Transient Stability Enhancement of Virtual Synchronous Generator Through Analogical Phase Portrait Analysis

Abstract

Virtual synchronous generator (VSG) control has been increasingly utilized for the grid integration of the voltage source inverter (VSI). Under large disturbances, such as voltage sags and grid faults, the VSG synchronization dynamic is highly nonlinear and cannot be evaluated by small-signal-based approaches. Conventionally, the equal area criterion (EAC) is utilized to analyze the transient stability of a synchronous machine or a VSG. However, it is found that the EAC is only valid under special scenarios when the damping effect is ignored. In this case, the EAC will provide conservative predictions and therefore put stringent requirements on the fault-clearing time. This paper reveals that the motion of a pendulum is essentially the same as the VSG swing equation. Due to this, the phase portrait approach, which was used to predict the pendulum motion, can be similarly applied for the VSG transient stability study. Based on the analogical phase portrait analysis, a damping coefficient tuning guideline is proposed, which always guarantees the synchronization stability as long as an equilibrium exists. The aforementioned theoretical findings are finally verified through a grid-connected VSG under the hardware-in-loop (HIL) environment.

1. Introduction

Driven by the urgent need for carbon reduction, fossil-based synchronous machines are gradually being replaced with power converter-based distributed generators. Though the high DG penetration level helps to alleviate the carbon emission issue, it also brings some challenges to the power grid. One prominent problem is the continuously declining power system inertia. Since the interfaced inverters are usually characterized by their fast responses and do not provide any inertia support, the grid frequency is easily subject to disturbances [1,2]. In addition, the reduced power system damping also brings low-frequency oscillations and threatens the power grid stability [3]. There is, accordingly, an urgent need to develop grid-friendly converters that are able to provide both inertia and damping support to the electrical power grid.

Inspired by the swing equation of a synchronous machine, VSG control is proposed and successfully implemented for DC/AC inverters [4,5,6]. On one hand, a VSG is able to perform the grid-forming function in the islanded operation mode, while on the other hand, a VSG also improves the frequency stability in the grid-connected mode [7]. To achieve a satisfactory VSG response, a number of small-signal modeling and VSG control parameter design procedures are reported in the literature [8,9,10,11,12,13,14]. Specifically, the line-frequency-average model of VSG is derived [8], which gives the decoupling condition between the active and reactive power control loops. In [9], the impact of a non-inductive grid impedance on VSG stability is investigated through a simplified third-order model, while a similar analysis and parameter tuning constraints are presented in [10]. Instead of building state-space models, the synchronizing and damping torques of VSG have been derived in [11], from which the low-frequency oscillation mechanism is clearly explained. In addition, the VSG small-signal stability under various scenarios are also studied, such as paralleling a VSG with a synchronous machine [12], paralleling multiple VSGs [13], integrating a VSG with an extremely weak AC grid [14], and having a limited DC-side energy storage capacity [15].

It should be mentioned that the above approaches are only applicable for small perturbations around the equilibrium point. However, this assumption does not hold for large disturbances, such as grid voltage sags, loss of a transformation line, and ground faults. Under such severe conditions, the VSG dynamic response is highly nonlinear and cannot be evaluated through the classical linear control theory. To precisely predict the transient response of a VSG or a synchronous machine, time-domain simulations are used in the literature [16]. Yet, the time-domain simulation is essentially a trial-and-error approach, which hardly comes to generalized conclusions. As an alternative, the EAC is preferred since it provides a clear and insightful physical interpretation of the transient response [17].

According to the EAC, a synchronous machine or a VSG can remain synchronized with the AC grid if the accelerating area (AA) is smaller than the deaccelerating area (DA). Based on this principle, a number of efforts have been devoted to analyze and improve the transient stability [18,19,20,21]. One intuitive method is to enlarge the DA by clearing the fault after a short period of time, where the critical clearing angle (CCA) can be theoretically calculated by the EAC [17]. In addition, the VSG inertia is intentionally designed to be adaptive during different stages [19,20,21]. Specifically, a larger inertia is synthesized for the accelerating stage to reduce the AA, while a smaller inertia is synthesized during the deaccelerating stage with an attempt to increase the recovery speed. Furthermore, the EAC is also extended to multiple-generator scenarios [22,23] and used in conjunction with time-domain simulations or Lyapunov’s approach to access the system stability [24,25].

