A VSG Power–Current Collaborative Control Strategy Based on Improved SOGI for Unbalanced Power Grid

Abstract

Large-scale grid integration of renewable generation is facilitated by resource abundance and advancements in power electronics. The high penetration of power electronics-based devices reduces system inertia, threatening grid stability. The Virtual Synchronous Generator (VSG) provides an effective solution. However, when the grid is unbalanced, problems such as unbalanced output current, overcurrent, and power fluctuations occur. Thus, a VSG power–current collaborative control strategy based on improved second-order generalized integrator (SOGI) is proposed. Firstly, the generation of power fluctuation and unbalanced current in VSG output under an unbalanced power grid is analyzed. Secondly, to suppress the DC component and high-order harmonics, an improved SOGI method is proposed for positive and negative sequence separation by introducing a difference node and adding an extra SOGI module on the basis of the traditional SOGI. To achieve the collaborative control of active/reactive power constant and current balance, a correlation coefficient μ is introduced and a unified equation is constructed. To prevent overcurrent during fault, virtual impedance and reactive power reference are introduced. Simulations have verified the effectiveness of the proposed method. It shows superiority in effective sequence separation, smooth power–current collaborative control, and safe operation without overcurrent.

1. Introduction

With the scheme of a “Double Carbon” goal and the construction of a new type of power system, renewable generation, primarily wind turbine and photovoltaic, has undergone rapid development [1]. However, with the increasing penetration of renewable generation, the security and stability of power systems face a huge threat. By emulating the operational mechanism of synchronous generators, Virtual Synchronous Generator (VSG) technology endows grid-connected inverters with virtual inertia and damping, thereby effectively enhancing system stability [2]. However, on account of asymmetric line parameters or asymmetric faults, unbalanced faults occur. This will lead to the imbalance of the grid-connected current and double-frequency oscillation of VSG output power. It seriously affects power quality and system stability [3]. To suppress power oscillation and balance output current, it is essential to study the VSG control strategy under an unbalanced grid.

Studies have been conducted to improve control strategies. In the studies on positive–negative sequence separation methods, reference [4] used notch filter for positive and negative sequence separation and reference current calculation. It achieved current balance. However, notch filter is sensitive to voltage unbalance and has transient overshoot. Reference [5] used dual second-order generalized integrator (DSOGI) to realize fast and effective sequence separation of Vienna rectifier grid voltage under an unbalanced, weak current network. Reference [6] used DSOGI to rapidly separate sequence components in photovoltaic storage VSG systems under unbalanced grid conditions. Though DSOGI shows advantages in frequency robustness and rapid dynamic response, the inhibiting ability for high-order harmonics is limited. Reference [7] adopts the Delayed DSOGI to achieve high-precision extraction of the fundamental positive sequence voltage signal under distorted and unbalanced grid conditions.

In the studies on power–current coordinated control and current-limiting control, reference [8] implemented VSG current balance control in unbalanced grids by designing a positive sequence current inner loop. Reference [9] proposed a dual dq-coordinate-based control method of positive and negative sequence current, aiming to achieve balanced currents and constant power output. Reference [10] ensured constant power output while maintaining current balance; however, its design did not account for current-limiting under fault conditions. References [11,12] proposed a voltage ride through control schemes for voltage sag faults, which can effectively suppress overcurrent. However, power fluctuation was not optimized. Numerical studies show that the existing control methods typically focus on single objectives. Moreover, faster response requires a more efficient method for sequence separation. Fault current reduction needs an improved current-limiting strategy.

In summary, to achieve the multi-objective coordinated optimal control of power and current, as well as a high-efficiency method for sequence separation and current-limiting, a VSG power–current collaborative control strategy based on improved second-order generalized integrator (SOGI) is put forward. The main contributions can be concluded as follows.

(1)

On the basis of traditional SOGI, a differential node is introduced and an additional SOGI module is incorporated, thus proposing an improved SOGI-based sequence separation method. It shows advantages in sequence separation and in suppressing DC components and high-order harmonics of the input signal.

(2)

By introducing a correlation coefficient μ, the collaborative control of three objectives, including current balance, constant active power output, and constant reactive power output and, is unified. The power–current collaborative control provides more flexibility for unbalanced scenarios.

