Virtual synchronous generators (VSGs) have considerable potential to be applied in renewable energy microgrid systems because they provide inertia and damping, thus improving system stability. However, the fault characteristics of renewable energy microgrids differ from those of conventional grids. Consequently, investigating the fault responses of VSGs and establishing the associated fault model would significantly enhance the fault calculation and protection mechanism design in renewable energy microgrids. This paper briefly introduces the VSG control principles and fault control strategy before establishing the corresponding transient- and steady-state VSG fault models and calculating the model parameters. For simplicity, the fault model is linearized at the steady-state estimation point and semi-decoupled for the computation of the voltage amplitude and power angle. Finally, the results from MATLAB/Simulink simulations are compared with those from the proposed fault models, and the influence of the system parameters on the model is discussed. The results demonstrate the validity, simplicity, intuitiveness, and broad applicability of our approach.

The rapid development of renewable energy sources and their control techniques has increased the penetration rate of inverter-interfaced distributed generators (IIDGs), including renewable energy sources and inverters in modern power systems. Grid-feeding control techniques, such as the traditional PQ control and constant current control, reduce the inertia and damping of the grid, potentially affecting its normal operation.1,2 By emulating the swing and excitation characteristics of conventional synchronous generators (SGs), virtual synchronous generators (VSGs) can be used to enhance the dynamic characteristics and stability of the system.3,4 However, the insufficient over-current capability of power electronic devices (typically only 1.2–2 × their rated current) limits the ability of VSGs to supply large-scale fault currents, thus potentially impacting the performance of protection devices. Hence, it is essential to examine VSG fault control strategies and establish the associated analytical fault models, as these will aid in the analysis of renewable energy microgrid stability and the design of protection schemes.

Existing research on VSGs has primarily centered on two aspects, i.e., steady-state stability analysis and large-perturbation stability analysis. The steady-state stability analysis focuses on disturbances caused by the load during normal operation,5–7 whereas the large-perturbation stability analysis focuses on large disturbances such as fault occurrence, also known as fault stability analysis. Fault stability is primarily analyzed in terms of power angle stability and current limitation. In terms of power angle stability, despite the presence of various active power-frequency fault control strategies, the fundamental objective of inductive grids is to stabilize the power angle within a certain range by compensating the active power reference value during a fault occurrence.8–12 Current limiting is mainly executed by controlling the output voltage, involving three primary methods of voltage compensation. The first method calculates the output voltage vector from the current limitation,8 while the second approach uses virtual impedance to adjust the voltage reference value.9–12 The third method involves changing the structure of the droop loop with the reactive current–voltage droop method,13 ensuring that the current amplitude remains within a specified range. This not only confines the fault current but also enables precise control of its reactive component, providing enhanced support for the grid voltage during grid faults.

There have been many studies on the fault modeling of grid-forming IIDGs. References 14 and 15 considered the transient- and steady-state fault modeling of constant voltage and frequency-controlled IIDGs. These studies extensively analyzed the fault response under various scenarios, considering the transfer functions of the controller, current limiter, and DC voltage fluctuations. Reference 16 modeled IIDGs under four distinct control strategies, namely constant current control, PQ control, V/f control, and droop control, and compared their fault responses. Reference 8 modeled a VSG with a direct voltage control strategy under symmetric fault conditions. The power angle was stabilized by reducing the active power reference, and the fault current was suppressed using the virtual impedance, with the calculated current phasors employed to adjust the output voltage. Reference 17 used an iterative algorithm to analyze the fault current of a VSG during asymmetric faults but did not consider the voltage–current inner loop and fault control strategy. Although several high-quality studies have proposed models for grid-forming IIDGs, they have mainly focused on V/f and droop-controlled generators. No viable analytical model for VSGs has yet been proposed besides iterative methods.16–18 As a result, there is an urgent need for a feasible analytical fault model for VSGs.

To fill this research gap, the present paper establishes an intuitive analytical model of a VSG under various fault conditions. As the fault responses of the VSG depend on its control strategy, different fault control strategies are first summarized. A simple and effective fault control strategy is then adopted to realize the decoupling of the VSG’s positive-sequence (PS), negative-sequence (NS), and zero-sequence (ZS) outputs. The sequence component models of the VSG are established, and the relevant parameters are calculated. To facilitate the computation process without introducing any significant transient- or steady-state errors, the fault model of the VSG is linearized at the equilibrium point, and the voltage amplitude and power angle are assumed to be semi-decoupled to aid their computation. Finally, the effectiveness of the proposed fault model is verified through MATLAB/Simulink simulations.

