)$, $H_{\mathrm{v}}=H_{\mathrm{v}}(s\mp j2\pi f_1)$, $H_{\mathrm{PLL}}=H_{\mathrm{PLL}}(s\mp j2\pi f_1)$.

4. Small-Signal Stability Analysis and Switching Criterion Derivation

The parameters of the wind turbine system configured with GFL-GFM switching control are listed in

Table 1. System parameters.

Parameter	Value	Parameter	Value
$u_{\mathrm{dc}}$	1200 V	$P^{\mathrm{ref}}$	1 MW
$V_1$	$690\sqrt{2}$ V	$Q^{\mathrm{ref}}$	0 Var
$L_{\mathrm{f}}$	0.15 mH	$K_{\mathrm{pi}}$	0.3
$C_{\mathrm{f}}$	0.4 mF	$K_{\mathrm{ii}}$	322
$R_{\mathrm{f}}$	0.2 $\mathsf{\Omega }$	$K_{\mathrm{pv}}$	1
$T_{\mathrm{s}}$	50 $\mathsf{\mu }$s	$K_{\mathrm{iv}}$	200
${\omega }^{\mathrm{ref}}$	$2\pi ·50$ rad/s	$K_{\mathrm{pPLL}}$	0.1
${\omega }_{\mathrm{v}}$	$2\pi ·5000$ rad/s	$K_{\mathrm{iPLL}}$	4.2
${\omega }_{\mathrm{i}}$	$2\pi ·5000$ rad/s

. The proportional and integral coefficients of PLL correspond to the PLL bandwidth $B{W}_{\mathrm{PLL}}$ = 20 Hz.

Under GFL control, the frequency responses of the positive-sequence impedance and their corresponding simulation measurements are depicted in Figure 5a. Theoretical derivation results for the positive-sequence impedance are shown as blue solid lines, while red crosses indicate the sweep verification results. The sweep verification shows good consistency with the impedance derivation, verifying the accuracy of the modeling.

According to the impedance-based stability criterion [21,30], the grid-connected converter’s stability requires $Z_{\mathrm{g}}\left(s\right)/Z_{\mathrm{p}}\left(s\right)$ satisfies sufficient phase margin, which means phase difference between the output impedance and the grid impedance must remain below 180° at all frequencies where their magnitudes intersect. For grids with inductive characteristics, instability risk arises when the output impedance phase is less than −90°. It should be noted that, based on the topology in Figure 3, $Z_{\mathrm{p}}\left(s\right)$ used for Bode plot generation includes the parallel combination of the original $Z_{\mathrm{p}}\left(s\right)$ and $1/s{C}_{\mathrm{f}}+R_{\mathrm{f}}$. Bode plots in Figure 6a highlight instability regions, indicating that an SCR below 1.5 leads to interaction between the GFL-controlled turbine’s impedance and the grid impedance, causing instability.

In GFM mode, the modeling accuracy is validated by the impedance sweep consistency shown in Figure 6b. Bode plots in Figure 6b suggest that instability arises when the SCR exceeds 23, due to interaction between the GFM-controlled turbine’s impedance and the grid impedance.

Owing to the rapid response and superior maximum power point tracking capabilities of GFL control, the switching criterion prioritizes GFL operation. The criterion is defined by the GFL instability boundary ${\lambda }_{\mathrm{b}}$; when it is detected to be below ${\lambda }_{\mathrm{b}}$, the system triggers switch to GFM mode.

Given the complementary instability characteristics of GFL control in weak grids and GFM control in strong grids, and considering the operational priority granted to GFL control, the switching boundary is predominantly governed by the critical stability limit of the GFL mode. Consequently, we focused on the GFL controller’s performance under varying parameters.

Figure 7a depicts the sequential impedance characteristics of GFL-controlled wind turbines with $P^{\mathrm{ref}}$ = 0.6 MW, 0.8 MW, 1.0 MW, and corresponding critical stable grid impedance. Other parameters were set as follows: $B{W}_{\mathrm{PLL}}$ = 20 Hz, the current loop bandwidth $B{W}_{\mathrm{I}}$ = 200 Hz. To ensure the accuracy of the derived impedance model, a sufficient spectral separation is maintained between the bandwidths of the PLL and the current control loop, thereby mitigating the effects of frequency coupling. As shown in Figure 7a, increasing $P^{\mathrm{ref}}$ reduces impedance magnitude in 1–100 Hz frequency band while maintaining phase characteristics. Consequently, critical stable grid impedance and switching boundary remains unchanged.

