Mechanism Analysis of Low-Frequency Oscillation Caused by VSG from the Perspective of Vector Motion

Abstract

Virtual synchronous generators (VSGs) have attracted widespread attention due to their advantage in supporting voltage and frequency of power systems. However, relevant studies have shown that a VSG has similar low-frequency oscillation as synchronous generators, which is more likely to occur under strong grid conditions. In this paper, the linearized mathematical model of a VSG is established by using small-signal analysis; based on this, the physical process of low-frequency oscillation of a VSG is explained from the perspective of vector motion. Firstly, the amplitude and phase motion of the current vector of a VSG under small disturbance are analyzed, then the mechanism of negative damping caused by terminal voltage control is revealed, and the reason why a VSG is more prone to instability under strong grid conditions is explained. Based on these, the influence of control and grid strength on the low-frequency oscillation of a VSG is analyzed. Studies show that the amplitude motion of the output current is the main cause of negative damping, and the oscillation can be suppressed by optimizing the value of key parameters.

1. Introduction

Due to the limited reserves and environmental pollution problems of fossil energy, renewable energy represented by wind power and photovoltaic power has been rapidly developed and applied in the power system, which is resulting in a large number of power electronic devices being connected to the grid [1]. As interfaces between the grid and renewable energies power generation, voltage source converters (VSCs) are widely used in the power system; their equipment characteristics are very different from traditional synchronous generators, which brings new challenges to stable operation of the power system [2,3,4,5]. There are two types of VSC currently used. One is the VSC that follows the grid through the phase-locked loop (PLL), and the other is the VSG that constructs the frequency by imitating the rotor dynamics of the synchronous generator [6].

The low-frequency oscillation of traditional VSCs has been studied by many scholars. The reason why a VSC is prone to low-frequency oscillation in weak grids is that a PLL reduces the oscillation damping by introducing a large phase lag around the frequency of the PLL bandwidth [7]. Similarly, the gains of the PLL, particularly at low SCRs, greatly affect the operation of the Voltage Source Converter-based High Voltage DC (VSC-HVDC) [8]. Relevant research shows that the effect of the interaction between a PLL and dc-link voltage control will damage the stability of the system by adding partial positive feedback to the dc-link voltage control in a weak grid [9]. The damping of the low-frequency oscillation mode of the system will be enhanced after the introduction of VSC-HVDC [10].

However, the low-frequency oscillation of a VSG is somewhat different from that of a VSC that follows a grid through PLL. Since the control strategy is different from that of synchronous generators, the influence of important control parameters on stability is studied [11,12]. Some scholars believe that the risk of low-frequency oscillations for a VSG is higher in strong grids [13]. In reference [14], a detailed eigenvalue analysis is conducted, showing that the strong-grid condition is harmful to the stability of a system with a VSG and SG. Similar results are derived in reference [15]. In a multi-VSG system, reference [16] found that the dominant poles move left when the system strength decreases. The same conclusion is obtained in a doubly fed induction generator (DFIG) and permanent magnet synchronous generator (PMSG) using virtual synchronous control [17,18]. The low-frequency oscillation of a VSG whose mathematical model is highly similar to that of the synchronous generator was analyzed in a single-machine infinite bus system [19]; then, the range of the damping coefficient was derived when the low-frequency oscillation of the VSG converged. In [20], a small-signal model of a single VSG infinite bus system is established, and the effect of a VSG on the low-frequency oscillation of the system is studied using the frequency domain analysis method. Reference [21] considered the influence of reactive power control and dq-axis voltage control, and the results showed that they had a great influence on the damping of low-frequency oscillation of a VSG. Reference [22] analyzes the small-signal stability of single-VSG and multi-VSG grid-connected systems by the eigenvalue analysis method, and the influence of parameter changes, including the virtual moment of inertia, virtual damping coefficient, line resistance, and line inductance, was discussed on system stability. Reference [23] further identifies the low-frequency oscillation of a multi-machine system using a variational mode decomposition and a Prony algorithm and finds two dominant oscillation modes without detailed models, and shows that the damping ratio decreases with the increase in short-circuit ratio. The interaction between the terminal voltage loop and the power synchronization on loop is regarded as the major cause of the low-frequency oscillation of a VSG in reference [24]. The existing research is not deep enough to understand the low-frequency oscillation of a VSG and cannot explain the generation mechanism of negative damping. Therefore, it is necessary to analyze the intermediate variables and find out the source of negative damping.

