Frequency Stability Analysis of Grid-Forming PMSG Based on Virtual Synchronous Control

A small-signal model of a virtual synchronous generator (VSG) control system for wind turbines, which considers the rotor speed regulation characteristics, has been established in this paper. Additionally, the damping torque coefficient expression of the VSG based permanent magnet synchronous generator (PMSG) was derived. The critical factors affecting the frequency stability of the wind turbine system in the maximum power point tracking (MPPT) mode were analyzed. Furthermore, it has been clarified that when the DC voltage is controlled by the generator-side-converter, the mechanical characteristics of the generator can weaken the wind turbine’s capability to damp the power oscillation. The minimum damping value of the VSG control system has been determined, which can provide a reference for the deployment of grid-forming PMSGs in industry. Finally, a typical VSG based PMSG system was established in the RT-LAB platform, and the correctness of the conclusions drawn in this paper was verified.

SECTION I. Introduction

Wind power has gained widespread attention in the energy field due to its advantages of safety, cleanliness, and low cost [1]. However, with the increasing penetration of wind power, the system with high proportion of wind power is gradually characterized by low inertia and weak damping [2]. Most of the existing wind turbines are grid-following controlled based on phase-locked loop (PLL) and operating in the maximum power point tracking (MPPT) mode [3]. However, grid-following wind turbines have low inertia, which makes them less capable of providing damping to the grid [4]. This is especially the case in scenarios where the weak grid is connected, the coupling between the PLL and the weak grid may cause instability [5].

To guarantee the stability of high-proportion wind power system, wind turbines are required the ability to dampen grid frequency and voltage fluctuations [6]. A grid-forming wind turbine self-synchronized control strategy without PLL has been proposed [7], [8], which provides the wind turbines with the equivalent grid-connected characteristics of a synchronous generator by establishing the virtual rotor mechanical equations. Compared to grid-following wind turbines, virtual synchronous generator (VSG) base grid-forming wind turbines have the ability to enhance the stability of weak grid [9]. Reference [10] analyzed the dynamic decoupling of the VSG control system by damping torque coefficient analysis (DTCA) and designed a virtual torque based system stability improvement method. Reference [11] designed a probability distribution based VSG control strategy for permanent magnet synchronous generator (PMSG), and proposed a method to coordinate VSG and MPPT scheduling to provide spare inertia for the system, however, this method will limit the output active power of the wind turbines.

Current research on VSG control has primarily focused on doubly-fed induction generator (DFIG) based wind turbines, with relatively few studies investigating VSG control strategies for PMSGs [12]. In fact, due to the structural characteristics of PMSGs, it is easy for PMSGs to realize the VSG control [13]. For PMSG operating in MPPT mode, VSG control implementation methods have been proposed in [14] and [15]. However, the effects of the source dynamics and the regulation characteristics of wind turbines on the VSG based PMSG system stability were not considered. The dynamic response characteristics of VSG based wind turbines connected to a weak grid are discussed in [16], which points out that there is a coupling effect between the source dynamics of wind turbine and the VSG control. However, the negative effects of MPPT control and pitch angle control on system inertia and damping are not fully investigated.

There are differences between the stabilities of the VSG based PMSG and conventional VSG, the factors such as source dynamics of wind turbines and reference input power will affect the stability characteristics of the VSG based PMSG [17]. Reference [18] proposed a VSG grid-connected system error measure based on the difference of motion trajectories using the parametric root trajectory and dominant state variable methods, and found that the primary frequency modulation capability of the VSG based PMSGs deteriorated after considering source dynamics of wind turbines. However, there is a lack of further research on the coupling mechanism between the source dynamics of VSG based PMSGs and the VSG control strategy.

The main contributions are summarized as follows:

1. Analyzed how source-side characteristics of wind turbines interact with the power control loop of VSG. Found that compared with the traditional VSG model that assume an ideal DC voltage source, the advanced VSG model, which integrates source-side characteristics of real-world wind turbine, exhibits weaker inertial response and reduced damping of power oscillations during grid-connection. This indicates that previous models may overestimate the performance capabilities of VSGs in actual grid operations.

2. Proposed a parameter design method for VSG based PMSG according to the wind turbine’s dynamic characteristics, and determined the minimum damping control value for the VSG-PMSG grid-connected system’s stability. This approach offers guidance for tuning control parameters in grid-forming wind turbines.

The rest of the paper is organized as follows: Section II discusses the mathematical model and critical parameters of PMSGs implementing VSG. The stability of VSG based PMSGs is analyzed in Section III. Section IV proposes a method to improve the stable operation range of VSG based PMSGs. Cases studies are carried out in Section V. The conclusions are provided in Section VI.

SECTION II. Virtual Synchronous PMSG Mathematical Model

The structure and control principles of the VSG based PMSG are illustrated in Fig. 1. The mechanical energy input by the PMSG into the wind turbine is converted into electrical energy, which is injected into the DC bus via a generator-side converter [19]. The grid-side converter using the VSG control and the LC filter are employed for the grid connection. The grid-side converter simulates the external characteristics of the synchronous generator through virtual rotor equations and virtual excitation, thus possessing active frequency and voltage support capabilities.

FIGURE 1.

Control block diagram of virtual synchronous direct drive wind turbine.

The generator-side converter of the PMSG, controlled in VSG mode, includes a dual closed-loop control loop for DC voltage and AC current. The DC voltage control maintains the DC voltage stability of the back-to-back converter, while the current control maintains the stator d-axis current at zero. $P_{\mathrm {w}}$ and $P_{\mathrm {e}}$ denote the wind energy captured by the turbine and the electromagnetic power output, respectively. $I_{\mathrm {sd}}$ and $I_{\mathrm {sdref}}$ represent the d-axis current and its reference, whereas $I_{\mathrm {sq}}$ and $I_{\mathrm {sqref}}$ are the q-axis current and its reference value.

The control of the grid-side converter is divided into two main parts: virtual synchronization and a dual closed-loop for AC voltage control and current control. The virtual synchronization includes both phase and amplitude control of the AC voltage, as shown in Fig. 1. The phase control, which is related to active power, sets the converter’s output voltage phase based on the power command $P_{\mathrm {w}}$ from MPPT, the active power output $P_{\mathrm {g}}$ of the converter, and the droop-controlled power command $P_{\mathrm {m}}$ . The amplitude control adjusts the converter’s output voltage amplitude according to the reactive power command $Q_{\mathrm {ref}}$ and the measured reactive power output $Q_{\mathrm {g}}$ .

