Oscillation Mechanism of SRF-PLL in Wind Power Systems Under Voltage Sags and Improper Control Parameters

Abstract

The synchronous reference frame phase-locked loop (SRF-PLL) is widely employed for grid synchronization in wind farms. However, it may exhibit oscillations under voltage sags or improper parameter settings. These oscillations may compromise the secure integration of large-scale wind power. Therefore, mitigating the oscillations of the SRF-PLL is crucial for ensuring stable and reliable operation. To this end, this paper investigates the underlying oscillation mechanism of the SRF-PLL from local and global perspectives. By taking into account the grid voltage and control parameters, it is revealed that oscillations of the SRF-PLL can be triggered by grid voltage sags and/or the improper control parameters. More specifically, from the local perspective, the SRF-PLL exhibits distinct qualitative behaviors around its stable equilibrium points under different grid voltage amplitudes. As a result, when grid voltage sags occur, the SRF-PLL may exhibit multiple oscillation modes and experience a prolonged transient response. Furthermore, from the global viewpoint, the large-signal analysis reveals that the SRF-PLL has infinitely many asymmetrical convergence regions. However, the sizes of these asymmetrical convergence regions shrink significantly under low grid voltage amplitude and/or small control parameters. In this case, even if the parameters in the small-signal model of the SRF-PLL are well-designed, a small disturbance can shift the operating point into other regions, resulting in undesirable oscillations and a sluggish dynamic response. The validity of the theoretical analysis is further supported by experimental verification.

1. Introduction

For grid-connected applications, synchronization is commonly required. Thus, a variety of phase-locked loop-based grid synchronization methods have been proposed in the literature [1,2], e.g., the second-order generalized integrator-based phase-locked loops (SOGI-PLL) [3], moving average filter based phase-locked loops (MAF-PLL) [4], and adaptive observer PLLs [5]. Among them, the synchronous reference frame PLL (SRF-PLL) remains the most prevalently adopted due to its simplicity and effectiveness [6,7,8,9]. In three-phase power systems, the SRF-PLL is employed to extract the phase angle, frequency, and voltage amplitude of the grid, and it plays a vital role in the control and monitoring of grid-connected converters in wind power systems.

With the rapid development of wind power generation, stability issues arising from the interaction between wind power systems and the grid have attracted increasing attention [10]. In this context, many recent studies have focused on improving power quality and synchronization, including advanced control and artificial intelligence (AI)-based control strategies [11,12,13,14]. In particular, synchronization is of critical importance, as the interaction between the SRF-PLL and grid stability has been widely explored in previous studies [15,16,17], and it has been shown that the SRF-PLL exerts a significant impact on the stability of AC grids. Meanwhile, the performance of SRF-PLL is strongly affected by the voltage conditions at the point of common coupling (PCC). To address these challenges, many enhanced PLL variants have been proposed for improved synchronization robustness and power quality under weak grids, such as Adaptive SRF-PLL [18], Extended State Observer Based PLL (ESO-PLL) [19], Dual Second-Order Generalized Integrator PLL (DSOGI-PLL) [20], and Enhanced PLL (EPLL) [21]. Each of these methods has its own advantages: Adaptive SRF-PLL dynamically tunes its parameters to accommodate different grid strengths; ESO-PLL enhances robustness against harmonics and disturbances by compensating unmodeled dynamics; DSOGI-PLL achieves accurate phase estimation under unbalanced and harmonic-distorted voltages; and EPLL provides a faster dynamic response and improved noise rejection. These improvements primarily target weak-grid and distorted-voltage conditions, while comparative studies under severe voltage sags are still limited in the literature. Although these advanced structures introduce additional adaptability or filtering components, they are generally built upon the conventional SRF-PLL framework. Therefore, a comprehensive evaluation of the SRF-PLL’s dynamic performance under different grid conditions remains essential, as it not only informs the design of wind turbine controllers but also provides valuable insights for the development of advanced PLL algorithms.

To this end, the SRF-PLL is modeled by a simplified small-signal model based on which the performance of the nonlinear SRF-PLL can be easily analyzed in the linear framework (linear analysis methods) [22]. However, the small-signal model can not accurately reflect its dynamics. The small-signal model-based methods are effective under the specific assumption that the estimated frequency $\widehat{\omega }$ closely matches the actual grid frequency $\omega$. However, when this assumption does not hold, ignoring their differences may obscure important dynamic characteristics of the system [23,24]. In fact, previous studies have indicated that large frequency deviations can lead to sluggish dynamic behavior in the SRF-PLL [22].