(3)

Considering the impact of overcurrent during fault conditions, virtual impedance is adopted, and the power reference is adjusted to realize effective suppression of overcurrent.

The remainder is organized as follows. Section 2 introduces the control strategy of the VSG. The generation of power fluctuation and unbalanced current in VSG control is analyzed. Section 3 designs the improved SOGI strategy and proposes a power–current collaborative control strategy. Section 4 presents case simulations. The effectiveness of the proposed method is verified under multiple conditions. Section 5 gives the conclusions.

2. Grid-Connected Characteristics of VSG in Unbalanced Power Grid

2.1. VSG Control Strategy

The main circuit and VSG control block diagram are shown in Figure 1.

Figure 1. The schematic diagram of VSG control.

Wherein, U_dc is the DC voltage. u_g = (u_ga,u_gb,u_gc)^T is the grid voltage. u = (u_a,u_b,u_c)^T is the voltage of the PCC node. i = (i_a,i_b,i_c)^T is the grid current. L_s, is the filter inductance. R_s is the parasitic resistance. C represents the filter capacitance. L_g and R_g are line inductance and resistance.

The VSG control incorporates both an active power–frequency (P–f) control loop and a reactive power–voltage (Q–V) control loop [13]. The P–f control loop can be displayed as

$$\left\{\begin{array}{l}J{\displaystyle \frac{d\omega }{dt}}={\displaystyle \frac{P_{\mathrm{ref}}}{{\omega }_N}}-{\displaystyle \frac{P_e}{{\omega }_N}}-D(\omega -{\omega }_N)\\ {\displaystyle \frac{d\delta }{dt}}=\omega -{\omega }_N\end{array}\right.$$

(1)

where J is the virtual moment of inertia. $\omega$ and ${\omega }_N$ are the output angular frequency and grid angular frequency. $\delta$ is the phase. D is the damping coefficient. P_ref is the reference value of active power. P_e is the actual value of active power.

The expression of the Q–V control loop can be displayed as

$$E={\displaystyle \frac{1}{Ks}}\left[Q_{\mathrm{ref}}-Q_e+D_q(U_{\mathrm{ref}}-U)\right]$$

(2)

where E is virtual internal electromotive force. K is the inertia coefficient of simulated excitation regulation. D_q is the reactive power regulation coefficient. U_ref and U are the reference voltage and output voltage of VSG, respectively. Q_ref and Q_e are the reference reactive power and output reactive power.

2.2. Fault Analysis Under Unbalanced Grid

For the main circuit shown in Figure 1, when unbalanced faults occur, the unbalanced grid voltage and current in a dq coordinate system can be expressed as

$$\left\{\begin{array}{l}{u}_{gdq}=u_{gdq}^{+}{e}^{j{\omega }_{g}t}+u_{gdq}^{-}{e}^{-j{\omega }_{g}t}\\ i_{dq}=i_{dq}^{+}{e}^{j{\omega }_{g}t}+i_{dq}^{-}{e}^{-j{\omega }_{g}t}\end{array}\right.$$

(3)

where $u_{gdq}$ and $i_{dq}$ are grid voltage and current, respectively. Superscript + and − represent the positive sequence component and negative sequence component. ${\omega }_g$ is the angular frequency on the grid side.

According to instantaneous power theory, under unbalanced grid voltage conditions, the instantaneous power injected into the grid can be derived as:

$$\left\{\begin{array}{l}P=P_0+P_{s2}\sin (2{\omega }_{g}t)+P_{c2}\cos (2{\omega }_{g}t)\\ Q=Q_0+Q_{s2}\sin (2{\omega }_{g}t)+Q_{c2}\cos (2{\omega }_{g}t)\end{array}\right.$$

(4)

where $P_0$ and $Q_0$ are the DC components of instantaneous active and reactive power, respectively. $P_{s2}$ and $P_{c2}$ are the sine and cosine amplitudes of active power fluctuation. $Q_{s2}$ and $Q_{c2}$ are the sine and cosine amplitudes of reactive power fluctuation, respectively. By comparing Equations (3) and (4), the amplitudes can be calculated.

