Stability Analysis and Enhanced Control of Wind Turbine Generators Based on Hybrid GFL-GFM Control

Abstract

With the proliferation of wind power generation, the receiving end grids exhibit unprecedented dynamic characteristics, imposing critical stability challenges on grid-connected wind turbine’s converter. To address this, wind turbine converter control strategies have evolved beyond traditional grid-following (GFL) methods to include grid-forming (GFM), mode-switching, and hybrid GFL-GFM controls. This paper establishes a small-signal model for hybrid GFL-GFM-controlled wind turbines to analyze stability at varying grid strengths, guiding the selection of coefficients in hybrid mode. Simulation tests validate the theoretical framework.

1. Introduction

The non-renewable nature of fossil energy and rising global energy demands are accelerating the development of renewable energy, especially wind and solar power. As wind power generation equipment penetrates power grids at increasing scales [1,2], the receiving end network’s dynamic characteristics undergo significant transformation: traditional strong mechanical inertia weakens, and grid strength diminishes [3,4]. Grid strength is typically categorized as very weak (SCR ≤ 2), weak (2 < SCR < 3), or strong (SCR > 3) according to the short-circuit ratio (SCR) [5]. Since the SCR is derived from grid impedance, the intermittent, stochastic and fluctuating nature of wind power generation—combined with the plug-and-play characteristics of distributed resources and loads—causes substantial grid impedance variations [6]. These impedance fluctuations consequently trigger SCR variations.

The traditional wind turbine’s converter control strategy is grid-following (GFL) control, which achieves maximum power point tracking and enables high-efficiency energy conversion. However, studies indicate that GFL control exhibits poor weak-grid stability, potentially triggering low-frequency harmonic resonance and even instability [7]. To enhance converter stability in weak grids and align with modern power system requirements, grid-forming (GFM) control has emerged as a critical research focus [8,9]. GFM control implemented in wind turbines generates a predefined voltage amplitude and angular frequency, providing inertia support during weak-grid operation and demonstrating robust stability in both weak-grid and islanded systems [10,11]. Nevertheless, classical GFM control adapts poorly to strong grids [12], which is analogous to the incompatibility of connecting two voltage sources in series. Furthermore, it suffers from coupled output active and reactive power [13,14].

Therefore, the GFL and GFM controls actually exhibit complementary characteristics. This complementarity motivates the development of integrated control schemes. In [15], a GFL/GFM switching control strategy enables converters to operate adaptively in either mode when grid conditions cross stability boundaries. A hybrid control strategy in the framework of model predictive control is proposed in [16] that evaluates GFL and GFM cost functions for each switching state and selects the state with minimal cost, improving stability during large grid impedance fluctuations. In [17], a hybrid dual-mode control strategy is proposed, utilizing parallel-connected converters with GFL and GFM control, where the minimum capacity for the GFM unit is determined through stability domain analysis, enabling optimal capacity allocation for stability and economy. Ref. [18] proposes a hybrid GFL-GFM control where modulation signals are weighted equally (coefficient: 0.5), utilizing the phase angle generated by GFM for synchronization. Ref. [19] further analyzes the advantages in small-signal stability and grid support capabilities of the hybrid control in [18], comparing with GFL and GFM control, based on the state space model. In [20], the authors implement weighted modulation signals combined with weighted phase angles, where phase angles are captured by the phase-locked loop (PLL) of GFL and the power synchronization loop (PSL) of GFM, and adjustable weighting coefficients can be selected.

Impedance-based stability analysis provides an effective methodology for assessing converter control stability. Ref. [21] establishes the impedance model of GFL converters with PLL-based grid synchronization, and ref. [22] studies the impedance model for GFM-controlled virtual synchronous generators, while ref. [12] systematically compares the impedance characteristics of GFL and GFM methods. Ref. [18] analyzes the traditional control proposed by [17] and concludes that the system’s stability is consistent with the GFM-controlled converter, but since the rated capacity is twice as large as that of GFM-controlled converter, the critical SCR is half. So the system is still unstable with strong grids and cannot adapt to SCR changes. For hybrid GFL-GFM converters with GFM synchronization, the authors of [18] quantify the stability domain at varying weighting coefficients, selecting a fixed coefficient of 0.5 to ensure stability across most grid strengths. However, this static weighting limits flexibility, particularly in strong grid conditions where pure GFL control is preferable. A more adaptive approach in [20] implements adjustable coefficients according to AC voltage levels. Nevertheless, this regulation may not ensure full stabilization.

Compared to doubly-fed induction generators, direct-drive permanent magnet synchronous generators (PMSGs) adopt GFM control more broadly given their simplifier controllers [23]; this work therefore concentrates on PMSG-based wind turbines. This paper establishes the sequential impedance model for wind turbine generators based on hybrid GFL-GFM control featuring adaptive weighting coefficients. The model enables small-signal stability analysis at varying grid strengths. The major contributions of this paper are as follows:

• Development of the impedance model for wind turbine generators under hybrid GFL-GFM control with tunable weighting coefficients, addressing the derivation of hybrid phase angle synthesis;
• Stability-guaranteed coefficient selection utilizing impedance-based stability criteria, providing a methodology for adaptive optimization hybrid coefficient adjustment under different SCR conditions.

The rest of this article is organized as follows. Section 2 presents the studied hybrid GFL-GFM control architecture. Section 3 develops the sequential impedance model for wind turbines under the hybrid GFL-GFM control using harmonic linearization theory. Section 4 presents small-signal stability analysis and determines optimal weighting coefficients across varying grid strengths. Section 5 provides simulation validation. Section 6 concludes the article.

