Dynamic Analysis of Virtual Synchronous Generator Control-Based PMSG Considering Low-Voltage Ride-Through Control

Abstract

Virtual synchronous generator control-based permanent magnetic synchronous generators (VSG-PMSGs) have been widely used for their stable operation in a weak grid and strong voltage and frequency support capacity. However, VSG-PMSGs have complex and time-varying dynamics due to control strategy switching, current limiters, and saturations. Additionally, they are prone to transient angle instability during voltage faults. A dynamic analysis method for VSG-PMSGs considering low-voltage ride-through (LVRT) control is proposed in this paper. First, an improved LVRT control strategy based on active power reference reduction and virtual electromagnetic force (EMF) reset is introduced to mitigate the instability risk of VSG-PMSGs. Then, the mechanisms by which initial power and fault voltages influence the dynamic responses are revealed. The dynamics of VSG-PMSGs under different conditions are classified into four types according to the current and EMF limiters’ state. To predict VSG-PMSG dynamics, we propose a method based on fault steady-state power flow for calculating the fault voltage. Using this approach, fault voltage dips in VSG-PMSGs within a wind farm are computed with an error of less than 0.002 p.u., and the dynamic behavior of each unit is accurately predicted within 10 s. To verify the validity of the proposed method, simulations were conducted across diverse scenarios. The results demonstrate that this method enables accurate and computationally efficient prediction of VSG-PMSG fault dynamics.

1. Introduction

In recent years, renewable energy sources, particularly wind and photovoltaics, have undergone rapid development in order to construct a clean and low-carbon new power system. Renewable generators are progressively replacing conventional synchronous generators as the primary power supply in the power grid. At present, most new energy power plants employ grid-following (GFL) control technology. GFL units are passively integrated into the grid as current sources, utilizing a phase-locked loop (PLL) to track the voltage phase at the point of common coupling (PCC) and adjusting the output current to achieve active and reactive power decoupled control. Under strong grid conditions, GFL units can effectively track power commands. However, in weak grids, they are prone to stability issues, such as loss of synchronization and wide-band oscillation [1,2,3]. Additionally, the time delay of a PLL hinders the units from responding to frequency changes instantaneously and providing necessary inertia support. The limited short-circuit current of GFL inverters also leads to inadequate voltage support. Thus, high penetration of GFL units poses great challenges to power system stability [4].

Grid-forming (GFM) control technology has emerged to mitigate the stability challenges imposed by GFL resources [5]. In contrast with GFL units, which operate as current sources synchronized to the grid via PLLs, GFM units function as voltage sources and do not need PLLs. They autonomously establish and regulate AC voltage and frequency, ensuring stable operation even under weak grid conditions. Moreover, GFM units can provide rapid frequency and voltage support, making them essential for critical applications such as islanded system operation and black-start restoration [6,7,8].

GFM control strategies mainly include droop control, matching control, and virtual synchronous generator (VSG) control [9]. Droop control mimics the active power–frequency (P-f) and reactive power–voltage (Q-U) droop characteristics of synchronous generators to establish voltage and frequency. While it features a simple structure, it lacks inertia support. Matching control is also known as power synchronization control. It establishes a synchronization mechanism by exploiting the dynamic relationship between the DC-link voltage and the virtual rotor angular speed. Although it provides inertia, it does not possess P-f droop functionality. The VSG control strategy emulates both the rotor swing equation and automatic voltage control (AVC) of synchronous generators. It adjusts the virtual rotor angle in response to active power variations and regulates the virtual electromotive force (EMF) based on reactive power and voltage measurements. The VSG can support voltage and frequency rapidly and effectively. It demonstrates favorable performance. Since VSG control-based permanent magnetic synchronous generators (VSG-PMSGs) are widely used in practice, they were chosen as the example to be investigated in this paper.

The control structure of a VSG-PMSG is relatively complex. To comply with grid code requirements [10], its control strategy needs to switch between normal operation and fault ride-through modes. Additionally, protective elements such as saturation blocks and current limiters are incorporated to protect the converter from overcurrent or overvoltage. These factors endow the VSG-PMSG with strongly nonlinear and discrete characteristics, leading to a high-dimensional model that exhibits complex and time-varying dynamic behaviors. It poses great challenges to the stability analysis of power systems with high penetration of renewable energy.

During voltage sags, the dynamic behavior of a VSG-PMSG depends on the low-voltage ride-through (LVRT) control strategy it has adopted. The LVRT control needs to simultaneously meet the requirements of maintaining transient angle stability, avoiding overcurrent, and providing reactive power support. This topic has been intensively studied. Existing control strategies primarily include control mode switching, adaptive adjustment of inertia and damping, virtual impedance control, and active power reference reduction. The study in [11,12] proposed an LVRT control strategy based on GFL/GFM control mode switching. The VSG transitions to GFL control upon the occurrence of a voltage sag. A key drawback of this approach is its dependence on the PLL for grid synchronization, which compromises its stability under weak grid conditions. The study in [13] proposed an adaptive LVRT method for VSGs based on voltage compensation and power angle abrupt variation compensation, which automatically restricts the current during various asymmetric faults at the onset of the fault and ensures a seamless recovery after the fault. In [14], an LVRT strategy for VSGs that utilizes a locked active power loop and a quantitatively designed virtual impedance during symmetrical grid voltage sags is proposed. The study in [15] investigates an LVRT control strategy that combines active power command optimization with adaptive virtual impedance regulation under various voltage dip depths. Furthermore, ref. [16] analyzes the transient voltage support mechanism of VSGs under current-limiting conditions and presents an improved LVRT strategy. This strategy effectively enhances voltage support capability through active power reference reduction and supplementary droop control. In [17], a virtual power compensation strategy combined with an improved current-limiting strategy is provided to enhance the transient stability and fault recovery capability. In [18], the critical clearing time for VSG transient angle stability during voltage faults is derived. Accordingly, an adaptive gain is introduced into the droop control to improve transient angle stability. In [19], a continuous high/low-voltage ride-through strategy for power conversion systems based on VSG technology is designed. The impact of voltage amplitude and phase jumps during voltage faults on conventional LVRT control strategies is investigated in [20]. An improved LVRT control strategy that incorporates amplitude switching and phase compensation is proposed to address these issues. For asymmetrical faults, ref. [21] introduces an adaptive LVRT control strategy, which significantly mitigates transient overcurrent caused by the delay inherent in the process of positive/negative sequence decomposition. In [22], an asymmetrical LVRT control strategy based on negative-sequence virtual voltage compensation is proposed.

The existing research mainly focuses on LVRT control strategies themselves, while studies on the mechanisms by which VSG dynamic responses under LVRT control are influenced remain insufficient. Most studies are based on single-machine systems and rely on numerical integration to obtain dynamic responses, yet multi-VSGs are used in wind farms. VSGs may exhibit distinct dynamic characteristics under different conditions. However, considering wind speed uncertainties and dispersed terminal voltages, there are numerous possible scenarios. Simulations under all possible scenarios would impose an unbearable computational burden, making it infeasible. Therefore, it is essential to investigate the underlying dynamic mechanisms of VSGs following voltage sags and summarize their behavioral patterns under different conditions. This provides a theoretical foundation for LVRT control strategy optimization [23], the dynamical equivalent of VSG-based wind farms [24], and transient stability analysis of power systems integrated with VSGs [25].