Figure 7b illustrates the impedance characteristics of GFL-controlled turbines under different $B{W}_{\mathrm{PLL}}$ of 20 Hz, 40 Hz, 60 Hz, with $P^{\mathrm{ref}}$ = 1 MW and $B{W}_{\mathrm{I}}$ = 200 Hz. Increasing PLL bandwidth reduces output impedance magnitude in 1–100 Hz frequency band, and induces phase drop in the negative damping region, with critical stable grid impedance increasing, which indicating higher susceptibility to instability in weak grids, necessitating an extension of ${\lambda }_{\mathrm{b}}$ to maintain stability in weak grids.

5. Switching Boundary Identification Based on EKF

EKF realizes state estimation through linearization of nonlinear input and measurement models, establishing itself as the predominant approach for nonlinear system filtering, particularly suitable for wind generation systems exhibiting complex nonlinear dynamics. This work employs EKF for grid impedance estimation within wind generation systems.

For the system in Figure 3, using the Clark transformation, the dynamic characteristics of the system are as follows:

$$\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{R_{\mathrm{g}}}{L_{\mathrm{g}}}}& 0\\ 0& -{\displaystyle \frac{R_{\mathrm{g}}}{L_{\mathrm{g}}}}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+\left[\begin{array}{cc}-{\displaystyle \frac{1}{L_{\mathrm{g}}}}& 0\\ 0& -{\displaystyle \frac{1}{L_{\mathrm{g}}}}\end{array}\right]\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]+\left[\begin{array}{cc}-{\displaystyle \frac{1}{L_{\mathrm{g}}}}& 0\\ 0& -{\displaystyle \frac{1}{L_{\mathrm{g}}}}\end{array}\right]\left[\begin{array}{c}{u}_{\mathrm{g}\alpha }\\ u_{\mathrm{g}\beta }\end{array}\right],$$

(26)

where $u_{\mathrm{g}\alpha }$, $u_{\mathrm{g}\beta }$ are grid voltages in two-phase stationary coordinates.

Assuming output voltage $u_{\mathrm{abc}}$ and grid voltage $u_{\mathrm{gabc}}$ remains constant during sampling periods, gives

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\\ u_{\alpha }\\ u_{\beta }\\ u_{\mathrm{g}\alpha }\\ u_{\mathrm{g}\beta }\end{array}\right]=\left[\begin{array}{cccccc}-{\displaystyle \frac{R_{\mathrm{g}}}{L_{\mathrm{g}}}}& 0& -{\displaystyle \frac{1}{L_{\mathrm{g}}}}& 0& {\displaystyle \frac{1}{L_{\mathrm{g}}}}& 0\\ 0& -{\displaystyle \frac{R_{\mathrm{g}}}{L_{\mathrm{g}}}}& 0& -{\displaystyle \frac{1}{L_{\mathrm{g}}}}& 0& {\displaystyle \frac{1}{L_{\mathrm{g}}}}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\\ u_{\alpha }\\ u_{\beta }\\ u_{\mathrm{g}\alpha }\\ u_{\mathrm{g}\beta }\end{array}\right].$$

(27)

Let $y={\left[\begin{array}{cccccc}{i}_{\alpha }& i_{\beta }& u_{\alpha }& u_{\beta }& u_{\mathrm{g}\alpha }& u_{\mathrm{g}\beta }\end{array}\right]}^{\mathrm{T}}$. Transforming these into discrete time-domain zero-order hold equivalent model and applying the simplifications, $\gamma =exp\left(-T_s{R}_{\mathrm{g}}/L_{\mathrm{g}}\right)\approx 1-T_s{R}_{\mathrm{g}}/L_{\mathrm{g}}$, $l_{\mathrm{g}}=1/L_{\mathrm{g}}$, to simplify the model to