2. Topology and Control Scheme of Wind Turbine System with Hybrid GFL-GFM Control

2.1. Circuit Topology

Figure 1 shows the PMSG-based wind generation system’s topology. For simplified modeling, internal turbine dynamics are neglected, and the DC-link voltage is considered as a constant $u_{\mathrm{dc}}$. Internal electric potentials $e_{\mathrm{a}}$, $e_{\mathrm{b}}$, $e_{\mathrm{c}}$; output currents $i_{\mathrm{a}}$, $i_{\mathrm{b}}$, $i_{\mathrm{c}}$; and output voltages $u_{\mathrm{a}}$, $u_{\mathrm{b}}$, $u_{\mathrm{c}}$ are denoted. The filter inductance $L_{\mathrm{f}}$, filter capacitor $C_{\mathrm{f}}$, and damping resistance $R_{\mathrm{f}}$ constitute the LC filter. $L_{\mathrm{g}}$ and $R_{\mathrm{g}}$ are the equivalent line inductance and resistance of the grid. $u_{\mathrm{ga}}$, $u_{\mathrm{gb}}$, $u_{\mathrm{gc}}$ stand for grid voltages. $i_{\mathrm{ga}}$, $i_{\mathrm{gb}}$, $i_{\mathrm{gc}}$ are the grid currents.

2.2. Hybrid Control Scheme

Figure 2 illustrates the hybrid GFL-GFM control. The three-phase output voltage $u_{\mathrm{abc}}$ and current $i_{\mathrm{abc}}$ undergo Park transformation to derive dq-axis components: d-axis voltage $u_{\mathrm{d}}$, q-axis voltage $u_{\mathrm{q}}$, d-axis current $i_{\mathrm{d}}$, and q-axis current $i_{\mathrm{q}}$. This transformation utilizes the hybrid phase angle $\theta$ as an input.

The hybrid phase angle $\theta$ is synthesized from both PLL and PSL outputs. A basic PLL is employed: the three-phase voltage $u_{\mathrm{abc}}$ is transformed to obtain dq-axis voltages $u_{\mathrm{GFL},\mathrm{d}}$, $u_{\mathrm{GFL},\mathrm{q}}$ corresponding to GFL control. The q-axis component $u_{\mathrm{GFL},\mathrm{q}}$ is processed through a PI regulator and integrator to generate the GFL phase angle ${\theta }_{\mathrm{PLL}}$. Concurrently, the PSL computes the GFM angular frequency ${\omega }_{\mathrm{GFM}}$ by subtracting instantaneous active power P from reference $P^{\mathrm{ref}}$, multiplying by active power droop coefficient $K_{\mathrm{P}}$ to obtain frequency deviation $\Delta \omega$, and subtracting reference frequency ${\omega }^{\mathrm{ref}}$. Integration of ${\omega }^{\mathrm{ref}}$ yields the GFM phase angle ${\theta }_{\mathrm{PSL}}$. The hybrid phase angle is defined as

$$\theta =k·{\theta }_{\mathrm{PLL}}+(1-k)·{\theta }_{\mathrm{PSL}},$$

(1)

where k and $1-k$ denote GFL and GFM weighting coefficients, respectively.

The hybrid modulation signal originates from GFL and GFM controllers. For GFL, active and reactive reference power $P^{\mathrm{ref}}$ and $Q^{\mathrm{ref}}$ are divided by $3/2$ times voltage magnitude $V_1$ to produce dq-axis current references $I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}$ and $I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}}$ corresponding to GFL. These references are compared with $i_{\mathrm{d}}$, $i_{\mathrm{q}}$, with the errors processed through PI regulator to generate GFL modulation signals $c_{\mathrm{GFL},\mathrm{dq}}$. For GFM, the deviation of reactive power reference $Q^{\mathrm{ref}}$ and instantaneous reactive power Q multiplied by reactive power droop coefficient $K_{\mathrm{Q}}$ yields voltage deviation $\Delta U$, which is subtracted from reference voltage $U^{\mathrm{ref}}$ to obtain d-axis voltage reference $U_{\mathrm{GFM},\mathrm{d}}$. Current references $I_{\mathrm{GFM},\mathrm{d}}^{\mathrm{ref}}$ and $I_{\mathrm{GFM},\mathrm{q}}^{\mathrm{ref}}$ corresponding to GFM are derived by regulating voltage deviations of $U_{\mathrm{GFM},\mathrm{d}}$, $u_{\mathrm{d}}$, and 0, $u_{\mathrm{q}}$, through PI controllers. The identical current loop then generates GFM modulation signals $c_{\mathrm{GFM},\mathrm{dq}}$. The hybrid modulation signal is as follows:

$$c_{\mathrm{dq}}=k·c_{\mathrm{GFL},\mathrm{dq}}+(1-k)·c_{\mathrm{GFM},\mathrm{dq}}+u_{\mathrm{dq}},$$

(2)

where $u_{\mathrm{dq}}$ represents the dq-axis output voltages. Finally, inverse Park transformation and PWM modulation convert $c_{\mathrm{dq}}$ to switching signals $S_{\mathrm{abc}}$.

3. Sequential Impedance Modeling for Hybrid GFL-GFM Control

The sequential impedance model for the wind turbine system utilizing hybrid GFL-GFM control is developed using harmonic linearization theory [24,25], with neglect of second and higher-order nonlinear terms in the small-signal linearization.

Assuming that small-signal perturbation is injected into the system, yielding time-domain converter output voltage and current expressions:

$$u_{\mathrm{a}}\left(t\right)=V_{1}cos\left(2\pi f_{\mathrm{l}}t\right)+V_{\mathrm{p}}cos(2\pi f_{\mathrm{p}}t+{\phi }_{\mathrm{vs}.\mathrm{p}})+V_{\mathrm{n}}cos(2\pi f_{\mathrm{n}}t+{\phi }_{\mathrm{vs}.\mathrm{n}}),$$

(3)

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

(4)

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

The frequency-domain expressions can be described as follows:

$${\mathbf{U}}_{\mathrm{a}}\left[f\right]=\left\{\begin{array}{cc}{\mathbf{V}}_1,\hfill & f=\pm f_{\mathrm{l}}\hfill \\ {\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,{\mathbf{I}}_{\mathrm{a}}\left[f\right]=\left\{\begin{array}{cc}{\mathbf{I}}_1,\hfill & f=\pm f_{\mathrm{l}}\hfill \\ {\mathbf{I}}_{\mathrm{p}},\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\mathbf{I}}_{\mathrm{n}},\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.,$$