Small-signal model methods focus on local behavior, whereas the large-signal model-based analysis provides a direct insight into the overall performance of the SRF-PLL. In [25], a large-signal model was established, based on which the global stability analysis of the SRF-PLL was conducted. Using the phase-portrait approach, it is shown in [25] that the SRF-PLL possesses infinitely many stable equilibrium points and corresponding convergence regions. However, the study in [25] does not explicitly address the underlying oscillation mechanisms under voltage sags and improper control parameters, which are common and critical in weak grid scenarios. In practical systems, voltage sags caused by grid faults are common and pose significant challenges to maintaining reliable synchronization and satisfactory control performance, as required by grid codes. Therefore, it is desirable to disclose the oscillation mechanism with respect to grid voltage and control parameters, since such insights can provide valuable guidance for parameter tuning and robust controller design in wind turbine applications.

Building on the above, this study aims to fill this gap by offering new insights into how grid voltage amplitudes and parameter settings give rise to undesirable oscillations in the SRF-PLL. Firstly, the nonlinear large-signal model for the SRF-PLL is derived, and subsequently, the phase-portrait method is adopted to investigate how the SRF-PLL behaves under different grid voltage amplitudes and control parameters. Based on the in-depth theoretical analysis, this study reveals that the performance of the SRF-PLL is highly sensitive to grid voltage amplitude and control parameter settings. In particular, undesirable oscillations may arise under voltage sags and/or improper control parameter, posing a serious threat to system stability. The contributions of this paper are summarized as follows:

• SRF-PLL oscillation analysis around stable equilibrium points. This paper demonstrates that the SRF-PLL has distinct qualitative behaviors around stable equilibrium points for different grid voltage amplitudes. In the vicinity of a stable equilibrium point, the SRF-PLL under a normal grid voltage converges directly to that point without oscillations. However, when the grid voltage amplitude drops significantly, the SRF-PLL may exhibit multiple oscillations, leading to an extended transient response.
• SRF-PLL oscillation analysis from a global viewpoint.This paper reveals that the sizes of the convergence regions of SRF-PLL are influenced by the grid voltage amplitude and control parameters. When the grid voltage amplitude and/or control parameters are small, these convergence regions shrink significantly. In this case, a small disturbance, e.g., a small frequency jump, can cause the state of the SRF-PLL to move outside the original convergence region. Consequently, undesirable oscillations and a prolonged transient process may occur, even if the control parameters $k_p$ and $k_i$ have been carefully designed according to the SRF-PLL’s small-signal model.

The rest of the paper is organized as follows. Section 2 reviews the large signal model of the SRF-PLL. Section 3 investigates the relationship between grid voltage, control parameters, and the oscillation mechanism of the SRF-PLL. The experimental validation of the analysis is presented in Section 4. Section 5 concludes this paper, while Section 6 provides further discussion and outlines potential future work.

2. Large-Signal Model of the SRF-PLL

Figure 1 illustrates the structure of the SRF-PLL, which includes three main components: a phase detector (PD), a loop filter (LF), and a voltage-controlled oscillator (VCO) [22]. The ideal grid voltages in a three-phase system, denoted as $u_a\left(t\right)$, $u_b\left(t\right)$, and $u_c\left(t\right)$, can be mathematically represented as follows:

$$\begin{array}{cc}\hfill u_a\left(t\right)& =Vcos\left(\theta \right)=Vcos(\omega t+\varphi )\hfill \\ \hfill u_b\left(t\right)& =Vcos\left(\theta -{\displaystyle \frac{2\pi }{3}}\right)=Vcos\left(\omega t+\varphi -{\displaystyle \frac{2\pi }{3}}\right)\hfill \\ \hfill u_c\left(t\right)& =Vcos\left(\theta +{\displaystyle \frac{2\pi }{3}}\right)=Vcos\left(\omega t+\varphi +{\displaystyle \frac{2\pi }{3}}\right)\hfill \end{array}$$

(1)

where V corresponds to the voltage magnitude, $\omega$ to the frequency, $\theta$ to the phase, and $\varphi$ to the initial phase angle of the grid voltage. To facilitate analysis in the rotating reference frame, the Clarke-Park transformation T is applied to convert $abc$ variables into $dq$ quantities:

$$T={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}cos\left(\widehat{\theta }\right)& cos(\widehat{\theta }-{\displaystyle \frac{2\pi }{3}})& cos(\widehat{\theta }+{\displaystyle \frac{2\pi }{3}})\\ -sin\left(\widehat{\theta }\right)& -sin(\widehat{\theta }-{\displaystyle \frac{2\pi }{3}})& -sin(\widehat{\theta }+{\displaystyle \frac{2\pi }{3}})\end{array}\right]$$

(2)

where $\widehat{\theta }$ denotes the estimated value of $\theta$, and the grid voltages defined in (1) are then converted into the $dq$ reference frame:

$$\begin{array}{ccc}\hfill u_d& =& Vcos(\theta -\widehat{\theta })\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill u_q& =& Vsin(\theta -\widehat{\theta })\hfill \end{array}$$

(4)

Taking the $u_q$ as the input of the LF, its output $\Delta \omega$ is given by the following:

$$\begin{array}{ccc}\Delta \omega & =& k_p{u}_q+k_i\int u_{q}dt\hfill \end{array}$$

(5)

where $k_p$ and $k_i$ are the control parameters in the SRF-PLL (see Appendix A. Nomenclature for descriptions and units). The output of the LF then drives the VCO, which produces the estimated frequency and estimated phase as follows:

$$\begin{array}{ccc}\widehat{\theta }& =& \int \widehat{\omega }dt\hfill \\ \widehat{\omega }& =& \Delta \omega +{\omega }_c=k_p{u}_q+k_i\int u_{q}dt+{\omega }_c\hfill \end{array}$$

(6)

in which ${\omega }_c$ represents the nominal grid frequency and remains constant throughout the operation.