Therefore, this paper proposes a dynamic analysis method for VSG-PMSGs considering the improved LVRT control strategy. It should be noted that this paper primarily focuses on electromechanical transients, while wide-band oscillations are outside of the scope of this paper. This paper is organized as follows. Section 2 presents an improved LVRT control strategy based on active power reference reduction and virtual EMF reset. In Section 3, the mechanism by which the initial power and fault voltage influence dynamic characteristics is elucidated, and the dynamic responses of the VSG-PMSG under different fault scenarios are classified into four types. Subsequently, Section 4 proposes a fault steady-state power flow calculation method to obtain the fault voltage, and the VSG-PMSG dynamics are predicted according to the state of current and EMF limiters. The effectiveness of the proposed method is validated through simulations in Section 5. Finally, the conclusions are given in Section 6.

This paper aims to address the challenge of predicting the LVRT dynamics of VSG-PMSGs accurately and efficiently, which is a critical issue for wind farm stability control. The key contributions of this study are as follows. First, the underlying mechanism by which initial power and fault voltage jointly influence the complex VSG-PMSG dynamics is revealed, a gap not fully explored in prior studies. Second, a dynamic prediction method grounded in fault steady-state power flow analysis is proposed. This method achieves a fault voltage dip error of less than 0.002 p.u., predicts unit dynamics within 10 s, and significantly outperforms numerical simulation-based approaches. Third, simulations under various voltage dips and wind speeds are conducted to validate the effectiveness of the proposed LVRT strategy and the dynamic analysis method. The novelty of this paper lies in two aspects. First, unlike existing studies that rely on transient simulations, the proposed method integrates dynamic analysis with steady-state power flow theory to balance accuracy and computational efficiency, which is an innovative framework not previously applied to VSG-PMSG dynamics prediction. Second, the dominant factors influencing the LVRT dynamics are identified, and the mechanisms by which they influence fault dynamics are also revealed.

2. Improved LVRT Control Strategy of the VSG-PMSG

2.1. Basic Control Principles of VSG-PMSG

A VSG-PMSG system is typically divided into two main parts: the machine-side part and the grid-side part. The machine-side part includes the wind turbine, the PMSG, the machine-side converter (MSC), and its controller. The objective of the machine-side part is to capture wind power according to the maximum power point tracking (MPPT) curve, transfer the power to the grid-side part via the MSC, and maintain a stable DC-link capacitor voltage. The grid-side part includes the grid-side converter (GSC), the GSC controller, and associated filter circuits. As the machine-side and grid-side parts are decoupled via back-to-back converters, the machine-side dynamics are often neglected during transient analysis for simplification. With the braking chopper absorbing fault-induced excess power, the DC-link voltage stabilizes, allowing the DC capacitor to be approximated as a constant voltage source. The schematic control diagram of the VSG-PMSG system is shown in Figure 1.

In Figure 1, $E$ and ${\theta }_{\mathrm{e}}$ are the virtual EMF and virtual rotor angle, respectively. $U_{\mathrm{w}}$ and ${\theta }_{\mathrm{w}}$ denote the terminal voltage and its phase angle. The virtual power angle $\delta$ is defined as the angle between the virtual EMF $E$ and the terminal voltage $U_{\mathrm{w}}$, i.e., $\delta ={\theta }_{\mathrm{e}}-{\theta }_{\mathrm{w}}$. Meanwhile, $\omega$ and ${\omega }_{\mathrm{n}}$ are the electrical angular velocity and the synchronous speed of the VSG-PMSG. H and D represent the inertia time constant and damping coefficient, respectively, and are set to 1 s and 60 p.u. in this study based on values from the literature [16]. $U_{\mathrm{g}}$, $U_{\mathrm{ref}}$ and $U_{\mathrm{N}}$ are the source voltage, the reference value of terminal voltage and the rated voltage. $P_{\mathrm{w}}$ and $Q_{\mathrm{w}}$ are the active and reactive power outputs. $P_{\mathrm{ref}}$ and $Q_{\mathrm{ref}}$ are the reference values of the active and reactive power; typically, $P_{\mathrm{ref}}$ is set to the MPPT value $P_{\mathrm{mppt}}$, and $Q_{\mathrm{ref}}$ is set to zero. $k_{\mathrm{q}}$ and $k_{\mathrm{u}}$ are the droop coefficients for Q-U control. $T_{\mathrm{E}}$ is the time constant of the reactive power control loop and is set to 0.02 s in this paper. m represents the modulation signal of the GSC.

As shown in Figure 1, the GSC controller includes the active power and frequency (P-f) control loop, the reactive power and voltage (Q-U) control loop, the current control loop, and the measurement modules. The P-f control emulates the rotor swing equations of the synchronous generator, as shown in Equations (1) and (2).

$${\dot{\theta }}_{\mathrm{e}}=(\omega -1){\omega }_{\mathrm{n}}$$

(1)

$$2H\dot{\omega }=P_{\mathrm{ref}}-P_{\mathrm{w}}-D(\omega -1)$$

(2)

The Q-U control emulates the regulation of EMF, as shown in Equation (3).

$$T_{\mathrm{E}}\dot{E}=k_{\mathrm{q}}(Q_{\mathrm{ref}}-Q_{\mathrm{w}})+k_{\mathrm{u}}(U_{\mathrm{ref}}-U_{\mathrm{w}})$$

(3)

Similar to the synchronous generator’s $P-\delta$ characteristic, the output powers of the VSG-PMSG are determined by the virtual power angle $\delta$ and can be calculated using the following formula:

$$P_{\mathrm{w}}=E{U}_{\mathrm{w}}\sin ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{w}})/(k_{\mathrm{z}}{X}_{\mathrm{v}})$$

(4)

$$Q_{\mathrm{w}}=E{U}_{\mathrm{w}}\cos ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{w}})/(k_{\mathrm{z}}{X}_{\mathrm{v}})-U_{\mathrm{w}}^2/(k_{\mathrm{z}}{X}_{\mathrm{v}})$$

(5)

$$U_{\mathrm{Z}}=\sqrt{E^2+U_{\mathrm{w}}^2-2E{U}_{\mathrm{w}}\cos ({\theta }_{\mathrm{e}}-{\theta }_{\mathrm{w}})}$$

(6)

$$k_z=\left\{\begin{array}{l}1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{U}_Z\le I_{\max }{X}_{\mathrm{v}}\\ U_Z/(I_{\max }{X}_{\mathrm{v}}), U_Z>I_{\max }{X}_{\mathrm{v}}\end{array}\right.$$

(7)

where $k_{\mathrm{z}}$ is the current-limiting coefficient, $Z_{\mathrm{v}}=R_{\mathrm{v}}+j{X}_{\mathrm{v}}$ is the virtual impedance, and $R_{\mathrm{v}}$ can be neglected in the analysis, as $R_{\mathrm{v}}\ll X_{\mathrm{v}}$. $U_{\mathrm{Z}}$ represents the voltage drop across $Z_{\mathrm{v}}$, when $U_Z>I_{\max }{X}_{\mathrm{v}}$, and the virtual impedance is increased to $k_{\mathrm{z}}{X}_{\mathrm{v}}$; thus, the current reference $I_{\mathrm{w},\mathrm{ref}}$ is limited to $I_{\max }$, preventing the converters from overcurrent.

As shown in Equations (1)–(7), the VSG-PMSG can be equivalent to a voltage source characterized by an internal voltage of $E\angle {\theta }_{\mathrm{e}}$ and an impedance of $j{k}_{\mathrm{z}}{X}_{\mathrm{v}}$. Under voltage fault conditions, the activation of the current and EMF limiters introduces discrete and switching characteristics to the VSG-PMSG system, which consequently exhibits complex dynamic behavior.