Nevertheless, the EAC does not take the damping effect into account. Since it is difficult to obtain the exact damping value of a synchronous machine, this simplification considers the worst scenario with zero damping. In contrast, the damping coefficient of a VSG can be flexibly designed through the digital controller. In this case, the conventional EAC gives overly conservative predictions about the VSG transient response. In fact, as the damping effect leads to energy dissipation, it is therefore possible to stabilize a grid-connected VSG even if the AA is larger than the DA.

To accurately analyze the VSG transient response, some new analytical methods are needed. Fortunately, it is revealed in this paper that the swing equation of a simple pendulum has the same format as that of a grid-connected VSG. Since the pendulum motion was well studied through phase portrait analyses [26,27], similar approaches can also be applied to providing a deep insight into the VSG transient stability. It should be mentioned that another phase portrait analysis was recently presented in [28] to analyze the transient stability of a first-order system. However, it cannot capture the dynamic of a second-order nonlinear system, i.e., a grid-connected VSG, accurately. Though Lyapunov-based approaches can also be adopted for transient stability analysis [29,30], it requires solving high-order differential equations and finding the appropriate Lyapunov function. Therefore, the computation burden will be considerable.

Based on our previous research [31], this paper attempts to analyze VSG transient stability through analogical pendulum phase portraits. Firstly, the VSG transient stability with a zero damming effect is studied and the marginally stable condition derived is verified by the conventional EAC. Then, the impact of a non-zero damping effect is also studied. Through the proper design of the damping coefficient, VSG can always be synchronized with the main grid as long as an equilibrium point exists. Moreover, for severe situations when an equilibrium point does not exist, it is found that the VSG is able to resynchronize with the grid after the fault clearance. One attractive feature is that this resynchronization ability has no requirement on the fault-clearing time. Compared with the conventional EAC demand that the fault must be cleared before the CCA, this feature significantly helps to release the post-fault operation stress. Finally, HIL simulation results from Typhoon 602 are provided for verification.

2. Modeling of Grid-Connected VSG and Pendulum

2.1. Modeling of a Grid-Connected VSG

Figure 1 shows the control block diagram for a grid-connected VSG. As can be seen from Figure 1a, the overall control scheme includes a voltage-loop and power-loop controller. The power-loop controller emulates the swing equation of a synchronous machine, where P_ref is the VSG reference active power at the nominal frequency ω₀, H is the inertia coefficient, and D is the VSG damping coefficient. The phase angle θ, along with the voltage magnitude E, is utilized to generate the three-phase reference voltages, which are subsequently tracked by the inner-loop controller with a much faster dynamic.

In order to analyze the VSG synchronization mechanism, the equivalent diagram of the active power loop is further depicted in Figure 1b, where δ is the angle difference between the VSG and grid, V_g refers to the voltage magnitude of the grid, ω_g denotes the grid frequency, and X_g = ωL_g is the line impedance. In some cases, the VSG output current may reach a pre-defined limit. Once it happens, a VSG will behave like a current source and its transient response is quite different from that under normal conditions. It should be mentioned that this phenomenon has been investigated in [31] and is not considered in this paper. From Figure 1b, the dynamic equation of the active power loop can be derived as

$${\displaystyle \frac{P_{ref}}{2H}} - {\displaystyle \frac{3E{V}_{g}sin \delta }{4H{X}_g}} - {\displaystyle \frac{D}{2H}} · {\displaystyle \frac{d\delta }{dt}}+{\displaystyle \frac{D· ({\omega }_0 -{ \omega }_g)}{2H}}={\displaystyle \frac{d^2\delta }{d{t}^2}}$$

(1)

Note that sampling noises for voltage and current sensors are not considered in this paper, as it makes little impact on Equation (1). For simplicity, the VSG steady-state output active power P₀ is denoted as

$$P_0={ P}_{ref}+D · ({\omega }_0 -{ \omega }_g)$$

(2)