In this paper, the low-frequency oscillation problem of a VSG grid-connected system is analyzed from the perspective of vector motion. The main contributions of this paper are:

The two-dimensional amplitude and phase motion of the current vector caused by the terminal voltage control is clarified.

The effect caused by the amplitude and phase motion of the current vector on stability is analyzed.

The impact of key parameters on stability is summarized.

The rest of this paper is organized as follows. In Section 2, the control strategy of the VSG is introduced, and a simplified model is constructed. In Section 3, the oscillation mechanism is analyzed, and influencing factors are discussed. Simulations are conducted for verification in Section 4. Conclusions are drawn in Section 5, and the contributions of this paper are summarized.

2. Modeling of VSG

The typical vector control structure of a VSG connected to an infinite power grid is shown in Figure 1. The VSG is filtered by an LCL filter and connected to an infinite power grid through a transmission line, whose resistance is not considered. In addition, the dynamics of the VSG’s DC side is not considered; no matter what the energy source of the DC side is, the research involved in this paper has universal applicability. A power synchronization control loop (PSC), reactive-voltage control outer loop, terminal voltage control loop (TVC) and AC current control loop (ACC) are included in the topological structure of the VSG; the PSC simulates the rotor swing process of synchronous generators, and the TVC and ACC improve the waveform of the VSG’s output voltage and current. Next, small-signal models of PSC, TVC and ACC will be introduced in turn.

2.1. Power Synchronization Control Loop

A PSC constructs the frequency of a VSG by simulating the rotor swing motion of synchronous generators and realizes the connection between the VSG and the power grid. The power synchronization control loop in Figure 1 can be expressed as

$$\begin{array}{ccccc}& & & & \end{array}\left\{\begin{array}{l}sJ\mathsf{\Delta }\omega =\mathsf{\Delta }{P}_{ref}-\mathsf{\Delta }{P}_e-D\mathsf{\Delta }\omega \\ s\mathsf{\Delta }{\theta }_s={\omega }_0\mathsf{\Delta }\omega \end{array}\right.\begin{array}{ccccc}& & & & \end{array}$$

(1)

where ω and ω₀, respectively, represent the per unit speed of the VSG and the reference speed, P_ref is the reference per unit value of the active power, P_e is its actual value calculated according to the measured terminal voltage and current, θ_s is the output power angle of the PSC, D is the virtual damping coefficient and J is the inertia coefficient. Unless otherwise specified, the physical signals involved in this paper are per unit value.

2.2. Reactive-Voltage Control Outer Loop

The reactive-voltage control outer loop combines reactive power control and AC terminal voltage control and sets the weight of them. This link can control a VSG’s reactive power and root mean square (RMS) value of the node voltage; its output will be transferred to the terminal voltage control loop as the d-axis voltage reference value. The expression for this procedure is

$$\begin{array}{cccccc}& & & & & \end{array}\mathsf{\Delta }{v}_{dref}=-{\displaystyle \frac{\mathsf{\Delta }{V}_t+\mathsf{\Delta }Q}{Ks}}\begin{array}{cccccc}& & & & & \end{array}$$

(2)

where Q_ref, V_tref, Q and V_t are the reference and calculated values of reactive power and node voltage, respectively; K_q is the voltage drop coefficient, whose magnitude determines the proportion of terminal voltage unbalance in the d-axis voltage reference value; and K is the integral coefficient of the reactive-voltage control outer loop; its value represents the speed of the integrator.

