A Method to Improve Both Frequency Stability and Transient Stability of Virtual Synchronous Generators during Grid Faults

Abstract

Transient stability and frequency stability of virtual synchronous generator (VSG)-controlled converters during grid faults are critical for the stable operation of high-share renewable energy power systems. However, the existing transient stability methods often overlook the consideration of frequency stability. This paper presents a method to improve both frequency stability and transient stability of VSGs during grid faults. The influence mechanism of inertia and damping on transient stability and frequency stability is comprehensively analyzed under two types of faults, with or without stable equilibrium points. To address the conflict between frequency stability and transient stability of VSGs with large inertia under the first type of fault, an improved VSG control with angular frequency deviation feedback is proposed. The transient response of this control method is analyzed using small-signal and large-signal models. Furthermore, to achieve optimal dynamic performance, this improved VSG control introduces the optimal damping ratio and presents a parameter design method. Finally, the correctness of the proposed method and its theoretical analysis is verified through experimental results.

1. Introduction

Since the signing of the Paris Climate Agreement, the development of green sustainable energy represented by wind power and photovoltaics has become a broad consensus of the international community. However, due to the lack of rotating parts in the renewable energy interface converter, the large-scale substitution of the converters for the traditional synchronous generators (SGs) reduces the inertia of the power systems, resulting in new problems, such as a worse anti-interference ability and weaker frequency stability. A VSG which imitates the inertia and damping characteristics of SGs has become the most promising method to tackle existing problems [1,2,3].

Compared to SGs, the VSG-controlled converter not only inherits the external characteristics of SGs but also has the flexible adjustment ability of inertia and damping parameters, which determines that the VSG will not only generate transient stability problems, as in SGs when large disturbances such as voltage drop, disconnection, and tripping occur in the power grid [4,5,6,7], but it will also generate frequency stability problems due to parameter changes [8,9]. The parameter values of VSG affect both frequency stability and transient stability [10]. At present, the research on VSGs mainly focuses on small-signal stability analysis under normal conditions, including modeling and simplification, parameter design, optimization and adaptive adjustment, operation control methods of single and multiple VSG, and strategy improvement [11,12,13]. There is less attention on the transient stability of VSGs under fault conditions, and in these studies, the transient stability and fault current limiting of VSGs are more studied, while the frequency stability is almost ignored.

Transient stability, also known as transient power angle stability or synchronous stability, refers to the ability of VSGs to maintain synchronization with the power grid under large disturbances. Once transient instability occurs, it will not only cause VSGs to lose synchronization but also generate a large fault current. Because the overcurrent capacity of the converter is far less than SG, if the current limiting is not carried out, the converter may be damaged, and the protection action will be triggered to remove the new energy from the power grid, posing a serious threat to system security. Therefore, the existing research mainly focuses on the transient stability and fault current limiting of VSGs. According to the current research, the transient stability problem can be classified into two types of solutions. One is to reduce the unbalanced power during the fault. The active power reference is reduced [7], and the output active power is increased by raising the output voltage of VSG [14] belonging to this category. The other is to delay the power angle increment during the fault and prolong the critical clearing time, but this approach can only delay the instability process and cannot fundamentally eliminate the transient instability. The inertial and damping co-adjustment method [15] and the mode switching method [16] for adjusting the damping during the fault belong to the second category.

To suppress the fault overcurrent problem accompanied by transient stability problems, a current limiting method based on the current limiter is developed, which switches the voltage-type control to the current-type control in the case of a grid fault [17]. However, the current-type control is not suitable for islanded operation, and there is a voltage overshoot during mode switching. Since the current limiter can easily cause transient power angle instability [18], a virtual impedance current limiting method without a current limiter is proposed [19]. Considering the inherent correlation between transient stability and fault overcurrent [20], the methods of stabilizing power angle and limiting overcurrent by adjusting active and reactive power commands during a fault are studied in [6,21]. However, reactive power loops of both methods need to be frozen, resulting in reactive power not being freely adjustable. A power angle stabilization and current limiting method with power feedback and virtual impedance is proposed [22], but the inrush current at the moment of fault clearing is not considered. In summary, although the above-mentioned literature solves the transient stability and fault overcurrent problems of the VSG, the frequency stability problem is ignored.

Frequency stability refers to the ability to maintain frequency stability, including frequency deviation and the rate of change of frequency (ROCOF). If frequency instability occurs, it will not only produce a large frequency deviation to threaten the transient stability of the VSG but also may produce a high ROCOF to trigger anti-islanding protection to make a large lumber of VSG disconnect from the grid, threatening the safety and stability of the grid operation. Therefore, the frequency stability of the VSG must be considered. To address the frequency stability problem of the VSG, the methods of adaptive inertia adjustment and its combined adjustment with the damping to optimize the frequency dynamic response of the VSG under load disturbance are proposed, respectively [23,24]. To adapt to multi-disturbance scenarios, a virtual impedance and adaptive inertia coordinated control method is studied to enhance the dynamic frequency response under disturbances such as power command and load switching [25]. However, these studies are confined to the frequency stability under small disturbances. Therefore, a multi-parameter cooperative adaptive strategy is investigated to enhance the frequency stability under grid faults [26]. However, the aforementioned studies are confined to frequency stability and ignore the influence of control parameter adjustment on VSG transient stability.

To properly balance the transient stability and frequency stability at the same time, the effect of inertia and damping on the transient stability and frequency stability of the VSG under fault with equilibrium points is analyzed, and a conclusion is drawn that the transient stability and frequency stability are contradictory [10], but the fault without any equilibrium point is not considered. A transient stability enhancement method of the VSG with frequency feedforward in a reactive power loop is proposed, which can enhance frequency stability at the same time [27]. However, it is doubtful whether this method is suitable for the fault without any equilibrium points. For this reason, two methods have been proposed to improve the transient stability and frequency stability of the VSG, namely only adjusting the inertia [28] and using the improved active loop to adjust the inertia and integral coefficient [29]. However, both of them ignore the change in the damping ratio, which makes it difficult to guarantee the optimal performance of the VSG. Furthermore, an improved VSG with additional power and auxiliary inertia has been proven to help enhance both transient stability and frequency stability [30], but the design method of parameters is not given. For the VSG control with an integral reactive power loop, the transient stability and frequency stability can be achieved simultaneously by jointly adjusting the inertia and the voltage regulation coefficient [13], but there is a parameter design problem as in [30]. A quantitative parameter design method considering both transient stability and frequency stability is proposed, but only the specific voltage drop is considered [31]. In general, although the above research takes into account both transient stability and frequency stability, there are still some shortcomings to be further studied.