2.2. Improved LVRT Control Strategy for VSG-PMSG

According to the grid’s technical codes for integrated wind generators, wind turbines must remain connected to the grid for a specified duration during voltage faults and provide a certain level of reactive power support. For GFL units, the reactive current $I_{\mathrm{q}}$ is increased to provide reactive power support. However, the active current $I_{\mathrm{p}}$ must be limited to less than $\sqrt{I_{\max }^2-I_{\mathrm{q}}^2}$ to protect the converters from overcurrent. Thus, GFL units can be treated as a voltage-controlled current source during voltage faults.

The LVRT control strategy for GFM units remains an active area under development. During a voltage fault, $E$ increases rapidly under Q-U control. When $E$ becomes large enough that the output current reaches $I_{\max }$, the current limiter will be activated to prevent further current increases. The operational state of the Q-U controller, wherein the integrator input is driven to zero, is defined as the equilibrium condition. When this equilibrium state is reached, the virtual EMF stops increasing and stabilizes at a constant value. According to Equation (3), the equilibrium-state reactive power $Q_{\mathrm{ws}}$ of the VSG-PMSG is given by

$$Q_{\mathrm{ws}}=Q_{\mathrm{ref}}+k_{\mathrm{u}}/k_{\mathrm{q}}(U_{\mathrm{ref}}-U_{\mathrm{wf}})$$

(8)

where $U_{\mathrm{wf}}$ refers to the fault voltage of the VSG-PMSG. $Q_{\mathrm{ws}}$ is a critical value for maintaining the equilibrium state. It can be seen that the deeper $U_{\mathrm{wf}}$ dips, the larger $Q_{\mathrm{ws}}$ will be. If the output reactive power $Q_{\mathrm{w}}$ of the VSG-PMSG remains below $Q_{\mathrm{ws}}$ when the current reaches $I_{\max }$, $E$ will continue to increase. However, due to the current limitation, the apparent power transferred by the inverters must be less than $U_{\mathrm{wf}}{I}_{\max }$. Any increase in $Q_{\mathrm{w}}$ will inevitably result in a reduction in $P_{\mathrm{w}}$. According to Equation (2), if the $P_{\mathrm{w}}$ remains consistently lower than the active power reference $P_{\mathrm{ref}}$, the virtual rotor angle ${\theta }_{\mathrm{e}}$ and power angle $\delta$ will continue to increase. This will lead to transient angle instability, accompanied by power and voltage oscillations, as illustrated in Figure 2.

To avoid transient angle instability, an improved LVRT control strategy is presented in this paper. The active power reference value $P_{\mathrm{ref}}$ is established based on two key constraints: (1) $P_{\mathrm{ref}}$ must not surpass the PMSG’s maximum generation power $P_{\mathrm{mppt}}$; (2) $P_{\mathrm{ref}}$ must also not exceed the converter’s maximum transferable power, which is governed by the maximum apparent power ($S_{\max }=U_{\mathrm{wf}}{I}_{\max }$) and reactive power ($Q_{\mathrm{w}}$). Based on these constraints, the active power reference value $P_{\mathrm{ref}}$ in Figure 1 is derived analytically as

$$P_{\mathrm{ref}}=\min \left\{P_{\mathrm{mppt}},\sqrt{U_{\mathrm{wf}}^2{I}_{\max }^2-Q_{\mathrm{w}}^2}\right\}$$

(9)

If the current limiter is inactive, the inverters can deliver the active power ($P_{\mathrm{mppt}}$) and reactive power ($Q_{\mathrm{w}}$) simultaneously. This indicates that $P_{\mathrm{mppt}}\le \sqrt{U_{\mathrm{wf}}^2{I}_{\max }^2-Q_{\mathrm{w}}^2}$ and the active power reference of the VSG-PMSG is set to $P_{\mathrm{mppt}}$. However, when the current limiter is activated, the active power P_w ceases to increase and is governed by $\sqrt{U_{\mathrm{wf}}^2{I}_{\max }^2-Q_{\mathrm{w}}^2}$. Meanwhile, the reference value P_ref is reduced to $\sqrt{U_{\mathrm{wf}}^2{I}_{\max }^2-Q_{\mathrm{w}}^2}$. Consequently, the gap between P_ref and P_w nearly vanishes, the virtual rotor angle remains constant, and the transient angle instability is prevented.

At the moment of fault clearance, the voltage $U_{\mathrm{w}}$ recovers rapidly, but ${\theta }_{\mathrm{e}}$ cannot change instantly due to the inertia. This will lead to a sharp increase in $P_{\mathrm{w}}$. Concurrently, the delayed withdrawal of the reactive power $Q_{\mathrm{w}}$ may cause a temporary reactive power surplus, resulting in a transient overvoltage. This may impose significant stress on the power system with large-scale wind integrations upon fault clearance. To address these issues, the superior controllability of the power electronic devices can be exploited by immediately resetting $E$ to the pre-fault EMF value $E_0$ upon fault clearance. The rationale for selecting $E_0$ as the reset value is that, with the network topology unchanged, $E_0$ closely approximates the post-fault equilibrium value. Adopting $E_0$ as the reset reference expedites the system recovery process and streamlines recovery operations. This helps to suppress the active power surge and transient overvoltage, thereby ensuring the safe and stable operation of the VSG-PMSG.

The power reduction and virtual EMF-reset-based improved LVRT control strategy is illustrated in Figure 3, where the state variable S is defined by

$$S=\left\{\begin{array}{l}1, \left|U_{\mathrm{w}}-U_{\mathrm{N}}\right|\le 0.1 \& \left|E^{\prime }-E_0\right|>0.03\\ 0, \mathrm{else}\end{array}\right.$$

(10)

At the time of fault clearance, S switches from 1 to 0, and $E$ is reset to $E_0$. The threshold value for $\left|U_{\mathrm{w}}-U_{\mathrm{N}}\right|$ is set to 0.1 because when $U_{\mathrm{w}}$ > 0.9 p.u., the PMSG will exit the LVRT control mode. Meanwhile, the threshold value for $\left|E^{\prime }-E_0\right|$ is set to 0.03 to guarantee the controller returns to the normal control mode when the virtual EMF is close enough to $E_0$.

3. Analysis of VSG-PMSG Dynamics Under LVRT Control

Section 2 introduces an improved LVRT control strategy for the VSG-PMSG. In this section, the dynamic responses of the VSG-PMSG under voltage sags are investigated, and especially, the mechanisms by which the fault voltage and initial operating power impact the dynamics are analyzed.

3.1. Dynamic Mechanisms of the VSG-PMSG Under Voltage Sags

For a given fault voltage, the VSG-PMSG dynamic behaviors are primarily determined by $E$ and $\delta$. The operation of current limiters or virtual EMF limiters, along with the reduction in active power reference, plays a critical role in shaping the system’s transient behavior. A minor variation in initial conditions may result in completely different dynamic responses. Therefore, it is essential to identify the dominant factors that affect the VSG-PMSG dynamics and study the mechanisms by which they impact the virtual EMF $E$ and the power angle $\delta$.