$$\begin{array}{cc}\hfill \underset{:=x(k+1)}{\underbrace{\left[\begin{array}{c}{i}_{\alpha }(k+1)\\ i_{\beta }(k+1)\\ u_{\alpha }(k+1)\\ u_{\beta }(k+1)\\ u_{\mathrm{g}\alpha }(k+1)\\ u_{\mathrm{g}\beta }(k+1)\\ R_{\mathrm{g}}(k+1)\\ l_{\mathrm{g}}(k+1)\end{array}\right]}}& =\underset{:=f}{\underbrace{\left[\begin{array}{cccccccc}\gamma & 0& -T_s{l}_{\mathrm{g}}& 0& T_s{l}_{\mathrm{g}}& 0& 0& 0\\ 0& \gamma & 0& -T_s{l}_{\mathrm{g}}& 0& T_s{l}_{\mathrm{g}}& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]}}\underset{:=x\left(k\right)}{\underbrace{\left[\begin{array}{c}{i}_{\alpha }\left(k\right)\\ i_{\beta }\left(k\right)\\ u_{\alpha }\left(k\right)\\ u_{\beta }\left(k\right)\\ u_{\mathrm{g}\alpha }\left(k\right)\\ u_{\mathrm{g}\beta }\left(k\right)\\ R_{\mathrm{g}}\left(k\right)\\ l_{\mathrm{g}}\left(k\right)\end{array}\right]}}\hfill \\ \hfill & =\left[\begin{array}{c}(1-R_{\mathrm{g}}{l}_{\mathrm{g}}{T}_s)i_{\alpha }\left(k\right)-T_s{l}_{\mathrm{g}}{u}_{\alpha }\left(k\right)+T_s{l}_{\mathrm{g}}{u}_{\mathrm{g}\alpha }\left(k\right)\\ (1-R_{\mathrm{g}}{l}_{\mathrm{g}}{T}_s)i_{\beta }\left(k\right)-T_s{l}_{\mathrm{g}}{u}_{\beta }\left(k\right)+T_s{l}_{\mathrm{g}}{u}_{\mathrm{g}\beta }\left(k\right)\\ u_{\alpha }\left(k\right)\\ u_{\beta }\left(k\right)\\ u_{\mathrm{g}\alpha }\left(k\right)\\ u_{\mathrm{g}\beta }\left(k\right)\\ R_{\mathrm{g}}\left(k\right)\\ l_{\mathrm{g}}\left(k\right)\end{array}\right]=\left[\begin{array}{c}{f}_1\\ f_2\\ f_3\\ f_4\\ f_5\\ f_6\\ f_7\\ f_8\end{array}\right]\hfill \end{array}.$$

(28)

The principle of EKF estimation has been widely described in the literature, as seen in [31], with key steps shown in Algorithm 1.

Algorithm 1: Recursive estimation algorithm for EKF

f, h represents the nonlinear system function, Z represents observation function, X and $\widehat{X}$, respectively, denote the system state vector and the filtered value, P denotes the covariance matrix, Q denotes the process noise covariance matrix, and R is the measurement noise covariance matrix. K is the filter gain matrix. The expressions of $\varphi (k+1|k)$, $H\left(k\right)$ are as follows:

$${\displaystyle \frac{\partial f}{\partial \widehat{X}\left(k\right)}}={\left.{\displaystyle \frac{\partial f}{\partial X\left(k\right)}}\right|}_{X\left(k\right)=\widehat{X}\left(k\right)}=\varphi (k+1|k),$$

(29)

$${\left.{\displaystyle \frac{\partial h}{\partial X\left(k\right)}}\right|}_{X\left(k\right)=\widehat{X}\left(k\right)}=H\left(k\right).$$

(30)

Using EKF recursive algorithms with derived $L_{\mathrm{g}}$ enables real-time grid impedance estimation and rapid identification of boundary condition attainment.

6. Simulation Verification

A simulation model of the wind generation system operating with GFL/GFM switching control is implemented in PLECS. Fluctuations in grid strength are emulated by varying the grid impedance. The system parameters are provided in Table 1, with pre-synchronization parameters $K_{\omega }=50$, $K_{\mathrm{U}}=50$.