(5)

where ${\mathbf{V}}_1$, ${\mathbf{V}}_{\mathrm{p}}$, ${\mathbf{V}}_{\mathrm{n}}$, ${\mathbf{I}}_1$, ${\mathbf{I}}_{\mathrm{p}}$, and ${\mathbf{I}}_{\mathrm{n}}$ correspond to phasors of fundamental voltage, positive-sequence voltage disturbance, negative-sequence voltage disturbance, fundamental current, positive-sequence current response, and negative-sequence current response. ${\mathbf{V}}_1=V_1/2$; ${\mathbf{V}}_{\mathrm{p}}=V_{\mathrm{p}}/2·e^{\pm j{\phi }_{\mathrm{vp}}}$; ${\mathbf{V}}_{\mathrm{n}}=V_{\mathrm{p}}/2·e^{\pm j{\phi }_{\mathrm{vn}}}$; ${\mathbf{I}}_1=I_1/2·e^{\pm j{\phi }_{\mathrm{i}1}}$; ${\mathbf{I}}_{\mathrm{p}}=I_{\mathrm{p}}/2·e^{\pm j{\phi }_{\mathrm{ip}}}$; ${\mathbf{I}}_{\mathrm{n}}=I_{\mathrm{n}}/2·e^{\pm j{\phi }_{\mathrm{in}}}$.

From Figure 1’s electrical circuit:

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

(6)

For phase-a, $e_{\mathrm{a}}=K_{\mathrm{PWM}}{u}_{\mathrm{dc}}{c}_{\mathrm{a}}+K_{\mathrm{f}}{v}_{\mathrm{a}}$, where $K_{\mathrm{PWM}}$ and $K_{\mathrm{f}}$ represent the PWM coefficient and the feedforward coefficient, respectively.

3.1. Small-Signal Modeling of the Hybrid Phase Angle

The hybrid phase angle $\theta$ incorporates both the positive-sequence angle ${\theta }_1$ derived from fundamental voltage and the phase angle perturbation component $\Delta \theta$ corresponding to small-signal voltage disturbances, i.e., $\theta ={\theta }_1+\Delta \theta$. Based on Equation (1), $\Delta \theta$ is composed of the phase angle perturbation component from PLL ($\Delta {\theta }_{\mathrm{PLL}}$) and the phase angle perturbation component from PSL ($\Delta {\theta }_{\mathrm{PSL}}$):

$$\Delta \theta =k·\Delta {\theta }_{\mathrm{PLL}}+(1-k)·\Delta {\theta }_{\mathrm{PSL}}.$$

(7)

The synchronous rotation coordinate transformation matrix $T\left(\theta \right)$ considering perturbations is

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

(8)

Since $\Delta \theta$ is a small perturbation, the following approximation can be derived:

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

(9)

3.1.1. Small-Signal Modeling of the PLL

When injection perturbation effects are neglected, $\theta ={\theta }_1$, the frequency-domain expressions for dq-axis output voltages $u_{\mathrm{d}1}$ and $u_{\mathrm{q}1}$ are derived as

$$\begin{array}{cc}\hfill & {\mathbf{U}}_{\mathrm{dl}}\left[f\right]=\left\{\begin{array}{cc}{V}_1,\hfill & \mathrm{dc}\hfill \\ G_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_p,\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ G_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_n,\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.\hfill \\ \hfill & {\mathbf{U}}_{\mathrm{ql}}\left[f\right]=\left\{\begin{array}{cc}0,\hfill & \mathrm{dc}\hfill \\ \mp j{G}_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_p,\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \pm j{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_n,\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.\hfill \end{array},$$

(10)

where $G_{\mathrm{vs}.}\left(s\right)$ represents the voltage sampling function modeling sampling delay, PWM delay, and low-pass filtering dynamics, $G_{\mathrm{vs}.}\left(s\right)=e^{-T_{\mathrm{s}}s}(1-e^{-T_{\mathrm{s}}s})/\left[\left(T_{\mathrm{s}}s\right)(1+s/{\omega }_{\mathrm{vs}.})\right]$. $T_{\mathrm{s}}$ is the sampling period, and ${\omega }_{\mathrm{vs}.}$ is the cut-off angular frequencies of the low-pass filter of the voltage signal.

Considering PLL dynamics, $u_{\mathrm{GFL},\mathrm{d}}$ and $u_{\mathrm{GFL},\mathrm{q}}$ are obtained from Equation (8) via

$$\left\{\begin{array}{cc}\hfill u_{\mathrm{GFL},\mathrm{d}}& =u_{\mathrm{d}1}+\Delta {\theta }_{\mathrm{PLL}}·u_{\mathrm{q}1}\hfill \\ \hfill u_{\mathrm{GFL},\mathrm{q}}& =-\Delta {\theta }_{\mathrm{PLL}}·u_{\mathrm{d}1}+u_{\mathrm{q}1}\hfill \end{array}\right..$$

(11)

The PLL structure in Figure 2 reveals

$$\Delta {\theta }_{\mathrm{PLL}}\left[f\right]=H_{\mathrm{PLL}}\left(s\right)·{\mathbf{U}}_{\mathrm{GFL},\mathrm{q}}\left[f\right],$$

(12)

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 $K_{\mathrm{i}\mathrm{P}\mathrm{L}\mathrm{L}}$ are the proportional and integral coefficients of PLL, respectively.

Assuming that the expression for $\Delta {\theta }_{\mathrm{PLL}}\left[f\right]$ is

$$\Delta {\theta }_{\mathrm{PLL}}\left[f\right]=\left\{\begin{array}{cc}{G}_{\mathrm{p}}\left(s\right)G_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_{\mathrm{i}})\hfill \\ G_{\mathrm{n}}\left(s\right)G_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_{\mathrm{i}})\hfill \end{array}\right.,$$

(13)

where $G_{\mathrm{p}}\left(s\right)$, $G_{\mathrm{n}}\left(s\right)$ is the frequency-domain transfer functions between small-signal voltage disturbance and $\Delta {\theta }_{\mathrm{PLL}}$, then the expressions for $u_{\mathrm{d}}$ and $u_{\mathrm{q}}$ in the frequency domain can be derived with Equation (11):

$$\begin{array}{cc}\hfill {\mathbf{U}}_{\mathrm{GFL},\mathrm{d}}\left[f\right]& =\left\{\begin{array}{cc}{V}_1,\hfill & \mathrm{dc}\hfill \\ G_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ G_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.\hfill \end{array},$$