Figure 4 depicts the phasor diagram of the VSG-PMSG under voltage fault conditions. Here, the phasor OT denotes the terminal fault voltage ${\dot{U}}_{\mathrm{w}}$; the phasor TA_i represents the voltage drop across the virtual impedance in the ith scenario; and phasor OA_i represents the virtual EMF ${\dot{E}}_i$, with B_i being the projection of A_i onto the x-axis (aligned with ${\dot{U}}_{\mathrm{w}}$). When A_i resides outside the current-limiting circle, A_i′ marks the intersection of TA_i with that circle, and B_i′ is the corresponding projection of A_i′ onto the x-axis. Geometrically, $\left|{\mathrm{A}}_i{\mathrm{B}}_i\right|$ equals $E\sin \delta$, while $\left|{\mathrm{TB}}_i\right|$ equals $E\cos \delta -U_{\mathrm{w}}$. According to Equations (4) and (5), the active and reactive power outputs are determined by $\left|{\mathrm{A}}_i{\mathrm{B}}_i\right|$ and $\left|{\mathrm{TB}}_i\right|$, respectively, when the current limiters remain inactive.

However, when the current limiters are activated, as indicated by phasor OA₂ in Figure 4b, the virtual impedance is increased to $k_{\mathrm{z}}{X}_{\mathrm{v}}$ (with $k_{\mathrm{z}}>1$). Consequently, the active and reactive power outputs are given by ${\left|{\mathrm{A}}_2{\mathrm{B}}_2\right|/k}_{\mathrm{z}}$ and ${\left|{\mathrm{TB}}_2\right|/k}_{\mathrm{z}}$, respectively. Owing to the geometric similarity between triangles ΔA₂B₂T and ΔA₂′B₂′T, it can be deduced that ${\left|{\mathrm{A}}_2{\mathrm{B}}_2\right|/k}_{\mathrm{z}}=\left|{{\mathrm{A}}_2}^{\prime }{{\mathrm{B}}_2}^{\prime }\right|$ and ${\left|{\mathrm{TB}}_2\right|/k}_{\mathrm{z}}=\left|{{\mathrm{TB}}_2}^{\prime }\right|$. Therefore, the active and reactive power outputs are ultimately determined by $\left|{{\mathrm{A}}_2}^{\prime }{{\mathrm{B}}_2}^{\prime }\right|$ and $\left|{{\mathrm{TB}}_2}^{\prime }\right|$.

Since the time constant $T_{\mathrm{E}}$ of the Q-U control loop is much smaller than the virtual inertia time constant H, the virtual EMF increases rapidly upon the occurrence of a voltage sag, whereas the virtual rotor angle ${\theta }_{\mathrm{e}}$ changes relatively slowly. Typically, $E_{\max }$, $I_{\max }$, and $X_{\mathrm{v}}$ are set to 2.0 p.u., 1.2 p.u., and 0.33 p.u., respectively. As shown in Figure 4, when the current reaches $I_{\max }$, the terminal point of phasor OA_i lies on the current-limiting circle, which is located inside the EMF-limiting circle. This indicates that the current limiter operates prior to the EMF limiter.

If the VSG-PMSG enters a fault steady state following a transient process with neither the current limiter nor the EMF limiter activated, both its P-f and Q-U control loops will reach equilibrium. Under this fault steady-state condition, the output active power of the VPMSG remains constant and equals the initial power $P_{\mathrm{w}0}$, while the reactive power equals $Q_{\mathrm{ws}}$, as given by Equation (8). Consequently, in Figure 4a, the terminal point of the virtual EMF phasor will lie on the line of AA₁ in the fault steady state, where $\left|{\mathrm{A}}_i{\mathrm{B}}_i\right|$ ($E\sin \delta$) remains unchanged. A lower fault voltage $U_{\mathrm{w}}$ results in a larger required reactive power $Q_{\mathrm{ws}}$. Thus, the terminal point A_i of the virtual EMF phasor will move closer to A₁ to yield a larger $Q_{\mathrm{w}}$, which is determined by $\left|{\mathrm{TB}}_i\right|$ ($E\cos \delta -U_{\mathrm{w}}$). When point A_i coincides with A₁, $Q_{\mathrm{w}}$ reaches its maximum value when the current limiter is inactive.

However, if $Q_{\mathrm{w}}$ remains less than $Q_{\mathrm{ws}}$ when phasor OA_i reaches the current-limiting circle, it can be seen from Equation (3) that $\dot{E}$ remains positive, and E will continue to increase. Consequently, the terminal point A_i moves outside the current-limiting circle, and the current limiter is activated. Under this condition, point A₂′ moves along the current-limiting circle as E increases, while $Q_{\mathrm{w}}$ is determined by $\left|{\mathrm{T}\mathrm{B}}_2^{\prime }\right|$. Since $\left|{\mathrm{T}\mathrm{B}}_2^{\prime }\right|<\left|{\mathrm{TA}}_3\right|$, the reactive power output $Q_{\mathrm{w}}$ that the VSG-PMSG can provide is also limited. If $Q_{\mathrm{w}}$ remains below the required level ($Q_{\mathrm{ws}}$) during the fault, the virtual EMF will keep increasing until it reaches $E_{\max }$. It is shown in Figure 4b that as E increases, point A₂′ will move downward along the current-limiting circle_, causing |A₂′B₂′| (corresponding to $P_{\mathrm{w}}$) to decrease. Meanwhile, the active power reference value $P_{\mathrm{ref}}$ will also be reduced according to Equation (9) under this condition. Since $P_{\mathrm{ref}}$ approximately equals $P_{\mathrm{w}}$, the rotor angle ${\theta }_{\mathrm{e}}$ will remain unchanged after the current limiter is activated.

The above analysis shows that the current limiting significantly affects the dynamic responses of the VSG-PMSG during a fault. The activation of current limiters primarily depends on the fault voltage and the initial active power. If the constraint described in Equation (11) is satisfied, the current limiter remains inactive; otherwise, it is activated.

$$\sqrt{P_{\mathrm{w}0}^{ 2}+{\left(Q_{\mathrm{ref}}+k_{\mathrm{u}}/k_{\mathrm{q}}(U_{\mathrm{ref}}-U_{\mathrm{wf}})\right)}^2}\le U_{\mathrm{wf}}{I}_{\max }$$

(11)

Under a given fault voltage, an increase in the initial active power means a larger $\left|{\mathrm{A}}_i{\mathrm{B}}_i\right|$, and point A₁ on the current-limiting circle in Figure 4 moves upward. This reduces the maximum reactive power ($\left|{\mathrm{TB}}_1\right|$) that the VSG-PMSG can deliver before current limiting engages, making the system more prone to enter the current-limiting state. For a specified initial active power, a lower fault voltage leads to a larger required reactive power $Q_{\mathrm{ws}}$, thereby also increasing the likelihood of entering the current-limiting state.

3.2. Classification of Fault Dynamic Responses for VSG-PMSGs Under LVRT Control

Section 3.1 analyzes the mechanisms by which fault voltage and initial power influence the VSG-PMSG dynamic responses under improved LVRT control. To validate the analysis, simulations are conducted under various voltage dips and initial power levels. Four typical dynamic responses, as illustrated in Figure 5, are identified from the simulation results.

The dynamic responses of type 1 correspond to VSG-PSMGs whose current and EMF limiters are not activated. The fault voltage and initial power are sufficiently small, such that the constraint of Equation (11) is satisfied. It can be observed that $E$ and ${\theta }_{\mathrm{e}}$ approach a fault steady state. During the fault, $P_{\mathrm{w}}$ returns to $P_{\mathrm{w}0}$ after an oscillation, while $Q_{\mathrm{w}}$ rises to the required value of $Q_{\mathrm{ws}}$.