$H_{\mathrm {v}}$ and $D_{\mathrm {v}}$ are the inertia and damping control parameters, respectively. According to the rotor motion equation of synchronous generator [20], the virtual rotor equation of VSG is

$$\begin{align*} \begin{cases} \displaystyle \frac {d\delta _{\textrm {s}}}{dt}=\omega _{\textrm {s}} -\omega _{\textrm {g}} \\ \displaystyle 2H_{\textrm {v}} \frac {d\omega _{\textrm {s}}}{dt}=P_{\textrm {m}} -P_{\textrm {g}} -D_{\textrm {v}} (\omega _{\textrm {s}} -\omega _{\textrm {g}} ) \end{cases} \tag {1}\end{align*}$$ $\delta _{\mathrm {s}}$ and $\omega _{\mathrm {s}}$ are the phase and angular frequency of VSG output voltage, while $\omega _{\mathrm {g}}$ is the system angular frequency. The active power of the wind turbine can be expressed as [21] $$\begin{equation*} P_{\textrm {g}} =\frac {V^{2}_{\textrm {fabc}}}{\left |{{Z_{\Sigma }}}\right |}\cos \theta -\frac {V_{\textrm {fabc}} \cdot V_{\textrm {gabc}} }{\left |{{Z_{\Sigma }}}\right |}\cos (\theta -\delta _{s} ) \tag {2}\end{equation*}$$ $V_{\mathrm {fabc}}$ and $V_{\mathrm {gabc}}$ are the voltage phasors of the VSG and the grid, $Z_{\mathrm {\Sigma }}$ is the impedance between the VSG and the grid.

For the inductive system, $\theta = 90^{\circ }$ , namely

$$\begin{equation*} P_{\textrm {g}} =\frac {V_{\textrm {fabc}} \cdot V_{\textrm {gabc}}}{X_{\Sigma }}\cdot \sin \delta _{\textrm {s}} \tag {3}\end{equation*}$$

The characteristic values of the small-signal model can be calculated by (3), and the related control parameters can be found in Appendix. We analyzed the effect of inertia change on the stability of the system for the sensory line, and the inertia change range was set to be: $0.7\sim 3.1$ . The characteristic root locus is shown in Fig. 2.

FIGURE 2.

Characteristic root locus of the inertia change system.

It can be seen that, with the increase of virtual inertia, the change of characteristic root has a larger effect on the stability of the system.

The virtual excitation of VSG simulates the reactive-voltage characteristic of the synchronous generator to obtain the output voltage amplitude of the VSG based PMSG,

$$\begin{equation*} V=V_{\textrm {ref}} +K_{\textrm {n}} (Q_{\textrm {ref}} -Q_{\textrm {g}} ) \tag {4}\end{equation*}$$ where $V_{\mathrm {ref}}$ is the reference value of the grid-side voltage amplitude, and $K_{\mathrm {n}}$ is the reactive droop coefficient.

In the VSG based PMSG, the command value of the output voltage amplitude is obtained by the virtual excitation,

$$\begin{equation*} V_{\textrm {ref}} =V_{\textrm {fabc0}} \tag {5}\end{equation*}$$

The virtual rotor equations directly determine the system inertia and damping characteristics, whereas the virtual excitation is only related to the output voltage amplitude. Thus, the voltage regulation process shown in (4) can be ignored. The small-signal model of the VSG based PMSG is appropriately simplified. $V_{\mathrm {fabc0}}$ is treated as a constant value, that is, $V_{\mathrm {fabc0}} = V_{\mathrm {fabc}}$ . (1) and (3) are linearized near the balance point, and the small-signal model of the VSG based PMSG system can be expressed as

$$\begin{align*} \begin{cases} \displaystyle \frac {d\Delta \delta _{\textrm {s}}}{dt}=\Delta \omega _{\textrm {s}} \\ \displaystyle \frac {d\Delta \omega _{\textrm {s}}}{dt}=\frac {1}{2H_{\textrm {v}}}(\Delta P_{\textrm {m}} -\Delta P_{\textrm {g}} -D_{\textrm {v}} \Delta \omega _{\textrm {s}} ) \\ \displaystyle \Delta P_{\textrm {g}} =\frac {V_{\textrm {fabc}} \cdot V_{\textrm {gabc}} }{X_{\Sigma }}\cdot \cos \delta _{\textrm {s(0)}} \cdot \Delta \delta _{\textrm {s}} \end{cases} \tag {6}\end{align*}$$ where $\Delta$ represents the small disturbance and $\delta _{\mathrm {s(0)}}$ is the initial value of the PMSG rotor phase angle. If the source dynamics of wind turbines are not considered, that is, the DC side of the converters is deemed as an ideal voltage source, and the input mechanical power disturbance satisfies $\Delta P_{\mathrm {m}}=0$ . The system dynamics equation can be expressed $$\begin{equation*} 2H_{\textrm {v}} \frac {d\Delta \omega _{\textrm {s}}}{dt}=-K_{\textrm {v}} \cdot \Delta \delta _{\textrm {s}} -D_{\textrm {v}} \Delta \omega _{\textrm {s}} \tag {7}\end{equation*}$$ where $K_{\mathrm {v}}$ is the equivalent synchronous torque parameter of the VSG control. $$\begin{equation*} K_{\textrm {v}} =\frac {V_{\textrm {fabc(0)}} \cdot V_{\textrm {gabc(0)}} }{X_{\Sigma }}\cos \delta _{\textrm {s}0} \tag {8}\end{equation*}$$ $V_{\mathrm {fabc(0)}}$ and $V_{\mathrm {gabc(0)}}$ are the initial voltage phasors of the VSG and grid, respectively.

Combining (6) and (7), the dynamic equation of the VSG based PMSG system can be expressed as,

$$\begin{equation*} 2H_{\textrm {v}} \frac {d^{2}\Delta \delta _{\textrm {s}} }{dt^{2}}+D_{\textrm {v}} \frac {d\Delta \delta _{\textrm {s}} }{dt}+K_{\textrm {v}} \Delta \delta _{\textrm {s}} =0 \tag {9}\end{equation*}$$

The dominant oscillation frequency $f_{\mathrm {v}}$ and damping ratio $\xi$ of the system can be obtained by (10) and (11), respectively. They are related to the VSG control parameters and the system operating point. A larger $H_{\mathrm {v}}$ results in a slower system response and weaker damping characteristics [22]. The damping ratio $\xi$ is directly proportional to $D_{\mathrm {v}}$ , and $D_{\mathrm {v}}$ plays a critical role in determining $\xi$ .

$$\begin{align*} f_{\textrm {v}} & =\frac {\omega _{\textrm {v}}}{2\pi }=\frac {1}{2\pi }\cdot \sqrt {\frac {K_{\textrm {v}}}{2H_{\textrm {v}}}-\left ({{\frac {D_{\textrm {v}} }{4H_{\textrm {v}}}}}\right )^{2}} \tag {10}\\ \xi & =\frac {D_{\textrm {v}}}{2\sqrt {2H_{\textrm {v}} K_{\textrm {v}}}} \tag {11}\end{align*}$$

In the preceding analysis, the DC voltage on the source side of the VSG was assumed to be under ideal conditions. However, when a direct-drive wind turbine is controlled by the VSG, the virtual speed regulation and virtual excitation modulate the turbine’s mechanical power output. Therefore, it is necessary to consider the impact of the turbine’s dynamics on the control efficacy of VSG based PMSG systems.