$$\left[\begin{array}{l}{P}_0\\ Q_0\\ P_{s2}\\ Q_{s2}\\ P_{c2}\\ Q_{c2}\end{array}\right]=\left[\begin{array}{cccc}{u}_{gd}^{+}& u_{gq}^{+}& u_{gd}^{-}& u_{gq}^{-}\\ u_{gq}^{+}& -u_{gd}^{+}& u_{gq}^{-}& -u_{gd}^{-}\\ u_{gq}^{-}& -u_{gd}^{-}& -u_{gq}^{+}& u_{gd}^{+}\\ -u_{gd}^{-}& -u_{gq}^{-}& u_{gd}^{+}& u_{gq}^{+}\\ u_{gd}^{-}& u_{gq}^{-}& u_{gd}^{+}& u_{gq}^{+}\\ u_{gq}^{-}& -u_{gd}^{-}& u_{gq}^{+}& -u_{gd}^{+}\end{array}\right]\left[\begin{array}{c}{i}_d^{+}\\ i_q^{+}\\ i_d^{-}\\ i_q^{-}\end{array}\right]$$

(5)

Under unbalanced conditions, the negative sequence components are not zero, resulting in double-frequency fluctuation of instantaneous power of VSG and imbalance of output current [14].

3. Power–Current Collaborative Control Strategy of VSG Based on Improved SOGI

3.1. An Improved SOGI for Positive and Negative Sequence Separation

The traditional SOGI can be used for sequence separation and filtering signals [15]. The transfer function of SOGI is as follows.

$$D_1(s)={\displaystyle \frac{u_1(s)}{u(s)}}={\displaystyle \frac{k{\omega }_{0}s}{s^2+k{\omega }_{0}s+{\omega }_0^2}}$$

(6)

$$Q_1(s)={\displaystyle \frac{qu(s)}{u(s)}}={\displaystyle \frac{k{\omega }_0^2}{s^2+k{\omega }_{0}s+{\omega }_0^2}}$$

(7)

where $u(s)$ is the input signal of SOGI. $u_1(s)$ and $qu(s)$ are the two mutually orthogonal output signals of SOGI. $qu(s)$ holds a leading phase of 90° compared with $u_1(s)$. ${\omega }_0$ is the angular frequency of the tracking AC signal. k is the gain of the integrator. $D_1(s)$ represents a second-order bandpass filter and $Q_1(s)$ represents a low-pass filter. The filtering effect and response characteristics of SOGI are determined by the value of k.

The Bode diagrams of SOGI are shown in Figure 2. SOGI shows good filter performance on high-order harmonics. However, when $u(s)$ contains a DC component, $qu(s)$ is easy to be affected by the DC component, which will introduce errors in signal detection [16]. Thus, DSOGI develops by the differential combination of two SOGIs [17].

Figure 2. Bode diagrams of $D_1(s)$ and $Q_1(s)$. The red dashed lines correspond to ω₀.

To further enhance the suppression of the DC component, SOGI is improved by introducing a differential node and adding a SOGI module. The schematic diagram of the improved SOGI is depicted in Figure 3.

Figure 3. Schematic diagram of the improved SOGI.

The transfer function of the improved SOGI can be expressed as.

$$D_2(s)={\displaystyle \frac{u_1(s)}{u(s)}}={\displaystyle \frac{k{\omega }_{0}s}{s^2+k{\omega }_{0}s+{\omega }_0^2}}$$

(8)

$$Q_2(s)={\displaystyle \frac{qu(s)}{u(s)}}={\displaystyle \frac{(k{\omega }_0^2-k{k}_1{\omega }_1^2)s^2+k{k}_1{\omega }_0^2{\omega }_{1}s+(k{\omega }_0^2{\omega }_1^2-k{k}_1{\omega }_0^2{\omega }_1^2)}{(s^2+k{\omega }_{0}s+{\omega }_0^2)(s^2+k_1{\omega }_{1}s+{\omega }_1^2)}}$$

(9)

where k is the gain of the improved integrator. k₁ is the gain of the added SOGI. ${\omega }_1$ is the internal angular frequency of the added SOGI. In general, ${\omega }_1$ = ${\omega }_0$.