The primary contributions of this work can be summarized as follows:

  1. A straightforward and effective simplified model of a VSG under different fault conditions is proposed.

  2. A detailed method for calculating every parameter of the proposed VSG equivalent model is provided.

  3. The influence of the VSG system parameters on the proposed fault model is analyzed.

The remainder of this paper is organized as follows: Section II provides a concise introduction to the fundamental control principles of VSGs and outlines the fault control strategy adopted herein. Section III focuses on modeling the VSG under various fault conditions and presents the calculation method for the model parameters. In Sec. IV, the control strategy and the fault model are validated. Section V analyzes the influence of the system parameters on the proposed model from a modeling perspective. Finally, Sec. VI concludes the paper.

The typical circuit and controller structure of a conventional VSG is illustrated in Fig. 1.12,13

FIG. 1.

Typical circuit and controller structure of VSG.

FIG. 1.

Typical circuit and controller structure of VSG.

Close modal

In Fig. 1, Udc is the voltage of the DC bus, Lf, Rf, and Cf represent the inductance, resistance, and capacitance of the filter, respectively, IL is the filter inductor current, Ug is the voltage at the point of common coupling (PCC), Io and Vo are the VSG’s current and voltage output, respectively, Zg = Rg + jXg is the line impedance between the VSG and PCC, and Um+ and Um are the PS and NS components of the modulation waveform, respectively.

The voltage and current controllers typically use proportional-integral regulators19 to regulate the VSG’s output voltage to its reference value. As the bandwidth of the voltage loop is typically designed to be about 1/10 of that of the current loop,19,20 the time constant of the current loop can be neglected. Hence, the transfer function of the inner loop can be simplified to
Vo=Gv(s)Vref,Gv(s)=kpvs+kivCfs2+kpvs+kiv,
(1)
where kpv and kiv are the proportional and integral coefficients of the voltage controller.
As shown in Fig. 1, when switches S1 and S2 are both closed at position 1, the VSG operates in normal mode. The corresponding transfer functions can be written as
Jdωdt=PrefPeD(ωω0),θ=ωdt,Vref=E0+kQ(Q0Q),Pe=pτfs+1,Q=qτfs+1,
(2)
where J is the virtual inertia, D is the damping coefficient, E0 is the nominal voltage amplitude, ω and ωg are the angular frequencies of the VSG and the grid, respectively, θ is the phase angle of the VSG’s output voltage, kQ is the reactive power-voltage droop coefficient, Q is the reactive power output, Q0 is the rated reactive power, τf is the time constant of the low-pass filter, Pe is the active power output, p and q are the instantaneous values of active and reactive power output, respectively, and Pref is the active power reference given by active power-frequency droop control,
Pref=P0+kω(ω0ω),
(3)
where kω is the droop coefficient of the active power-frequency loop, P0 is the rated active power, and ω0 is the rated angular frequency of the VSG.
The power angle δ represents the phase difference between the grid voltage and the terminal voltage of the VSG. By combining Eqs. (2) and (3), the power angle δ can be expressed as
δ=P0(Js+D+kω)sp(Js+D+kω)(τfs+1)s,p=1.5Vo[Rg(VoUgcosδ)+XgUgsinδ]Zg2.
(4)
When a severe fault occurs, switches S1 and S2 are closed at position 2, and the VSG operates in fault mode. In the active power loop, we adopt the method of directly altering the active power reference to restrict the power angle deviation. The rated active power should be changed from P0 to Pf, where Pf is calculated according to
Pf=1.5UgfId0,
(5)
in which Ugf is the post-fault grid voltage amplitude and Id0 is the d-axis component of the pre-fault output current. In this paper, Id0 = IN, where IN is the rated current of the VSG.
As for the reactive power loop, to fulfill the grid voltage support or reactive current compensation requirements, we use the method described in Ref. 13. After a severe fault, the VSG switches to the reactive current–voltage droop control mode, and the voltage amplitude reference can be calculated from
Vref=E0+kis(Iq0ioq),
(6)
where ki is the integral coefficient, Iq0 is the reactive current reference, and ioq is the reactive current output.
According to the low-voltage ride-through requirements,21 IIDGs should generate a specific amount of reactive current according to the voltage sag after fault inception. The required reactive current iq0 is expressed as follows:
iq0=0,Ugpu0.9,k1(0.9Ugpu)IN,0.2Ugpu<0.9,0.7k1IN,Ugpu<0.2,
(7)
where Ugpu is the per unit grid voltage and k1 is a proportional coefficient. In this paper, k1 = 1.5.
For asymmetric faults, the sequence component method is employed to independently control the VSG’s PS, NS, and ZS components. The NS controller is tailored to specific objectives, such as boosting the PS voltage,22 reducing active or reactive power fluctuations, and suppressing the NS current or voltage.20,23 To suppress the current caused by excessive NS voltage in the grid, we adopt the method of regulating the NS current of the inverter to zero through direct current control,24 i.e.,
iLref=0,
(8)
where iLref is the reference value of the NS current. Additionally, as the NS components are controlled in the synchronous reference frame, a phase angle θ is assigned during the NS Park transformation. In this paper, this angle is the opposite of the PS phase angle, i.e., θ = −θ.