From (1) and (2), the system dynamic equation can be further derived as

$${\displaystyle \frac{d^2\delta }{d{t}^2}}+{\displaystyle \frac{D}{2H}}·{\displaystyle \frac{d\delta }{dt}}+{\displaystyle \frac{3E{V}_g}{4H{X}_g}}·sin \delta ={\displaystyle \frac{P_0}{2H}}$$

(3)

It can be clearly seen from (3) that the mathematical model of VSG synchronization is essentially a second-order nonlinear system. The presence of the trigonometric function makes it very difficult to analyze the VSG transient stability through the classical linear control theory.

2.2. Modeling of a Simple Pendulum

Figure 2 shows the force analysis diagram of a pendulum [26], where l is the rope length, m is the mass of pendulum, δ is the pendulum angular position with respect to the vertical line, and ω represents the angular speed. Several different types of forces are applied on the pendulum, such as the gravity mg, the tension of the rope T, an external driving force F_d, and the air friction f. It is widely accepted that the friction f is proportional to the pendulum speed, i.e.,

$$f=\gamma v=\gamma \omega l$$

(4)

where γ is the friction coefficient. Based on Newton’s second law, it can be obtained that

$$F_d - \gamma \omega l - mg sin \delta =ma$$

(5)

where a refers to the tangential acceleration speed. From (5), the angular acceleration speed can be calculated as

$${\displaystyle \frac{d^2\delta }{d{t}^2}}={\displaystyle \frac{a}{l}}={\displaystyle \frac{F_d}{ml}} - {\displaystyle \frac{\gamma \omega }{m}} - {\displaystyle \frac{g}{l}}sin \delta$$

(6)

Reorganizing (6) yields

$${\displaystyle \frac{d^2\delta }{d{t}^2}}+{\displaystyle \frac{\gamma }{m}}·{\displaystyle \frac{d\delta }{dt}}+{\displaystyle \frac{g}{l}}sin \delta ={\displaystyle \frac{F_d}{ml}}$$

(7)

which forms a generalized dynamic equation for describing the pendulum motion [26]. Comparing (3) and (7), it is interesting to find that the pendulum motion equation has the same format compared with that of the grid-connected VSG. This standard form can be expressed as

$${\displaystyle \frac{d^2\delta }{d{t}^2}}+a_1·{\displaystyle \frac{d\delta }{dt}}+a_0 sin \delta =C$$

(8)

where a₁, a₀, and C are positive constants.

3. Preliminaries on Pendulum Phase Portrait

The above similarity on dynamic equation makes it possible to analyze the VSG transient stability through the analogy with a pendulum. Prior to this, some preliminary knowledge about pendulum phase portraits is provided in this section.

It has been quite a long time since physicists started to study the motion of a pendulum. Over the past centuries, a powerful analytical tool known as phase portrait analysis has been developed. The phase portrait is a geometric representation of pendulum motion trajectory, where the angular position δ is the horizontal coordinate and angular speed ω serves as the vertical coordinate. From a phase trajectory in the δ-ω plane, it is easy to predict whether the pendulum angular position will become stable or unstable (δ becomes unbounded).

Figure 3 shows typical phase portraits of a simple pendulum. With different system parameters and initial points, the corresponding phase trajectories are also completely different. For example, for the initial point A₀, the angular speed ω is always positive in its forward path. Therefore, the pendulum is unstable since δ continuously increases and never reaches an equilibrium point. Similarly, the initial point A₁ is also unstable, though its forward phase trajectory has marginally approached the δ-axis. In comparison, the corresponding phase trajectory of the initial point A₂ will circulate around the equilibrium point E₂. As a consequence, A₂ is a stable initial point, according to the system stability definition.