SECTION III. Stability Analysis of VSG Based PMSG in MPPT Mode

A. PMSG-VSG System Model Considering Source Characteristics

The PMSG and VSG control are interlinked via the power-speed ($P_{\mathrm {w}}$ /$\omega _{\mathrm {r}}$ ) curve and the dynamics of the DC capacitor [23]. Grid frequency disturbances are propagated to the PMSG’s DC side through unbalanced power, causing disturbances in the DC voltage. These fluctuations subsequently impact the wind turbine’s rotational speed. Changes in rotational speed then perturb the power output on the grid side as dictated by the power-speed curve. The speed control capabilities of a DFIG surpass those of a PMSG because the DFIG can adjust the rotor speed via stator control, offering a broader control range. In contrast, the PMSG’s rotor speed is subject to a smaller range. Moreover, during inertial control process, the VSG based PMSG must prevent the rotor speed from exceeding the safe operational limits. Additionally, once the system frequency has stabilized, the PMSG must return to its optimal speed in order to maximize wind energy capture.

When the PMSG operates in the MPPT mode, the maximum output $P_{\mathrm {w}}$ of the wind turbine is expressed as [24]

$$\begin{equation*} P_{\textrm {w}} =K_{\textrm {m}} \cdot \omega _{\textrm {r}}^{3} \tag {12}\end{equation*}$$ $\omega _{\mathrm {r}}$ is the rotor speed and $K_{\mathrm {m}}$ is the MPPT factor of the PMSG. $$\begin{equation*} K_{\textrm {m}} =\frac {\pi R^{5}\rho C_{\max }}{2\lambda _{\textrm {opt}} ^{3}} \tag {13}\end{equation*}$$ $C_{\max }$ is the maximum wind energy utilization factor, $\rho$ is the air density, R is the turbine blade radius, and $\lambda _{\mathrm {opt}}$ is the optimal tip speed ratio.

The rotor motion equation of PMSG is expressed as [25]

$$\begin{equation*} 2H_{\textrm {p}} \omega _{\textrm {r}} \frac {d\omega _{\textrm {r}} }{dt}=P_{\textrm {w}} -P_{\textrm {e}} -D_{\textrm {p}} \omega _{\textrm {r}} \tag {14}\end{equation*}$$ $H_{\mathrm {p}}$ and ${D} _{\mathrm {p}}$ denote the inertia and damping constants of the wind turbine, respectively. $P_{\mathrm {e}}$ is the output electromagnetic power of the wind turbine.

Considering the source characteristics of PMSG and the primary frequency modulation from VSG control

$$\begin{equation*} P_{\textrm {m}} =P_{\textrm {w}} +m\Delta \omega _{\textrm {s}} \tag {15}\end{equation*}$$ $P_{\mathrm {m}}$ is the mechanical power output of the wind turbine and m is the droop coefficient of the system. $\Delta \omega _{\mathrm {s}}$ is the system frequency deviation.

The wind speed remained constant during small disturbances. Owing to the nonlinear characteristics of $C_{\max }$ , the wind power deviation when the PMSG operates in MPPT mode is small [26]. Therefore, $\Delta P_{\mathrm {m}}$ can be ignored.

Because the DC voltage control bandwidth of PMSG is smaller than the frequency of low-frequency oscillations, it can be assumed that the generator-side converter can achieve fast DC voltage control [27]. The power on both AC sides of the back-to-back converter equal at any moment, that is, $P_{\mathrm {e}}=P_{\mathrm {g}}$ . Ignoring the losses, the output active power of the VSG based PMSG in MPPT mode can be gotten from (1), (14), and (15), namely,

$$\begin{equation*} P_{\textrm {g}} =K_{\textrm {m}} \omega _{\textrm {r}}^{3} -(m-D_{\textrm {v}} )(\omega _{\textrm {s}} -\omega _{\textrm {g}} )-2H_{\textrm {v}} \frac {d\omega _{\textrm {s}}}{dt} \tag {16}\end{equation*}$$

In combination with (7), the linearization of (14) is

$$\begin{equation*} 2H_{\textrm {p}} \omega _{\textrm {r}0} \frac {d\Delta \omega _{\textrm {r}} }{dt}=-\Delta P_{\textrm {m}} =-K_{\textrm {v}} \Delta \delta _{\textrm {s}} \tag {17}\end{equation*}$$ $\omega _{\mathrm {r0}}$ is the initial rotor speed of the PMSG. The linearization of (16) is $$\begin{align*} K_{\textrm {v}} \Delta \delta _{\textrm {s}} =3D_{\textrm {p}} K_{\textrm {m}} \omega _{\textrm {r}0}^{2} \Delta \omega _{\textrm {r}} -2H_{\textrm {v}} \frac {d\omega _{\textrm {s}}}{dt}-(m-D_{\textrm {v}} )\Delta \omega _{\textrm {s}} \tag {18}\end{align*}$$

Combining (6), (17), and (18), the state-space model of the VSG based PMSG system can be expressed as

$$\begin{align*} \frac {d}{dt}\left [{{\begin{array}{l} \Delta \delta _{\textrm {s}} \\ \Delta \omega _{\textrm {s}} \\ \Delta \omega _{\textrm {r}} \\ \end{array}}}\right ]=\left [{{\begin{array}{ccccc} 0 & 1& 0 \\ -\frac {K_{\textrm {v}}}{2H_{\textrm {v}}}& \frac {D_{\textrm {v}} -m}{2H_{\textrm {v}}}& \frac {3K_{\textrm {m}} D_{\textrm {p}} \omega _{\textrm {r}0}^{2}}{2H_{\textrm {v}}} \\ -\frac {K_{\textrm {v}}}{2H_{\textrm {p}} \omega _{\textrm {r}}}& 0& \textrm {0} \\ \end{array}}}\right ]\left [{{\begin{array}{l} \Delta \delta _{\textrm {s}} \\ \Delta \omega _{\textrm {s}} \\ \Delta \omega _{\textrm {r}} \\ \end{array}}}\right ] \tag {19}\end{align*}$$