To facilitate fault analysis, we make the following assumptions: (1) the grid frequency remains constant before and after fault inception; and (2) the grid voltage reaches the steady state instantaneously upon fault occurrence.

Similar to SGs, VSGs can function as controlled voltage sources during normal operation. Their output power is determined by the voltage amplitude and power angle. When a fault occurs, the fault control strategy is activated, and the fault model in Fig. 2 is established.

FIG. 2.

Equivalent model of VSG. (a) PS model, (b) NS model, and (c) ZS model.

FIG. 2.

Equivalent model of VSG. (a) PS model, (b) NS model, and (c) ZS model.

Close modal

When an asymmetric fault occurs, the system contains PS, NS, and ZS components. According to the VSG fault control strategy described in Sec. II, the reference value of the NS current of the VSG output is zero,10 as shown in Fig. 1. Considering that the current inner-loop responds very quickly, the response time of the NS current of the system can be ignored, i.e., iL=iLref. At this time, the NS current of the VSG output is zero. Considering the main circuit shown in Fig. 1, the equivalent circuit corresponding to its NS component is shown in Fig. 2(b). As for the ZS component, the three-phase three-bridge system adopted in this paper is ungrounded, so the output ZS current of the VSG system is zero, and its corresponding equivalent circuit is as shown in Fig. 2(c).

In Fig. 2(a), the PS model is represented as a voltage source. Vo+ is the PS voltage amplitude, δ is the PS power angle, Io+ is the PS output current, and Ug+ is the PS voltage of the grid. In Figs. 2(b) and 2(c), the NS and ZS models consist of a capacitor and infinite impedance, respectively. Io is the NS current of the VSG, and Vo, Ug and Vo0, Ug0 are the NS and ZS voltages of the VSG terminal and the grid, respectively.

The VSG remains in normal operation mode under high-impedance symmetric faults. The active power-frequency and reactive power-voltage control loops can be analyzed independently.

1. Calculation of voltage amplitude

The control strategy described in Sec. II causes a slow and minuscule change in the power angle after fault inception. Therefore, its influence on reactive power-voltage control can be neglected.25 The reactive power-voltage control loop is analyzed first, with the power angle characterized as a constant.

Combining Eqs. (1) and (2), the transfer function of the reactive power-voltage controller can be written as
Vo=E0kQGv(s)qτfs+1,
(9)
where q can be expressed as follows:9 
q=1.5Vo[Xg(VoUgcosδ)RgUgsinδ]Zg2=f(Vo).
(10)

Figure 3(a) shows a block diagram of reactive power control during high-impedance faults.

FIG. 3.

Block diagram of reactive power control of VSG during high-impedance fault. (a) Normal and (b) simplified.

FIG. 3.

Block diagram of reactive power control of VSG during high-impedance fault. (a) Normal and (b) simplified.

Close modal

In Eq. (10), f(Vo) contains a quadratic term in the voltage amplitude Vo. Considering that the voltage amplitude and power angle of the VSG do not change significantly during high-impedance faults, f(Vo) can be linearized to help derive an analytical model. To avoid any steady-state error, the model is linearized at its post-fault steady-state equilibrium point. The VSG’s steady-state fault voltage amplitude Vest and power angle δest should be estimated first. The estimation can be performed by a Newton iteration according to Eqs. (10) and (4). Then, f(Vo) is linearized at this estimated point, i.e., Vo2Vest2+2Vest(VoVest).