The VSG-PSMG exhibits type-2 dynamic responses when the current limiter is active, while the EMF limiter remains inactive during the fault. As shown in Figure 5a, the virtual EMF continues to rise. However, due to the relatively slight voltage dip, $E$ rises slowly and remains below $E_{\max }$ by the time of fault clearing. It is worth noting that $E$ would reach $E_{\max }$ if the fault duration were sufficiently long. Throughout the fault, the virtual rotor angle ${\theta }_{\mathrm{e}}$ stays nearly unchanged as a result of the reduction in the active power reference $P_{\mathrm{ref}}$. It can also be observed that during the fault, $Q_{\mathrm{w}}$ increases exponentially, while $P_{\mathrm{w}}$ decreases gradually, and the apparent power $U_{\mathrm{wf}}{I}_{\max }$ remains unchanged.

As the fault voltage sag deepens, the VSG-PMSG exhibits type-3 dynamic responses. Under this condition, both the current and EMF limiters are activated. Owing to the moderate fault voltage, the rate of increase in $E$ is also moderate. Consequently, $E$ undergoes a noticeable period of rise before eventually reaching $E_{\max }$, while ${\theta }_{\mathrm{e}}$ remains unchanged shortly after the fault. During the fault, $Q_{\mathrm{w}}$ initially grows exponentially but ceases to increase and remains constant once $E$ reaches $E_{\max }$. Simultaneously, $P_{\mathrm{w}}$ decreases gradually and then stabilizes after the activation of the EMF limiter.

When the fault voltage sags sufficiently deeply, the VSG-PMSG shows type-4 dynamic responses, which are very similar to the type-3 pattern. The main distinction is that for type 4, E increases so rapidly that the transition from the pre-fault steady state to the fault steady state is accomplished almost instantaneously.

It should be mentioned that the fault duration primarily affects VSG-PMSG systems with type-2 dynamics. As the fault persists, the active power decreases, while the reactive power gradually increases. For systems with type-1, type-3, or type-4 dynamics, the impact of fault duration is relatively negligible. All the above analysis mainly focuses on the dynamic responses of the VSG-PMSG during the fault. After fault clearance, the terminal voltage $U_{\mathrm{w}}$ recovers to nearly 1.0 p.u., and the virtual EMF E is reset to E₀ according to the improved LVRT control strategy. Thus, the post-fault dynamics are mainly governed by ${\theta }_{\mathrm{e}}$. As shown in Figure 5b, ${\theta }_{\mathrm{e}}$ returns to the post-fault steady state gradually after a duration of damped oscillations. It can be observed that the post-fault dynamic responses of type 2, type 3, and type 4 are largely similar. However, the oscillation magnitude depends on the extent of active power reduction during the fault. A larger reduction in $P_{\mathrm{ref}}$ leads to a larger post-fault oscillation magnitude, and vice versa.

4. Fault Dynamics Analysis of VSG-PMSGs in the Wind Farm

Due to the spatial distribution of wind, VSG-PMSGs in a wind farm experience different wind speeds and, thus, different initial powers. Variations in grid connection impedances and transmitted powers further lead to distributed fault voltages. Section 3 demonstrates that the fault voltage and initial power significantly influence the VSG-PMSG dynamic characteristics. Although the fault voltage $U_{\mathrm{wf}}$ is assumed to be known in previous analyses, it practically remains unknown without time-consuming numerical integrations. Therefore, determining the VSG-PMSG fault voltages is essential for the investigation of their dynamics. To address this issue, a fault voltage calculation method based on fault steady-state power flow is developed.

4.1. Fault Steady-State Power Flow Calculation Based on Power Sensitivities

Though the fault voltage of a VSG-PMSG fluctuates during a fault, it rapidly approaches a constant value due to the rapid electromagnetic transients. Therefore, the fault voltage can be approximated as constant, which can be calculated via fault steady-state power flow analysis.

In the pre-fault steady-state, the jth VSG-PMSG can be handled as a static load with an active power of $-P_{\mathrm{w}0,j}$ and a reactive power of $-Q_{\mathrm{w}0,j}$. Here, $j=1,\cdots ,n_{\mathrm{w}}$, and $n_{\mathrm{w}}$ is the number of VSG-PMSGs in the wind farm; $P_{\mathrm{w}0,j}$ equals $P_{\mathrm{mppt},j}$, while $Q_{\mathrm{w}0,j}$ is calculated by

$$Q_{\mathrm{w}0,j}=Q_{\mathrm{ref}}+k_{\mathrm{u}}/k_{\mathrm{q}}(U_{\mathrm{ref}}-U_{\mathrm{w}0, j})$$

(12)

where $U_{\mathrm{w}0, j}$ is the pre-fault steady-state voltage. Based on this property, the pre-fault power flow can be solved. The VSG-PMSG is equivalent to a Thevenin-type circuit, where the ideal voltage source ($E_{0, j}\angle {\theta }_{\mathrm{e}0,j}$) is connected to the terminal through a virtual impedance of Z_v. Thus, $E_{0, j}$ and ${\theta }_{\mathrm{e}0, j}$ are given by

$${\dot{E}}_{0,j}={\dot{U}}_{\mathrm{w}0, j}+{\dot{I}}_{\mathrm{w}0, j}{Z}_{\mathrm{v}}$$

(13)

In the fault steady state, the output power of VSG-PMSGs is determined by distinct functions depending on their dynamic types. For a VSG-PMSG displaying type-1 dynamics, the active power is given by $P_{\mathrm{w}0}$ and its reactive power by $Q_{\mathrm{ws}}$. In contrast, supposing the voltage fault lasts sufficient time, the VSG-PMSG exhibiting type-2, type-3, or type-4 dynamics can be equivalent to a voltage source $E_{\max }\angle {\theta }_{\mathrm{e}0,j}$ in series with an impedance of $j{k}_{\mathrm{z}, j}{X}_{\mathrm{v}}$, and $k_{\mathrm{z}, j}$ is determined by Equations (6) and (7). Consequently, the active and reactive power outputs are given as follows:

$$P_{\mathrm{w},j}=f_{\mathrm{p}, j}(U_{\mathrm{w}, j},{\theta }_{\mathrm{w}, j})={\displaystyle \frac{E_{\max }{U}_{\mathrm{w}, j}}{k_{\mathrm{z}, j}{X}_{\mathrm{v}}}}\sin ({\theta }_{\mathrm{e}0,j}-{\theta }_{\mathrm{w},j})$$

(14)

$$Q_{\mathrm{w},j}=f_{\mathrm{q}, j}(U_{\mathrm{w}, j},{\theta }_{\mathrm{w}, j})={\displaystyle \frac{E_{\max }{U}_{\mathrm{w}, j}}{k_{\mathrm{z}, j}{X}_{\mathrm{v}}}}\cos ({\theta }_{\mathrm{e}0,j}-{\theta }_{\mathrm{w},j})-{\displaystyle \frac{U_{\mathrm{w}, j}^2}{k_{\mathrm{z}, j}{X}_{\mathrm{v}}}}$$

(15)

It can be seen that $P_{\mathrm{w},j}$ and $Q_{\mathrm{w},j}$ are determined by $f_{\mathrm{p}, j}(U_{\mathrm{w}, j},{\theta }_{\mathrm{w}, j})$ and $f_{\mathrm{q}, j}(U_{\mathrm{w}, j},{\theta }_{\mathrm{w}, j})$, respectively. Based on the aforementioned properties, the fault steady-state power flow can be calculated.