The Bode plots of the improved SOGI with different controller parameter settings are displayed in Figure 4. When the value of k is fixed and k₁ is 1, the improved SOGI achieves the optimal filtering performance for DC components and high-order harmonic components. When the value of k₁ is fixed, the filtering performance improves with decreasing k. The performance of sequence separation with different k and k₁ values are also tested and shown in Figure 5. When k and k₁ both equal 1, the system shows the shortest separation time and the smoothest waveform. Thus, the values of k and k₁ are selected as 1.

Figure 4. Bode diagrams of $D_2(s)$ and $Q_2(s)$ with different k and k₁ values. (a) Bode diagrams of $D_2(s)$ with different k₁ values. (b) Bode diagrams of $Q_2(s)$ with different k₁ values. (c) Bode diagrams of $D_2(s)$ with different k values. (d) Bode diagrams of $Q_2(s)$ with different k values.

Figure 5. Comparison of sequence separation performance with different k and k₁ values.

The improved SOGI can achieve rapid separation and effectively suppress harmonics. The overall control structure is displayed in Figure 6.

Figure 6. Overall control structure of the proposed sequence separation method.

3.2. Power–Current Collaborative Control Strategy

Ignoring the influence of filter capacitor shown in Figure 1, the voltage–current relationship at the inverter output port is described as

$$u_{abc}^{*}-u_{gabc}=R{i}_{abc}+L{\displaystyle \frac{d{i}_{abc}}{dt}}$$

(10)

where $u_{abc}^{*}$ is the voltage of the inverter output port. $u_{gabc}$ is the grid voltage. $i_{abc}$ is the grid-connected current. R and L are the equivalent resistances and inductances, respectively.

Following the dq decomposition of the three-phase inverter output voltages oriented to the grid voltage axis, the positive sequence reference current is derived.

$$\left\{\begin{array}{l}i{d}^{+*}={\displaystyle \frac{R(u{d}^{*+}-ug{d}^{+})+\omega L(u{d}^{*+}-ug{d}^{+})}{R^2+{(\omega L)}^2}}\\ i{q}^{+*}={\displaystyle \frac{-\omega L(u{q}^{*+}-ug{q}^{+})+R(u{q}^{*+}-ug{q}^{+})}{R^2+{(\omega L)}^2}}\end{array}\right.$$

(11)

where $i{d}^{+*}$ and $i{q}^{+*}$ are the dq components of the positive sequence reference current. $u{d}^{+*}$ and $u{q}^{+*}$ are the dq components of the VSG output voltage.

As analyzed in Section 2.2, under unbalanced conditions, the instantaneous power of the VSG will show double-frequency fluctuation. The output current will be unbalanced. To reduce the fluctuation and imbalance, it is essential to adjust the corresponding negative sequence current reference [18]. Three control objectives have been set for the adjustment. Objective (a) is constant active power. Objective (b) is three-phase current balance. Objective (c) is constant reactive power.

To achieve objective (a), the sine and cosine amplitudes of active power fluctuation in Equation (5) should be set to 0. It is $P_{s2}=P_{c2}=0$.

$$\left[\begin{array}{c}{P}_0\\ Q_0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{cccc}{u}_{gd}^{+}& u_{gq}^{+}& u_{gd}^{-}& u_{gq}^{-}\\ u_{gq}^{+}& -u_{gd}^{+}& u_{gq}^{-}& -u_{gd}^{-}\\ u_{gq}^{-}& -u_{gd}^{-}& -u_{gq}^{+}& u_{gd}^{+}\\ u_{gd}^{-}& u_{gq}^{-}& u_{gd}^{+}& u_{gq}^{+}\end{array}\right]\left[\begin{array}{c}{i}_d^{+}\\ i_q^{+}\\ i_d^{-}\\ i_q^{-}\end{array}\right]$$

(12)

Assuming that the negative sequence currents $i_d^{-}$ and $i_q^{-}$ are only used to suppress power fluctuations and do not participate in the provision of total power $P_0$ and $Q_0$, the solution can be obtained.

$$\left[\begin{array}{l}{i}_d^{-*}\\ i_q^{-*}\end{array}\right]={\displaystyle \frac{-2}{3({(u_{gd}^{+})}^2+{(u_{gq}^{+})}^2)}}\left[\begin{array}{cc}{u}_{gd}^{-}& -u_{gq}^{-}\\ u_{gq}^{-}& u_{gd}^{-}\end{array}\right]\left[\begin{array}{c}{P}_0\\ Q_0\end{array}\right]$$

(13)

To achieve objective (b), set $i_d^{-*}=i_q^{-*}=0$.