Therefore, Eq. (10) can be simplified as
f(Vo)1.5[(2XgVestXgUgcosδestRgUgsinδest)VoXgVest2]Zg2=kvVoq1,
(11)
where
kv=1.5(2XgVestXgUgcosδestRgUgsinδest)Zg2,q1=1.5XgVest2Zg2.
(12)
Therefore, Fig. 3(a) can be replaced by Fig. 3(b), from which we obtain
q=kQkv(kpvs+kiv)Q0+E0kQ(τfs+1)(Cfs2+kpvs+kiv)q1τfCfs3+(Cf+kpvτf)s2+(kpv+kivτf+kpvkQkv)s+(kiv+kivkQkv).
(13)
By substituting Eq. (13) into Eq. (9), the terminal voltage amplitude of the VSG is derived as
Vo=kQ(kpvs+kiv)Q0+E0kQ+q1τfCfs3+(Cf+kpvτf)s2+(kpv+kivτf+kpvkQkv)s+(kiv+kivkQkv).
(14)
Equation (14) is still somewhat complex and requires further simplification. The fault voltage of the VSG under high-impedance faults is related to various parameters, including the filter parameters, the voltage and current controller parameters, the line impedance, the droop coefficient, and the grid voltage amplitude. In most cases, Cf is much smaller than the proportional coefficient kpv of the voltage controller and the filter time constant τf.17 Therefore, it can be ignored without introducing significant errors,
Vo=kQQ0+E0kQ+q1τfs+kQkv+1.
(15)

2. Calculation of power angle

Similarly, the instantaneous active power of the VSG under high-impedance faults can be expressed as9 
p=1.5Vo[Rg(VoUgcosδ)+XgUgsinδ]Zg2=f(δ).
(16)
The linearization procedure described in Sec. III B 1 is used to simplify this model. First, the steady-state power angle δest is estimated. Then, f(δ) is linearized at this equilibrium point,
sinδsinδest+(δδest)cosδest,cosδcosδest(δδest)sinδest.
(17)
Substituting Eq. (17) into Eq. (16) yields the simplified expression for instantaneous active power,
p=1.5VoZg2(UgRgsinδest+UgXgcosδest)δ+(VoRgUgRgcosδest+UgXgsinδest)(UgRgsinδest+UgXgcosδest)δest=kδδ+p1.
(18)
Substituting Eq. (18) into Eq. (4) yields the following expression for the power angle:
δ=P0p1Jτfs3+[(D+kω)τf+J]s2+(D+kω)s+kδ.
(19)
The corresponding control block diagram is shown in Fig. 4,
P=P0p1,p1=1.5VoZg2(VoRgUgRgcosδest+UgXgsinδest)(UgRgsinδest+UgXgcosδest)δest,kδ=1.5VoUgZg2(Rgsinδest+Xgcosδest).
(20)
FIG. 4.

Block diagram of VSG active power control.

FIG. 4.

Block diagram of VSG active power control.