Taking the derivatives of the estimated frequency and phase defined in (6) yields the following:

$$\begin{array}{c}\dot{\widehat{\theta }}=\widehat{\omega }\hfill \\ \dot{\widehat{\omega }}=k_p{\dot{u}}_q+k_i{u}_q\hfill \end{array}$$

(7)

To proceed with the derivation, we define the phase and frequency estimation errors as the differences between the estimated and actual values, i.e., $\tilde{\theta }=\widehat{\theta }-\theta$ and $\tilde{\omega }=\widehat{\omega }-\omega$. Calculating the derivatives of these estimation errors leads to

$$\begin{array}{c}\dot{\tilde{\theta }}=\widehat{\omega }-\omega =\tilde{\omega }\hfill \\ \dot{\tilde{\omega }}=k_p{\dot{u}}_q+k_i{u}_q\hfill \end{array}$$

(8)

Substituting the expression of $u_q$ from Equation (4) into the above Equation (8) yields

$$\begin{array}{ccc}\dot{\tilde{\theta }}& =& \tilde{\omega }\hfill \\ \dot{\tilde{\omega }}& =& -k_{i}Vsin\tilde{\theta }-k_{p}V\tilde{\omega }cos\tilde{\theta }\hfill \end{array}$$

(9)

which provides a large-signal representation that reflects the nonlinear dynamics of the SRF-PLL. Equation (9) clearly indicates that the SRF-PLL’s performance is affected by grid voltage amplitude V and control parameters $k_p$ and $k_i$. In the next section, the underlying oscillation mechanisms of the SRF-PLL are investigated in-depth from both local and global perspectives.

3. Oscillations Mechanism of the SRF-PLL

In this section, the undesirable oscillations of the SRF-PLL are further analyzed under different grid voltage amplitudes and control parameters. The relationships among the grid voltage amplitudes, control parameters, and oscillations of the SRF-PLL are revealed as follows.

3.1. SRF-PLL Oscillation Around Stable Equilibrium Points

From the large-signal model described in (9), its corresponding equilibrium points satisfy the following condition:

$$\begin{array}{c}0=\tilde{\omega }\hfill \\ 0=-k_{i}Vsin\tilde{\theta }-k_{p}V\tilde{\omega }cos\tilde{\theta }\hfill \end{array}$$

(10)

which implies that the SRF-PLL possesses an infinite number of equilibrium points located at $(n\pi ,0),n=0,\pm 1,\pm 2,\cdots$. To facilitate analysis, we define the transformed variables ${\tilde{\theta }}_n=\tilde{\theta }-n\pi$ and ${\tilde{\omega }}_n=\tilde{\omega }-0$. By performing Taylor series expansion of the nonlinear system described by Equation (9) around the equilibrium points $(n\pi ,0)$, we obtain a linearized model as follows:

$$\left(\begin{array}{c}{\dot{\tilde{\theta }}}_n\\ {\dot{\tilde{\omega }}}_n\end{array}\right)=A_n\left(\begin{array}{c}{\tilde{\theta }}_n\\ {\tilde{\omega }}_n\end{array}\right)$$

(11)

where

$$A_n=\left[\begin{array}{cc}0& 1\\ -k_{i}Vcos\left(n\pi \right)& -k_{p}Vcos\left(n\pi \right)\end{array}\right]$$

The theoretical foundation of this linearization method can be found in [26].

Around each equilibrium points, the SRF-PLL exhibits behavior analogous to the linearized system described by (11). Clearly, when $n=2k+1$, it follows that $cos\left(n\pi \right)=-1$. A straightforward calculation then shows that the corresponding linearized system of (11) is unstable. In this case, the equilibrium points $\left(\right(2k+1)\pi ,0)$ are unstable due to the presence of one eigenvalue of $A_n$ with a positive real part and another with a negative real part. In addition, if $n=2k$, $cos\left(n\pi \right)=1$. In this case, the linear system of (11) is stable, and hence, the equilibrium points $(2k\pi ,0)$ are stable points. This indicates that within a small neighborhood of a stable point, the SRF-PLL will converge to that stable point eventually. However, the eigenvalues of $A_n$ rely on the grid voltage V and control parameters $k_p$ and $k_i$. $A_n$ has two complex eigenvalues:

$$-{\displaystyle \frac{k_{p}V}{2}}\pm j\sqrt{k_{i}V-{\displaystyle \frac{{\left(k_{p}V\right)}^2}{4}}}$$