To achieve objective (c), the sine and cosine amplitudes of reactive power fluctuation in Equation (5) should be set to 0. It is $Q_{s2}=Q_{c2}=0$.

$$\left[\begin{array}{c}{P}_0\\ Q_0\\ 0\\ 0\end{array}\right]=\left[\begin{array}{cccc}{u}_{gd}^{+}& u_{gq}^{+}& u_{gd}^{-}& u_{gq}^{-}\\ u_{gq}^{+}& -u_{gd}^{+}& u_{gq}^{-}& -u_{gd}^{-}\\ -u_{gd}^{-}& -u_{gq}^{-}& u_{gd}^{+}& u_{gq}^{+}\\ u_{gq}^{-}& -u_{gd}^{-}& u_{gq}^{+}& -u_{gd}^{+}\end{array}\right]\left[\begin{array}{c}{i}_d^{+}\\ i_q^{+}\\ i_d^{-}\\ i_q^{-}\end{array}\right]$$

(14)

It can be solved as follows.

$$\left[\begin{array}{l}{i}_d^{-*}\\ i_q^{-*}\end{array}\right]={\displaystyle \frac{2}{3({(u_{gd}^{+})}^2+{(u_{gq}^{+})}^2)}}\left[\begin{array}{cc}{u}_{gd}^{-}& -u_{gq}^{-}\\ u_{gq}^{-}& u_{gd}^{-}\end{array}\right]\left[\begin{array}{c}{P}_0\\ Q_0\end{array}\right]$$

(15)

The above three control objectives are mutually independent. To achieve the collaborative control of current and active/reactive power, a correlation coefficient $\mu$ is introduced with a value range of [−1, 1]. The unified equation of negative sequence current command is established.

$$\left[\begin{array}{l}{i}_d^{-*}\\ i_q^{-*}\end{array}\right]={\displaystyle \frac{-2\mu }{3({(u_{gd}^{+})}^2+{(u_{gq}^{+})}^2)}}\left[\begin{array}{cc}{u}_{gd}^{-}& -u_{gq}^{-}\\ u_{gq}^{-}& u_{gd}^{-}\end{array}\right]\left[\begin{array}{c}{P}_0\\ Q_0\end{array}\right]$$

(16)

When $\mu =-1$, the control objective is constant reactive power. When $\mu =0$, the control objective is the balance of three-phase currents. When $\mu =1$, the control objective is constant active power. By adjusting the value of $\mu$, continuous changes among the three objectives can be achieved. The control performance can be adjusted according to the requirement of concrete cases.

3.3. Current-Limiting Control Strategy

Figure 7 illustrates the vector diagram before and after grid fault. Wherein, U is the VSG output voltage. $U_{g.0}$ and $U_{g.F}$ are the amplitude of grid voltages before and after fault, respectively. ${\delta }_0$ and ${\delta }_F$ are the phase shift of the VSG output voltage relative to the grid voltage before and after fault, respectively. When the grid voltage changes from $U_{g.0}$ to $U_{g.F}$ after a fault, due to the voltage source characteristics, VSG maintains the same port output voltage as that before fault. A large voltage difference $\Delta U_F$ between $U$ and $U_{g.F}$ generates. Because the system impedance R and X are small, a large overcurrent generates. If current-limiting measures are not taken, then stable operation will be influenced [19].

Figure 7. Voltage vector diagram before and after fault.

Virtual impedance offers an effective means to reduce the overcurrent after fault [20]. The voltage drops caused by virtual impedance in the dq coordinate system is calculated in Equation (17).

$$\left[\begin{array}{c}{u}_{dv}\\ u_{qv}\end{array}\right]=\left[\begin{array}{cc}{R}_V& -\omega L_V\\ \omega L_V& R_V\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]$$

(17)

wherein, $R_v$ and $L_v$ are the virtual resistance and virtual inductance. i_d and i_q are the dq-axis components of grid current. $u_{dv}$ and $u_{qv}$ are the dq-axis components of voltage drop by adding virtual impedance, respectively. The parameter tuning of the virtual impedance can be referenced in reference [21]. In this study, $R_v$ and $L_v$ are set to 0.1 and 3.14 mH, respectively.