Let the power flow correction equations [26] in the fault steady-state be

$$\left[\begin{array}{l}\Delta P\\ \Delta Q\end{array}\right]=\left[\begin{array}{cc}{H}_{10}+\Delta H_1& N_{10}+\Delta N_1\\ J_{10}+\Delta J_1& L_{10}+\Delta L_1\end{array}\right]\left[\begin{array}{l}\Delta \theta \\ \Delta U\end{array}\right]$$

(16)

where $U$ and $\theta$ are the vectors of node voltages and phase angles of the system, respectively; $\Delta U$ and $\Delta \theta$ are the corrections of $U$ and $\theta$; $\Delta P$ and $\Delta Q$ are the active and reactive power imbalances of the nodes; $H_{10}$, $N_{10}$, $J_{10}$, and $L_{10}$ are the corresponding submatrices of the Jacobian matrix without considering the influence of VSG-PMSGs; and $\Delta H_1$, $\Delta N_1$, $\Delta J_1$, and $\Delta L_1$ are the correction submatrices accounting for the influences of VSG-PMSGs. The correction submatrices are determined by the power sensitivities of the VSG-PMSG as $\Delta H_{1,m_j{m}_j}=\mathsf{\partial }{P}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$, $\Delta N_{1,m_j{m}_j}=\mathsf{\partial }{P}_{\mathrm{w},j}/\mathsf{\partial }{U}_{\mathrm{w},j}$, $\Delta J_{1,m_j{m}_j}=\mathsf{\partial }{Q}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$, and $L_{1,m_j{m}_j}=\mathsf{\partial }{Q}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$, where m_j denotes the bus number to which the jth VSG-PMSG is connected.

For VSG-PMSGs with different types of dynamics, their power sensitivity calculation methods also differ. Let $U_{\mathrm{w},j}^{(k)}$ and ${\theta }_{\mathrm{w},j}^{(k)}$ be the voltage magnitude and phase angle of the jth VSG-PMSG at the kth iteration. First, determine the dynamic types of this unit in the fault steady state. If $U_{\mathrm{w},j}^{(k)}$ satisfies the constraint of Equation (17),

$$\sqrt{P_{\mathrm{w}0,j}^{ 2}+{\left(Q_{\mathrm{ref}}+k_{\mathrm{u}}/k_{\mathrm{q}}(U_{\mathrm{ref}}-U_{\mathrm{w},j}^{(k)})\right)}^2}\le U_{\mathrm{w},j}^{(k)}{I}_{\max }$$

(17)

the jth VSG-PMSG shows type-1 dynamics; otherwise, it will exhibit dynamics of type 2/type 3/type 4. For the former, the power sensitivities $\mathsf{\partial }{P}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$, $\mathsf{\partial }{P}_{\mathrm{w},j}/\mathsf{\partial }{U}_{\mathrm{w},j}$, and $\mathsf{\partial }{Q}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$ equal zero, while $\mathsf{\partial }{Q}_{\mathrm{w},j}/\mathsf{\partial }{U}_{\mathrm{w},j}$ equals $-k_{\mathrm{u}}/k_{\mathrm{q}}$. For the latter, the power sensitivities $\mathsf{\partial }{P}_{\mathrm{w},j}/\mathsf{\partial }{U}_{\mathrm{w},j}$ and $\mathsf{\partial }{Q}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$ can be calculated as follows:

$${\displaystyle \frac{\mathsf{\partial }{P}_{\mathrm{w},j}}{\mathsf{\partial }{U}_{\mathrm{w},j}}}={\displaystyle \frac{f_{\mathrm{p}, j}(U_{\mathrm{w}, j}^{(k)}+{\zeta }_{\mathrm{u}},{\theta }_{\mathrm{w}, j}^{(k)})-f_{\mathrm{p}, j}(U_{\mathrm{w}, j}^{(k)},{\theta }_{\mathrm{w}, j}^{(k)})}{{\zeta }_{\mathrm{u}}}}$$

(18)

$${\displaystyle \frac{\mathsf{\partial }{Q}_{\mathrm{w},j}}{\mathsf{\partial }{U}_{\mathrm{w},j}}}={\displaystyle \frac{f_{\mathrm{q}, j}(U_{\mathrm{w}, j}^{(k)}+{\zeta }_{\mathrm{u}},{\theta }_{\mathrm{w}, j}^{(k)})-f_{\mathrm{q}, j}(U_{\mathrm{w}, j}^{(k)},{\theta }_{\mathrm{w}, j}^{(k)})}{{\zeta }_{\mathrm{u}}}}$$

(19)

where ${\zeta }_{\mathrm{u}}$ is a voltage perturbation of the jth VSG-PMSG, generally set to 0.02 p.u. Similarly, the power sensitivities $\mathsf{\partial }{P}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$ and $\mathsf{\partial }{Q}_{\mathrm{w},j}/\mathsf{\partial }{\theta }_{\mathrm{w},j}$ can also be calculated. Hence, the Jacobian matrix at the kth iteration can be updated.

Supposing the source voltage drops to $U_{\mathrm{gf}}$ when the fault occurs, the initial values of node voltages can be set to $U_{\mathrm{gf}}$ with zero phase angles. The Jacobian matrix is updated based on the power sensitivities calculated by Equations (17)–(19). Substituting it into Equation (16), $\Delta U$ and $\Delta \theta$ can be solved, and then $U$ and $\theta$ are corrected. This process iterates until the power flow converges. Therefore, the fault steady-state power flow is solved, and the fault voltage and phase angle for each VSG-PMSG are obtained.

4.2. Fault Dynamics Predictions of VSG-PMSGs in the Wind Farm

As is demonstrated in Section 3.2, the fault dynamics of the VSG-PMSG depend on the state of its current and EMF limiters. According to Equation (17), the state of the current limiters can be determined based on the fault voltages obtained from fault steady-state power flow. However, the state of the EMF limiter remains undetermined.

To judge its state, the virtual EMF at the fault-clearing time, which is denoted as $E_{\mathrm{c}}$, is estimated. From Equation (3), it can be derived that

$$E_{\mathrm{c}}=E_0+{\displaystyle \frac{1}{T_{\mathrm{E}}}}{\displaystyle {\int }_{t_0}^{t_{\mathrm{c}}}\left(k_{\mathrm{q}}(Q_{\mathrm{ref}}-Q_{\mathrm{w}})+k_{\mathrm{u}}(U_{\mathrm{ref}}-U_{\mathrm{wf}})\right)} \mathrm{d}t$$

(20)

where $t_0$ and $t_{\mathrm{c}}$ denote the fault occurrence and clearance instants, respectively. In Equation (20), $U_{\mathrm{wf}}$ has been obtained by fault steady-state power flow calculation, while $Q_{\mathrm{w}}$ is approaching the fault steady-state value $Q_{\mathrm{ws}}$ during the fault. Therefore, $E_{\mathrm{c}}$ can be approximated by

$$E_{\mathrm{c}}=E_0+{\displaystyle \frac{1}{T_{\mathrm{E}}}}\left(\beta k_{\mathrm{q}}(Q_{\mathrm{ref}}-Q_{\mathrm{ws}})+k_{\mathrm{u}}(U_{\mathrm{ref}}-U_{\mathrm{w}})\right)\left(t_{\mathrm{c}}-t_0\right)$$

(21)

where $\beta$ is a discount coefficient accounting for the growth process of $Q_{\mathrm{w}}$ and can be determined through numerical tests, typically taking a value around 0.92. If $E_{\mathrm{c}}<E_{\max }$, the EMF limiter is not activated during the fault; otherwise, the EMF limiter will be activated. If $E_{\mathrm{c}}$ is much greater than $E_{\max }$, the EMF limiter accomplishes state switching almost instantaneously upon fault occurrence. Consequently, the dynamics can be predicted based on the states of the current limiter and the EMF limiter, as described in Section 3.2.