It should be mentioned that the equilibrium point E₂ in Figure 3 is one specific type of equilibrium point known as the center. In addition, a more generalized classification is shown in Figure 4, where the equilibrium points are classified as the center, the spiral, and the node. A center is an equilibrium point around which the phase portrait circulates rather than converges. It usually occurs when the pendulum has zero damping, i.e., γ = 0. A spiral is an equilibrium point occurring at the end of a phase trajectory. It indicates an insufficient damping effect and hence exhibits oscillatory behaviors in the neighborhood of the equilibrium point. Lastly, a node appears as the result of over-damping and makes sure that there is no oscillation in the phase portrait trajectory. Clearly, an initial point of the pendulum is stable if a center, a spiral, or a node exists in its forward phase portrait trajectory. Otherwise, the pendulum phase angle δ will become unbounded as time goes by.

In addition, an analytical tool can be used to judge whether an equilibrium point belongs to a center, a spiral, or a node. Without loss of generality, the standard form of (8) can be reorganized as

$$\left\{\begin{array}{c}|{\displaystyle \frac{d\omega }{dt}}=- a_1· \omega - a_0 sin \delta +C\\ |{\displaystyle \frac{d\delta }{dt}} =\omega \end{array}\right.$$

(9)

Linearizing (9) at the equilibrium point, i.e., δ = δ₀ and ω = 0, yields

$$\left[\begin{array}{c}d\omega /dt\\ d\delta /dt\end{array}\right]= J· \left[\begin{array}{c}\omega \\ \delta \end{array}\right]=\left[\begin{array}{cc}-a_1& -a_0 cos {\delta }_0\\ 1& 0\end{array}\right]· \left[\begin{array}{c}\omega \\ \delta \end{array}\right]$$

(10)

where J is the Jacobin matrix, from which the characteristic equation is derived as

$${\lambda }^2 - tr(J)·\lambda +det(J)=0$$

(11)

$$tr\left(J\right)= -a_1, det\left(J\right)={ a}_0·cos {\delta }_0$$

(12)

where tr(J) is the trace of the Jacobin matrix and det(J) refers to the determinant of the Jacobin matrix. The types of equilibria can be classified and related in the det-tr plane, as shown in Figure 5.

As can be seen from Figure 5, the entire det-tr plane is divided into several sub-regions. Among them, a stable region can be obtained if tr(J) > 0 and det(J) ≥ 0 are satisfied simultaneously. Moreover, the hyperbola tr²(J) = 4det(J) further determines the critical damping condition and distinguishes between the spiral and the node.

4. Transient Stability of a Grid-Connected VSG

The above knowledge is applied in this section to analyze the VSG transient stability. Discussions concern two different scenarios: without the damping effect and with the damping effect.

4.1. Transient Stability Without the Damping Effect

First, the scenario that a synchronous machine or a VSG has no damping effect is investigated. Considering that D = 0, the previously derived equation, i.e., (3), can be further simplified to characterize the VSG post-fault dynamic, i.e.,

$${\displaystyle \frac{d^2\delta }{d{t}^2}}+{\displaystyle \frac{3E{V}_{g-II}}{4H{X}_{g-II}}}·sin \delta ={\displaystyle \frac{P_0}{2H}}$$

(13)

where V_g-II and X_g-II are the post-fault grid voltage magnitude and line impedance, respectively. Additionally, the post-fault active power transmission limit P_II is defined as

$$P_{II}={\displaystyle \frac{3E{V}_{g-II}}{2X_{g-II}}}$$

(14)

Combining (13) and (14),

$${\displaystyle \frac{d^2\delta }{d{t}^2}}={\displaystyle \frac{1}{2H}}· (P_0 - P_{II}sin \delta )$$

(15)

Multiplying dt/dδ to both sides of (15) gives

$${\displaystyle \frac{d\omega }{dt}}· {\displaystyle \frac{dt}{d\delta }}={\displaystyle \frac{dt}{d\delta }}· [{\displaystyle \frac{P_0}{2H}} - {\displaystyle \frac{P_{II}}{2H}}·sin \delta ]$$

(16)

Rearranging (16) yields

$$\omega · d\omega =({\displaystyle \frac{P_0}{2H}} - {\displaystyle \frac{P_{II}}{2H}}sin \delta ) · d\delta$$

(17)

The VSG phase portrait can be obtained by integrating both sides of (17):

$${\int }_0^{\omega }\omega · d\omega ={\int }_{{\delta }_0}^{\delta }({\displaystyle \frac{P_0}{2H}} - {\displaystyle \frac{P_{II}}{2H}}·sin \delta )· d\delta$$