By converting (19) into the Laplace form, the dynamic equation of the VSG based PMSG can be expressed as

$$\begin{align*} & \hspace {-.1pc}2H_{\textrm {v}} \left ({{1-\frac {2K_{\textrm {m}} H_{\textrm {p}} \omega _{\textrm {r}0} s^{2}}{2K_{\textrm {m}} H_{\textrm {p}} \omega _{\textrm {r}0} s^{2}+m}}}\right )\frac {d\Delta \omega _{\textrm {s}}}{dt} \\ & = -K_{\textrm {v}} \Delta \delta _{\textrm {s}} -\left ({{m+D_{\textrm {v}} +\frac {3K_{\textrm {m}} K_{\textrm {v}} D_{\textrm {p}} \omega _{\textrm {r}0} }{2H_{\textrm {p}} s^{2}}}}\right )\Delta \omega _{\textrm {s}} \tag {20}\end{align*}$$

By combining (10) and substituting s $=j\omega _{\mathrm {v}}$ , the expressions for inertia and damping torque coefficients of the VSG based PMSG, considering the source characteristics, can be expressed as

$$\begin{align*} \begin{cases} \displaystyle {H}'_{\textrm {v}} =H_{\textrm {v}} -\frac {2K_{\textrm {m}} H_{\textrm {p}} \omega _{\textrm {r}0} s^{2}}{2K_{\textrm {m}} H_{\textrm {p}} \omega _{\textrm {r}0} s^{2}+m} \\ \displaystyle D_{\textrm {v}}^{\prime }=D_{\textrm {v}} -\frac {3mK_{\textrm {m}} K_{\textrm {v}} D_{\textrm {p}} H_{\textrm {v}}^{2} \omega _{\textrm {r}0} }{2H_{\textrm {p}} (2mK_{\textrm {v}} H_{\textrm {v}} -D_{\textrm {v}} )} \end{cases} \tag {21}\end{align*}$$

Comparing equations (9) and (21), it can be observed that the power regulation characteristics of the PMSG are coupled with $D_{\mathrm {v}}$ and $H_{\mathrm {v}}$ . Additionally, $H'_{\mathrm {v}} \lt ~{H} _{\mathrm {v}}$ , $D'_{\mathrm {v}}\lt ~{D} _{\mathrm {v}}$ . This indicates that the source side mechanical characteristics of the VSG based PMSG weaken the damping support capability and inertia response performance of the system.

The key factors determining $H'_{\mathrm {v}}$ and $D'_{\mathrm {v}}$ are related to two categories: one is related to the VSG control parameters, including the virtual inertia coefficient $H_{\mathrm {v}}$ , virtual damping coefficient $D_{\mathrm {v}}$ , frequency modulation coefficient m, and initial rotational speed $\omega _{\mathrm {r0}}$ ; the other is related to the inherent characteristics of PMSG, including rotor inertia constant $H_{\mathrm {p}}$ and damping constant $D_{\mathrm {p}}$ . Therefore, further research is needed to study the mechanism of the effect of the key factors on the frequency stability of VSG based PMSG system.

B. Effects of Key Parameters of VSG Based PMSG System

In Section III-A, we established the small-signal model for the VSG-based PMSG system and analyzed the effects of key parameters such as inertia, damping, frequency modulation coefficient, and system impedance on PMSG stability. Note that the analysis excludes the influence of wind turbine parameters $H_{\mathrm {p}}$ and $D_{\mathrm {p}}$ . Details of the PMSG parameters can be found in Table 2 of the Appendix.

TABLE 1 Initial power configuration

TABLE 2 PMSG parameters

Fig. 3 illustrates how changes in $H_{\mathrm {v}}$ and $D_{\mathrm {v}}$ affect the stability of the VSG-based PMSG system. $H_{\mathrm {v}}$ ranges from -1.5 to 2.5 and $D_{\mathrm {v}}$ from -4 to 10.8. The arrows indicate the movement pattern of the system’s poles with increasing parameters. An increase in $H_{\mathrm {v}}$ causes poles $S_{1}$ , $S_{2}$ , and $S_{3}$ to move towards the origin, indicating a slower response to disturbances and reduced stability. Conversely, as Dv increases, poles $S_{2}$ and $S_{3}$ approach the real axis and move away from the imaginary axis. This indicates that the capability of the system to suppress frequency disturbance improves, but the transient recovery speed of the system decreases.

FIGURE 3.

Stability analysis under varying $H_{\mathrm {v}}$ and $D_{\mathrm {v}}$ .

Fig. 4 illustrates the effects of altering $Z_{\mathrm {\Sigma }}$ and m on the VSG-based PMSG system’s stability. The system impedance $Z_{\mathrm {\Sigma }}$ changes from resistive to inductive, with the resistance-to-inductance ratio varying from 0.023 to 38. With $Z_{\mathrm {\Sigma }}$ changing from resistive to inductive, there is a increase in oscillation frequency coupled with a decline in the damping ratio. When the resistance-to-inductance ratio is reduced to 0.023, pole $S_{1}$ moves into the right half-plane, rendering the system unstable. The modulation coefficient m ranges from -857 to 2176. The trend of change for m is similar to that of $D_{\mathrm {v}}$ , however, variations in system impedance influence the transient recovery speed.

FIGURE 4.

Stability analysis under varying Z$_{\mathrm {\Sigma }}$ and m.

The above stability analysis suggests that while adjustments to $H_{\mathrm {v}}$ , $D_{\mathrm {v}}$ , or m can enhance the stability of the VSG based PMSG, they do not simultaneously address the stability and recovery speed of the system following a disturbance.

SECTION IV. Impact of Wind Speed on Frequency Stability of the VSG Based PMSG System

According to the small-signal model of the VSG based PMSG established in Section III-A, the initial wind speed will affect the synchronization stability of the system. Assuming that the wind turbine is operating at the optimal operating point when the disturbance occurs and ignoring the minor change in input mechanical power, the system operates in equilibrium, namely,

$$\begin{equation*} \left .{{\frac {\partial P_{\textrm {m}}}{\partial \omega _{\textrm {r}}}}}\right |_{x=x_{0}} =0 \tag {22}\end{equation*}$$ $x_{0}$ is the initial operating point, $x_{0} =$ [$\omega _{\mathrm {r0}}$ , $v_{\mathrm {w0}}$ , $\beta _{0}$ ]^T, $\omega _{\mathrm {r0}}$ is the initial wind turbine rotational speed, $v_{\mathrm {w0}}$ is the initial wind speed, $\beta _{0}$ is the initial pitch angle.

The equivalent inertia $M_{\mathrm {eq}}$ of the VSG based PMSGs system can be obtained by combining (20) and (22),

$$\begin{equation*} M_{\textrm {eq}} (s)=\frac {2H_{\textrm {p}} \lambda _{opt} v_{\textrm {w}0} s^{2}}{2H_{\textrm {p}} \lambda _{opt} v_{\textrm {w}0} s^{2}+RK_{\textrm {m}} (k_{\textrm {pv}} s+k_{\textrm {iv}} )} \tag {23}\end{equation*}$$ $K_{\mathrm {pv}}$ and $K_{\mathrm {iv}}$ are the proportional and integral coefficients of the rotational speed control. For the Bode diagram in Fig. 5, $K_{\mathrm {pv}} =5$ , $K_{\mathrm {iv}} =0.3$ , ${J} =2$ .