To compare the performance of the aforementioned existing methods more clearly and intuitively, four aspects of improvement of transient stability, suppression of fault current, improvement of frequency stability, and implementation complexity are selected for comparison and are listed in

Table 1. Comparison of different VSG transient stability methods.

Methods	Improvement of Transient Stability	Suppression of Fault Current	Improvement of Frequency Stability	Implementation Complexity
Reduce active power reference [7]	Good	No considered	No considered	Low
Reduce output active power [14]	Good	No considered	No considered	Medium
Adjust inertial and damping [15]	Medium	No considered	No considered	Medium
Adjust damping [16]	Medium	No considered	No considered	Medium
Mode-switching and current limiter [17]	Good	Good	No considered	High
Virtual impedance [18,19]	No considered	Good	No considered	Medium
Reduce active and reactive power reference and virtual impedance [6,20,21,22]	Good	Good	No considered	Medium
Adjust inertial and damping [23,24]	No considered	No considered	Good	Medium
Parameter cooperative adaptive [25,26]	No considered	No considered	Good	High
Adjust inertial and improve RPCL [10,27]	Good	No considered	Medium	High
Adjust inertial and improve APCL [28,29]	Good	No considered	Medium	High
Improve APCL [30]	Good	No considered	Medium	High
Adjust inertial and voltage regulation coefficient [13]	Medium	No considered	Medium	Medium
Adjust inertial and damping [31]	Medium	No considered	Medium	Medium
The proposed method in this paper	Good	No considered	Good	Medium

. It is not difficult to find that the proposed method is a good compromise between transient stability, frequency stability, and implementation complexity, and its voltage and current double closed-loop structure provides convenience for further fault current suppression.

In response to the aforementioned problems, the main contributions of this paper are described in the following three points:

(1)

The effect of inertia and damping on VSG frequency stability and transient stability under two fault types is analyzed comprehensively, and the influence mechanism of parameter variation is deeply revealed, which fills the gap and deficiency of [10].

(2)

Compared with [27,28,29,30,31], the proposed method achieves both transient stability and frequency stability of the VSG under two fault types, which are analyzed and verified from the perspectives of a small signal and large signal.

(3)

The proposed method considers the optimal damping ratio and gives the parameter design method, which not only enhances the frequency stability and transient stability but also has better dynamic performance, which makes up for the shortcomings of [27,28,29,30,31].

The rest of this paper is arranged as follows. In Section 2, the influence mechanism of parameters on the transient performance of VSG is analyzed. In Section 3, the theoretical analysis and simulation verification of the proposed method are implemented. In Section 4, a parameter design method considering the optimal damping ratio is proposed. In Section 5, the theoretical analysis and proposed method are validated. In Section 6, important conclusions are drawn.

2. VSG Model and Influence Mechanism Analysis

2.1. Main Circuit and VSG Model

Figure 1 shows the main circuit and traditional VSG control, the VSG-controlled converter port is connected to the LC filter with inductance and capacitance of L_f and C_f, respectively, and then connected to the power grid through the transmission line with impedance of Z_g after the PCC point. u and u_g are the voltage of capacitance C_f and grid voltage, i is the output current of VSG, and U_dc is the DC side voltage. The VSG model consists of the active power control loop (APCL) and the reactive power control loop (RPCL), as shown in Formulas (1) and (2), respectively.

For the grid-connected converter topology in Figure 1, the specific topology type [32,33,34,35] can be selected according to the application scenarios requirements, such as a two-level bridge or full-bridge circuit, three-level NPC circuit, three-level T-type circuit, or even other higher-level circuits, etc. In order to facilitate theoretical analysis and simplify controller design, this paper takes the two-level three-phase bridge converter topology as an example to research.

$$E=U_0+k_{\mathrm{q}}\left(Q_{\mathrm{m}}-Q_{\mathrm{e}}\right)\left\{\begin{array}{l}{P}_{\mathrm{m}}-P_{\mathrm{e}}-D{\omega }_0(\omega -{\omega }_0)=J{\omega }_0\frac{\mathrm{d}(\omega -{\omega }_0)}{\mathrm{d}t}\\ \frac{\mathrm{d}\delta }{\mathrm{d}t}=\mathsf{\Delta }\omega =\omega -{\omega }_0\end{array}\right.$$

(1)

$$E=U_0+k_{\mathrm{q}}\left(Q_{\mathrm{m}}-Q_{\mathrm{e}}\right)$$

(2)

where P_m and Q_m are active and reactive power reference values. P_e and Q_e are the output active and reactive power actual values. ω₀, ω, δ, and Δω are rated angular frequency, angular frequency, power angle, and angular frequency deviation, respectively. E and U₀ are the output voltage and rated voltage magnitude. J and D are the inertia and the damping coefficient. k_q is the regulation coefficient.

2.2. Transient Power Angle Characteristics of VSG

Let the VSG output voltage be e = E ∠ δ, u_g = U_g ∠ 0. Considering the power decoupling, Z_g is considered to be inductive, namely, Z_g = X_g = ωL_g. P_e and Q_e are derived as

$$\left\{\begin{array}{l}{P}_{\mathrm{e}}=\frac{3E{U}_{\mathrm{g}}}{2X_{\mathrm{g}}}\sin \delta \\ Q_{\mathrm{e}}=\frac{3E^2-3E{U}_{\mathrm{g}}\cos \delta }{2X_{\mathrm{g}}}\end{array}\right.$$

(3)

Considering the influence of RPCL, the quadratic equation of E can be derived by substituting Q_e into (3), as shown in (4).

$$1.5k_{\mathrm{q}}{E}^2+\left(X_{\mathrm{g}}-1.5k_{\mathrm{q}}{U}_{\mathrm{g}}\cos \delta \right)E-\left(U_0+k_{\mathrm{q}}{Q}_{\mathrm{m}}\right)X_{\mathrm{g}}=0$$

(4)