(18)

$${\omega }^2= f(\delta )={\displaystyle \frac{P_0}{H}}· (\delta - {\delta }_0) +{\displaystyle \frac{P_{II}}{H}}(cos \delta - cos {\delta }_0)$$

(19)

In (19), f (δ) is a mixed function made up of a linear function P₀/H·δ, a trigonometric function P_II/H·cos δ, and also a constant term determined by the initial condition. Figure 6 illustrates the typical post-fault VSG phase portraits. When the fault happens, the initial state immediately changes from point a to point b. Since P₀ is larger than P_II·sin δ between δ₀ and δ₁, the phase angle difference increases until point c is reached. Oppositely, a deceleration stage occurs between δ₁ and δ₂ because P₀ is smaller than P_II sin δ. From the phase portrait perspective, the VSG synchronization stability depends on whether the phase trajectory will cross the δ-axis before δ₂. Figure 6a and Figure 6b show the stable and unstable scenarios, respectively. To ensure that the trajectory will cross the δ-axis before δ₂, the condition below shall be satisfied:

$$f({\delta }_2)<0$$

(20)

From (19) and (20),

$$P_0 <{ P}_{II}· {\displaystyle \frac{cos {\delta }_2 - cos {\delta }_0}{{\delta }_2 - {\delta }_0}}$$

(21)

which gives the condition for maintaining the synchronization. To verify the correctness of the above phase portrait analysis, the stability condition is also derived through the conventional EAC, which defines the area S_abc in Figure 6 as the AA and S_cod in Figure 6 as the DA. Their values are calculated as

$$S_{abc}={\int }_{{\delta }_0}^{{\delta }_1}(P_0 - P_{II}sin \delta ) · d\delta ={ P}_0 ({\delta }_1 - {\delta }_0)+{ P}_{II} (cos {\delta }_1-cos {\delta }_0)$$

(22)

$$S_{cod}={\int }_{{\delta }_1}^{{\delta }_2}(P_{II}sin \delta - P_0) · d\delta ={ P}_0({\delta }_1 - {\delta }_2)+{ P}_{II} (cos {\delta }_1- cos {\delta }_2)$$

(23)

According to the EAC, the AA should be smaller than the DA such that a VSG, or a synchronous machine, is able to maintain the grid synchronization [17]. Therefore, it is required that

$$S_{abc}-{ S}_{cod}={ P}_0({\delta }_2 - {\delta }_0)+{ P}_{II} (cos {\delta }_2 - cos {\delta }_0) < 0$$

(24)

A comparison indicates the same condition as (21) and validates the feasibility of the phase portrait analysis.

Notice that the damping effect of a synchronous machine is influenced by many factors, such as frictions and frequency-dependent loads in the system [17]. Owing to this reason, it is difficult to obtain the exact value of the damping effect, while assuming that D = 0 indicates the worst scenario. According to the EAC, if the AA is calculated to be larger than the DA, some efforts should be devoted to reverting the situation. Among them, one commonly adopted method is to clear the fault before the CCA. Yet, this strategy imposes requirements on the response speed of relaying and fault clearance.

4.2. Transient Stability with the Damping Effect

The VSG damping coefficient D can be flexibly designed within a certain range, which helps to stabilize the VSG under large disturbances. In view of this, some questions arise inherently: Is it possible for a VSG to maintain grid synchronization even if the AA is larger than the DA? If so, how much damping is sufficient to reach this target? To properly answer these questions, analogical phase portrait analyses with damping effect are presented in this section. But before this, some clarifications are presented:

(1.) The scenarios where the AA is smaller than the DA are not considered, since the VSG synchronization stability can be easily guaranteed even without the damping effect.

(2.) It is assumed that there exists an equilibrium point after the fault. In other words, P_II > P₀. (It should be mentioned that the transient response without a post-fault equilibrium, i.e., P_II < P₀, will also be discussed later.)