(14)

$$\begin{array}{cc}\hfill {\mathbf{U}}_{\mathrm{GFL},\mathrm{q}}\left[f\right]& =\left\{\begin{array}{cc}0,\hfill & \mathrm{dc}\hfill \\ \left[-G_{\mathrm{p}}\left(s\right)V_1\mp j\right]G_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \left[-G_{\mathrm{n}}\left(s\right)V_1\pm j\right]G_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.\hfill \end{array}.$$

(15)

Based on Simultaneous Equations (12) and (15), $G_{\mathrm{p}}\left(s\right)$ and $G_{\mathrm{n}}\left(s\right)$ are solved as

$$G_{\mathrm{p}}\left(s\right)=\mp j{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{1+V_1{H}_{\mathrm{PLL}}\left(s\right)}},G_{\mathrm{n}}\left(s\right)=\pm j{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{1+V_1{H}_{\mathrm{PLL}}\left(s\right)}}.$$

(16)

Substituting to Equation (13), we get the perturbation component $\Delta {\theta }_{\mathrm{PLL}}\left[f\right]$:

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

(17)

3.1.2. Small-Signal Modeling of the PSL

Based on Equation (5), three-phase stationary coordinates are converted to two-phase stationary coordinates with the Clarke transform:

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

(18)

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

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

(19)

P and Q in the frequency domain are derived using power formula $P=1.5\left(u_{\alpha }{i}_{\alpha }+u_{\beta }{i}_{\beta }\right)$, $Q=1.5\left(u_{\beta }{i}_{\alpha }-u_{\alpha }{i}_{\beta }\right)$, and the frequency-domain convolution theorem:

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

(20)

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

(21)

In the GFM controller, ${\theta }_{\mathrm{PSL}}$ is calculated by

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

(22)

In the frequency domain, substituting Equation (20) into Equation (22) yields the perturbation variable $\Delta {\theta }_{\mathrm{GFM}}$:

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

(23)

3.1.3. Hybrid Phase Angle Synthesis

Combining Equation (7), Equation (17), and Equation (23), the hybrid phase angle perturbation $\Delta \theta$ in the frequency domain is obtained:

$$\Delta \theta \left[f\right]=\left\{\begin{array}{cc}\begin{array}{c}-{\displaystyle \frac{3(k-1)}{2}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left[G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{p}}{V}_1+G_{\mathrm{v}}(s\pm j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{p}}{e}^{\mp j{\phi }_{\mathrm{i}1}}\right]\hfill \\ {\displaystyle \frac{\mp jk{H}_{\mathrm{PLL}}\left(s\right)}{1+V_1{H}_{\mathrm{PLL}}\left(s\right)}}{G}_v(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{p}},\hfill \end{array}\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \begin{array}{c}-{\displaystyle \frac{3(k-1)}{2}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left[G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}}{V}_1+G_{\mathrm{v}}(s\mp j2\pi f_1)I_1{\mathbf{V}}_{\mathrm{n}}{e}^{\pm j{\phi }_{\mathrm{i}1}}\right]\hfill \\ {\displaystyle \frac{\pm jk{H}_{\mathrm{PLL}}\left(s\right)}{1+V_1{H}_{\mathrm{PLL}}\left(s\right)}}{G}_v(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}},\hfill \end{array}\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right..$$

(24)

Thus, $sin\theta$ and $cos\theta$ in the frequency domain are as follows:

$$\begin{array}{c}\begin{array}{cc}& sin\theta \left[f\right]=\left\{\begin{array}{cc}\mp {\displaystyle \frac{j}{2}},\hfill & f=\pm f_1\hfill \\ \mp {\displaystyle \frac{jk}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\mp j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\mp j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}}-{\displaystyle \frac{3(1-k)}{4}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{p}}{V}_1+G_{\mathrm{v}}\left(s\right)I_1{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right),\hfill & f=\pm f_{\mathrm{p}}\hfill \\ \pm {\displaystyle \frac{jk}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\pm j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\pm j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{n}}-{\displaystyle \frac{3(1-k)}{4}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{n}}{V}_1+G_{\mathrm{v}}\left(s\right)I_1{e}^{\pm j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right),\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.\hfill \end{array},\end{array}$$

(25)

$$\begin{array}{c}\begin{array}{cc}& cos\theta \left[f\right]=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & f=\pm f_1\hfill \\ {\displaystyle \frac{k}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\mp j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\mp j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{p}}\mp {\displaystyle \frac{3(1-k)}{4}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{p}}{V}_1+j{G}_{\mathrm{v}}\left(s\right)I_1{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right),\hfill & f=\pm f_{\mathrm{p}}\hfill \\ {\displaystyle \frac{k}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}(s\pm j2\pi f_1)}{V_1{H}_{\mathrm{PLL}}(s\pm j2\pi f_1)+1}}{G}_{\mathrm{v}}\left(s\right){\mathbf{V}}_{\mathrm{n}}\pm {\displaystyle \frac{3(1-k)}{4}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left(G_{\mathrm{i}}\left(s\right){\mathbf{I}}_{\mathrm{n}}{V}_1+j{G}_{\mathrm{v}}\left(s\right)I_1{e}^{\pm j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right),\hfill & f=\pm f_{\mathrm{n}}\hfill \end{array}\right.\hfill \end{array}.\end{array}$$

(26)

3.2. Small-Signal Modeling of the Hybrid Modulation Signal

Substituting Equations (25) and (26) into Equation (8), yielding the expression of the synchronous rotation coordinate transformation matrix $T\left(\theta \right)$.