The solution of Equation (4) can be obtained as

$$E=\frac{1.5k_{\mathrm{q}}{U}_{\mathrm{g}}\cos \delta -X_{\mathrm{g}}+\sqrt{{\left(X_{\mathrm{g}}-1.5k_{\mathrm{q}}{U}_{\mathrm{g}}\cos \delta \right)}^2-6k_{\mathrm{q}}{X}_{\mathrm{g}}\left(U_0+k_{\mathrm{q}}{Q}_{\mathrm{m}}\right)}}{3k_{\mathrm{q}}}$$

(5)

Substituting E and P_e into Formula (1), the second-order differential equation about δ when RPCL is considered is

$$J{\omega }_0\frac{{\mathrm{d}}^2\delta }{\mathrm{d}{t}^2}+D{\omega }_0\frac{\mathrm{d}\delta }{\mathrm{d}t}=P_{\mathrm{m}}-\frac{3U_{\mathrm{g}}^2}{8X_{\mathrm{g}}}\sin 2\delta +\frac{U_{\mathrm{g}}\sin \delta }{2k_{\mathrm{q}}}-\frac{U_{\mathrm{g}}\sqrt{{\left(X_{\mathrm{g}}-1.5k_{\mathrm{q}}{U}_{\mathrm{g}}\cos \delta \right)}^2-6k_{\mathrm{q}}{X}_{\mathrm{g}}\left(U_0+k_{\mathrm{q}}{Q}_{\mathrm{m}}\right)}}{2k_{\mathrm{q}}{X}_{\mathrm{g}}}\sin \delta$$

(6)

Let $\mathit{x}={\left[x_1,x_2\right]}^T={\left[\delta ,\mathsf{\Delta }\omega \right]}^T$. Since the bandwidth of APCL is usually much smaller than that of the voltage and current inner loop, ignoring the influence of the dynamic characteristics of the inner loop, the reduced-order state-space model of VSG, also known as the large-signal model, can be expressed as Formula (7).

$$\left[\stackrel{·}{\begin{array}{l}{x}_1\\ \stackrel{·}{x_2}\end{array}}\right]=\left[\begin{array}{c}{x}_2\\ \frac{P_{\mathrm{m}}}{J{\omega }_0}-\frac{3E{U}_{\mathrm{g}}}{2J{\omega }_0{X}_{\mathrm{g}}}\sin \delta -\frac{D{x}_2}{J{\omega }_0}\end{array}\right]$$

(7)

Considering the influence of RPCL, combined with the P_e of Formula (3), the power angle curve of the VSG under different fault degrees k can be drawn as shown in Figure 2, where k is the ratio of the grid fault voltage to the rated voltage. It can be seen from Figure 2 that there are two intersections between P_e and P_m when k = 0.8 and k = 0.6, while there is no intersection when k = 0.3. Points a and c are called stable equilibrium points (SEPs), and points b and d are called unstable equilibrium points (UEPs). If the system crosses the UEP during the fault period, it will lead to transient instability of the system. Therefore, the power angle of the UEP is also critical. To facilitate the analysis, the fault with SEPs during the fault is defined as fault I, and the fault without SEPs during the fault is defined as fault II. Therefore, this paper will focus on two types of faults, k = 0.6 and k = 0.3.

2.3. Transient Stability Analysis of VSG under Fault I

To observe the effect of the change of J and D on the transient stability of VSG under fault I, phase portraits of different J and D at k = 0.6 are drawn according to Equation (6), as shown in Figure 3. δ₀, δ_s, and δ_cr represent the steady-state power angle before the fault and the power angle of the SEP and UEP after the fault, respectively. Δω_max represents the maximum deviation of angular frequency, and Δδ_max represents the maximum deviation of power angle, which is also called the power angle overshoot. Note that all variables are expressed in the per-unit (p.u.) values unless this paper specifies otherwise.

Figure 3a,b show that when only J changes, the smaller J is, the smaller Δδ_max is, but the larger Δω_max and ROCOF are. When only D changes, the larger the D, the smaller the Δδ_max and Δω_max, but the ROCOF remains unchanged. Since the larger Δδ_max is closer to δ_cr, VSG is more prone to instability, and the larger Δω_max and ROCOF are more unfavorable to frequency stability. Therefore, when VSG suffers from fault I, low inertia conduces to transient stability but not to frequency stability, while high damping conduces to both transient stability and frequency stability.

2.4. Transient Stability Analysis of VSG under Fault II

To observe the effect of the change of J and D on the transient stability of VSG under fault II, phase portraits of different J and D at k = 0.3 are also drawn as shown in Figure 4. t_cr represents the critical clearing time, and its power angle δ_cr is called the critical clearing angle.

Figure 4a,b show that the VSG has transient instability under fault II, and if the fault is cleared before t_cr, it can restore stability and return to δ₀. Specifically, when only J changes, the larger J, the longer t_cr (from 0.18 s to 0.32 s), the smaller Δω_max and ROCOF; when only D changes, the larger the D is, the longer the t_cr (from 0.08 s to 0.61 s) is, and the smaller the Δω_max is, but the ROCOF remains unchanged. Because the longer t_cr is more conducive to delaying transient instability, high inertia and high damping are both beneficial to transient stability and frequency stability under fault II.

Based on the analysis of Section 2.3 and Section 2.4, high damping conduces to both transient stability and frequency stability of VSG regardless of the fault type, and high inertia conduces to the frequency stability of VSG. However, the influence of high inertia on the transient stability depends on the fault type, which proves the correctness of the conclusion that there is a contradiction between the transient stability and frequency stability of the low inertia VSG described in [10], but this conclusion is only applicable to fault I.

3. VSG Control Method with Angular Frequency Deviation Feedback and Its Performance Analysis

In Section 2, it has been proven that J and D will affect the transient stability and frequency stability of the VSG at the same time, and the active power imbalance is pointed out to be the main reason for the VSG transient instability [6]. To improve both transient stability and frequency stability of the VSG and make them consistent under two types of faults, this section proposes a VSG control method that introduces angular frequency deviation feedback in the active loop and performs performance analysis.

3.1. VSG Control Method with Angular Frequency Deviation Feedback

Figure 5 shows the VSG active loop control with angular frequency deviation feedback.