The generalized phase portrait trajectories of (8) are plotted and illustrated in Figure 7, where the damping effect gradually increases from Figure 7a to f. δ₀ is the initial phase angle difference and δ_∞ is the steady-state phase angle difference, if one ever exists. A comparatively small damping effect is applied in Figure 7a, and the corresponding trajectory (marked in a solid line) is compared with that without a damping effect (marked in a dashed line). It can be seen that the damping effect decreases the angular frequency difference ω and the nadir. This happens because the kinetic energy accumulated during the acceleration stage, which is proportional to ω², is partially dissipated by the damping effect. Nevertheless, the system is still unstable as the phase portrait trajectory becomes unbounded and never crosses the δ-axis.

In Figure 7b, the increase in D, consequently, leads to more energy dissipations and helps to further decrease the nadir. A critical stability condition is met when D = D_cir1 and the nadir falls exactly on the δ-axis. Figure 7c depicts this scenario where the phase portrait trajectory reaches and then crosses the δ-axis. Due to the characteristics of a phase portrait, the crossing with the δ-axis is accomplished vertically with an infinite slope. Once ω becomes negative, the phase angle difference δ will decrease, even though the trajectory will not recover to the initial point δ₀ due to the damping effect. Instead, it oscillates and finally converges to another value δ_∞, indicating that the system is stable. According to the equilibrium point definition, the equilibrium point δ = δ_∞ and ω = 0 is a spiral.

Then, the damping effect keeps increasing and the oscillation around the equilibrium point becomes smaller, as shown in Figure 7d. When the damping effect is sufficiently large to reach the critical damping condition, i.e., D = D_cir2, the final equilibrium point in Figure 7e turns from a spiral to a node. As the damping effect becomes even stronger, the system is over-damped and free from oscillations, as shown in Figure 7f. In spite of this, the transient response becomes sluggish, accordingly.

The above phase portrait analysis indicates that it is possible to maintain synchronization when D > D_cir1. Nevertheless, it is not wise to select D as D_cir1 since the system will still suffer from serious oscillations. Alternatively, selecting D as D_cir2 provides superior performance in terms of system transient stability and oscillation damping. Recalling from the hyperbola in Figure 5, the value of D_cir2 can be calculated as

$$t{r}^2(J) - 4 det(J)={ a}_1^2 - 4a_0· cos {\delta }_{\infty }=0$$

(25)

From (3) and (8),

$$a_1={\displaystyle \frac{D_{cir2}}{2H}}, a_0={\displaystyle \frac{3E{V}_{g-II}}{4H{X}_{g-II}}}$$

(26)

From (25) and (26),

$$D_{cir2}={\left({\displaystyle \frac{12E·V_{g-II}·H·cos {\delta }_{\infty }}{X_{g-II}}}\right)}^{1/2}$$

(27)

Moreover, it is important to make sure that the VSG transient stability is always guaranteed regardless of the initial and final states. Figure 8 depicts phase portrait trajectories with different final states (under the critical damping condition D = D_cir2). It is observed that the VSG transient stability can be ensured for various P₀/P_II ratios, even for the extreme condition that P₀/P_II is close to unity. This indicates that synchronization is easily maintained as long as an equilibrium point exists, i.e., P₀/P_II < 1.

For worse scenarios when the equilibrium does not exist, the VSG is not able to remain synchronized with the grid unless the fault has been cleared. If so, the initial phase angle difference at the recovery instant may vary according to the fault-clearing time. Figure 9 shows the phase portrait trajectories with different initial states. It was found that the trajectory always converges to the final state for any initial phase angle difference between −π and π. This attractive feature helps to release the stress of fault clearance. (Note that the conventional EAC requires fault to be cleared before the CCA; otherwise, synchronization will be lost.)