The addition of virtual impedance changes the impedance characteristics of the system, which may cause the instability of the VSG. Thus, virtual impedance only works when overcurrent generates. When a fault occurs, the virtual impedance is introduced. The virtual impedance is deactivated when the fault is cleared.

Reactive power support is important for the grid during asymmetric voltage drop [22,23,24]. According to the national grid connection requirements, the required reactive current is expressed as follows.

$$I_{qref}=\left\{\begin{array}{ll}1.5(0.9-\Delta U)I_N\hfill & 0.2\le \Delta U\le 0.9\hfill \\ 1.5I_N\hfill & \Delta U<0.2\hfill \\ 0\hfill & \Delta U>0.9\hfill \end{array}\right.$$

(18)

wherein, $\Delta U$ is the per-unit value of the voltage at PCC. $I_N$ is the rated value of the grid current. $I_{qref}$ is the reference value of reactive current provided. When $0.2\le \Delta U\le 0.9$, the VSG outputs additional reactive current as $1.5\left(0.9-\Delta U\right)I_N$. When $\Delta U<0.2$, the VSG outputs additional reactive current as $1.5I_N$. When $\Delta U>0.9$, the VSG does not output additional reactive current.

The reference value of active current is calculated as follows.

$$I_{dref}=\sqrt{{(1.1I_N)}^2-{I_{qref}}^2}$$

(19)

The reference values of power are calculated as

$$\left\{\begin{array}{l}{P}_{ref.F}=1.5u_d{I}_{dref}\\ Q_{ref.F}=1.5u_d{I}_{qref}\end{array}\right.$$

(20)

wherein, $u_d$ is the d-axis component of the voltage at PCC. $P_{ref.F}$ and $Q_{ref.F}$ are the reference values of the VSG output active power and reactive power during fault, respectively.

To sum up, the overall structure diagram of the proposed control strategy is shown in Figure 8. It contains four parts of VSG control, power–current collaborative control, sequence separation based on improved SOGI, and virtual impedance overcurrent-limiting control. During the voltage asymmetry fault period, the active power, reactive power, and three-phase current are controlled by adjusting the negative sequence current reference value. In addition, switch S1 is closed. VSG provides reactive power support. If the grid current exceeds the limit, switch S2 is closed and the overcurrent is limited through the virtual impedance. After the fault is removed, switches S1 and S2 open and the system returns to normal operation.

Figure 8. Overall structure diagram of the proposed control strategy.

4. Case Study

4.1. Case Settings

The main circuit model displayed in Figure 1 was constructed by MATLAB 2023b/Simulink. The system parameter settings are presented in Table 1. The case settings are presented in Table 2. The rated current of the power grid is set as I_n = 64.3 A. The upper limit of the current is set as I_lim = 1.1I_n.

Table 1. Parameter settings.

Parameters	Value
Udc/V	800
Un/V	311
Ls/mH	1.8
Rs/Ω	0.01
Lg/mH	1
Rg/Ω	0.01
C/uF	20
J/(kg·m−2)	0.15
D/(N·m·s·rad−1)	40
Pref0/kW	30
Qref0/kvar	0
k	1
k1	1
Rv/Ω	0.1
Lv/mH	3.14
ω0/(rad/s)	314

Table 2. Case settings.