Specifically, the angular frequency deviation feedback is implemented in the APCL through the proportional integral element during the fault, where the proportional integral element is represented by a red box. The improved APCL of the VSG is expressed as

$$\begin{array}{cc}J\frac{\mathrm{d}\mathsf{\Delta }\omega }{\mathrm{d}t}& =\frac{P_{\mathrm{m}}-P_{\mathrm{e}}}{{\omega }_0}-D\mathsf{\Delta }\omega -\left(K_{\mathrm{P}}+\frac{K_{\mathrm{I}}}{s}\right)\mathsf{\Delta }\omega \\ & =\frac{P_{\mathrm{m}}-P_{\mathrm{e}}}{{\omega }_0}-\frac{K_{\mathrm{I}}}{s}\mathsf{\Delta }\omega -\left(D+K_{\mathrm{P}}\right)\mathsf{\Delta }\omega \\ & =\frac{P_{\mathrm{m}}-P_{\mathrm{e}}}{{\omega }_0}-K_{\mathrm{I}}\mathsf{\Delta }\delta -\left(D+K_{\mathrm{P}}\right)\mathsf{\Delta }\omega \\ & =\frac{P_{\mathrm{m}}-K_{\mathrm{I}}{\omega }_0\mathsf{\Delta }\delta }{{\omega }_0}-\frac{P_{\mathrm{e}}}{{\omega }_0}-\left(D+K_{\mathrm{P}}\right)\mathsf{\Delta }\omega \end{array}$$

(8)

where K_P and K_I are the proportional and integral coefficients, respectively. Combined with Formula (8), it can be seen that the role of K_P is the same as that of D, which increases the system damping and strengthens the anti-interference ability. Because the transient power angle will increase during the fault, according to Formula (8), the increased power angle deviation Δω is introduced into APCL as a feedback term through K_I, which can be regarded as reducing the equivalent active reference P_m − K_Iω₀Δδ, thereby reducing the active power imbalance and enhancing the transient stability. It is completely consistent with the method proposed in [6,20,21] to reduce the active command during the fault to enhance VSG transient stability.

3.2. Performance Analysis of Proposed Method Based on Small-Signal Model

Under large-signal disturbances such as grid faults, the VSG is generally difficult to stabilize near the original steady-state equilibrium point. It is inaccurate to evaluate transient stability by the small-signal stability analysis method, but it can provide a clear physical perspective and intuitive explanation for understanding stability. If the VSG is stable at a certain point of large signal, it will eventually stabilize near this point, that is, the large-signal and small-signal model system have the same response trend. For example, a high damping ratio will shrink the overshoot of linear systems, which is also applicable to nonlinear systems. Therefore, the small signal linearization model can be applied to qualitative analysis during the faults. In this section, the qualitative analysis of the proposed method will be carried out with the help of the small-signal model method.

3.2.1. Frequency Stability Analysis of Proposed Method

Let $f\left(\delta \right)=E\cdot \sin \delta$. The linearized model of $f\left(\delta \right)$ around the SEP can be obtained.

$$\mathsf{\Delta }{P}_{\mathrm{e}}=\frac{3U_{\mathrm{g}}}{2X_{\mathrm{g}}}{f}^{\prime }\left(\delta \right){|}_{\delta ={\delta }_s}\cdot \mathsf{\Delta }\delta =G\cdot \mathsf{\Delta }\delta$$

(9)

where Δ represents the small disturbance, δ_s is the power angle of SEP after fault, and G is the approximate gain between P_e and δ.

Substitute (5) into $f\left(\delta \right)$, then

$$f^{\prime }\left(\delta \right)=E_{{\delta }_s}\cos {\delta }_s-\frac{3E_{{\delta }_s}{k}_{\mathrm{q}}{U}_{\mathrm{g}}\sin {\delta }_s}{2X_{\mathrm{g}}+3k_{\mathrm{q}}\left(2E_{{\delta }_s}-U_{\mathrm{g}}\cos {\delta }_s\right)}$$

(10)

where E_s represents the VSG output voltage after the fault. $f^{\prime }\left(\delta \right)$ can be derived by calculation, and G can be obtained from Formulas (9) and (10).

$$G=\frac{3U_{\mathrm{g}}}{2X_{\mathrm{g}}}\left[E_{{\delta }_s}\cos {\delta }_s-\frac{3E_{{\delta }_s}{k}_{\mathrm{q}}{U}_{\mathrm{g}}\sin {\delta }_s}{2X_{\mathrm{g}}+3k_{\mathrm{q}}\left(2E_{{\delta }_s}-U_{\mathrm{g}}\cos {\delta }_s\right)}\right]$$

(11)

According to Formula (11) and Figure 5, Figure 6 shows the small-signal model of the proposed method.

Ignoring the grid frequency disturbance in Figure 6, that is, Δω_g = 0, the small-signal transfer function between Δδ and Δω can be obtained

$$\left\{\begin{array}{l}\frac{\mathsf{\Delta }\delta }{\mathsf{\Delta }{P}_{\mathrm{m}}}=\frac{1}{J{\omega }_0{s}^2+\left(D+K_{\mathrm{P}}\right){\omega }_{0}s+K_{\mathrm{I}}{\omega }_0+G}\\ \frac{\mathsf{\Delta }\omega }{\mathsf{\Delta }{P}_{\mathrm{m}}}=\frac{s}{J{\omega }_0{s}^2+\left(D+K_{\mathrm{P}}\right){\omega }_{0}s+K_{\mathrm{I}}{\omega }_0+G}\end{array}\right.$$

(12)

When ΔP_m unit step changes, Δω(s) can be obtained

$$\begin{array}{c}\mathsf{\Delta }\omega \left(s\right)=\frac{1}{J{\omega }_0{s}^2+\left(D+K_{\mathrm{P}}\right){\omega }_{0}s+K_{\mathrm{I}}{\omega }_0+G}\\ =\frac{\frac{1}{J{\omega }_0}}{s^2+\frac{D+K_{\mathrm{P}}}{J}s+\frac{K_{\mathrm{I}}{\omega }_0+G}{J{\omega }_0}}\end{array}$$

(13)

Comparing Formula (13) with the standard second-order system transfer function, it can be obtained that

$$\mathsf{\Delta }\omega \left(s\right)=\frac{{\omega }_{\mathrm{n}}^2/\left(K_{\mathrm{I}}{\omega }_0+G\right)}{s^2+2\zeta {\omega }_{\mathrm{n}}s+{\omega }_{\mathrm{n}}^2}$$

(14)

Comparing (13) and (14), the natural oscillation frequency ω_n and damping ratio ζ are

$$\left\{\begin{array}{l}{\omega }_{\mathrm{n}}=\sqrt{\frac{K_{\mathrm{I}}{\omega }_0+G}{J{\omega }_0}}\\ \zeta =0.5\left(D+K_{\mathrm{P}}\right)\sqrt{\frac{{\omega }_0}{\left(K_{\mathrm{I}}{\omega }_0+G\right)J}}\end{array}\right.$$