Inverter currents $i_{\mathrm{a}}$, $i_{\mathrm{b}}$, and $i_{\mathrm{c}}$ transform into $i_{\mathrm{d}}$, $i_{\mathrm{q}}$, via $T\left(\theta \right)$. Hence, the frequency-domain expressions for $i_{\mathrm{d}}$, $i_{\mathrm{q}}$ are

$${\mathbf{I}}_{\mathrm{d}}\left[f\right]=\left\{\begin{array}{cc}{I}_{1}cos{\phi }_{\mathrm{i}1},\hfill & \mathrm{dc}\hfill \\ \begin{array}{c}\mp jk{I}_{1}sin{\phi }_{\mathrm{i}1}{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{p}}+G_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{p}}\hfill \\ -{\displaystyle \frac{3(1-k)}{2}}{\displaystyle \frac{K_{\mathrm{P}}sin{\phi }_{\mathrm{i}1}}{s}}\left(G_{\mathrm{i}}(s\mp j2\pi f_1)I_1{V}_1{\mathbf{I}}_{\mathrm{p}}+G_{\mathrm{v}}(s\mp j2\pi f_1)I_1^2{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right)\hfill \end{array},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \begin{array}{c}\pm jk{I}_{1}sin{\phi }_{\mathrm{i}1}{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{n}}+G_{\mathrm{i}}(s\pm j2\pi f_1){\mathbf{I}}_{\mathrm{n}}\hfill \\ -{\displaystyle \frac{3(1-k)}{2}}{\displaystyle \frac{K_{\mathrm{P}}sin{\phi }_{\mathrm{i}1}}{s}}\left(G_{\mathrm{i}}(s\pm j2\pi f_1)I_1{V}_1{\mathbf{I}}_{\mathrm{n}}+G_{\mathrm{v}}(s\pm j2\pi f_1)I_1^2{e}^{\pm j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right)\hfill \end{array},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.,$$

(27)

$${\mathbf{I}}_{\mathrm{q}}\left[f\right]=\left\{\begin{array}{cc}{I}_{1}sin{\phi }_{\mathrm{i}1},\hfill & \mathrm{dc}\hfill \\ \begin{array}{c}\pm jk{I}_{1}cos{\phi }_{\mathrm{i}1}{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{p}}\mp j{G}_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{p}}\hfill \\ +{\displaystyle \frac{3(1-k)}{2}}{\displaystyle \frac{K_{\mathrm{P}}cos{\phi }_{\mathrm{i}1}}{s}}\left(G_{\mathrm{i}}(s\mp j2\pi f_1)I_1{V}_1{\mathbf{I}}_{\mathrm{p}}+G_{\mathrm{v}}(s\mp j2\pi f_1)I_1^2{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right)\hfill \end{array},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \begin{array}{c}\mp jk{I}_{1}cos{\phi }_{\mathrm{i}1}{\displaystyle \frac{H_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}}\pm j{G}_{\mathrm{i}}(s\mp j2\pi f_1){\mathbf{I}}_{\mathrm{n}}\hfill \\ +{\displaystyle \frac{3(1-k)}{2}}{\displaystyle \frac{K_{\mathrm{P}}cos{\phi }_{\mathrm{i}1}}{s}}\left(G_{\mathrm{i}}(s\pm j2\pi f_1)I_1{V}_1{\mathbf{I}}_{\mathrm{n}}+G_{\mathrm{v}}(s\pm j2\pi f_1)I_1^2{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right)\hfill \end{array},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right..$$

(28)

Meanwhile, transforming $u_{\mathrm{a}}$, $u_{\mathrm{b}}$, $u_{\mathrm{c}}$ to $u_{\mathrm{d}}$, $u_{\mathrm{q}}$ via $T\left(\theta \right)$. The frequency-domain expressions for $u_{\mathrm{d}}$, $u_{\mathrm{q}}$ are

$${\mathbf{U}}_{\mathrm{d}}\left[f\right]=\left\{\begin{array}{cc}{V}_1,\hfill & \mathrm{dc}\hfill \\ G_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{p}},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ G_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right.,$$

(29)

$${\mathbf{U}}_{\mathrm{q}}\left[f\right]=\left\{\begin{array}{cc}0,\hfill & \mathrm{dc}\hfill \\ \begin{array}{c}\pm jk{\displaystyle \frac{V_1{H}_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{p}}\mp j{G}_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{p}}\hfill \\ +{\displaystyle \frac{3(1-k)}{2}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left(G_{\mathrm{i}}\left(s\right)V_1^2{\mathbf{I}}_{\mathrm{p}}+G_{\mathrm{v}}\left(s\right)I_1{V}_1{e}^{\mp j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{p}}\right)\hfill \end{array},\hfill & f=\pm (f_{\mathrm{p}}-f_1)\hfill \\ \begin{array}{c}\mp jk{\displaystyle \frac{V_1{H}_{\mathrm{PLL}}\left(s\right)}{V_1{H}_{\mathrm{PLL}}\left(s\right)+1}}{G}_{\mathrm{v}}(s\mp j2\pi f_1){\mathbf{V}}_{\mathrm{n}}\pm j{G}_{\mathrm{v}}(s\pm j2\pi f_1){\mathbf{V}}_{\mathrm{n}}\hfill \\ +{\displaystyle \frac{3(1-k)}{2}}{\displaystyle \frac{K_{\mathrm{P}}}{s}}\left(G_{\mathrm{i}}\left(s\right)V_1^2{\mathbf{I}}_{\mathrm{n}}+G_{\mathrm{v}}\left(s\right)I_1{V}_1{e}^{\pm j{\phi }_{\mathrm{i}1}}{\mathbf{V}}_{\mathrm{n}}\right)\hfill \end{array},\hfill & f=\pm (f_{\mathrm{n}}+f_1)\hfill \end{array}\right..$$

(30)

In the GFL controller, with current loop $c_{\mathrm{GFL},\mathrm{d}}=(I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}})H_{\mathrm{i}}\left(s\right)-K_{\mathrm{d}}{i}_{\mathrm{q}}$, $c_{\mathrm{GFL},\mathrm{q}}=(I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}}-i_{\mathrm{q}})H_{\mathrm{i}}\left(s\right)+K_{\mathrm{d}}{i}_{\mathrm{d}}$, the GFL modulation signal $c_{\mathrm{GFL},\mathrm{dq}}$ is deviated, with detailed expressions given in Equations (A1) and (A2) in Appendix A.

In GFM controller, with the following voltage loop and current loop,

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

(31)

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

(32)

where $H_{\mathrm{vs}.}\left(s\right)=(K_{\mathrm{pvs}.}+K_{\mathrm{i}\mathrm{v}}/s)/s$. $K_{\mathrm{pvs}.}$ and $K_{\mathrm{ivs}.}$ are the proportional and integral coefficients of the voltage loop in GFM control, respectively. The dq-frame modulation signal in the frequency domain are shown in Equations (A3) and (A4) in Appendix A.