FIGURE 5.

Influence of wind speed on inertia characteristics of VSG based PMSG.

The wind speed significantly affects the inertia of the PMSG system in the low and middle frequency range. As the initial wind speed increases, the amplitude-frequency curve in the low-frequency range shifts upward, and the phase-frequency curve in the middle frequency range declines. This indicates that, in the MPPT mode, the initial wind speed affects the inertia response performance of the PMSG system. The influence of the wind speed on the inertia is relatively small in the high frequency range.

It cannot be ignored that the energy required for wind power frequency regulation comes from the wind turbine control [28]. When the speed recovery is too slow after the primary frequency regulation of the PMSGs, it may cause a secondary drop in frequency, leading to the disconnection of the PMSGs [29]. Therefore, the source characteristics of the VSG based PMSG system make a complex disturbance transmission mechanism, and detailed analysis of the impact of wind speed on the system frequency stability is required.

The VSG based PMSG controls the rotor to store or release kinetic energy for the frequency regulation by adjusting the VSG power reference value [30]. When the angular speed of the wind turbine is adjusted to the optimum value, the speed of the wind turbine in the frequency regulation can be obtained from (12) and (13),

$$\begin{equation*} \omega _{\textrm {r0}} =\frac {\lambda _{\textrm {opt}}}{R}v_{\textrm {r0}} \tag {24}\end{equation*}$$ $\lambda _{\mathrm {opt}}$ is the optimum tip speed ratio, $v_{\mathrm {r0}}$ is the initial wind speed.

Combining (14) and (24), the linearization of the wind turbine rotor motion equation can be obtained

$$\begin{equation*} 2H_{\textrm {p}} \omega _{\textrm {r0}} s\Delta \omega _{\textrm {r}} =\Delta P_{\textrm {w}} -\Delta P_{\textrm {e}} -D_{\textrm {p}} \Delta \omega _{\textrm {r}} \tag {25}\end{equation*}$$

Substituting (24) into (25) and the rotor motion equation becomes

$$\begin{equation*} \Delta P_{\textrm {w}} -\Delta P_{\textrm {e}} =\left ({{\frac {2H_{\textrm {p}} \lambda _{\textrm {opt}} v_{\textrm {r0}} s}{R}+D_{\textrm {p}} }}\right )\Delta \omega _{\textrm {r}} \tag {26}\end{equation*}$$

Assuming the wind speed is constant during the system disturbance, ignoring the losses. Combining (26) and (15), the following is derived,

$$\begin{equation*} \Delta P_{\textrm {w}} =-\frac {Rm}{2H_{\textrm {v}} \lambda _{\textrm {opt}} v_{\textrm {r0}} s^{2}+RD_{\textrm {p}}}\Delta P_{\textrm {g}} \tag {27}\end{equation*}$$

The above equation is the small-signal model of the VSG based PMSG that considers the effect of wind speed, as shown in Fig. 6.

FIGURE 6.

Small-signal model of the VSG based PMSG considering the wind speed.

Combining (1), (3), and (27) can give,

$$\begin{equation*} -\frac {Rm}{2H_{\textrm {v}} \lambda _{\textrm {opt}} v_{\textrm {r0}} s^{2}+RD_{\textrm {p}}}=\frac {(H_{\textrm {v}} s+D_{\textrm {v}} )Z_{\Sigma } }{V_{\textrm {fabc}} V_{\textrm {gabc}} \cos \delta _{\textrm {s}} \Delta \delta _{\textrm {s}}}\Delta \omega _{\textrm {s}} \tag {28}\end{equation*}$$ This simplification provides the small-signal equation of the VSG based PMSG with respect to the state variable $\Delta \delta _{\mathrm {s}}$ , $$\begin{align*} & \hspace {-.1pc}s^{4}\Delta \delta _{\textrm {s}} +\frac {D_{\textrm {V}} +D_{\textrm {P}}}{H_{\textrm {V}}}s^{3}\Delta \delta _{\textrm {s}} +\frac {V_{\textrm {fabc}} V_{\textrm {gabc}} \cos \delta _{\textrm {s}} }{H_{\textrm {v}} Z_{\Sigma }}s^{2}\Delta \delta _{\textrm {s}} \\ & \quad + \frac {mRV_{\textrm {fabc}} V_{\textrm {gabc}} \cos \delta _{\textrm {s}} +D_{\textrm {p}}}{2H_{\textrm {p}} \lambda _{\textrm {opt}} v_{\textrm {r0}} H_{\textrm {v}} Z_{\Sigma }}s\Delta \delta _{\textrm {s}} \\ & \quad +\frac {mR+D_{\textrm {p}}}{H_{\textrm {v}} \lambda _{\textrm {opt}} v_{\textrm {r0}} Z_{\Sigma }}\Delta \delta _{\textrm {s}} =0 \tag {29}\end{align*}$$

The system characteristic values corresponding to wind speeds of 6–14 m/s are derived from the small-signal model, and the influence of the characteristic value on the stability of the VSG based PMSG is analyzed.

The analysis of the characteristic root traces of the system at different wind speeds is shown in Fig. 7. It indicates that the PMSG stability is influenced by the wind speed. As the wind speed increases, $S_{1}$ and $S_{2}$ continue to move to the left of the initial position of the negative real axis, exhibiting a stable attenuated oscillation mode. $S_{3}$ and $S_{4}$ gradually shift from the initial position of the positive real axis to the left and gradually switch to the attenuated oscillation mode. The change in wind speed significantly affects the stability of the system.

FIGURE 7.

Characteristic root traces of the PMSG system at different wind speeds.

According to (11), a damping ratio at zero indicates that the PMSG system is marginally stable. In this scenario, the wind speed corresponding to marginally stable state is

$$\begin{align*} {v}'_{\textrm {r0}} =\frac {mH_{\textrm {v}} RV_{\textrm {fabc}} V_{\textrm {gabc}} \cos \delta _{\textrm {s}}}{2H_{\textrm {p}} \lambda _{\textrm {opt}} (D_{\textrm {v}} +D_{\textrm {p}} )(mV_{\textrm {fabc}} V_{\textrm {gabc}} \cos \delta _{\textrm {s}} -D_{\textrm {v}} Z_{\Sigma } )} \tag {30}\end{align*}$$

It can be seen that the key factors affecting $v'_{\mathrm {r0}}$ include the VSG control parameters $H_{\mathrm {v}}$ and $D_{\mathrm {v}}$ , and system impedance. When the operating wind speed $v_{\mathrm {r0}}\gt {v}'_{\mathrm {r0}}$ , the PMSG system is in the attenuated oscillation mode and the system is stable.