(15)

In general, VSG control operates in an under-damped state, i.e., ζ < 1, and the time-domain response of Δω(s) can be obtained from Formulas (14) and (15) as follows:

$$\begin{array}{cc}\mathsf{\Delta }\omega \left(t\right)& =\frac{{\omega }_{\mathrm{n}}}{\left(K_{\mathrm{I}}{\omega }_0+G\right)\sqrt{1-{\zeta }^2}}\cdot e^{-\zeta {\omega }_{\mathrm{n}}t}\sin \left({\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}t\right)\\ & =\frac{2\zeta }{\left(D+K_{\mathrm{P}}\right){\omega }_0}\cdot \frac{1}{\sqrt{1-{\zeta }^2}}\cdot e^{-\zeta {\omega }_{\mathrm{n}}t}\sin \left({\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}t\right)\end{array}$$

(16)

By deriving Formula (16), ROCOF can be derived as follows:

$$\begin{array}{cc}\mathrm{ROCOF}& =\frac{\mathrm{d}\mathsf{\Delta }\omega \left(t\right)}{\mathrm{d}t}\\ & =\frac{{\omega }_{\mathrm{n}}^2}{\left(K_{\mathrm{I}}{\omega }_0+G\right)\sqrt{1-{\zeta }^2}}{e}^{-\zeta {\omega }_{\mathrm{n}}t}\sin \left(\theta -{\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}t\right)\\ & =\frac{1}{J{\omega }_0}\cdot \frac{1}{\sqrt{1-{\zeta }^2}}{e}^{-\zeta {\omega }_{\mathrm{n}}t}\sin \left(\theta -{\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}t\right)\end{array}$$

(17)

where θ in Formula (17) can be calculated by $\sin \theta =\sqrt{1-{\zeta }^2}$.

According to Formula (17), when t = 0, ROCOF takes the maximum value, then

$${\mathrm{ROCOF}|}_{\max }=\frac{1}{J{\omega }_0}$$

(18)

When the derivative of Δω(t) is equal to 0, that is, ROCOF = 0, Δω(t) obtains the maximum value, and $t=\frac{\arcsin \sqrt{1-{\zeta }^2}}{{\omega }_{\mathrm{n}}\sqrt{1-{\zeta }^2}}$ can be obtained. Substituting t into Δω(t), Δω_max can be obtained as follows:

$$\begin{array}{cc}\mathsf{\Delta }{\omega }_{\max }& =\frac{2\zeta }{\left(D+K_{\mathrm{P}}\right){\omega }_0}\cdot \frac{1}{\sqrt{1-{\zeta }^2}}{e}^{-\frac{\varsigma \arcsin \sqrt{1-{\zeta }^2}}{\sqrt{1-{\zeta }^2}}}\sin \theta \\ & =\sqrt{\frac{1}{\left(K_{\mathrm{I}}{\omega }_0+G\right)J{\omega }_0}}{e}^{-\zeta }\\ & \approx \frac{2\zeta }{\left(D+K_{\mathrm{P}}\right){\omega }_0}{e}^{-\zeta }\end{array}$$

(19)

According to Formulas (15) and (19), the smaller the damping ratio ζ is, the smaller the Δω_max is, that is, reducing ζ helps to reduce Δω_max. However, the larger the K_I is, the smaller the ζ is. Therefore, increasing K_I helps to reduce Δω_max.

According to the Formulas (15), (18), and (19), if K_p = 0 and K_I = 0, the proposed control becomes the traditional VSG control, then

$$\left\{\begin{array}{l}{\mathrm{ROCOF}|}_{\max }=\frac{1}{J{\omega }_0}\\ \mathsf{\Delta }{\omega }_{\max }\approx \frac{2\zeta }{D{\omega }_0}{e}^{-\zeta }\\ \zeta =0.5D\sqrt{\frac{{\omega }_0}{GJ}}\end{array}\right.$$

(20)

From Formula (20), it is not difficult to see that Δω_max and ROCOF_max are only affected by J and D in APCL. When D increases, Δω_max decreases; when J is increased, both Δω_max and ROCOF_max can be kept small. Increasing J improves the frequency stability of the VSG but deteriorates the transient stability under fault I, and the increase in power angle cannot be contained during the fault. Compared to the traditional VSG control, Δω_max and ROCOF_max of the proposed control are not only related to J and D but also related to K_p and K_I. Increasing K_p and K_I can reduce Δω_max. In addition, the analysis of Formula (8) shows that K_I can also automatically adjust the equivalent active power reference value P_eq to effectively suppress the increase in the transient power angle. In summary, by adjusting K_p and K_I, the VSG with larger J can have stronger transient stability and frequency stability under two types of faults, which will be further verified and analyzed in the next section.

3.2.2. Transient Stability Analysis of Proposed Method

According to Formula (12), when ΔP_m unit step changes, Δδ(s) can be obtained.

$$\mathsf{\Delta }\delta \left(s\right)=\frac{\frac{1}{J{\omega }_0}}{s\left(s^2+\frac{D+K_{\mathrm{P}}}{J}s+\frac{K_{\mathrm{I}}{\omega }_0+G}{J{\omega }_0}\right)}$$

(21)

By implementing the Laplace inverse transformation, the time-domain response of expression of (21) is

$$\mathsf{\Delta }\delta \left(t\right)=\frac{1}{K_{\mathrm{I}}{\omega }_0+G}-\frac{1}{K_{\mathrm{I}}{\omega }_0+G}\frac{1}{\sqrt{1-{\zeta }^2}}{e}^{-\zeta {\omega }_{n}t}\sin \left(\sqrt{1-{\zeta }^2}{\omega }_{n}t+\theta \right)$$

(22)

Further, the derivative of Δδ(t) can be obtained.

$$\frac{\mathrm{d}\mathsf{\Delta }\delta }{\mathrm{d}t}=\frac{1}{K_{\mathrm{I}}{\omega }_0+G}\frac{{\omega }_{\mathrm{n}}{e}^{-\zeta {\omega }_{\mathrm{n}}t}}{\sqrt{1-{\zeta }^2}}\sin \left(\sqrt{1-{\zeta }^2}{\omega }_{\mathrm{n}}t\right)$$