Substituting $c_{\mathrm{GFL},\mathrm{dq}}\left[f\right]$, $c_{\mathrm{GFM},\mathrm{dq}}\left[f\right]$ to Equation (2), then multiply the result $c_{\mathrm{dq}}\left[f\right]$ by inverse coordinate transformation matrix $T^{-1}\left(\theta \right)$, the a-phase modulating waveform $c_{\mathrm{a}}$ in the frequency domain is obtained.

Combine with Equation (6), the positive/negative sequence impedance $Z_p\left(s\right)$, $Z_n\left(s\right)$ of wind turbines under hybrid control can be derived, and the detailed expressions are represented by

$$Z_p\left(s\right)=-{\displaystyle \frac{{\mathbf{V}}_{\mathrm{p}}\left(s\right)}{{\mathbf{I}}_{\mathrm{p}}\left(s\right)}}={\displaystyle \frac{\begin{array}{c}{L}_{f}s+K_{\mathrm{PWM}}{u}_{\mathrm{dc}}\times \left(\begin{array}{c}k{G}_{\mathrm{i}}{H}_{\mathrm{i}}+(1-k)G_{\mathrm{i}}{H}_{\mathrm{i}}-j{K}_{\mathrm{d}}{G}_{\mathrm{i}}+j{\displaystyle \frac{3K_Q}{4}}(1-k)G_{\mathrm{i}}{H}_{\mathrm{i}}{H}_v{V}_1\hfill \\ +{\displaystyle \frac{3K_P}{4(s-j2\pi f_1)}}k(1-k)G_{\mathrm{i}}{H}_{\mathrm{i}}{V}_1(j{I}_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}-I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\hfill \\ +j{\displaystyle \frac{3K_P}{4(s-j2\pi f_1)}}{(1-k)}^2{G}_{\mathrm{i}}{H}_{\mathrm{i}}{H}_v{V}_1^2\hfill \end{array}\right)\end{array}}{\begin{array}{c}1-K_m{V}_{\mathrm{dc}}\times \left(\begin{array}{c}{G}_{\mathrm{v}}{K}_f-(1-k)G_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v+{\displaystyle \frac{k^2}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}}{1+V_1{H}_{\mathrm{PLL}}}}{G}_{\mathrm{v}}{H}_{\mathrm{i}}(I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}+j{I}_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\hfill \\ +{\displaystyle \frac{k(1-k)}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}}{1+V_1{H}_{\mathrm{PLL}}}}{G}_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v{V}_1+j{\displaystyle \frac{3K_Q}{4}}(1-k)G_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v{I}_1{e}^{-j{\phi }_{\mathrm{i}1}}\hfill \\ +{\displaystyle \frac{3K_P}{4(s-j2\pi f_1)}}k(1-k)G_{\mathrm{v}}{H}_{\mathrm{i}}{I}_1{e}^{-j{\phi }_{\mathrm{i}1}}(-j{I}_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}+I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\hfill \\ -j{\displaystyle \frac{3K_P}{4(s-j2\pi f_1)}}{(1-k)}^2{G}_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v{I}_1{V}_1{e}^{-j{\phi }_{\mathrm{i}1}}\hfill \end{array}\right)\end{array}}},$$

(33)

$$Z_n\left(s\right)=-{\displaystyle \frac{{\mathbf{V}}_{\mathrm{n}}\left(s\right)}{{\mathbf{I}}_{\mathrm{n}}\left(s\right)}}={\displaystyle \frac{\begin{array}{c}{L}_{f}s+K_{\mathrm{PWM}}{u}_{\mathrm{dc}}\times \left(\begin{array}{c}k{G}_{\mathrm{i}}{H}_{\mathrm{i}}+(1-k)G_{\mathrm{i}}{H}_{\mathrm{i}}+j{K}_{\mathrm{d}}{G}_{\mathrm{i}}-j{\displaystyle \frac{3K_Q}{4}}(1-k)G_{\mathrm{i}}{H}_{\mathrm{i}}{H}_v{V}_1\hfill \\ +{\displaystyle \frac{3K_P}{4(s+j2\pi f_1)}}k(1-k)G_{\mathrm{i}}{H}_{\mathrm{i}}{V}_1(-j{I}_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}-I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\hfill \\ -j{\displaystyle \frac{3K_P}{4(s+j2\pi f_1)}}{(1-k)}^2{G}_{\mathrm{i}}{H}_{\mathrm{i}}{H}_v{V}_1^2\hfill \end{array}\right)\end{array}}{\begin{array}{c}1-K_m{V}_{\mathrm{dc}}\times \left(\begin{array}{c}{G}_{\mathrm{v}}{K}_f-(1-k)G_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v+{\displaystyle \frac{k^2}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}}{1+V_1{H}_{\mathrm{PLL}}}}{G}_{\mathrm{v}}{H}_{\mathrm{i}}(I_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}-j{I}_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\hfill \\ +{\displaystyle \frac{k(1-k)}{2}}{\displaystyle \frac{H_{\mathrm{PLL}}}{1+V_1{H}_{\mathrm{PLL}}}}{G}_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v{V}_1-j{\displaystyle \frac{3K_Q}{4}}(1-k)G_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v{I}_1{e}^{j{\phi }_{\mathrm{i}1}}\hfill \\ +{\displaystyle \frac{3K_P}{4(s+j2\pi f_1)}}k(1-k)G_{\mathrm{v}}{H}_{\mathrm{i}}{I}_1{e}^{j{\phi }_{\mathrm{i}1}}(j{I}_{\mathrm{GFL},\mathrm{d}}^{\mathrm{ref}}+I_{\mathrm{GFL},\mathrm{q}}^{\mathrm{ref}})\hfill \\ +j{\displaystyle \frac{3K_P}{4(s+j2\pi f_1)}}{(1-k)}^2{G}_{\mathrm{v}}{H}_{\mathrm{i}}{H}_v{I}_1{V}_1{e}^{j{\phi }_{\mathrm{i}1}}\hfill \end{array}\right)\end{array}}},$$

(34)

where simplification is used, and complete expressions are $G_{\mathrm{i}}=G_{\mathrm{i}}\left(s\right)$, $G_{\mathrm{v}}=G_{\mathrm{v}}\left(s\right)$, $H_{\mathrm{i}}=H_{\mathrm{i}}(s\mp j2\pi f_1)$, $H_{\mathrm{v}}=H_{\mathrm{v}}(s\mp j2\pi f_1)$, $H_{\mathrm{PLL}}=H_{\mathrm{PLL}}(s\mp j2\pi f_1)$.