Close modal
The voltage amplitude Vo in Eq. (20) should be calculated using the estimated value Vest. According to Eq. (19), the power angle response of the VSG under a high-impedance fault is determined by a range of parameters, including the virtual inertia, damping, droop coefficient, filter time constant, terminal voltage amplitude, grid voltage amplitude, and line impedance. Considering that the filter time constant τf is generally much smaller than (D + kω), and the virtual inertia is also much smaller than the damping and droop coefficient,26 Eq. (19) can be simplified as
δ=P0p1(D+kω)s+kδ.
(21)
Based on the above-mentioned analysis, the output current of the VSG during a high-impedance symmetric fault is
Io+=V0UgZg,
(22)
where Vo = Voe denotes the terminal voltage phasor of the VSG, and Ug = Ugej 0 denotes the grid voltage phasor.
When a low-impedance symmetric fault occurs and current limiting is not considered, the system output voltage is referenced to the high-impedance fault. However, when considering current limits, the VSG adopts the fault control strategy described in Sec. II. For consistency with the high-impedance case, the voltage amplitude response is analyzed first. By substituting Eq. (1) into Eq. (6), the voltage amplitude is expressed as
Vo=E0+kisGv(s)(Iq0ioq).
(23)
As the d-axis of the synchronous reference frame is aligned with the voltage phasor of the VSG and the response of the inner control loop is sufficiently fast, the q-axis voltage component is approximately zero during any disturbance. Therefore, the reactive power satisfies
q=1.5Vo[(VoUgcosδest)XgUgRgsinδest]Zg2=1.5Voioq.
(24)
The reactive current can be obtained as
ioq=(VoUgcosδest)XgUgRgsinδestZg2=(kqVobq),
(25)
where kq=XgZg2, bq=Ug(Xgcosδest+Rgsinδest)Zg2.
Substituting Eq. (25) into Eq. (23), the voltage amplitude of the VSG during a low-impedance fault can be expressed as
Vo=(Cfs3+kpvs2+kivs)E0+ki(kpvs+kiv)(Iq0bq)Cfs3+kpvs2+(kivkpvkikq)skivkikq.
(26)
According to the Routh–Hurwitz criterion,27 the characteristic equation of this third-order system is
Cfs3+kpvs2+(kivkpvkikq)skivkikq=0.
(27)
A sufficient and necessary condition for its stabilization is
Cf>0,kpv>0,kivkpvkikq>0,kivkikq>0,Cfkpvkivkpvkikqkivkikq>0.
(28)
For the voltage in Eq. (26) to be stable, the integral coefficient must satisfy ki < 0. As the filter capacitance Cf is small and E0 is constant, the higher-order terms in Eq. (26) can be ignored, leading to
Vo=ki(Iq0bq)skikq.
(29)
Regarding the power angle response, the analysis carried out for the high-impedance case is applicable in this scenario. The only discrepancy lies in the compensation of the active power reference. Consequently, the fault response of the power angle can be expressed in the following simplified form:
δ=(τfs+1)Pfp1(D+kω)s+kδ,
(30)
where Pf is given in Eq. (5), kδ and p1 are given in Eq. (20), and the voltage amplitude Vo in Eq. (20) is calculated using Vest.

In summary, the fault response of the VSG during low-impedance symmetric faults can be described by Eq. (22), while the voltage amplitude and the power angle are given by Eqs. (29) and (30).

When an asymmetric fault occurs, the sequence components of the VSG will be independently controlled. The PS model comes directly from the symmetric fault cases. As for the NS components, Fig. 1 yields the following expression for the NS voltage and current:
Vo=Ug+IoZg,
(31)
where Vo and Io are the NS output voltage and current phasors of the VSG, respectively, and Ug is the NS voltage phasor of the grid.
The current phasor can be calculated as follows:
Io=UgZg+1/jωCf.
(32)
As the neutral point of the VSG is considered to be ungrounded, its ZS voltage and current outputs are
Vo0=Ug0,Io0=0.
(33)
In summary, the total current of the VSG can be expressed as
Io=Io++Io,
(34)
where Io+ and Io are given by Eqs. (22) and (32), respectively.

To verify the effectiveness and accuracy of the proposed equivalent model, the grid-connected VSG model shown in Fig. 1 was implemented on the MATLAB/Simulink platform. The relevant model parameters are listed in Table I.

TABLE I.

Parameters of VSG system.

ParameterValueParameterValue
P0 10 kW kQ 0.003 V/Var 
J 0.2 kg m2 τf 0.01 
D 30 W s/rad ki −100 
ω0 314 rad/s kpv 
Q0 0 kVar kiv 
E0 311 V kpi 10 
Udc 800 V kii 50 
Lf, Rf 2 mH, 0.01 Ω kpn 
Cf 100 µF kin 100 
Zg 0.7 + j * 1.57 Ω kω 3000 W s/rad 
ParameterValueParameterValue
P0 10 kW kQ 0.003 V/Var 
J 0.2 kg m2 τf 0.01 
D 30 W s/rad ki −100 
ω0 314 rad/s kpv 
Q0 0 kVar kiv 
E0 311 V kpi 10 
Udc 800 V kii 50 
Lf, Rf 2 mH, 0.01 Ω kpn 
Cf 100 µF kin 100 
Zg 0.7 + j * 1.57 Ω kω 3000 W s/rad 

When t = 2 s, a three-phase short-circuit fault occurs at PCC with a fault resistance of 30 Ω. According to the proposed modeling method, the steady-state values of the VSG’s voltage amplitude and power angle are estimated first. The results and corresponding simulation values are listed in Table II. The voltage and current waveforms of the VSG obtained from the calculation model and the simulations are shown in Fig. 5.