(23)

When the derivative of Δδ(t) is equal to 0, Δδ(t) takes the maximum value Δδ_max. Combined with the swing equation of the VSG, $\sqrt{1-{\zeta }^2}{\omega }_{\mathrm{n}}t=\pi$ can be obtained. Substituting t into (22), then

$$\mathsf{\Delta }{\delta }_{\max }=\frac{1}{K_{\mathrm{I}}{\omega }_0+G}\left(1+e^{-\frac{\pi \zeta }{\sqrt{1-{\zeta }^2}}}\right)$$

(24)

It can be seen from (24) that Δδ_max is related to the two quantities of K_I and ζ. Substituting (15) into (24), then

$$\mathsf{\Delta }{\delta }_{\max }=\frac{4J{\zeta }^2}{{\omega }_0{\left(D+K_{\mathrm{P}}\right)}^2}\left(1+e^{-\frac{\pi \zeta }{\sqrt{1-{\zeta }^2}}}\right)$$

(25)

To study the monotonicity of Δδ_max when ζ < 1, that is, the VSG is under-damped, the derivation of (25) can be obtained.

$$\frac{\mathrm{d}\mathsf{\Delta }{\delta }_{\max }}{\mathrm{d}\zeta }=\frac{8J\zeta }{{\omega }_0{\left(D+K_{\mathrm{P}}\right)}^2}\left[1+\left(1-\frac{0.5\pi \zeta }{\sqrt{1-{\zeta }^2}}\frac{1}{1-{\zeta }^2}\right)e^{-\frac{\pi \zeta }{\sqrt{1-{\zeta }^2}}}\right]>0$$

(26)

When K_p = 0 and K_I = 0, the proposed control becomes the traditional VSG control, then

$$\mathsf{\Delta }{\delta }_{\max }=\frac{1}{G}\left(1+e^{-\frac{\pi \zeta }{\sqrt{1-{\zeta }^2}}}\right)$$

(27)

Further, the derivative of (27) can be obtained.

$$\frac{\mathrm{d}\mathsf{\Delta }{\delta }_{\max }}{\mathrm{d}\zeta }=-\frac{1}{G}\cdot \frac{\pi }{\left(1-{\zeta }^2\right)\sqrt{1-{\zeta }^2}}{e}^{-\frac{\pi \zeta }{\sqrt{1-{\zeta }^2}}}<0$$

(28)

According to Formulas (26) and (28), Δδ_max increases monotonically with the increase in ζ for the proposed control, while it is monotonically decreasing for the traditional VSG control. In other words, the smaller the ζ, the stronger the transient stability of the proposed control and the worse the traditional VSG control. Since increasing K_I reduces ζ, increasing K_I can improve both frequency stability and transient stability of VSG.

3.3. Performance Analysis of Proposed Method Based on Large-Signal Model

Substituting (8) into (7), the large-signal model of the proposed control is built:

$$\left[\stackrel{·}{\begin{array}{l}{x}_1\\ \stackrel{·}{x_2}\end{array}}\right]=\left[\begin{array}{c}{x}_2\\ \frac{P_{\mathrm{m}}}{J{\omega }_0}-\frac{3E{U}_{\mathrm{g}}}{2J{\omega }_0{X}_{\mathrm{g}}}\sin \delta -\frac{K_{\mathrm{I}}{x}_1}{J{\omega }_0}-\frac{\left(D+K_{\mathrm{P}}\right)x_2}{J{\omega }_0}\end{array}\right]$$

(29)

To intuitively analyze the effect of the increase in K_I on the transient stability of the VSG, the power angle curve of the proposed control can be drawn according to (3) and (29) as shown in Figure 7. Since K_p and D play the same role in formula (8), K_p = 0 is assumed and only the change in K_I is considered. In Figure 7, the black dotted line represents the power angle curve before the fault, and the solid line represents the power angle curve during fault I. It can be seen that when K_I increases from 0 to 15.78 p.u. along the arrow direction, the maximum output active power P_e increases, and the equivalent active power command P_eq and the transient stability power angle δ_s decrease along the respective arrow direction. Therefore, the transient stability of the proposed VSG is enhanced, which agrees well with the conclusions in Section 3.1 and Section 3.2.

To further observe the transient performance of the proposed VSG control and validate the aforementioned small-signal model and the power angle characteristic analysis, according to the large-signal model shown in (29), the transient responses of the traditional VSG control and the proposed VSG control under two types of faults are plotted utilizing the phase portrait method, as shown in Figure 8, Figure 9, Figure 10 and Figure 11.

Figure 8 and Figure 9 show that increasing J reduces ROCOF_max, and increasing D and K_I reduces Δω_max. In addition, it can be found that when J is the same, the frequency response of the proposed VSG is faster, the transient stability power angle δ_s is smaller, and Δδ_max is smaller, that is, the power angle overshoot is smaller, so the transient stability is stronger. In addition, with the increase in K_I, δ_s decreases but intensifies the oscillation of Δω, which is due to the decrease in ζ, while the δ_s of the traditional VSG remains unchanged and is not affected by the change in D. Therefore, the simulation results indicate that the proposed control can simultaneously enhance the frequency stability and transient stability of the VSG with high inertia under fault I, which fully verifies the conclusions in Section 3.1 and Section 3.2.

Figure 10 and Figure 11 indicate that increasing J reduces ROCOF_max, and increasing D and K_I reduces Δω_max, which is consistent with the parameter influence mechanism under fault I. In addition, it can also be found that when J is the same, the frequency response of the proposed control is faster, and the transient stability is significantly enhanced, that is, from transient instability to transient stability. In addition, as K_I increases, the transient power angle δ_s is smaller, Δδ_max is smaller, that is, the power angle overshoot is smaller, but it also aggravates the oscillation of Δω. However, the traditional VSG always maintains the transient instability state. Only with the increase in D, the critical switching time t_cr is greatly prolonged, which implies that the transient stability is improved equivalently, but the transient instability is not fundamentally eliminated, and the frequency response is greatly reduced. Therefore, the simulation results indicate that the proposed control can improve both frequency stability and transient stability of the VSG with high inertia under fault II, which fully verifies the conclusions in Section 3.1 and Section 3.2.