2.3. Terminal Voltage Control Loop

As can be seen from Figure 1, the terminal voltage loop tracks the voltage from two branches based on a dq reference frame. The PI controllers of two branches adopt the same proportional control parameter K_p1 and integral control parameter K_i1. The outputs of the integrators of the dq axis voltage integrators are expressed by x₁ and x₂, respectively. The state-space equation as shown below can be obtained as follows:

$$\begin{array}{ccc}& & \end{array}\left\{\begin{array}{l}s\mathsf{\Delta }{x}_1=K_{i1}(\mathsf{\Delta }{v}_{tdref}-\mathsf{\Delta }{v}_{td})\\ s\mathsf{\Delta }{x}_2=K_{i1}(\mathsf{\Delta }{v}_{tqref}-\mathsf{\Delta }{v}_{tq})\\ \mathsf{\Delta }{i}_{dref}=K_{p1}(\mathsf{\Delta }{v}_{tdref}-\mathsf{\Delta }{v}_{td})+\mathsf{\Delta }{x}_1-{\omega }_0{C}_f\mathsf{\Delta }{v}_{tq}\\ \mathsf{\Delta }{i}_{qref}=K_{p1}(\mathsf{\Delta }{v}_{tqref}-\mathsf{\Delta }{v}_{tq})+\mathsf{\Delta }{x}_2+{\omega }_0{C}_f\mathsf{\Delta }{v}_{td}\end{array}\right.\begin{array}{ccc}& & \end{array}$$

(3)

where v_tdref, v_tqref and v_td, v_tq represent the reference value and the instantaneous value of the dq axis component of the terminal voltage, respectively; i_tdref, i_tqref represent the reference value of the dq axis component of the output current, respectively; and C_f is the capacitance in the LCL filter.

2.4. AC Current Control Loop

The outputs of the integrators of the dq axis current integrators are expressed by x₃ and x₄, respectively. Two PI controllers of the ACC adopt the same proportional control parameter K_p2 and integral control parameter K_i2. The state-space equation of ACC can be expressed as

$$\begin{array}{cc}& \end{array}\left\{\begin{array}{l}s\mathsf{\Delta }{x}_3=K_{i2}(\mathsf{\Delta }{i}_{dref}-\mathsf{\Delta }{i}_d)\\ s\mathsf{\Delta }{x}_4=K_{i2}(\mathsf{\Delta }{i}_{qref}-\mathsf{\Delta }{i}_q)\\ \mathsf{\Delta }{e}_d=\mathsf{\Delta }{v}_{td}+K_{p2}(\mathsf{\Delta }{i}_{dref}-\mathsf{\Delta }{i}_d)+\mathsf{\Delta }{x}_3-{\omega }_0{L}_{f1}\mathsf{\Delta }{i}_q\\ \mathsf{\Delta }{e}_q=\mathsf{\Delta }{v}_{tq}+K_{p2}(\mathsf{\Delta }{i}_{qref}-\mathsf{\Delta }{i}_q)+\mathsf{\Delta }{x}_4+{\omega }_0{L}_{f1}\mathsf{\Delta }{i}_d\end{array}\right.\begin{array}{cc}& \end{array}$$

(4)

where i_gd, i_gq and i_d, i_q represent the instantaneous value of the dq axis component of the current flowing into the grid and the output current of a VSG, respectively; e_d, e_q represent the instantaneous value of the dq axis component of the internal voltage, respectively; and L_f1 is the inductance near the VSG side of the LCL filter.

2.5. Coordinate Transformation

Assuming that the frequency of grid voltage V_g remains constant, taking the grid voltage as the X-axis of the common rotating coordinate frame, the grid voltage is converted to dq axis voltage v_gd and v_gq at the synchronous rotating coordinate frame based on the output power angle θ_s of PSC. Its expression can be written as

$$\begin{array}{ccccc}& & & & \end{array}\left[\begin{array}{l}\mathsf{\Delta }{v}_{gd}\\ \mathsf{\Delta }{v}_{gq}\end{array}\right]=-V_g\left[\begin{array}{l}\sin {\theta }_{s0}\\ \cos {\theta }_{s0}\end{array}\right]\mathsf{\Delta }{\theta }_s\begin{array}{ccccc}& & & & \end{array}$$

(5)