TABLE II.

Estimated and simulated steady-state values.

ParameterEstimationCalculationSimulation
δ/rad 0.11015 0.11016 0.1102 
Vo/V 305.47 305.46 305.5 
ParameterEstimationCalculationSimulation
δ/rad 0.11015 0.11016 0.1102 
Vo/V 305.47 305.46 305.5 
FIG. 5.

Calculation and simulation results in case 1. (a) Output voltage amplitude, (b) power angle, (c) phase A current, and (d) d-q axis currents.

FIG. 5.

Calculation and simulation results in case 1. (a) Output voltage amplitude, (b) power angle, (c) phase A current, and (d) d-q axis currents.

Close modal

Table II and Fig. 5 indicate that the proposed model exhibits negligible steady-state errors in the voltage amplitude and power angle when subjected to a high-impedance symmetric fault. However, during the transient process, clear errors can be observed in Figs. 5(a) and 5(b). These errors primarily arise from the simplifications made during the model analysis. Figures 5(c) and 5(d) demonstrate that the output characteristics of the proposed fault model are generally aligned with the simulation results, affirming its equivalence for fault calculation and characteristics analysis under high-impedance symmetric faults.

When t = 2 s, a phase-to-phase grounding fault occurs at PCC with a fault resistance of 30 Ω. The calculation and simulation results of the PS components are the same as in Fig. 5, whereas the NS and ZS components are calculated using the fault models in Figs. 2(b) and 2(c), respectively. The model calculations and simulation results are shown in Fig. 6.

FIG. 6.

Calculation and simulation results in case 2. (a) d-q axis NS voltages, (b) d-q axis NS currents, (c) ZS voltages, and (d) ZS currents.

FIG. 6.

Calculation and simulation results in case 2. (a) d-q axis NS voltages, (b) d-q axis NS currents, (c) ZS voltages, and (d) ZS currents.

Close modal

For the NS components, the errors during the transient process are mainly induced by the power angle estimation error for the NS components, although these are not obvious, as shown in Figs. 6(a) and 6(b). Figure 6(c) confirms that the ZS voltages of both the VSG and the grid are identical, and Fig. 6(d) confirms that there is no ZS current. Overall, the fault output characteristics of the proposed model remain consistent with those of the simulations in this case. Consequently, the proposed model is deemed suitable for fault calculation and fault characteristic analysis in high-impedance asymmetric fault cases.

When t = 2 s, a three-phase short-circuit fault occurs at PCC with a fault resistance of 1 Ω. The steady-state values of the voltage amplitude and power angle are estimated first. The results and the simulation values are listed in Table III. The fault responses calculated with the proposed fault model and the simulation results are shown in Fig. 7.

TABLE III.

Estimated and simulated steady-state values.

ParameterEstimationCalculationSimulation
δ/rad 0.1202 0.1207 0.1206 
Vo/V 230.69 230.65 230.5 
ParameterEstimationCalculationSimulation
δ/rad 0.1202 0.1207 0.1206 
Vo/V 230.69 230.65 230.5 
FIG. 7.

Calculation and simulation results in case 3. (a) Output voltage amplitude, (b) power angle, (c) phase A current, and (d) d-q axis current.

FIG. 7.

Calculation and simulation results in case 3. (a) Output voltage amplitude, (b) power angle, (c) phase A current, and (d) d-q axis current.

Close modal

Table III and Fig. 7 illustrate that the steady-state error between the calculations and simulation results is remarkably small; the moderate error during the transient state comes from neglecting the time constant of the inner control loop. For the voltage amplitude, the fast response speed of the voltage loop ensures that Vq = 0 throughout the transient process. Consequently, the error between the model and the simulation results is insignificant, as shown in Fig. 7(a). Conversely, a certain degree of error is introduced to the power angle response by the model linearization and order reduction, as illustrated in Fig. 7(b). Finally, Figs. 7(c) and 7(d) demonstrate that the fault currents from the calculations and simulations are consistent under a low-impedance symmetric fault.

When t = 2 s, a phase-to-phase grounding fault occurs at PCC with a fault resistance of 1 Ω. The calculation and simulation results for the PS components align with those depicted in Fig. 7, and the NS and ZS components are illustrated in Fig. 8.