4. Parameter Design Method of VSG Control with Angular Frequency Deviation Feedback

The above analysis has proven the effectiveness of the proposed method. Increasing D conduces to improve transient stability and frequency stability, but too large D will reduce the frequency response, and the influence of K_p is the same as that of D. Similarly, increasing K_I is beneficial to enhance transient stability, but too large K_I will reduce VSG output power and aggravate frequency oscillation. Therefore, how to comprehensively weigh and design appropriate K_p and K_I to make the VSG obtain excellent performance is very important.

The analysis in Section 3.3 indicates that once the transient power angle δ of the VSG crosses the power angle δ_cr of the UEP, the VSG will undergo transient instability, that is, as long as δ > δ_cr, the VSG will inevitably undergo transient instability. To facilitate calculation, π is usually used instead of δ_cr. Therefore, it is necessary to calculate the minimum K_I of VSG transient critical stability, namely K_Imin, to ensure that the VSG does not undergo transient instability under two types of faults. However, although the adjustable K_I makes K_I > K_Imin to maintain transient stability, the increase in K_I will reduce the damping ratio and deteriorate the dynamic response. Therefore, a parameter design method considering the optimal damping ratio is proposed to comprehensively adjust K_P and K_I to obtain the optimal dynamic response of the VSG.

The second-order differential equation of the proposed VSG for δ is obtained by combining Formulas (6) and (8).

$$\begin{array}{cc}J{\omega }_0\frac{{\mathrm{d}}^2\delta }{\mathrm{d}{t}^2}+\left(D+K_{\mathrm{P}}\right){\omega }_0\frac{\mathrm{d}\delta }{\mathrm{d}t}& =P_{\mathrm{m}}-K_{\mathrm{I}}\left(\delta -{\delta }_0\right)-\frac{3U_{\mathrm{g}}^2}{8X_{\mathrm{g}}}\sin 2\delta +\frac{U_{\mathrm{g}}\sin \delta }{2k_{\mathrm{q}}}\\ & -\frac{U_{\mathrm{g}}\sqrt{{\left(X_{\mathrm{g}}-1.5k_{\mathrm{q}}{U}_{\mathrm{g}}\cos \delta \right)}^2-6k_{\mathrm{q}}{X}_{\mathrm{g}}\left(U_0+k_{\mathrm{q}}{Q}_{\mathrm{m}}\right)}}{2k_{\mathrm{q}}{X}_{\mathrm{g}}}\sin \delta \end{array}$$

(30)

According to Formula (30), δ is solved, and K_Imin and δ_s at the critical stability of the VSG under different k are obtained by iterative calculation. Then, the deviation between δ_s and δ₀ is compared, and the K_I satisfying the allowable deviation range is obtained by iterative calculation so that ζ can be calculated by Formula (15). Finally, it is judged whether ζ is equal to the optimal damping ratio, namely, 0.707. Considering the calculation error, the allowable error range of ζ can be given, and K_P and K_I can be obtained by iterative calculation. The specific iterative calculation process is shown in Figure 12.

It is noteworthy that the smaller the δ_s is, the larger the transient stability margin is, and the smaller the fault current is. However, the smaller δ_s requires a larger K_I, which will aggravate the frequency oscillation. Therefore, through comprehensive trade-offs and simulation tests, this paper assumes that the deviation between δ_s and δ₀ is 0.2 rad. In practical engineering, the iterative operation in Figure 12 puts forward higher requirements on the computing ability of the controller including the efficiency, accuracy, and calculation speed of the algorithm. Even if the computing ability is strong enough, it is necessary to consider the influence of the sampling delay, control delay, and changes in system parameters and operating conditions. Therefore, to ensure the control effect of the system while greatly reducing the system construction cost and improving the operating economy, it is suggested to use off-line calculation, that is, according to the proposed parameter design method in this paper, calculate the K_p and K_I under different k, and make them into a table. In practical applications, the lookup table method can be easily used to obtain and greatly reduce the calculation amount.

According to Figure 12, the critical stable boundary of the proposed VSG under different k at D = 24.65 p.u. is drawn as shown in Figure 13, where the red line represents the boundary curve formed by each K_Imin at J = 0.25 p.u., and the blue line denotes the boundary curve formed by each K_Imin at J = 9.86 p.u. The region with K_I > K_Imin is the stable region, and the region with K_I < K_Imin is the unstable region. In practice, due to the high requirement of the controller’s operation ability for online operation, the optimal parameter table corresponding to different fault k can be made according to the critical stability boundary, and the real-time requirement can be satisfied only by looking up the table.

5. Experimental Verification

Based on the RT-LAB hardware-in-the-loop platform shown in Figure 14, further verification is implemented. The model of the main circuit shown in Figure 1 is established and implemented in RT-LAB. The RT-LAB is connected to a TMS320F28335 DSP controller through an interface card. When the proposed method is successfully programmed and run in the DSP, the generated PWM signals are sent to the RT-LAB model for implementation and the corresponding waveforms are output to the oscilloscope. The transient performance of the traditional and the proposed VSG control under two types of faults k = 0.6 and k = 0.3 is simulated, respectively. The sampling time T_s = $1\times {10}^{-5}$ and the experimental parameters are listed in

Table 2. Experimental parameters.

Parameter	Value	p.u.
Pm	20 kW	1.0
Qm	0	0
Udc	800 V	-
Lf	3 mH	0.13
Cf	40 μF	0.09
Lg	11.5 mH	0.50
U0	311 V	1.0
UgN	311 V	1.0
ω0	314 rad·s−1	-
J	0.05 kg/m2	0.25
D	5 N.m.s/rad	24.65
kq	$1\times {10}^{-3}$ V/kvar	0.064

.

5.1. Transient Performance of Proposed Method under Fault I

To validate the proposed method under fault I, the effects of changes in J and K_I on the transient performance of the VSG control are compared and analyzed in this section.

5.1.1. Influence of Variable J on Transient Performance of VSG Control

Assuming that D = 24.65 p.u., K_p = 0, K_I = 4.93 p.u., J takes four values, namely 0.25 p.u., 0.5 p.u., 4.93 p.u., and 9.86 p.u., respectively. The transient performance of the VSG control is shown in Figure 15. It can be found that the VSG is transiently stable under fault I. With the increase in J, ζ decreases, which leads to the decrease in ROCOF and Δω_max, so δ_max decreases from 0.99 rad to 0.91 rad. The experimental results indicate that when the proposed method is adopted, increasing J can enhance the VSG frequency stability and transient stability at the same time.