When the PMSG operates at $v'_{\mathrm {r0}}$ , the corresponding initial rotational speed $\omega _{\mathrm {r0}}$ for the wind turbines is

$$\begin{equation*} \omega _{\textrm {r0}} =\frac {\lambda _{\textrm {opt}}}{R}v_{\textrm {r}0} ^{\prime } \tag {31}\end{equation*}$$

By substituting (31) into (21), and eliminating $K_{\mathrm {v}}$ and the frequency modulation coefficient, the damping torque coefficient of the PMSG system can be obtained,

$$\begin{equation*} f(D_{\textrm {v}} )=D_{\textrm {v}} -\frac {a}{b-D_{\textrm {v}}^{2}} \tag {32}\end{equation*}$$ where a and b are defined as follows, $$\begin{align*} \begin{cases} \displaystyle a=8K_{\textrm {m}}^{(2/3)}K_{\textrm {v}}^{(1/2)}H_{\textrm {v}}^{2} /H_{\textrm {p}} \\ \displaystyle b=2K_{\textrm {v}} H_{\textrm {v}} /D_{\textrm {p}} \end{cases} \tag {33}\end{align*}$$

In practice, the parameter $D_{\mathrm {v}}$ should satisfy the requirements of system stability. Therefore, $f(D_{\mathrm {v}})$ must be greater than 0, namely,

$$\begin{equation*} f(D_{\textrm {v}} )\gt 0 \tag {34}\end{equation*}$$

The minimum value of the damping coefficient of the VSG based PMSG system can be derived as

$$\begin{equation*} D_{\textrm {v}\min } =5H_{\textrm {v}} K_{\textrm {m}}^{(2/3)}/H_{\textrm {p}} \tag {35}\end{equation*}$$

When the operating wind speed satisfies that $v_{\mathrm {r0}}$ is greater than ${v}'_{\mathrm {r0}}$ , the PMSG system is in the attenuated oscillation mode, and the system operates stably. At this time, the damping parameter of the VSG system must satisfy $D_{\mathrm {v}} \gt D_{\mathrm {vmin}}$ to ensure the stable operation of the system.

SECTION V. Hardware-in-the-Loop Experimental Verification

The full-power converter is the core component of the VSG based PMSG system. When studying the impact of the source characteristics on the dynamic performance of PMSG using virtual synchronous control, the simulation of the controllers should be based on strict experimental foundations. A semi-physical simulation based on a combination of the RT-ALB physical controller and a virtual controlled object reflects the true response characteristics of the converter. This experiment can reflect the actual situation onsite and is in line with a practical project.

A hardware-in-the-loop (HIL) experimental platform for a VSG-PMSG wind farm grid-connected system is constructed in our laboratory, utilizing RT-ALB of the OPRT5600 series. The simulation system, as shown in Fig. 8, is composed of three 800MW synchronous machines (labeled G1, G2 and G3), two local loads, and a PMSG-based wind farm with a rated capacity of 800MW. The initial power configuration of the test system is presented in Table 1. The wind farm comprises 400 turbines, each with a rated power of 2MW, equipped with standard MPPT control and pitch angle control. The transmission line in the system is modeled as a 200 km double-circuit line, using LGJ-$4\times 400.35$ conductors.

FIGURE 8.

Main circuit of the HIL simulation platform.

The digital simulation model including the main circuit of the PMSG system was built using Simulink. The VSG control algorithm is implemented using a DSP28335 chip of TMS320F28335, with a switching frequency of 3 kHz. The digital signal processor (DSP) controls the grid-side converter of the PMSG through an optical fiber connection to the main circuit. The physical part of the hardware is a controller used in practical project. The two parts interact via an AC/DC interface. The simulation machine was run at a speed of 2.5 GHz in steps of $15~\mu$ s. The system parameters are listed in Table 3 in Appendix.

TABLE 3 VSG cntroller parameters

A. Hardware-in-the-Loop Simulation Experiment for Verification of Wind Power Generation

First, the accuracy of the platform combining the Simulink offline simulation and RT-LAB HIL experiment was tested. A Simulink digital simulation with variable step sizes was conducted to ensure high solution accuracy. The models of the pneumatic, mechanical, and other main circuits were consistent with those of the HIL simulation. The HIL simulation was performed by connecting it to practical external controllers. Both simulations adopted a control strategy based on VSG technology with the same wind conditions that have an average wind speed of 8 m/s. What is more, the delay modules have been incorporated to emulate the actual detection and feedback delays.

The steady-state waveform of the HIL co-simulation is shown in Fig. 9. It can be seen that the generator speed n can accurately track the reference value, the DC voltage $u_{\mathrm {dc}}$ is stable, and the output current $i_{\mathrm {sa}}$ lags behind the grid line voltage $u_{\mathrm {sab}}$ by 30°, achieving unit power factor operation. Figs. 10–​12 show comparisons of the generator speeds and wind power from the HIL and offline simulations. Fig. 10 shows the horizontal component of the wind speed at the hub. The analysis indicated that the wind turbine could operate under at all the wind speeds selected in this study. The two sets of simulation results are essentially identical, validating the HIL simulation. Additionally, it is found that the delays did not affect the stability of the DC side of the power source.

FIGURE 9.

HIL steady-state waveform.

FIGURE 10.

Horizontal component of the wind speed.

FIGURE 11.

Wind-turbine speed.

FIGURE 12.

Output power of wind turbine.

B. Verification of Effects of PMSG Source Characteristics on System Dynamic Performance

The differences between the VSG based PMSG system model considering the source characteristics presented in Section III-A and the conventional VSG model were verified. The conventional VSG is controlled using a static ideal DC voltage source. When the DC side is a direct-drive wind turbine, the MPPT control is coupled with the VSG control through the dynamic mechanical characteristics of the rotor, affecting the stable operation of the PMSG system, according to (20). In this sub-Section, the dynamic response of two VSGs with a source controlled by an energy storage system and a direct-drive wind power system was analysed. The energy storage in the simulation platform was modeled using a method described in the [31]. The parameters of the simulation system are listed in Table 4 in Appendix. At 0.5s of the simulation, a line of the grid was disconnected due to a three-phase fault, causing a change in the equivalent impedance of the grid from 0.039+j0.133 pu to 0.042+ j0.218 pu and a reduction of the short circuit ratio from 4 to 2.5 [32].

TABLE 4 Enery storage system parameters

The parameters of the two systems remained consistent with the VSG control parameters of $H_{\mathrm {v}}=0.6$ and $D_{\mathrm {v}}=2.5$ , and the input virtual mechanical power $P_{\mathrm {m}}=0.7$ p.u.