FIG. 8.

Calculation and simulation results in case 4. (a) d-q axis NS voltages, (b) d-q axis NS currents, (c) ZS voltages, and (d) ZS currents.

FIG. 8.

Calculation and simulation results in case 4. (a) d-q axis NS voltages, (b) d-q axis NS currents, (c) ZS voltages, and (d) ZS currents.

Close modal

According to Fig. 8, the calculation results are generally consistent with the simulation results. However, an error does exist during the transient process for the same reason described in the high-impedance case. Furthermore, the ZS voltage and current follow a similar pattern as in case 2.

The response of the fault model is clearly affected by a variety of factors, including the control parameters, circuit parameters, and severity of the fault. To further demonstrate the applicability of the proposed model, the influence of these factors is now analyzed.

When a high-impedance fault occurs, the output voltage and power angle are related to the line impedance, as described in Eqs. (15) and (21). Assume that the Xg/Rg ratio of the line impedance remains constant. As the line length increases, the line impedance is enhanced. Then, as the voltage drop is small under a high-impedance fault, the change in the numerator of Eq. (15) is smaller than that of the denominator. As a result, the voltage amplitude increases with increasing line impedance, and the dynamic time constant of the reactive power-voltage loop becomes larger, leading to a longer transient period. According to Eq. (21), when the line impedance increases, the active power component p1 and the power angle coefficient kδ decrease simultaneously. Therefore, the steady-state power angle and the dynamic time constant increase, leading to a longer transient period. The simulation results in Fig. 9 consider scenarios L1 and L2 (L2 > L1), in which the line lengths are different.

FIG. 9.

Influence of line impedance on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

FIG. 9.

Influence of line impedance on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

Close modal

In Fig. 9, the steady-state values of the voltage amplitude and power angle increase as the line impedance rises. The voltage amplitude stabilizes after 2.013 s in scenario L1 and after 2.016 s in scenario L2. Similarly, the power angle stabilizes after 2.332 s in scenario L1 and after 2.521 s in scenario L2. This indicates a prolongation of the dynamic response time for both the voltage amplitude and power angle due to the increased time constants, which agrees with our previous analysis. Furthermore, the discrepancy between the fault responses of the proposed model and the simulated model is remarkably small.

When a low-impedance fault occurs, according to Eqs. (29) and (30), the steady-state fault voltage amplitude and power angle increase with increasing line impedance. The simulation results for this scenario are also given in Fig. 9 and are consistent with the theoretical analysis.

When designing the inner-loop control parameters, the LC filter parameters and bandwidth of the inner control loop are typically taken into account.19,20 As the response time of the inner loop is much shorter than that of the power outer loop, the dynamic process of the current loop cannot make any significant impact on the fault responses. Therefore, the analysis of the control parameters and filter inductance Lf can be skipped. However, the influence of filter capacitance Cf should be examined because the model reduction process omits the filtering capacitance.

From Eqs. (15), (21), (29), and (30), it is apparent that the fault responses of the proposed model are independent of Cf. However, the fault responses of the detailed VSG model are related to the capacitance value Cf through Eqs. (14), (19), and (26); the influence of Cf is exclusive to the transient-state responses, while the steady-state fault responses are independent of Cf per the final value theorem. When Cf increases, the dynamic response time of the inner loop increases, which leads to an increase in the power angle response time. The waveforms from the simulations and calculations are shown in Fig. 10, where the capacitance value Cf1 is less than Cf2.

FIG. 10.

Influence of LC filter and inner loop control parameters on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

FIG. 10.

Influence of LC filter and inner loop control parameters on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

Close modal

According to Fig. 10, an increase in the filter capacitance Cf does not change the proposed model. Nevertheless, the transient error and dynamic response time of the detailed model increase, which agrees with our previous analysis.

During high-impedance faults, the VSG responses are related to the power filtering time constant τf. When τf increases, the voltage amplitude time constant and corresponding dynamic response time of the proposed model and the detailed model increase according to Eqs. (14) and (15). The power angle of the proposed model is independent of τf, although the power angles in the detailed model are associated with τf through Eq. (19), and the dynamic response time of the power angle increases with increasing τf. Conversely, when a low-impedance fault occurs, the proposed model is independent of τf. The power angle dynamic response time of the detailed model increases as τf rises, according to Eq. (19). The waveforms given by the simulations and calculations are shown in Fig. 11, where the power filtering time constant τf1 is less than τf2.