5. Case Studies

To validate the effectiveness of the proposed method, it was applied to a test system integrating with a wind farm based on VSG-PMSGs, as illustrated in Figure 6.

5.1. Test System

The wind farm in the test system comprises twelve VSG-PMSGs. For each unit, the grid side employs the VSG control and improved LVRT control strategy module, as described in Section 2. Each unit has a rated power of 10.0 MW and a rated capacity of 11.11 MVA. The parameters of the VSG-PMSG are listed in

Table 1. Parameters of the VSG-PMSG.

Parameter	Value	Parameter	Value
Rated capacity (MVA)	11.11	Inertia time constant H (s)	2
Rated voltage (V)	575	Damping coefficient D (p.u.)	60
Rated power (MW)	10	Droop coefficient of reactive power Kq	0.1
Voltage reference Uref (p.u.)	1.0	Droop coefficient of voltage Ku	0.9
Filter inductance LF (mH)	0.0118	Q-U controller time constant TE (s)	0.02
Filter capacitance CF (µF)	6000	Reactive power reference Qref	0
Virtual EMF lower limit Emin (p.u.)	0.5	Virtual resistance Rv (p.u.)	0.01
Virtual EMF upper limit Emax (p.u.)	2.0 p.u.	Virtual inductance Lv (p.u.)	0.33
Virtual EMF initial value E0 (p.u.)	1.0 p.u.	DC-link voltage UDC (V)	1100

. The units are connected to the grid at Bus 3 through two step-up transformers of 575 V/35 kV and 35 kV/220 kV. Transformers T₁–T₁₂ have a rated capacity of 15 MVA and a short-circuit impedance of 0.005 + 0.05j p.u. Transformer T_pcc has a rated capacity of 300 MVA and a short-circuit impedance of 0.004 + j 0.06 p.u. The external grid is modeled by a voltage source with series impedances. The base capacity and rated frequency of the test system are 100 MVA and 60 Hz, respectively. The line parameters are listed in

Table 2. Line impedances of the test system.

	Zs1	Zs2	Z10, Z20, Z30	Z11–Z13, Z21–Z23, Z31–Z33
Length (km)	—	65	5	2
Impedances (Ω)	0.8929	7.4945 + 25.7296j	0.5765 + 1.9792j	0.2306 + 0.7917j

, while the wind speed and steady-state power output for each VSG-PMSG are given in

Table 3. Wind speed and power of the VSG-PMSG in pre-fault steady state.

	W1	W2	W3	W4	W5	W6	W7	W8	W9	W10	W11	W12
Wind speed (m/s)	12.0	10.0	8.5	9.5	10.0	11.0	7.5	9.0	9.5	15.0	8.5	11.0
Active power (p.u.)	0.7822	0.4942	0.3089	0.4248	0.4936	0.6632	0.2130	0.3667	0.4319	0.9020	0.3089	0.6632

.

A source voltage dip fault is assumed to occur at 0.5 s and be cleared at 1.0 s, with the voltage recovering to 1.0 p.u. To validate the effectiveness of the proposed method under different voltage dip depths, two scenarios are configured: (1) Scenario A, where the voltage dips to 0.6 p.u.; (2) Scenario B, where the voltage dips to 0.4 p.u. Simulations are conducted with an Intel Core i7/8.0 GHz computer using MATLAB/Simulink 2019b software, and the simulation time step is set to 2 µs.

5.2. Results of Fault Steady-State Power Flow Calculation

The fault steady-state power flows for Scenarios A and B are calculated utilizing the methods introduced in Section 4.1, and the fault voltages of the VSG-PMSGs are given in

Table 4. Fault steady-state voltages of the VSG-PMSGs under Scenarios A and B.

Scenario	Voltage	W1	W2	W3	W4	W5	W6	W7	W8	W9	W10	W11	W12
A	Simulated (p.u.)	0.8499	0.8774	0.8920	0.8985	0.8519	0.8754	0.8909	0.8975	0.8534	0.8745	0.8920	0.8990
	Calculated (p.u.)	0.8501	0.8791	0.8944	0.8997	0.8525	0.8770	0.8927	0.8993	0.8535	0.8753	0.8939	0.9013
B	Simulated (p.u.)	0.6612	0.6907	0.7109	0.7187	0.6633	0.6897	0.7107	0.7193	0.6622	0.6864	0.7087	0.7157
	Calculated (p.u.)	0.6589	0.6886	0.7080	0.7168	0.6610	0.6876	0.7085	0.7172	0.6603	0.6842	0.7066	0.7136

. For comparison, fault voltages obtained by numerical simulations are also listed. As shown in Table 4, VSG-PMSGs provide significant voltage support to the system during the fault, with the terminal voltage effectively enhanced. The voltages within the wind farm exhibit a maximum difference of 0.05 p.u., which demonstrates the necessity of considering the voltage differences in dynamics analysis. The calculated fault voltages coincide well with the simulated values, with a maximum error below 0.002 p.u. These results indicate that the proposed fault steady-state power flow analysis method achieves satisfactory accuracy and meets the requirements of VSG-PMSG dynamic analysis.

5.3. Dynamics Analysis Based on the Fault Steady-State Power Flow

To analyze the dynamics of VSG-PMSGs in the wind farm, the state of the current and EMF limiters should first be judged. The current limiter operating boundary line in the $P_{\mathrm{w}0}-U_{\mathrm{wf}}$ plane (determined by Equation (11)) is depicted in Figure 7. If the point $\left(P_{\mathrm{w}0}, U_{\mathrm{wf}}\right)$ of a VSG-PMSG lies above the boundary line, the current limiter remains inactive; otherwise, it triggers. The simulation results under Scenarios A and B have proved the law. In Scenario A, the current limiters of W₁, W₂, W₅, W₆, W₉, and W₁₀ are activated with the corresponding points $\left(P_{\mathrm{w}0}, U_{\mathrm{wf}}\right)$ lying under the line, whereas in Scenario B, all current limiters are activated with the points lying under the line.

The virtual EMF at the time of fault clearance, $E_{\mathrm{c}}$, is calculated by Equation (21). Table 4 lists the EMF limiter state and the corresponding E_c of each VSG-PMSG under Scenarios A and B. It should be noted that for VSG-PMSGs with inactive current limiters, their EMF limiters will not trigger, so there is no need to calculate $E_{\mathrm{c}}$.

Table 5. The EMF limiter state and E_c of each VSG-PMSG under Scenarios A and B.

Scenario		W1	W2	W3	W4	W5	W6	W7	W8	W9	W10	W11	W12
A	Ec (p.u.)	2.0839	1.4107	-	-	2.0123	1.4685	-	-	1.9858	1.5235	-	-
	EMF limiter 1	T	F	F	F	T	F	F	F	F	F	F	F
B	Ec (p.u.)	6.8673	6.0965	5.5981	5.3791	6.7967	6.1322	5.5842	5.3679	6.8104	6.2348	5.6318	5.4707
	EMF limiter	T	T	T	T	T	T	T	T	T	T	T	T

¹ ‘T’ denotes the EMF limiter activated; ‘F’ denotes the EMF limiter not activated.

shows that the virtual EMF E_c of W₁ and W₅ exceeds E_max in Scenario A; therefore, their EMF limiters are activated. Meanwhile, in Scenario B, all units have an E_c far greater than E_max, and all EMF limiters are activated as a result of the lower fault voltage.