2.6. LCL Filter

The state-space equation of LCL filter can be expressed as

$$\begin{array}{cccc}& & & \end{array}\left\{\begin{array}{l}s{L}_{f1}\mathsf{\Delta }{i}_d=\mathsf{\Delta }{e}_d-\mathsf{\Delta }{v}_{td}+{\omega }_0{L}_{f1}\mathsf{\Delta }{i}_q\\ s{L}_{f1}\mathsf{\Delta }{i}_q=\mathsf{\Delta }{e}_q-\mathsf{\Delta }{v}_{tq}-{\omega }_0{L}_{f1}\mathsf{\Delta }{i}_d\\ s{L}_g\mathsf{\Delta }{i}_{gd}=\mathsf{\Delta }{v}_{td}-\mathsf{\Delta }{v}_{gd}+{\omega }_0{L}_g\mathsf{\Delta }{i}_{gq}\\ s{L}_g\mathsf{\Delta }{i}_{gq}=\mathsf{\Delta }{v}_{tq}-\mathsf{\Delta }{v}_{gq}-{\omega }_0{L}_g\mathsf{\Delta }{i}_{gd}\\ s{C}_f\mathsf{\Delta }{v}_{td}=\mathsf{\Delta }{i}_d-\mathsf{\Delta }{i}_{gd}+{\omega }_0{C}_f\mathsf{\Delta }{v}_{tq}\\ s{C}_f\mathsf{\Delta }{v}_{tq}=\mathsf{\Delta }{i}_q-\mathsf{\Delta }{i}_{gq}-{\omega }_0{C}_f\mathsf{\Delta }{v}_{td}\end{array}\right.\begin{array}{cccc}& & & \end{array}$$

(6)

where L_g is the sum of L_f2 and L_tg.

2.7. Power Calculation

The expressions of active and reactive power in the dq axis are, respectively

$$\begin{array}{ccc}& & \end{array}\left\{\begin{array}{l}\mathsf{\Delta }{P}_e=v_{td0}\mathsf{\Delta }{i}_d+v_{tq0}\mathsf{\Delta }{i}_q+i_{d0}\mathsf{\Delta }{v}_{td}+i_{q0}\mathsf{\Delta }{v}_{tq}\\ \mathsf{\Delta }Q=v_{tq0}\mathsf{\Delta }{i}_d-v_{td0}\mathsf{\Delta }{i}_q+i_{d0}\mathsf{\Delta }{v}_{tq}-i_{q0}\mathsf{\Delta }{v}_{td}\end{array}\right.\begin{array}{ccc}& & \end{array}$$

(7)

The complete state-space model of a VSG can be obtained by combining the above equations, as follows:

$$s\mathsf{\Delta }X=A\mathsf{\Delta }X$$

(8)

Equation (8) introduces the mathematical model of a VSG in detail, but it is too complex for a theoretical analysis of low-frequency oscillations and increases the difficulty of the study.

2.8. Simplification of Model

A detailed mathematical model of a VSG is given above, which covers multiple time scales. In the study of low-frequency oscillations of a VSG, the dynamics of some timescales may not be necessary to consider. Therefore, this paper compares the dynamic process of a VSG under different models after a small disturbance, as shown in Figure 2. As can be seen from Figure 2, the frequency waveform of a VSG under a small disturbance after ignoring the dynamics of an AC current loop almost coincides with that before ignoring the dynamics, while a VSG hardly oscillates at a low frequency under small disturbances when the dynamics of the terminal voltage loop are ignored. The relative error when ignoring the ACC dynamic is within 0.0005%. It can be concluded that the dynamic process of the AC current time scale can be ignored when studying the low-frequency oscillation of a VSG, but the effect of the terminal voltage loop must be taken into account. Therefore, in the following research, this paper will no longer consider the dynamics of an AC current time scale, but will consider the instantaneous value of a VSG’s current equal to the command value; at this time, the VSG is equivalent to a current source. The VSG is then connected to the infinity grid by an equal inductor, and the filter inductance in the LCL filter is no longer considered; thus, i_gd = i_d, i_gq = i_q. The control structure and grid-connected topology of the VSG shown in Figure 1 can be represented as equivalent to Figure 3.