Fig. 13 illustrates the comparative virtual angular frequency responses to a disturbance observed in the two systems. Following the disturbance, the PMSG system exhibited a more obvious rate of frequency change (d$\omega _{\mathrm {s}}$ /dt) compared to the energy-storage VSG system. Furthermore, the PMSG system showed a larger oscillation amplitude and endured a longer oscillation period. Therefore, a VSG system accompanied by a PMSG-based wind turbine has a reduced damping capacity in the grid-connected system, which in turn influences system stability adversely.

FIGURE 13.

Frequency comparison of different VSG grid-connected systems.

Fig. 14 shows the output power of both systems. In reference to (16), when a disturbance occurs, the speed of wind turbine in the MPPT mode undergoes adjustment. This adjustment propagates the mechanical dynamics to the PMSG’s VSG power control loop, resulting in a variable mechanical power $P_{\mathrm {m}}$ . Conversely, the energy storage based VSG system’s power source does not exhibit any dynamic adjustment effect, hence the power reference value remains constant.

FIGURE 14.

Comparison of source-side output power of different VSG grid-connected systems.

Fig. 15 and 16 present comparative dynamic response curves of the DC voltage and the terminal voltage of different VSG grid-connected systems. The analysis reveals that both systems remain stable following disturbance. Notably, when the VSG utilizes energy storage as its source of DC side, the fluctuations in both DC voltage and terminal voltage are minimal, enabling a quick return to stability after disturbance. Conversely, when wind energy captured by turbines serves as the VSG’s energy source, the DC voltage exhibits more substantial fluctuations. The DC voltage can return to its initial operational state within approximately 1.6s after disturbance, with a voltage amplitude fluctuation within 2%. These observations follow the requirements for the safe and stable operation of the system.

FIGURE 15.

Comparison of DC voltage of different VSG grid-connected systems.

FIGURE 16.

Comparison of terminal voltage of different VSG grid-connected systems.

Fig. 17 provides a comparison of the frequency response curves for the two systems. After the disturbance, the maximum frequency deviation in both systems is less than 0.01Hz. The system frequency for each respectively recovers within 3.5s and 2.5s, meeting the standards prescribed in power system safety and stability guidelines.

FIGURE 17.

Comparison of frequency of different VSG grid-connected systems.

C. Verification of Effects of Inertia and Damping Parameters on Virtual Synchronous PMSG

The influence of two key parameters, specifically virtual inertia and virtual damping, on the frequency stability of the VSG based PMSG system was investigated. The simulation was set up with a sudden load reduction of 16MW at 0.5s for $L_{1}$ and a sudden load increase of 21MW at 3s for $L_{2}$ .

Figs. 18 and 19 show comparisons of the effects of the conventional and PMSG based VSG on the AC frequency. In Figs. 15 and 16, the virtual inertia $H_{\mathrm {v}} =2$ , the virtual damping $D_{\mathrm {v}} =2.5$ . Under the same disturbance, both the systems exhibited anti-disturbance capabilities. However, compared with the VSC system, due to the source characteristics of the PMSG system, its dynamic response capability decreased and speed for restoring stability slowed down, which validates the analysis of the impact of the wind power source characteristics on the stability of VSG.

FIGURE 18.

Effects of two systems on the AC-side transient frequency under the same inertia.

FIGURE 19.

Effects of two systems on the AC-side transient frequency under the same damping coefficient.

Figs. 20 and 21 illustrate the DC voltage response curves of different VSG grid-connected systems when inertia parameter $H_{v}$ and damping parameter $D_{v}$ are altered. At the initial configuration, the values are set to $H_{v}=3.0$ and $D_{v}=3.5$ . Then $H_{v}$ is increased to 5.0 and $D_{v}$ to 5.5 at 0.5s, and subsequently, $H_{v}$ is decreased back to 3.0 and $D_{v}$ to 3.5 at 1.5s. All these adjustments take place under a wind speed of 10m/s. It can be seen that changes in the inertia and damping parameters result in fluctuations of the grid-forming PMSG DC voltage by approximately 0.4% and 2%, respectively. It is noteworthy that damping parameters have a more substantial impact on the grid-forming PMSG’s DC voltage fluctuation.

FIGURE 20.

DC voltage response of different VSG grid-connected systems when the inertia parameter is changed.

FIGURE 21.

DC voltage response of different VSG grid-connected systems when the damping parameter is changed.

Compared to traditional VSG with an ideal DC voltage source at the DC side, VSG using the wind turbine as the DC power source would diminish the grid-connected system’s inertial response and the ability to dampen power oscillations. However, both VSGs remain within the safe and stable operational requirements of the system. Consequently, to ensure safe and stable system operation, it is imperative to consider the actual DC source and to optimize the inertia and damping parameters of the VSG-PMSG grid-connected system accordingly.

The differences in the stable operation capabilities of the two systems with same control parameters were analyzed. The effects of the control parameter changes on the stable operation of the PMSG based VSG system were also studied. Additionally, the effects of the VSG control parameters, $H_{\mathrm {v}}$ and $D_{\mathrm {v}}$ were examined.

The effects of inertia $H_{\mathrm {v}}$ and damping $D_{\mathrm {v}}$ on the frequency stability of the VSG based PMSG system are shown in Figs. 22 and 23, respectively. The operating wind speed was set to 10 m/s. With the $H_{\mathrm {v}}$ increasing, the overshoot of frequency reduced, but the speed for restoring after the disturbance decreased. The increase of $D_{\mathrm {v}}$ can further reduce the system frequency fluctuations and shorten the transient recovery time after disturbance. Therefore, adjusting the inertia and damping coefficients of the VSG system can improve the stability of PMSG system and robustness to disturbances.

FIGURE 22.

Effect of inertia on the AC-side transient frequency of the PMSG system.

FIGURE 23.

Effect of damping on the AC-side transient frequency of the PMSG system.

D. Verification of Effect of Wind Speed on Frequency Stability of Virtual Synchronous PMSG System

The effect of wind speed on the stability characteristics of the VSG based PMSG system, as presented in Section IV, was verified. The analysis confirms that the operational wind speed of the PMSG influences the stability of the VSG system. The system parameters are listed in Appendix. The wind speed corresponding to marginally stable state $v'_{\mathrm {r}}$ is 8 m/s and the corresponding minimum damping of the VSG $D_{\mathrm {vmin}}$ is 2.5. The simulation was set up with a sudden load increase of 18 MW at 0.5 s for $L_{1}$ and a sudden load reduction of 25 MW at 3 s for $L_{2}$ . The effects of the operating wind speed and control parameters on the stability characteristics of the VSG based PMSG system were verified.