FIG. 11.

Influence of power filter coefficients on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

FIG. 11.

Influence of power filter coefficients on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

Close modal

As shown in Fig. 11, during high-impedance faults, the dynamic response times of both the proposed and detailed models increase with τf. In contrast, during low-impedance faults, the amplitudes of the output voltage and power angle in the proposed model remain largely unaffected by τf. Only a small increase in the power angle transient response time of the detailed model can be observed, which agrees with the earlier analysis.

The equations describing the proposed and detailed models indicate that the virtual inertia and damping coefficients significantly impact the power angle response but have a relatively minor influence on the voltage amplitude.

For analytical convenience, high-order terms relating to the virtual inertia and damping coefficients are neglected in the proposed model, as described by Eqs. (21) and (30). Therefore, the power angle of the proposed model exhibits minimal sensitivity to changes in the virtual inertia. As for the detailed model, its steady-state power angle remains constant with increasing virtual inertia, but the dynamic response time increases. The waveforms given by the simulations and calculations are shown in Fig. 12, where the virtual inertia J1 is less than J2.

FIG. 12.

Influence of virtual inertia on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

FIG. 12.

Influence of virtual inertia on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

Close modal

Figure 12 illustrates the independence of the proposed model from virtual inertia. In the detailed model, the output voltage is slightly influenced by the virtual inertia. As the virtual inertia increases, the steady-state power angle of the detailed model remains unchanged while the transient period becomes longer. This result is consistent with the previous analysis.

As the damping coefficient increases, both the proposed model and the simulation model exhibit unaltered steady-state power angles. However, the dynamic time constant increases according to Eqs. (21) and (30). Figure 13 displays the waveforms from the proposed and detailed models under varying damping coefficients, where D1 is less than D2.

FIG. 13.

Influence of damping coefficient on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

FIG. 13.

Influence of damping coefficient on the model. (a) Comparison of output voltage amplitude waveforms and (b) comparison of power angle waveforms.

Close modal

The output voltage amplitude is essentially independent of the damping coefficient. However, the dynamic process of the power angle in both the proposed and detailed models becomes longer as the damping coefficient increases, which is consistent with the previous analysis.

In summary, while the line impedance can change the steady-state fault responses of the model, other system parameters only influence the transient-state responses. The simulation results show that the error between the responses given by the proposed model and the simulation results is tiny, verifying the correctness and effectiveness of the proposed VSG model. The model can be applied to fault calculation and fault analysis in renewable energy microgrid systems containing VSGs.

This paper began by introducing the fault control strategy of VSGs used for modeling. Subsequently, a comprehensive fault model for VSG was developed. The effectiveness and accuracy of the proposed fault model were then validated through simulations, and the influence of different parameters was comprehensively studied. The following conclusions can be drawn:

  1. The adopted fault control strategy for VSGs provides reactive power compensation per the grid code while suppressing negative sequence currents.

  2. The fault equivalent model proposed in this paper accurately captures the transient- and steady-state fault responses of VSGs. It is applicable to the fault analysis and protection scheme design of renewable energy microgrids under different scenarios.

  3. The equivalent model proposed in this paper is intuitive and straightforward. The linearization and model reduction processes do not induce significant errors. As a result, the model can be employed for accurate fault calculations in real renewable energy microgrid applications.

  4. The impact of the control parameters and circuit parameters on the fault model was analyzed and found to be consistent with the simulation results, which confirms the correctness and applicability of the proposed model.

In practical engineering applications, the proposed VSG fault model may still be a little overcomplicated. Our current research focus is on further simplifying the fault model based on the dominant parameters of specific VSGs. Hopefully, a more simplified fault model that is better suited to industrial applications can be developed in the future.

This work was supported by the Science and Technology Commission of Shanghai Municipality (Grant No. 21142201200).

The authors have no conflicts to disclose.

G.C. is responsible for paper writing, L.M. is responsible for paper checking and correcting, and C.F. and M.Z. are responsible for offering help in theory and practice. All authors read and approved the final paper.

Guozhen Chen: Writing – original draft (equal). Longhua Mu: Writing – review & editing (equal). Chongkai Fang: Writing – review & editing (equal). Mi Zhang: Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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