(12)

From (12), the real part of the eigenvalues is $-k_{p}V/2$, which represents the decay rate of the system’s dynamics. A more negative real part indicates faster attenuation of oscillations. Hence, a smaller grid voltage V or a smaller control parameter $k_p$ leads to slow dynamic for the SRF-PLL. In addition, the ratio between the imaginary part and real part is

$$\sqrt{{\displaystyle \frac{4k_i}{k_p^{2}V}}-1}$$

(13)

which is closely related to the damping ratio, a measure of how quickly oscillations decay. When the grid voltage amplitude is small, this ratio increases, which means the damping ratio of the SRF-PLL decreases, resulting in more pronounced and sustained oscillations.

In summary, even if the control parameters $k_p$ and $k_i$ are well-designed for the normal grid voltage, a small voltage amplitude, like a grid voltage sag, may result in oscillations and slow dynamics of the SRF-PLL. To clearly illustrate the relationship between grid voltage amplitude and the oscillation characteristics of the SRF-PLL, the phase portraits corresponding to grid voltage amplitudes of 1 p.u. and $0.1$ p.u. are shown in Figure 2. The control parameters $k_p$ and $k_i$ are kept identical in both cases. When the amplitude of the grid voltage is the normal value (1 p.u.), the SRF-PLL settles at the stable point $(0,0)$ without oscillations. In contrast, when the voltage amplitude drops to $0.1$ p.u., the SRF-PLL exhibits significant oscillations according to Figure 2. It confirms the above analysis.

3.2. Global Oscillation Analysis for the SRF-PLL

The above analysis addresses the local stability condition. In the following, undesirable oscillations of the SRF-PLL are discussed from a global viewpoint.

Firstly, a rigorous Lyapunov analysis is performed to demonstrate that, given any initial states $(\tilde{\theta }\left(t_0\right),\tilde{\omega }\left(t_0\right))$ within a convergence region, the trajectory $(\tilde{\theta }\left(t\right),\tilde{\omega }\left(t\right))$, for $t>t_0$, will stay confined within this convergence region and approach its corresponding stable point asymptotically.

To simplify the analysis, we first define the intermediate variables as

$$\begin{array}{c}\hfill {\tilde{\theta }}_{2k}=\tilde{\theta }-2k\pi ,{\tilde{\omega }}_{2k}=\tilde{\omega }\end{array}$$

(14)

Based on the defined intermediate variables, the original model in (9) is transformed into the following equivalent large-signal expression:

$$\begin{array}{c}{\dot{\tilde{\theta }}}_{2k}={\tilde{\omega }}_{2k}\hfill \\ {\dot{\tilde{\omega }}}_{2k}=-k_{i}Vsin{\tilde{\theta }}_{2k}-k_{p}V{\tilde{\omega }}_{2k}cos{\tilde{\theta }}_{2k}\hfill \end{array}$$

(15)

The nonlinear characteristics of the SRF-PLL make the Lyapunov function construction and subsequent stability analysis difficult. Therefore, the variable gradient approach is adopted to facilitate the construction of a suitable Lyapunov function. Specifically, define $U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ be a scalar function of ${({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})}^T$ and assume that its gradient vector with respect to the state vector can be expressed as

$$\nabla U=\left(\begin{array}{c}\partial U/\partial {\tilde{\theta }}_{2k}\\ \partial U/\partial {\tilde{\omega }}_{2k}\end{array}\right)=\left(\begin{array}{c}{g}_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})\\ g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})\end{array}\right)=g({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k}).$$

(16)

The derivative of $U(·)$ along the trajectories of system (15) is given by

$$\begin{array}{c}\dot{U}(·)={\displaystyle \frac{\partial U}{\partial {\tilde{\theta }}_{2k}}}{\dot{\tilde{\theta }}}_{2k}+{\displaystyle \frac{\partial U}{\partial {\tilde{\omega }}_{2k}}}{\dot{\tilde{\omega }}}_{2k}\hfill \\ =g_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k}){\tilde{\omega }}_{2k}+g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})(-k_{i}Vsin{\tilde{\theta }}_{2k}-k_{p}V{\tilde{\omega }}_{2k}cos{\tilde{\theta }}_{2k})\hfill \end{array}$$

(17)

The variable gradient method is to determine a suitable function $g({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ that serves as the gradient of a positive definite function $U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ and, at the same time, ensures that the derivative of U along the system trajectories satisfies the Lyapunov stability conditions, i.e.,

$$\begin{array}{c}\dot{U}(·)=g_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k}){\tilde{\omega }}_{2k}+g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})(-k_{i}Vsin{\tilde{\theta }}_{2k}-k_{p}V{\tilde{\omega }}_{2k}cos{\tilde{\theta }}_{2k})\le 0\hfill \end{array}$$