Based on the results of the above analysis, the dynamic pattern of VSG-PMSGs in the wind farm can be predicted as follows:

• Scenario A: Type 1 for W₃, W₄, W₇, W₈, W₁₁, and W₁₂ since their current and EMF limiters are both inactive; type 2 for W₂, W₆, W₉, and W₁₀ because their current limiters are activated, while their EMF limiters remain inactive; and type 3 for W₁ and W₅, as their current limiters are activated, and the EMF limiters operate after a noticeable period after fault occurrence.
• Scenario B: Type 4 for all units in the wind farm, since their current and EMF limiters are activated immediately upon fault occurrence.

Simulations are conducted to validate the accuracy of the predictions. The dynamic responses of VSG-PMSGs in the wind farm under Scenario A are shown in Figure 8. It can be observed that W₁ and W₅ exhibit type-3 dynamics, as depicted in Figure 5, while W₂, W₆, W₉, and W₁₀ show typical type-2 dynamics. The remaining VSG-PMSGs (W₃, W₄, W₇, W₈, W₁₁, and W₁₂) display type-1 dynamics behavior. The simulation results under Scenario A are in good agreement with the predictions.

Similarly, the dynamic responses of VSG-PMSGs under Scenario B are simulated, with the results depicted in Figure 9. This validates the conclusion that all VSG-PMSGs demonstrate type-4 dynamics under severe voltage sags. Indeed, the simulation results further indicate that VSG-PMSGs exhibit dynamics identical to those in Figure 9 even under more extreme scenarios, such as voltage drops to 0 p.u. or 0.1 p.u.

The simulation results under various voltage dips demonstrate that the dynamic analysis method based on fault steady-state power flow can accurately predict the dynamic patterns of VSG-PMSGs within 10 s. It shows high efficiency compared with the time-consuming methods relying on numerical integration. Additionally, it has been shown that the VSG-PMSG with improved LVRT control can be equivalent to a voltage source. The internal voltage increases rapidly after the occurrence of voltage faults, while the phase angle ${\theta }_{\mathrm{e}}$ remains nearly unchanged after the current limiter triggers. Thus, the VSG-PMSG has a strong voltage support capacity.

5.4. Impacts of H, D, and SCR on LVRT Dynamics of VSG-PMSG

The aforementioned simulations were conducted under a specific grid condition, where the inertia time constant H, damping coefficient D, and short-circuit ratio (SCR) are set to 1 s, 60 p.u., and 8.0, respectively. Given the significant influences of H, D, and SCR on the VSG-PMSG dynamics, their impacts are subsequently analyzed. To focus on the effect of the LVRT control strategy, the wind farm in the test system shown in Figure 6 is replaced with a single equivalent VSG-PMSG of equal capacity. The grid voltage is assumed to dip to 0.2 p.u. at 0.5 s, while the fault is cleared at 1.0 s.

Simulations are conducted using various values of H and D, with the corresponding results presented in Figure 10. It is evident from Figure 10 that the impacts of H and D on the transient dynamics are predominantly manifested during the post-fault-clearing period. Referring to [26], the damping ratio $\zeta$ of a synchronous generator is given by $\zeta =\frac{D}{2\sqrt{K_{\mathrm{S}}(2H){\omega }_{\mathrm{n}}}}$, and the undamped natural frequency ${\omega }_{\mathrm{nat}}$ is expressed as ${\omega }_{\mathrm{nat}}=\sqrt{\frac{K_{\mathrm{S}}{\omega }_{\mathrm{n}}}{2H}}$, where $K_{\mathrm{S}}$ denotes the synchronizing torque coefficient. The VSG-PMSG exhibits a similar property. As illustrated in Figure 10, the damping ratio diminishes when H grows or D decreases; meanwhile, the system response speed slows down with the increase in H.

To investigate the impact of the SCR on the transient dynamics, simulations are conducted under SCRs of 8.0, 6.0, 4.0, and 2.5, respectively. The simulation results are presented in Figure 11. As the grid weakens (i.e., with a decreasing SCR), the impact of a source voltage sag on the VSG-PMSG diminishes due to the increased grid impedance. Consequently, both the reactive power support and the VSG-PMSG fault voltage increase. However, when the SCR is below 2.5, the VSG-PMSG in the test system exhibits high-frequency oscillations, which may stem from the interaction between the control bandwidth and weak grid impedance. It should be noted that the impact of decreasing the SCR on the VSG-PMSG is multifaceted. Wide-band oscillation may arise from impedance mismatch between the external system and the VSG, while low-frequency oscillations can emerge from virtual power angle dynamics. Thus, further in-depth analysis of this complex issue remains necessary.

6. Conclusions

This paper proposes an LVRT dynamic analysis method for a VSG-PMSG. To suppress transient angle instability and overcurrent of the VSG-PMSG during a voltage fault, an improved LVRT control strategy combining active power reference reduction and virtual EMF reset is presented. The VSG-PMSG exhibits four typical dynamics under various conditions. It has been found that the initial active power and fault voltage play crucial roles in shaping the post-fault dynamics responses, and the activations of the current and EMF limiters lead to more complex dynamics. The corresponding relationship between the dynamic patterns and the limiters’ state is revealed.

To predict the dynamics of VSG-PMSGs in a wind farm, a fault steady-state power flow-based method is proposed. The power sensitivities of VSG-PMSGs are deduced according to their virtual EMF and rotor angle variation patterns, and the fault steady-state power flow is calculated. The current limiter state is identified by the boundary line in the $P_{\mathrm{w}0}-U_{\mathrm{wf}}$ plane, while the EMF limiter state is judged by the estimated virtual EMF at the fault-clearing time. Thus, the dynamic pattern of VSG-PMSGs in the wind farm during a fault can be predicted according to the limiters’ states, avoiding time-consuming numerical integration. The simulation results under various voltage dip depth scenarios demonstrate the accuracy of the power flow calculation and fault dynamics prediction.

The proposed method enables accurate and rapid prediction of LVRT dynamics for VSG-PMSGs in wind farms. However, it should be noted that this analysis method and the derived results are strictly limited to VSG-PMSGs employing the active power reference reduction and the virtual EMF reset-based LVRT strategy. For VSG-PMSGs employing other LVRT methods, such as mode-switching control, virtual impedance control, adaptive inertia and damping tuning control, the dynamics and analysis methods still require further investigation. Another limitation of this study is that the validation is based on simulations. While simulations provide theoretical insights, the conclusions still lack validation against hardware-in-the-loop (HIL) tests or field data, which may reveal practical challenges (e.g., converter switching delays and sensor noise). The simplified VSG-PMSG model, which neglects the machine-side converter dynamics and DC-link voltage fluctuations, also diminishes the fidelity to the real system, thereby restricting the credibility of the results. Promising future research directions may include hardware-in-the-loop (HIL) tests to further validate the effectiveness and feasibility of LVRT control strategies in real-time environments, particularly under extremely weak grids and unbalanced faults, coordinated optimization of LVRT control accounting for multi-machine interactions in renewable power plants [27], and the dynamic analysis method for VSGs employing other control strategies.