In the simplified model, the terminal voltage vector of a VSG moves forward and merges with the internal voltage. The phase of the grid voltage is taken as the X-axis of the common coordinate system; the active power and reactive power emitted by the VSG can be expressed in a more concise way in the common coordinate system.

$$\left\{\begin{array}{l}{P}_e=V_{g}I\cos {\theta }_I\\ Q=-V_{g}I\sin {\theta }_I+I^2{X}_g\end{array}\right.$$

(9)

Equation (9) can be linearized as

$$\begin{array}{ll}& \end{array}\left\{\begin{array}{ll}\mathsf{\Delta }{P}_e& =V_g\cos {\theta }_{I0}\mathsf{\Delta }I-V_g{I}_0\sin {\theta }_{I0}\mathsf{\Delta }{\theta }_I\\ & =K_{IP}\mathsf{\Delta }I+K_{\theta P}\mathsf{\Delta }{\theta }_I\\ \mathsf{\Delta }{Q}_e& =(2I_0{X}_g-V_g\sin {\theta }_{I0})\mathsf{\Delta }I-V_g{I}_0\cos {\theta }_{I0}\mathsf{\Delta }{\theta }_I\\ & =K_{IQ}\mathsf{\Delta }I+K_{\theta Q}\mathsf{\Delta }{\theta }_I\end{array}\right.\begin{array}{cc}& \end{array}$$

(10)

where θ_I represents the phase of the current vector relative to the grid voltage.

3. Mechanism Analysis

3.1. Mechanism Explanation without the Dynamic of v_tdref

In this paper, the process of low-frequency oscillation of a VSG will be analyzed from the perspective of vector motion. Assuming that the output phase of power synchronization control (θ_s) is greater than the steady-state value, since the steady-state value of θ_s (θ_s0) is generally a positive number slightly greater than zero, it can be obtained that the dq-axis component (v_td and v_tq) of the terminal voltage vector will be less than their reference value; then, the difference will pass through the PI controller of the terminal voltage loop, causing the dq-axis component of the current vector to increase, which further causes amplitude and phase motion of the current vector. When the amplitude and phase of the current vector change, the voltage drop on the network inductor will also change, thus affecting the terminal voltage vector. Therefore, the change of the terminal voltage vector will affect its own change by affecting the change of the current vector. This process described above is depicted in Figure 4. The amplitude and phase motion of the current vector will also cause the active power output of a VSG to vary according to Equation (10), then affect the damping of low-frequency oscillation.

Based on the above-mentioned physical processes, the damping provided by amplitude and phase motion are studied in this paper. As shown in Figure 5, the linearized model of a VSG has three feedback paths, where G_θsI(s) is the transfer function from Δθ_s to the amplitude increment (ΔI), and G_θsθdq(s) is the transfer function from Δθ_s to the phase increment (Δθ_dq) in the dq-axis frame. K_IP and K_θP represent the impact of the amplitude and phase of a VSG’s current vector on active power, respectively. Path 1 reflects the influence of phase variation of the current vector on the active power caused by the change of θ_s. Since K_θP is a coefficient less than zero, there is no quantity that is orthogonal to Δθ_s in ΔP_e1. This path does not provide damping torque, but only provides negative synchronous torque, whereas the path in synchronous generators provides positive synchronous torque because K₁ is positive, which is a difference between a VSG and synchronous generators.

The damping torque and synchronous torque provided by paths 2 and 3 can be expressed by the following equation.

$$\begin{array}{c}\begin{array}{ccc}& & \end{array}\end{array}\left\{\begin{array}{l}{D}_{path2}=\mathrm{Re}(K_{IP}{\displaystyle \frac{{\omega }_{base}}{s}}{G}_{\theta sI}(s){|}_{s=j{\omega }_d})\\ S_{path2}=\mathrm{Re}(K_{IP}{G}_{\theta sI}(s){|}_{s=j{\omega }_d})\\ D_{path3}=\mathrm{Re}(K_{\theta P}{\displaystyle \frac{{\omega }_{base}}{s}}{G}_{\theta s\theta dq}(s){|}_{s=j{\omega }_d})\\ S_{path3}=\mathrm{Re}(K_{\theta P}{G}_{\theta s\theta dq}(s){|}_{s=j{\omega }_d})\end{array}\right.\begin{array}{c}\begin{array}{ccc}& & \end{array}\end{array}$$

(11)

where f_d is the frequency of low-frequency oscillation, and D_path2, S_path2, D_path3 and S_path3 represent the damping torque and synchronization torque provided by the two paths, respectively.