Case	Aim	Simulation Condition
1	Verify the performance of the positive and negative sequence separation	The simulation duration is 1.5 s. During 0.8~1.4 s, Uga and Ugb both drop 50%.
2	Verify the filtering performance of the improved SOGI on high-order harmonics	Simulation duration: 1.5 s. At 0.8 s, −17th and +19th harmonics are injected.
3	Verify the control strategies with three individual objectives	Simulation duration: 1.5 s. During 0.6~1 s, Uga and Ugb both drop 50%.
4	Verify the power–current collaborative control performance	The simulation duration is 3 s. During 0.5~2.5 s, Uga and Ugb both drop 50%. During 0.5~1 s, constant reactive power control. During 1.2~1.7 s, three-phase currents balance control. During 1.9~2.5 s, constant active power control.
5	Verify the overcurrent-limiting effect	The simulation duration is 1.5 s. During 0.5~0.9 s, Uga and Ugb both drop 50%.
6	Verify the effectiveness under weak grid conditions	The same simulation time settings as Case 4 with different Lg.
7	Verify the effectiveness under unbalanced grid conditions	The same simulation time settings as Case 5 with (1) Uga drop 30%; (2) Uga drop 30%, Ugb drop 50% and Ugc drop 70%.
8	Verify the effectiveness under unbalanced grid conditions	The same simulation time settings as Case 4 with (1) Uga drop 30%; (2) Uga drop 30%, Ugb drop 50% and Ugc drop 70%.

4.2. Verification of Improved SOGI

To prove the advantages of the sequence separation method by the proposed improved SOGI, three methods of notch filter (NF), DSOGI, and improved SOGI are used and compared. Case 1 is simulated. The waveforms are shown in Figure 9 and the separation time comparison is presented in Table 3. NF, DSOGI, and improved SOGI can effectively separate the positive sequence component. The separation time of the improved SOGI is the shortest among the three methods. In addition, the waveform during the separation process is smoother and more stable than the other two methods. It indicates that the proposed improved SOGI method is more efficient and stable for positive and negative sequence separation.

Figure 9. Waveform comparison of the three sequence separation methods.

Table 3. Comparison of Separation Time.

Separation Methods	Separation Time/ms (At the Beginning of the Fault)	Separation Time/ms (At the End of the Fault)
NF	26	25
SOGI	20	19.8
Improved SOGI	17.5	18.7

To verify the filtering performance of the improved SOGI on high-order harmonics, case 2 is simulated. The power grid current waveform and FFT analysis of the traditional VSG control and the VSG control with improved SOGI are displayed in Figure 10. Without the improved SOGI strategy, the THD of the current reaches 2.74%. The amplitudes of the −17th and +19th harmonics exceed 0.9%. When the improved SOGI is adopted, the THD of the output current is reduced to 1.57%. The amplitudes of the −17th and +19th harmonics are less than 0.6%. The results indicates that the control with the improved SOGI has better filtering performance on high-order harmonics and suppression effect on noise.

Figure 10. Current waveform and FFT analysis of the traditional VSG control (a) and the VSG control with improved SOGI (b).

4.3. Verification of Power–Current Collaborative Control Strategy

To verify the validation of the control strategies with the three individual objectives, case 3 is simulated. The results are displayed in Table 4. When control objective (a) is achieved, the peak value of active power fluctuation is obviously decreased, realizing constant output of active power. When control objective (b) is achieved, the unbalance rate of the output current is greatly decreased, and the three-phase output current maintains balanced operation. When control objective (c) is achieved, the peak value of reactive power fluctuation is substantially reduced, achieving constant reactive power output.

Table 4. Comparison of Simulation Results of Three Control Objectives.

Control Objectives	Active Peak/kW	Reactive Peak/kvar	Current Amplitude/A
			Imax	Imin
Objective (a)	30.7	7.5	121.4	79.5
Objective (b)	31.2	1.3	97.5	96.9
Objective (c)	36.5	0.72	119.5	81.2

To further verify the power–current collaborative control performance, case 4 is simulated. The simulation waveforms are displayed in Figure 11. The values −1, 0, and 1 of the correlation coefficient $\mu$ correspond to the control objectives of reactive power control, three-phase current balance control, and active power control. By unifying the control strategy in one equation as Equation (16), continuous operation and smooth switching among the three control objectives are achieved.

Figure 11. Simulation results of power–current collaborative control strategy.

4.4. Verification of Overcurrent-Limiting Control Strategy

To verify the overcurrent-limiting effect of virtual impedance and the supporting of VSG output reactive power on the power grid, case 5 is simulated on the basis of three-phase current balance control. Figure 12 and Figure 13 illustrate the waveforms without the overcurrent-limiting strategy and with the overcurrent-limiting strategy.

Figure 12. Waveforms without overcurrent-limiting strategy.

Figure 13. Waveforms with overcurrent-limiting strategy.