(18)

Let us consider

$$\begin{array}{c}\hfill g({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})=\left[\begin{array}{c}\alpha \left({\tilde{\theta }}_{2k}\right){\tilde{\theta }}_{2k}+\beta {\tilde{\omega }}_{2k}\\ \gamma {\tilde{\theta }}_{2k}+\delta {\tilde{\omega }}_{2k}\end{array}\right]\end{array}$$

(19)

where $\alpha \left({\tilde{\theta }}_{2k}\right)$ denotes the scalar function, while $\beta ,\gamma$ and $\delta$ are constants to be determined. Note that $U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ is a scalar function. Thus,

$${\displaystyle \frac{{\partial }^{2}U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})}{\partial {\tilde{\theta }}_{2k}\partial {\tilde{\omega }}_{2k}}}={\displaystyle \frac{\partial g_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})}{\partial {\tilde{\omega }}_{2k}}}={\displaystyle \frac{\partial g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})}{\partial {\tilde{\theta }}_{2k}}}.$$

(20)

It implies that $\beta =\gamma$. Submitting (19) into (18) results in

$$\begin{array}{c}{g}_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k}){\tilde{\omega }}_{2k}+g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})(-k_{i}Vsin{\tilde{\theta }}_{2k}-k_{p}V{\tilde{\omega }}_{2k}cos{\tilde{\theta }}_{2k})\hfill \\ =\alpha \left({\tilde{\theta }}_{2k}\right){\tilde{\theta }}_{2k}{\tilde{\omega }}_{2k}+\beta {\tilde{\omega }}_{2k}^2-k_{i}V\beta {\tilde{\theta }}_{2k}sin{\tilde{\theta }}_{2k}-k_{p}V\beta {\tilde{\theta }}_{2k}\hfill \\ \times {\tilde{\omega }}_{2k}cos{\tilde{\theta }}_{2k}-k_{i}V\delta {\tilde{\omega }}_{2k}sin{\tilde{\theta }}_{2k}-k_{p}V\delta {\tilde{\omega }}_{2k}^{2}cos{\tilde{\theta }}_{2k}\hfill \\ =-k_{i}V\beta {\tilde{\theta }}_{2k}sin{\tilde{\theta }}_{2k}-(k_{p}V\delta cos{\tilde{\theta }}_{2k}-\beta ){\tilde{\omega }}_{2k}^2+{\tilde{\omega }}_{2k}\hfill \\ \times \left[\alpha \left({\tilde{\theta }}_{2k}\right){\tilde{\theta }}_{2k}-k_{p}V\beta {\tilde{\theta }}_{2k}cos{\tilde{\theta }}_{2k}-k_{i}V\delta sin{\tilde{\theta }}_{2k}\right].\hfill \end{array}$$

Choosing $\alpha \left({\tilde{\theta }}_{2k}\right){\tilde{\theta }}_{2k}$ as $k_{p}V\beta {\tilde{\theta }}_{2k}cos{\tilde{\theta }}_{2k}+k_{i}V\delta sin{\tilde{\theta }}_{2k}$ leads to

$$\begin{array}{c}{g}_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k}){\tilde{\omega }}_{2k}+g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})(-k_{i}Vsin{\tilde{\theta }}_{2k}-k_{p}V{\tilde{\omega }}_{2k}cos{\tilde{\theta }}_{2k})\hfill \\ =-k_{i}V\beta {\tilde{\theta }}_{2k}sin{\tilde{\theta }}_{2k}-(k_{p}V\delta cos{\tilde{\theta }}_{2k}-\beta ){\tilde{\omega }}_{2k}^2.\hfill \end{array}$$

(21)

Consequently,

$$\begin{array}{c}{g}_1({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})=k_{p}V\beta {\tilde{\theta }}_{2k}cos{\tilde{\theta }}_{2k}+k_{i}V\delta sin{\tilde{\theta }}_{2k}+\beta {\tilde{\omega }}_{2k},\hfill \\ g_2({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})=\beta {\tilde{\theta }}_{2k}+\delta {\tilde{\omega }}_{2k}\hfill \end{array}$$

(22)

The function $U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ is then obtained as

$$\begin{array}{c}U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})={\int }_0^{{\tilde{\theta }}_{2k}}{g}_1(y_1,0)d{y}_1+{\int }_0^{{\tilde{\omega }}_{2k}}{g}_2({\tilde{\theta }}_{2k},y_2)d{y}_2\hfill \\ =k_{p}V\beta {\int }_0^{{\tilde{\theta }}_{2k}}{y}_{1}cos{y}_{1}d{y}_1+\beta {\tilde{\theta }}_{2k}{\tilde{\omega }}_{2k}+{\displaystyle \frac{1}{2}}\delta {\tilde{\omega }}_{2k}^2+k_{i}V\delta {\int }_0^{{\tilde{\theta }}_{2k}}sin{y}_{1}d{y}_1\hfill \end{array}$$