4. Impedance Characteristics and Coefficient Optimization Method

This section analyzes the stabilization mechanism of hybrid GFL-GFM-controlled wind turbines based on the sequential impedance model. System parameters are listed in

Table 1. System configuration.

Parameter	Value	Parameter	Value
$u_{\mathrm{dc}}$	700 V	$P^{\mathrm{ref}}$	20,000 W
$V_1$	220 V	$Q^{\mathrm{ref}}$	0 Var
$L_{\mathrm{f}}$	3 mH	$K_{\mathrm{pi}}$	1
$C_{\mathrm{f}}$	20 $\mathsf{\mu }$F	$K_{\mathrm{ii}}$	270
$R_{\mathrm{f}}$	1.5 $\Omega$	$K_{\mathrm{pv}}$	1
$T_{\mathrm{s}}$	50 $\mathsf{\mu }$s	$K_{\mathrm{iv}}$	200
${\omega }^{\mathrm{ref}}$	$2\pi ·50$ rad/s	$K_{\mathrm{pPLL}}$	0.27
${\omega }_{\mathrm{v}}$	$2\pi ·5000$ rad/s	$K_{\mathrm{iPLL}}$	11
${\omega }_{\mathrm{i}}$	$2\pi ·5000$ rad/s

.

4.1. Impedance Characteristics of Hybrid GFL-GFM Control

When k = 1 (complete GFL mode), the positive-sequence and negative-sequence impedance frequency responses and their simulation measurements are shown in Figure 3a. Blue solid lines and black dashed lines represent theoretical derivation results for positive- and negative-sequence impedance, respectively, while red crosses and black circles denote sweep verification results. Sweep verification shows good agreement with impedance derivation, validating the modeling process.

When k = 0 (complete GFM mode), corresponding results are shown in Figure 3b. When k = 0.5 (averaging mixed mode), results are shown in Figure 3c. Both cases confirm modeling accuracy through impedance sweep consistency.

Figure 4a presents Bode diagrams for k = 0.2, 0.4, 0.6, and 0.8. Comparison with Figure 3a,b reveals

• The GFL-controlled system exhibits capacitive behavior in 9–150 Hz bands, increasing oscillatory risk with inductive weak grids. Hybrid control impedance resembles GFM control and shows localized negative impedance near fundamental frequency (40–50 Hz), potentially coupling with grid impedance to induce instability.
• As k decreases (increased GFL weighting), the system progressively exhibits GFL characteristics with larger magnitude in low-frequency bands; as k decreases (increased GFM weighting), GFM characteristics dominate.
• Hybrid control’s negative impedance region is not bounded by GFL/GFM limits, and the area exhibits a non-monotonic relationship with k. Take the k = 0.5 case, for example; this region is smaller than that of GFL and GFM control.

The negative impedance region in the output impedance spectrum indicates potential instability due to negative damping behavior. However, this characteristic alone cannot predict system instability; the dynamic interaction between converter and grid must be assessed.

4.2. Small-Signal Stability Analysis and Coefficient-Adaptive Hybrid Control

For grid-connected wind turbines, impedance-based small-signal stability can be assessed via the Nyquist criterion, which involves partitioning the entire system into two distinct subsystems: the hybrid controlled wind turbine and the grid. Then the impedance ratio (grid impedance divided by output impedance) is constructed, and the Nyquist stability criterion is applied to this transfer function; thus, the converter–grid interaction stability is evaluated. Figure 5a illustrates the small-signal representation of the hybrid GFL-GFM-controlled wind turbine system. Based on this model, the ouput current expression is derived as

$$\tilde{i}={\tilde{i}}_{\mathrm{n}}{\displaystyle \frac{Z}{Z+Z_{\mathrm{g}}}}-{\tilde{u}}_{\mathrm{g}}{\displaystyle \frac{1}{Z+Z_{\mathrm{g}}}}=({\tilde{i}}_{\mathrm{n}}-{\displaystyle \frac{{\tilde{u}}_{\mathrm{g}}}{Z}}){\displaystyle \frac{1}{1+Z_{\mathrm{g}}/Z}}.$$

(35)

Critical to this analysis is the prerequisite that both subsystems operate independently in stable conditions. ${\tilde{i}}_{\mathrm{n}}$, $1/Z$ represent the ideal current source and the equivalent conductance in the Norton equivalent circuit, and ${\tilde{u}}_{\mathrm{g}}$ denotes the grid ideal voltage source. These components are stable by definition. Consequently, ${\tilde{i}}_{\mathrm{n}}-{\tilde{u}}_{\mathrm{g}}/Z$ is stable. As evidenced by Equation (35), the small-signal stability depends on $1/\left(1+Z_{\mathrm{g}}/Z\right)$. Figure 5b depicts the equivalent feedback control structure for stability analysis. As shown in Figure 3c, the negative-sequence impedance phase under hybrid control consistently exceeds −90°, indicating that negative-sequence interaction does not affect the small-signal stability of the system. The closed-loop transfer function $Z_{\mathrm{g}}\left(s\right)/Z_{\mathrm{p}}\left(s\right)$ determines stability: the phase difference between the amplitude-frequency curves of the output impedance and the grid impedance is less than 180° at crossing frequencies [18,26]. For inductive grids, the output impedance phase below −90° defines instability risk regions. Note that based on the topology in Figure 1, $Z_{\mathrm{p}}\left(s\right)$ employed for generating the Bode diagram is actually $Z_{\mathrm{p}}\left(s\right)\Vert \left[1/s{C}_{\mathrm{f}}+R_{\mathrm{f}}\right]$. Bode plots (Figure 4b) with shaded instability regions indicate that when the SCR is less than 2, GFL-controlled wind turbine’s positive-sequence impedance interacts with grid impedance, inducing instability.