When the overcurrent-limiting control strategy is not adopted, the steady-state overcurrent reaches 97.2 A during the fault period. This is 1.51 times the rated current. The transient current reaches 321.2 A at the voltage drop beginning, which is 5 times the rated current. The transient current reaches 395.3 A at the voltage recovery beginning, which is 6.1 times the rated current. In addition, the power exhibits large fluctuations and overshoot.

When the overcurrent-limiting control strategy is adopted, the transient current is 96.3 A at the voltage drop beginning. The fault duration is only 7 ms. Then, the fault current is limited below the upper limit during fault and recovery. In addition, the fluctuation and overshoot of power are small at the time of fault occurrence and recovery. The converter increased reactive power output during the fault, which ensured the reactive power support for the power grid.

The comparison of simulation data without and with the current-limiting strategy are listed in Table 5. Wherein, $I_{F.s}$ represents the amplitude of steady-state overcurrent. $I_{F.t1}$ and $I_{F.t2}$ indicate the amplitude of transient overcurrent at fault beginning and fault ending, respectively. $t_{F1}$ and $t_{F2}$ indicate the duration of transient overcurrent at fault beginning and fault ending, respectively. When the overcurrent-limiting control strategy is adopted, the output current amplitude and transient overcurrent duration are significantly reduced, ensuring stable operation under an unbalanced power grid.

Table 5. The comparison of simulation results without and with overcurrent-limiting strategy. (System rated current I_n = 64.3 A).

Comparison		Without Overcurrent-Limiting	With Overcurrent-Limiting
Steady state	$I_{F.s}$/A	97.2	65.5
Transient	$I_{F.t1}$/A	321.2	96.3 A
	$t_{F1}$/ms	35	7
	$I_{F.t2}$/A	395.3	75.5
	$t_{F2}$/ms	44	5

4.5. Verification of the Proposed Strategy in Weak Grid

In the cases above, the inductance parameter L_g = 1 mH, i.e., SCR = 10. According to the IEEE std 1204-1997(R2003) standard [25], it corresponds to a strong grid scenario. To verify the validation of the proposed control strategy under weak grid conditions, case 6 is simulated. A weak grid (L_g = 2 mH, i.e., SCR = 2.5) and an extremely weak grid (L_g = 3 mH, i.e., SCR = 1.14) is simulated. The results are displayed in Figure 14.

Figure 14. The three-phase currents in case 6 with different L_g values.

Compared with the waveforms in Figure 11a, with the reduction in grid strength, the transient period lasts longer. However, under both of the weak grid scenarios, the system can regain stability, which demonstrates the applicability of the proposed control approach.

4.6. Applicability of the Proposed Strategies Under Different Unbalanced Scenarios

To verify the validation of the proposed strategy under different unbalanced conditions, case 7 and case 8 are simulated. The corresponding simulation results are depicted in Figure 15 and Figure 16. The proposed power–current collaborative control strategy is effective under both the unbalanced scenario of single-phase voltage drop and the unbalanced scenario of three-phase voltage drop to a different extent. Thus, the application of the proposed strategy to more unbalanced scenarios has been verified.

Figure 15. Waveforms under the scenario of U_ga drop 30%.

Figure 16. Waveforms under the scenario of U_ga drop 30%, U_gb drop 50% and U_gc drop 70%.

5. Conclusions

In an unbalanced power grid, the VSG shows the problems of output current imbalance, power fluctuation, and overcurrent. Thus, a VSG power–current collaborative control strategy based on an improved SOGI is proposed. To overcome the limitations of SOGI in sequence separation, an improved SOGI is proposed by adding a differential node and an extra SOGI module to the traditional SOGI. Both the DC component and high-order harmonic components are effectively suppressed. Then, collaborative control of the three objectives is achieved by introducing a correlation coefficient μ. In addition, virtual impedance and the reactive power reference value are introduced to prevent overcurrent and improve transient stability. Simulations have verified the effectiveness of the proposed strategy. It shows advantages in improving power quality and ensuring safe operation in an unbalance power gird. In our future research, the analysis of the overall system stability, parameter security domain, and the verification of hardware experiments, will be focused on. The applicability of the proposed strategy to the parallel operation of multiple VSGs will also be discussed.