4.3. Parameter Design Procedure

Based on the above analyses, a parameter design procedure is also provided to properly tune the VSG inertia coefficient H and the damping coefficient D. In order to guarantee the critical damping condition for various post-fault conditions, the value of D is determined as

$$D ={ \left( {\displaystyle \frac{12E{V}_{g}H}{X_g}}\right)}^{1/2} > {\left({\displaystyle \frac{12E·V_{g-II}·H·cos {\delta }_{\infty }}{X_{g-II}}}\right)}^{1/2}$$

(28)

Figure 10 shows a simple flowchart for control parameter design. To start, the inertia coefficient is tuned as an initial value H₀, which is the maximum inertia support that can be provided by the converter and the DC-side energy storage unit. Next, the damping coefficient D is tuned according to (28). Though the critical damping condition has been satisfied, the grid code also requires that the steady-state VSG active power P₀ = P_ref + D(ω₀ − ω_g) cannot exceed its capacity limit P_max when grid frequency ω_g varies between ω_min and ω_max [8]. If this requirement is not satisfied, the value of H is decreased and a new damping coefficient value is calculated. This process will be repeated until the power capacity limit condition is satisfied.

5. Hardware-in-Loop Simulation Results

In order to validate the aforementioned theoretical findings, a grid-connected VSG was built under Typhoon HIL 602 and the digital controls were implemented by a dSPACE MicroLabBox. Figure 11a shows the laboratory setup and Figure 11b shows the fault scenario, where an open-circuit fault happens and cuts off one power transmission line. The key system parameters are provided in

Table 1. Key system parameters.

Parameters	Descriptions	Values
Vg, E0	Nominal voltage magnitude	155 V
ω0	Nominal grid frequency	50 Hz
H	VSG inertia coefficient	100
D	VSG damping coefficient	3050 s·W
Pref	VSG nominal power at ω0	2 kW
Lg1, Lg2	Line impedances	20 mH
PII	Post-fault active power limit	11.56 kW

. Two different situations have been studied, and the corresponding results are shown below.

5.1. VSG Responses with Post-Fault Equilibrium

Figure 12 shows VSG dynamic responses with different steady-state power P₀ values. In Figure 12a, P₀ equals 5 kW. The associated AA and DA are calculated as 282 rad·W and 9611 rad·W, respectively. As the DA is much larger than the AA, the VSG grid synchronization is easily guaranteed and δ smoothly increases from 12.4° to 25.4°. In Figure 12b, P₀ equals 10 kW. According to the EAC, the VSG will be out of step because the DA (1088 rad·W) is smaller than the AA (1356 rad·W). However, it can be seen that the VSG transient stability is still well guaranteed. A similar result is also observed in Figure 12c, where P₀ = 11.5 kW is very close to the power transfer limit P_II. Though the DA (29.5 rad·W) is significantly smaller than the AA (1990 rad·W), δ finally converges to its equilibrium value 84.2° without losing synchronization. From the above results, it is clear that VSG synchronization stability can be easily guaranteed as long as an equilibrium exists after the fault.

5.2. VSG Responses Without Post-Fault Equilibrium

For some worse scenarios where post-fault equilibrium does not exist, the VSG transient stability was tested, as shown in Figure 13. In Figure 13a, no recovery is achieved after the fault occurs. Since P₀ = 15 kW already exceeds the active power transfer limit P_II, by no means can a VSG maintain the grid synchronization. Hence, the VSG becomes out of step after the fault. As a result, the phase angle difference δ, the angular speed difference ω, and the delivered active power P oscillate periodically. In Figure 13b, the fault is cleared after the CCA. According to the conventional EAC, the synchronization cannot be maintained. Yet, it can be seen that the VSG is able to resynchronize with the grid after the fault clearance. An even worse situation test is shown in Figure 13c, where the fault clearance is very slow and the VSG has gone through several rounds of oscillation. Nonetheless, the VSG resynchronization ability is not affected, as clearly seen in Figure 13c. The testing results are in good accordance with the previously presented phase portrait analysis.

6. Conclusions

This paper studies the transient stability of the grid-connected VSG through analogical phase portraits. It is revealed that with the proper tuning of the damping coefficient, the synchronization can always be maintained as long as an equilibrium point exists. Additionally, for severe faults scenarios where there is no equilibrium point, the VSG can still resynchronize with the grid after clearing the fault. This resynchronization ability is found to be independent of the fault-clearing time and therefore releases the system post-fault operation stress. The above theoretical findings are then verified by both phase portrait trajectories and HIL simulation results.