Figure 8a illustrates the waveform pertaining to the wind turbine functioning in GFL mode. $L_{\mathrm{g}}$ in Figure 8 represents the actual value, while $L_{\mathrm{g}}$ in Figure 9 is the estimated result of EKF. Prior to impedance change at t = 1.04 s, the wind turbine operates stably under SCR = 30.3 with $L_{\mathrm{g}}$ = 0.1 mH. At t = 1.04 s, 1.09 s, inductor 0.5 mH, 2 mH are inserted, with SCR changing to 5.1, 1.2. After 1.09 s, output current $i_{\mathrm{abc}}$ exhibits pronounced oscillations, validating former small-signal stability analysis which predicted loss of stability when SCR < 1.5.

Figure 8b illustrates GFM operation waveform. Before impedance change at t = 1.04 s, stable operation persists under SCR = 1.2. At t = 1.04 s, 1.09 s, inductor 0.5 mH, 2 mH are removed, with SCR changing to 1.4, 30.3. This transition pushes the system into the instability region predicted by the impedance-based criterion for SCR > 23. After t = 1.09 s, output current demonstrates oscillations, aligning with former analysis.

In Figure 9a, the waveform of GFL-to-GFM transition and estimated grid impedance are displayed, where 0 denotes the GFM mode signal and 1 denotes the GFL mode signal. At t = 1.05 s, inductor 2 mH insertion induces potential instability risk for the turbine under GFL control. At t = 1.062 s, EKF recognition recognizes the grid impedance change and triggers switching command with recognition time 12 ms. The subsequent transition to GFM control stabilizes the system, demonstrating the practical effectiveness of the derived switching boundary in preventing instability.

In Figure 9b, the waveform of GFM-to-GFL transition and estimated grid impedance are displayed. At t = 1.05 s, inductor 2 mH is removed. At t = 1.053 s, EKF recognition triggers switching command with recognition time 3 ms, restoring operation to GFL control in accord with the switching boundary ${\lambda }_{\mathrm{b}}=1.5$, determined by analysis in Section 4 and Figure 6a.

Seamless mode transitions without amplitude/phase discontinuity, indicates that the switching process has little perturbation to the grid. This smooth transition is attributed to the accurate and timely identification of grid impedance, combined with the precisely designed stability boundary.

These results confirm the validity of the derived switching criterion under SCR variations, and EKF applicability for GFL/GFM switching due to rapid and non-invasive identification capability.

7. Experimental Validation Plan

To strengthen the practical relevance of the proposed method, we plan to conduct experimental validation in our future work. A laboratory-scale test platform will be developed to evaluate the GFL/GFM switching strategy under grid strength variations. The setup will include a Chroma grid simulator with grid impedance emulation capability, a B-Box rapid control prototyping (RCP) system, a two-level converter, a PMSG, filters, and voltage/current sensors. The grid simulator will be used to reproduce grid strength variations, where the ability to emulate dynamic impedance is crucial, since physically switching inductors in experiments is impractical. The B-Box RCP will process voltage and current signals in real time and implement the proposed control algorithms, including EKF-based impedance identification, GFL/GFM control with designed switching criterion. Key validation objectives will include: verification of impedance identification accuracy, evaluation of switching stability and effectiveness under varying grid strengths, and assessment of dynamic performance under grid disturbances.

8. Conclusions

This paper develops small-signal sequence impedance models for wind turbine systems operating under both GFL and GFM control modes. A small-signal stability analysis across varying grid strengths is conducted, leading to the derivation of a quantitative stability boundary for GFL/GFM switching. This boundary ensures system stability under diverse grid conditions. Validation via impedance sweep tests in PLECS simulations confirms a precise alignment between the theoretical models and practical results. The derived switching criterion guarantees stable system operation, while EKF-based identification demonstrates effective applicability for fast mode transitions.

This work enables wind turbines to dynamically adapt their control strategy to dynamic grid conditions, showing potential for practical implementation. Nevertheless, existing impedance modeling does not yet consider the effect of frequency coupling, and applying the method to large-scale wind farms with multiple turbines introduces further challenges, such as designing coordinated switching boundaries and multi-turbine control strategies. From a hardware perspective, one prerequisite for effective switching is that converters must operate reliably under both GFL and GFM controls, which may introduce additional hardware design considerations. For instance, GFM control can be more susceptible to overcurrent under fault conditions. Moreover, accurate and fast estimation of grid impedance is required, relying on high-precision voltage and current sensors. Future work will focus on extending the switching criterion to multi-turbine systems, impedance modeling considering frequency coupling, and experimental validation.