(23)

It is clear that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial U}{\partial {\tilde{\theta }}_{2k}}}& =& k_{p}V\beta {\tilde{\theta }}_{2k}cos{\tilde{\theta }}_{2k}+k_{i}V\delta sin{\tilde{\theta }}_{2k}+\beta {\tilde{\omega }}_{2k},\hfill \\ \hfill {\displaystyle \frac{\partial U}{\partial {\tilde{\omega }}_{2k}}}& =& \beta {\tilde{\theta }}_{2k}+\delta {\tilde{\omega }}_{2k}.\hfill \end{array}$$

(24)

Thus, $g({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ is the gradient of $U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$.

Clearly, choosing $\beta =0$ and $\delta >0$ implies

$$\begin{array}{c}U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})={\displaystyle \frac{1}{2}}\delta {\tilde{\omega }}_{2k}^2+k_{i}V\delta {\int }_0^{{\tilde{\theta }}_{2k}}sin{y}_{1}d{y}_1\hfill \\ \dot{U}({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})=-k_{p}V\delta cos{\tilde{\theta }}_{2k}{\tilde{\omega }}_{2k}^2\hfill \end{array}$$

(25)

It means that in the region $\mathrm{\Omega }=\left\{({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})\right|U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})\le \delta k_{i}V\}$, $U({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})$ is positive definite and its derivative

$$\dot{U}({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})=-k_{p}V\delta cos{\tilde{\theta }}_{2k}{\tilde{\omega }}_{2k}^2\le 0$$

Hence, the SRF-PLL is stable within this region. Given an initial state $({\tilde{\theta }}_{2k}\left(t_0\right),{\tilde{\omega }}_{2k}\left(t_0\right))$ inside the region $\mathrm{\Omega }$, the system trajectory $({\tilde{\theta }}_{2k}\left(t\right),{\tilde{\omega }}_{2k}\left(t\right))$, for all $t>t_0$, will stay within this convergence region $\mathrm{\Omega }$.

In the following, it is further shown that ${lim}_{t\to \infty }{\tilde{\theta }}_{2k}\left(t\right)=0$ and ${lim}_{t\to \infty }{\tilde{\omega }}_{2k}\left(t\right)=0$ by applying LaSalle’s invariance theorem [26]. Define S to denote the set of all points where $\dot{U}=0$, i.e.,

$$\dot{U}=0\Rightarrow -k_{p}V\delta cos{\tilde{\theta }}_{2k}{\tilde{\omega }}_{2k}^2=0\Rightarrow {\tilde{\omega }}_{2k}=0$$

(26)

Hence, $S=\{({\tilde{\theta }}_{2k},{\tilde{\omega }}_{2k})\in \mathrm{\Omega }|{\tilde{\omega }}_{2k}=0\}$. Let $({\tilde{\theta }}_{2k}\left(t\right),{\tilde{\omega }}_{2k}\left(t\right))$ be a trajectory that remains entirely within S, i.e.,

$$\begin{array}{c}\hfill {\tilde{\omega }}_{2k}\equiv 0\Rightarrow {\dot{\tilde{\omega }}}_{2k}\equiv 0\Rightarrow k_{i}Vsin\left({\tilde{\theta }}_{2k}\right)\equiv 0\end{array}$$

(27)

which leads to ${\tilde{\theta }}_{2k}\left(t\right)\equiv 0$. Consequently, the only trajectory that remains entirely within the set S corresponds to the trivial solution $({\tilde{\theta }}_{2k}\left(t\right),{\tilde{\omega }}_{2k}\left(t\right))\equiv (0,0)$. By invoking the LaSalle’s invariance theorem [26], it follows that every solution starting from an initial state $({\tilde{\theta }}_{2k}\left(t_0\right),{\tilde{\omega }}_{2k}\left(t_0\right))\in \mathrm{\Omega }$ converges to $(0,0)$ as $t\to \infty$. This guarantees that the SRF-PLL asymptotically converges to its stable equilibrium point inside the corresponding convergence region.

The preceding analysis rigorously proves the existence and properties of convergence regions, showing that every stable equilibrium point is associated with a corresponding convergence region, and for any initial state within this convergence region, the trajectory of SRF-PLL will eventually converge to the corresponding stable point.

Importantly, the size of each convergence region is determined by the grid voltage amplitude V and the parameters $k_p$ and $k_i$. Specifically, lower grid voltage amplitudes and/or smaller control parameters lead to reduced convergence regions. This relationship is further illustrated by the phase portraits in Figure 3, which depict the convergence region of the SRF-PLL under different control parameters and grid voltage amplitudes. It is worth noting that the boundary of the convergence region is determined by the stable manifold of the saddle points, which specifies the set of initial conditions from which the system can achieve stable locking. States outside this region will not converge to the locked equilibrium, although the overall system may still remain stable, potentially resulting in degraded dynamic performance. As shown, either decreased controller parameters $k_p$, $k_i$ (Figure 3b), or a reduced grid voltage amplitude V (Figure 3c) result in a noticeably smaller convergence region, corroborating the theoretical analysis presented above.