5.1.2. Influence of Variable K_I on Transient Performance of VSG Control

To analyze the influence of variable K_I on the transient performance of the VSG control and verify the control effect of the proposed method considering the optimal damping ratio in Case 4, different parameter settings are shown in

Table 3. Different parameter settings under fault I.

Scheme	J	D	Kp	KI
Case 1	0.25	24.65	0	0
Case 2	9.86	24.65	0	9.86
Case 3	9.86	24.65	0	19.72
Case 4	9.86	24.65	97.2	9.86

. In addition, to obtain a better frequency response, the J of the last three cases is increased from 0.25 p.u. to 9.86 p.u. Figure 16 shows all experimental results.

Comparing Figure 16a with Figure 16b–d, it can be seen that ROCOF and Δω_max decrease significantly with the increase in J, which indicates that increasing J improves the frequency stability of the VSG. Comparing Figure 16b–d with Figure 16d, it can be seen that when J is the same, Δω_max and δ_max decrease with the increase in K_I, which indicates that increasing K_I can improve both frequency stability and transient stability of the VSG, which agrees well with the above analysis.

However, although Figure 16b,c obtains better transient performance, the increase in J and K_I reduces ζ. The excessive K_I of the high-inertia VSG not only aggravates the decrease in ζ but also reduces P_e and deteriorates the dynamic performance of the VSG. Compared with Figure 16b, Figure 16d is reduced by 40% when the optimal damping ratio is adopted, which improves the frequency stability. Compared with Figure 16c, although the δ_max of Figure 16d increases slightly, the power waveform is more stable, and the dynamic performance of the VSG is better. The experimental results indicate that the proposed method considering the optimal damping ratio can improve both transient stability and frequency stability of the VSG with high inertia under fault I.

5.2. Transient Performance of Proposed Method under Fault II

To demonstrate the proposed method under fault II, the effects of changes in J and K_I on the transient performance of the VSG control are also compared and analyzed.

5.2.1. Influence of Variable J on Transient Performance of VSG Control

The parameter setting is the same as in Section 5.1.1, and the transient performance of the VSG control is shown in Figure 17. It can be found that the VSG is transiently stable under fault II due to the existence of K_I. With the increase in J, ζ decreases, resulting in the decrease in ROCOF, Δω_max, and δ_max, which agrees well with the results in Section 5.1.1. The results again indicate that when the proposed method is adopted, increasing J can improve both the frequency stability and transient stability of the VSG.

5.2.2. Influence of Variable K_I on Transient Performance of VSG Control

To analyze the influence of variable K_I on the transient performance of the VSG control, different parameter settings are shown in

Table 4. Different parameter settings under fault II.

Scheme	J	D	Kp	KI
Case 1	0.25	24.65	0	0
Case 2	0.25	24.65	0	4.93
Case 3	0.25	24.65	0	9.86
Case 4	0.25	24.65	0	19.72
Case 5	9.86	24.65	0	4.93
Case 6	9.86	24.65	0	9.86
Case 7	9.86	24.65	0	19.72
Case 8	9.86	24.65	123	19.72

. The J is set to 0.25 p.u. in Case 1–Case 4 and 9.86 p.u. in Case 5–Case 8 to observe the influence of K_I change on the transient performance under low inertia and high inertia, respectively. Case 8 is the proposed VSG control method considering the optimal damping ratio. The experimental results of Case1–Case 4 and Case 5–Case 8 are shown in Figure 18 and Figure 19, respectively.

Figure 18a shows that when the traditional VSG control shown in Case 1 is used, transient instability will occur under class II faults. Due to the effect of K_I, the VSG will change from transient instability to transient stability in Figure 18b–d. It can also be found that P_e is less than P_m during the fault, which indicates that K_I can automatically reduce P_eq, which is completely consistent with the analysis of Figure 7. As shown in Figure 18b–d, with the increase in K_I, δ_max decreases from 2.13 rad to 0.62 rad, which indicates that the smaller the power angle overshoot Δδ_max is, the stronger the transient stability is. In addition, Δω_max decreases from 7.0 rad·s⁻¹ to 5.4 rad·s⁻¹, which indicates that the smaller the Δω_max, the better the frequency stability, but the ROCOF is large due to the small J. The results show that increasing K_I can improve both the transient stability and frequency stability of the VSG with low inertia.

To further reduce the ROCOF of Case1–Case4 and enhance the frequency stability, Figure 19 shows the experimental results of the VSG with high inertia. Compared with Figure 18, Figure 19a–c show that ROCOF decreases significantly with the increase in J, and with the increase in K_I, Δω_max decreases from 1.6 rad·s⁻¹ to 1.2 rad·s⁻¹, and δ_max decreases from 2.03 rad to 0.62 rad, which indicates that increasing K_I can simultaneously enhance the transient stability and frequency stability of the VSG with high inertia.

However, although Figure 19a–c obtains better transient performance, the increase in J and K_I leads to the decrease in ζ, and the excessive K_I of the high-inertia VSG aggravates the decrease in ζ. To obtain the optimal dynamic performance, Figure 19d gives the VSG control transient performance waveform considering the optimal damping ratio, which is reduced by 50% compared with Figure 19c, and the frequency stability is further improved. The results show that the proposed method considering the optimal damping ratio can improve both transient stability and frequency stability of the VSG with high inertia under fault II.

In summary, regardless of the inertia of the VSG, the proposed method considering the optimal damping ratio can improve both the transient stability and frequency stability of the VSG under two types of faults.

6. Conclusions

Aiming at the issues in existing research that it is hard to balance the transient stability and frequency stability of the VSG during grid faults, and there is a lack of systematic analysis of the effect of parameter changes on the transient stability under two types of faults, this paper proposes a method to improve both the frequency stability and transient stability of the VSG in grid faults. The proposed method and its parameter design method considering optimal damping are verified by theoretical analysis and experiments. The following conclusions are obtained:

(1)

For traditional VSGs, high inertia is beneficial to the frequency stability, but the influence on the transient stability depends on the fault type, that is, high inertia deteriorates the transient stability under fault I, while it conduces to the transient stability under fault II.

(2)

Regardless of the type of fault, high damping is beneficial to both the transient stability and frequency stability of VSGs.

(3)

Regardless of the inertia of VSGs, the proposed method considering the optimal damping ratio can improve both the transient stability and frequency stability of VSGs under two types of faults.