Under the same operating wind speed, the VSG based PMSG system with larger inertia coefficients has a better ability to regulate the frequency, but it can only provide a short period of power support, and there will be a serious power drop in the rotational speed recovery process at a low wind speed. Therefore, this paper establishes the bounds for virtual inertia and damping values through a detailed parameter analysis under varying wind speeds. By optimally tuning the output active power, the PMSGs dynamically offset the system power shortage, mitigate the electromagnetic torque oscillation, and reduce the rotational speed recovery time. These adjustments enhance the dynamic stability of the system.

Fig. 24 shows the frequency variation of the VSG based PMSG system at different wind speeds. When the system load suddenly increased, as the wind speed increased from 7 to 12 m/s, the lowest value of the system frequency increased from 49.653 to 49.879 Hz, and the frequency drop decreased by 66%. The frequency drop time increases, resulting in a small frequency change rate. When the system load suddenly decreased, the frequency rise time increased by 2.15 s. Therefore, a big initial operating wind speed reduces the frequency change rate and drop amplitude of the system and enhances the inertia response capability of the system.

FIGURE 24.

Effect of the wind speed on the system frequency.

Fig. 25 shows the dynamic responses of the PMSG output power undergoing a sudden increase in the load at different wind speeds. With the VSG control, the PMSG can reduce the rotor speed and release kinetic energy to exhibit a frequency regulation capability. When the VSG based PMSG operates at low wind speeds, due to its insufficient frequency regulation capability, a secondary drop in the system frequency may occur. As the operating wind speed increases, the kinetic energy from the rotor can be released gradually increases. Thus, the frequency regulation capability of the VSG based PMSG varies at different wind speeds.

FIGURE 25.

Comparison of unit output power responses under different wind speeds.

Figs. 26 and 27 show the dynamic responses of the system with different virtual damping coefficient $D_{\mathrm {v}}$ under the wind speed of 10 m/s. It can be seen that, as $D_{\mathrm {v}}$ increased, the system changed from instability with $D_{\mathrm {v}} \lt D_{\mathrm {vmin}}$ to small-disturbance stability with $D_{\mathrm {v}} \gt D_{\mathrm {vmin}}$ . Moreover, when the system reached a stable operating state, with the increase of $D_{\textrm {v}}$ , the power angle, angular frequency, and first swing amplitude of the PMSG output power were significantly reduced. The time for the system to resume stable operation was also shortened. This indicates that, within the stable operating scenarios, the VSG based PMSG system with a larger $D_{\mathrm {v}}$ has better damping characteristics and dynamic stability capability.

FIGURE 26.

Dynamic responses of the virtual synchronous PMSG system with different damping coefficients.

FIGURE 27.

Effects of different damping coefficients on the system output power.

E. Comparison of Grid-Forming VSG Based PMSG and Conventional Grid-Following PMSG

The grid-forming VSG based PMSG is compared with a VSG with an ideal DC voltage source, and a PLL based grid-following PMSG. Fig. 28 to 31 display dynamic response curves under different scenarios, where the disturbance is set to reduce the effective short-circuit ratio at the system from 3.9 to 1.1 by increasing line impedance at 0.5s. The PLL’s bandwidth set at 15Hz.

FIGURE 28.

Output voltage of the converters.

FIGURE 29.

Output power of the converters.

FIGURE 30.

DC voltage of the converters.

FIGURE 31.

Frequencies of the systems.

Fig. 28 to 31 indicate that under the disturbance caused by changes in the system’s short-circuit ratio, the traditional grid-following PMSG fails to maintain stable operation. There are oscillations in its DC voltage, output active power, and system frequency. This suggests that traditional grid-following PMSG based on PLL exhibit poor damping characteristics under weak grid conditions, which can lead to instability. But, the grid-forming PMSG based on VSG, regains a new stable point after a short oscillation period under the disturbance. Its DC voltage amplitude fluctuates by less than 5%, and active power amplitude by less than 15%, which are within the limits required for safe and stable system operation.

The comparison shows that, in weak grid scenarios, the grid-forming PMSG based on VSG plays an active role in supporting the grid, in contrast to PLL-based grid-following PMSG. Noted that the VSG with an ideal DC voltage source exhibits even stronger stability in operation compared to the grid-forming PMSG.

Fig. 32 provides a comparison of L1’s power response curves following a disturbance in different VSG grid-connected systems. The GFL system exhibits significant load power fluctuation, whereas the load power in both the VSG and PMSG-VSG systems remains relatively stable. Due to the steady power source on the DC side of VSG, it shows the smallest power fluctuation in L1. The power response of L2 has a similar behavior.

FIGURE 32.

Power responses of L1 in different VSG grid-connected systems.

Fig. 33 to Fig. 36 show the power response of G1, G2, G3 and wind farms in different VSG grid-connected systems. Observations of these transient process response curves indicate that the VSG, equipped with an ideal DC voltage source, offers superior system support compared to the PMSG-VSG.

FIGURE 33.

Power responses of G1 in different VSG grid-connected systems.

FIGURE 34.

Power responses of G2 in different VSG grid-connected systems.

FIGURE 35.

Power responses of G3 in different VSG grid-connected systems.

FIGURE 36.

Power responses of wind farm in different VSG grid-connected systems.

Fig. 33 to 35 demonstrate that the grid-forming PMSG-VSG promptly responds to changes in grid frequency, significantly suppressing power oscillations in traditional synchronous generators following a disturbance, which notably improves the stability of weak grids with renewable energy sources integrated. Fig. 36 shows that fluctuation in the active power of the grid-forming wind farm are also effectively mitigated. Therefore, incorporating grid-forming PMSG-VSG in the system provides substantial support for the stable operation of both wind farm and traditional synchronous generators, thereby enhancing the stability and reliability of the power system.

SECTION VI. Conclusion

A VSG based PMSG grid-connected system model, considering the source characteristics, was developed. Through small-signal analysis, the key factors affecting the frequency stability of the VSG based PMSG operating in the MPPT mode were clarified. The following conclusions were drawn from the theoretical analysis and digital-analog hybrid simulations:

1. Compared to the conventional VSG connected to an energy-storage system on the DC side, the connection to the wind turbines on the DC side will weaken the system’s inertial response and its capability to dampen power oscillations, thereby increasing the risk of instability due to small disturbances.

2. An analysis of the interplay between the wind turbine’s operational state and the VSG control revealed that a higher initial operating wind speed enhances the system’s equivalent inertia and frequency regulation capacity. With the equivalent damping primarily influenced by the wind turbines’ source characteristics, a minimum damping value for the VSG was determined, taking into account the stability constraints given by the wind speed at the marginally stable state. The results of this study offer valuable guidance for the deployment of grid-forming VSG based PMSGs in industrial applications.

Appendix

See Tables 2–​4.