To determine optimal weighting coefficients for different grid strengths, we analyze the phase margin of the closed-loop transfer function with different weighting coefficients and SCRs. Considering the adoption of GFL control for SCR > 2, Figure 6 shows the minimum phase difference between $Z_{\mathrm{p}}\left(s\right)$ and $Z_{\mathrm{g}}\left(s\right)$ for k varying from 0 to 1 at intervals of 0.2 and the SCR from 1 to 3 at intervals of 0.1.

For turbines utilizing GFL control (i.e., k = 1), instability occurs when SCR ≤ 2 due to the negative phase margin as shown in Figure 6. Similarly, under hybrid control with k = 0.8, instability emerges when SCR < 1.7. To ensure stability under varying SCR conditions, a coefficient-adaptive hybrid GFL-GFM control is proposed in this article. The general idea of the method is illustrated in Figure 7a, where $u_{\mathrm{dq}}$, $i_{\mathrm{dq}}$ are dq-axis voltage and current, respectively. Given GFL control’s fast response and maximum power tracking advantages, our optimal weighting selection maximizes the GFL coefficient while ensuring stability. The derived relationship between weighting coefficient and SCR satisfying this principle is as follows: for SCR > 2, k = 1; for 1.7 < SCR ≤ 2, k = 0.8; for 1 < SCR ≤ 1.7, k = 0.6. As described above, the adaptive control flowchart shown in Figure 7b can be obtained. The detection principle for selecting the weighting coefficients is the grid impedance measurement (via non-characteristic harmonic injection, the recursive least squares method, etc., without elaboration here).

Through this coefficient-adaptive hybrid control, the wind turbine operates in GFL-controlled mode in a strong grid, ensuring wind energy conversion efficiency and maximum power tracking capability. In a weak grid, the weighting coefficients dynamically adapt to actual grid strength, activating hybrid control to provide active grid support capabilities. This ensures stable and economic operation under various grid conditions, demonstrating engineering sense and practice value for wind generation systems.

5. Simulation Verification

A simulation of the wind turbine operating under hybrid GFL-GFM control with adjustable coefficients was built in PLECS to verify the preceding analysis, with k = 1, 0.8, and 0.6 cases. Grid strength fluctuations were emulated by varying grid impedance. The converter parameters are given in Table 1.

In scenario k = 1 (shown in Figure 8a), initial grid impedance is 5.51 mH, and the system is stable. At t = 1.1 s, inductance 2.75 mH is added, driving the SCR from 3 to 2. The system then becomes unstable, aligning with the preceding inferences. This instability occurs because pure GFL control lacks inertia support. When the SCR drops below 2, the phase margin decreases below 0° (shown in Figure 6), causing negative damping in the converter–grid interaction.

In scenario k = 0.8 (shown in Figure 8b), initial grid impedance $L_{\mathrm{g}}$ = 8.26 mH ensures stable operation. Inserting additional inductance 2.75 mH at t = 1.1 s reduces the SCR from 2 to 1.5, after which instability emerges. Compared to $k=1$, the $k=0.8$ case extends stability boundary to SCR = 1.7. The 20% GFM component provides virtual inertia, delaying instability onset.

In scenario k = 0.6 (shown in Figure 8c), initial grid impedance is 7.71 mH. At t = 1.1 s, inductance 3.85 mH is added, setting the SCR from 1.5 to 1. The system maintains stability, validating the preceding analysis. Here, 40% GFM weighting elevates the phase difference between the converter and grid impedance to above 0° across the range 1 ≤ SCR ≤ 3. This phase margin enhancement satisfies the Nyquist stability criterion.

The current performance and harmonic spectrum analysis of these scenarios are shown in Figure 9. As illustrated with k = 1, the total harmonic distortion (THD) rises from 1.2070% to 13.0537% during the process of decreasing the SCR from 3 to 2, while the THD is only 0.6969% with k = 0.8, SCR = 2. This THD reduction results from the active damping introduced by the GFM component, which enhances the phase margin at SCR = 2 conditions. This mechanism effectively avoids resonance during converter–grid interaction, mitigating harmonic amplification risks. This proves that k = 0.8 is more adapted to this grid condition than k = 1, with higher power quality, and that flexible adjustment of k according to the grid conditions is an effective means.

Under weak grid conditions, wind turbines require robust stability against severe disturbances, including harmonic perturbations, frequency fluctuations, and sudden load changes. The hybrid control performance under these conditions is validated. For harmonic disturbances, Figure 10 depicts stable operation at k = 0.8 and SCR = 2 with 3rd/5th harmonics at 10% fundamental amplitude and k = 0.6 and SCR = 1.5 with 7th/11th harmonics at 10% fundamental amplitude. These results are further supported by broadband harmonic sweep tests (1–8 kHz, 5% fundamental amplitude) in Figure 3c, where the system maintains stability under wide-spectrum harmonic injection conditions, obtaining the proper impedance sweep verification results, which confirm the reliability of the impedance characterization. Figure 11 examines frequency fluctuation scenarios showing stable operation during ±0.5 Hz deviations for identical ks and SCRs. As can be seen in Figure 11a,c, the rate of frequency change is lower for higher GFM weighting coefficient. Additionally, Figure 12 demonstrates stable operation with 5000W load insertion/removal transients. Collectively, these tests confirm the hybrid control’s resilience to grid anomalies.

6. Conclusions

With the rapid development of renewable energy, modern power grids increasingly exhibit weak grid characteristics. Wind power systems face stability control challenges under fluctuating SCR conditions. A small-signal sequential impedance model was constructed in this work for PMSG-based wind turbines under hybrid GFL-GFM control with adjustable weighting coefficients. Through small-signal stability analysis at varying grid strengths, we derived and validated optimal weighting factors that ensure system stability across different grid conditions. The proposed impedance model was verified via impedance sweep results in PLECS simulations, demonstrating precise alignment between theoretical predictions and practical results. This work provides a method for hybrid control with enhancing grid adaption in wind generation systems. By introducing the proposed adaptive hybrid control strategy with adjustable coefficients, the system achieves stable operation across wide SCR ranges, demonstrating practical applicability in real-world grids. However, the study has not validated coefficient adjustment during SCR fluctuations, smooth transition control between different coefficients requires further research to prevent power disturbances, and experimental verification is needed for field deployment. Future work will focus on developing flexible control algorithms for dynamic SCR variations and implementing experimental validation.