The above analysis reveals how the size of each convergence region depends on the control parameters $k_p$, $k_i$ and the grid voltage amplitude V. Since the SRF-PLL has infinitely many such regions, the overall stability characteristic is largely governed by these parameters. In particular, when the grid voltage amplitude or the control parameters are small, the corresponding convergence regions shrink significantly. This reduction in region size decreases the system’s robustness; as a result, even minor disturbances can cause the state of the SRF-PLL to move out of its original convergence region and enter another region. Consequently, the system settles at a new stable equilibrium point that is far from the original one, resulting in significant oscillations and a prolonged transient response.

To illustrate this phenomenon vividly, Figure 4 presents a case with grid voltage amplitude $V=0.1$ p.u., and control parameters are $k_p=46$, $k_i=1058$, representing a condition of sagged grid voltage and moderate control parameters. The control parameters of the SRF-PLL were selected based on its small-signal model [2]. Following widely used design practice [2,22], the parameters were tuned to achieve a damping ratio of about 0.707. This ensures a suitable compromise between dynamic response and robustness against disturbances. It is worth noting that the focus of this paper is on investigating how parameter variations influence the oscillation behavior and dynamic performance of the SRF-PLL rather than on proposing optimal design rules. When subjected to a $4.5$ Hz frequency jump, the reduced size of the convergence region causes the state of SRF-PLL to jump into Region III. As shown by the red line, this transition triggers significant oscillations, further confirming the theoretical analysis above.

4. Experimental Results

This section presents the experimental validation of the preceding analysis. The power stage hardware consists of a three-phase two-level voltage source converter built with MOSFETs. The converter is controlled by a TI TMS320F28335 digital signal processor (DSP) platform, which executes the entire SRF-PLL control algorithm and implements pulse width modulation at a switching frequency of 10 kHz. All experiments were conducted under normal laboratory conditions. The grid simulator used was the STR3060A, which is capable of generating frequency jumps and simulating various grid voltage disturbances.

4.1. Local Oscillations

In this subscetion, the SRF-PLL is tested under a 1 Hz frequency jump to verify the oscillation around stable equilibrium points. Firstly, the relationship between the grid voltage amplitude and the SRF-PLL oscillations is examined through Cases A to C.

Table 1. Experimental parameters and key performance indicators for Cases A–C.

Cases	Voltage Amplitudes	Control Parameters	With/Without Oscillations	Convergence Time
Case A	$V=1$ p.u.	$k_p=46,k_i=1058$	without oscillations	140 ms
Case B	$V=0.5$ p.u.	$k_p=46,k_i=1058$	without oscillations	300 ms
Case C	$V=0.1$ p.u.	$k_p=46,k_i=1058$	with oscillations	2000 ms

shows the experimental parameters and the compared experimental results of these cases. Figure 5 present the phase portrait comparisons and experimental results for when the voltage amplitude $V=1$ p.u., $V=0.5$ p.u., and $V=0.1$ p.u., respectively. From Figure 5, it is observed that the small voltage amplitude may cause significant oscillations and a long transient process in the SRF-PLL, validating the theoretical analysis in Section 3.

Furthermore, under a 1 Hz frequency jump, the relationship between the control parameters and the SRF-PLL oscillations is investigated through Cases D to F.

Table 2. Experimental parameters and key performance indicators for Cases D–F.

Cases	Voltage Amplitudes	Control Parameters	With/Without Oscillations	Convergence Time
Case D	$V=0.5$ p.u.	$k_p=46,k_i=1058$	without oscillations	300 ms
Case E	$V=0.5$ p.u.	$k_p=46\times 0.7,k_i=1058\times 0.49$	without oscillations	450 ms
Case F	$V=0.5$ p.u.	$k_p=46\times 0.4,k_i=1058\times 0.16$	without oscillations	750 ms

summarizes the experimental conditions and corresponding results for each case, while Figure 6 presents both the phase portraits and experimental waveforms. It can be observed that the small control parameters may lead to prolonged transient responses of the SRF-PLL. This observation supports the theoretical analysis presented in Section 3.

4.2. Global Oscillations

The above discussion validates the SRF-PLL’s oscillation behavior around stable equilibrium points. In this subsection, the SRF-PLL is subjected to a $4.5$ Hz frequency jump to verify the oscillation characteristics of the SRF-PLL from a global viewpoint. Specifically, Cases G–I investigate the influence of the grid voltage amplitude on the SRF-PLL’s global behavior.

Table 3. Experimental parameters and key performance indicators for cases G–I.

Cases	Voltage Amplitudes	Control Parameters	With/Without Oscillations	Convergence Time	Convergence Region Size
Case G	$V=1\mathrm{p}.\mathrm{u}.$	$k_p=46,k_i=1058$	without oscillations	$140\mathrm{ms}$	Large
Case H	$V=0.5\mathrm{p}.\mathrm{u}.$	$k_p=46,k_i=1058$	without oscillations	$300\mathrm{ms}$	Medium
Case I	$V=0.1\mathrm{p}.\mathrm{u}.$	$k_p=46,k_i=1058$	with oscillations	$2800\mathrm{ms}$	Small

summarizes the experimental parameters and the corresponding experimental results. Figure 7 presents the phase portraits and the experimental results of the SRF-PLL in Cases G–I. Among them, Case I features the smallest voltage amplitude, which leads to the smallest convergence region. To quantify this effect, we experimentally determined the maximum frequency jump that keeps the state within the convergence region of the equilibrium point $(0,0)$. Specifically, this maximum tolerable frequency jump is 15.9 Hz at 1.0 p.u., decreases to 10.0 Hz at 0.5 p.u., and further drops to 3.7 Hz at 0.1 p.u. As a result, under the $4.5$ Hz frequency jump, the state of the SRF-PLL moves out of the original region (Region I). Consequently, the SRF-PLL in Case I experiences significant oscillations and a slow transient response, thereby confirming the theoretical analysis from a global viewpoint in Section 3.

Furthermore, the influence of control parameters on the oscillation behavior of the SRF-PLL is investigated in Cases J–L.

Table 4. Experimental parameters and key performance indicators for Cases J–L.

Cases	Voltage Amplitudes	Control Parameters	With/Without Oscillations	Convergence Time	Convergence Region Size
Case J	$V=0.5$ p.u.	$k_p=46,k_i=1058$	without oscillations	300 ms	Large
Case K	$V=0.5$ p.u.	$k_p=46\times 0.7,k_i=1058\times 0.49$	without oscillations	450 ms	Medium
Case L	$V=0.5$ p.u.	$k_p=46\times 0.4,k_i=1058\times 0.16$	with oscillations	1500 ms	Small

presents the experimental parameters and the comparison of the experimental results across these cases. Figure 8 illustrates the phase portraits and experimental results of the SRF-PLL. Among the three cases, Case L adopts the smallest control parameters, resulting in the smallest convergence region. To quantify this effect, we also conducted experiments to identify the largest frequency jump the system can tolerate without leaving the convergence region of the equilibrium point $(0,0)$. Specifically, this maximum tolerable frequency jump is 10 Hz for Case J, decreases to 7 Hz for Case K, and further drops to 4 Hz for Case L. Consequently, when subjected to a $4.5$ Hz frequency jump, the state of the SRF-PLL moves to Region II, which induces significant oscillations and a prolonged transient response. These results agree well with the theoretical analysis in Section 3.

5. Conclusions

This paper investigated the underlying oscillation mechanism of the SRF-PLL. Building upon the established large-signal model, the dynamic behavior of the SRF-PLL was analyzed in-depth from both local and global perspectives, leading to the following key findings:

(1)

The SRF-PLL exhibits distinct dynamic behaviors around its stable equilibrium points under different grid voltage amplitudes. Consequently, during grid voltage sags, the SRF-PLL may undergo multiple oscillation modes and experience a prolonged transient response.

(2)

From the global viewpoint, the SRF-PLL possesses infinitely many asymmetrical convergence regions. However, their sizes shrink significantly under a low grid voltage amplitude and/or small control parameters, which increases the risk of undesirable oscillations and sluggish dynamics even when parameters are well tuned based on small-signal criteria.

These findings provide unique insights into the nonlinear dynamics of the SRF-PLL under voltage sags and improper control parameters, thereby offering valuable guidance for the design of advanced PLLs and control strategies for wind turbines.

6. Discussion and Future Work

In practical design, an important implication of this study is that the parameters initially tuned by small-signal criteria should be further checked from the perspective of a convergence region under worst-case grid disturbances specified in relevant grid codes. For instance, the Chinese standard [27] requires wind turbines to withstand a minimum voltage of 0.2 p.u. during fault-ride-through (FRT) events and remain connected for at least 625 ms. If the trajectory of the SRF-PLL exits the convergence region under such conditions, parameter adjustments are necessary until both stability and FRT duration requirements are satisfied.

Building on these findings, future research will be guided by the mechanisms revealed in this study. In particular, future research will focus on developing comprehensive parameter tuning guidelines under various grid codes, designing advanced PLL structures with improved dynamic performance and reduced sensitivity to parameter variations, as well as exploring integrated synchronization strategies to enhance robustness under weak grid conditions. These directions are expected to bridge the gap between theoretical analysis and engineering practice, ultimately supporting the reliable and large-scale integration of wind energy into power systems.