Path 2 reflects the influence of the change of θ_s on active power through amplitude motion of the current vector. As can be seen from Figure 6, the phase of G_θsI(s) ranges from −90° to 0° in the frequency band of low-frequency oscillation, so ΔI has a phase lag of 0–90° relative to Δθ_s, and the phase of ΔI on the ω-θ plane is shown in Figure 7. Since K_IP is greater than 0, ΔP_eI is also 0–90° behind Δθ_s in phase, and the amplitude motion provides negative damping. Path 3 represents the influence of phase motion of the current vector relative to the dq-axis frame on active power caused by the change of θ_s. Although K_θP is less than 0, the phase of G_θsθdq(s) is greatly affected by control parameters and grid strength, and this path may provide positive or negative damping.

When the short-circuit ratio (SCR) is 2.5 and typical control parameters are adopted [21], synchronous torque and damping torque provided by the three paths are shown in Figure 8 and Figure 9, respectively. It can be seen from Figure 8 that the positive synchronous torque provided by path 2 is much larger than the negative synchronous torque provided by path 1 and path 3, so it is not necessary to consider the problem of synchronous torque instability. It can be seen from Figure 9 that the absolute damping value provided by amplitude motion is greater than that provided by phase motion. This is because a VSG mainly outputs active power under normal circumstances, resulting in the absolute value of K_IP being far greater than the absolute value of K_θP in the steady state of operation. Therefore, the amplitude motion of the current vector is the main cause of negative damping.

3.2. Mechanism Explanation with the Dynamic of v_tdref

When the reactive-voltage outer loop is taken into account, v_dref is no longer a fixed constant. Compared to the case without the reactive-voltage outer loop, when θ_s is disturbed, it will directly cause changes in the output reactive power and the terminal voltage vector of a VSG, resulting in Δv_dref, Δv_dref will further affect the amplitude and phase of the current vector. At this time, the form of small signal model shown in Figure 5 for analyzing low-frequency oscillation of a VSG remains unchanged, and only the expressions for G_θsI(s) and G_θsθdq(s) become more complex, so the low-frequency oscillation mechanism of a VSG will not change. G_θsI(s) reflects the influence of the change of θ_s on the amplitude motion of the current vector by changing the reactive power and the terminal voltage. G_θsθdq(s) reflects the influence of the change of θ_s on the phase motion of the current vector through reactive power and the terminal voltage. In order to explore the influence of the reactive-voltage outer loop on the low-frequency oscillation of a VSG, the following two figures show the comparison of the bode chart of G_θsI(s) and G_θsθdq(s) with or without the reactive-voltage outer loop.

Since the actual oscillation frequency is mostly in the range of 1–1.5 Hz, it can be seen from the above two figures that after the addition of the reactive-voltage outer loop, the gain and phase lag of G_θsI(s) are increased, and the gain and phase lead of G_θsθdq(s) are also increased, so the negative damping caused by the amplitude motion and phase motion of the current vector are increased. Therefore, the introduction of the reactive-voltage outer loop will intensify the amplitude and phase motion of a VSG’s current vector, resulting in a greater negative damping component in the electromagnetic power returned by path 2 and path 3, which further threatens the safe and stable operation of the system. However, the reactive-voltage outer loop does not change the mechanism of the VSG’s low-frequency oscillation. Compared with Figure 10 and Figure 11, the gain of G_θsI(s) is much greater than that of G_θsθdq(s), which is one reason why the amplitude motion of the current vector plays a decisive role in the generation of negative damping. If you want to highlight the role of the reactive-voltage outer loop, its generated action can be extracted as a separate path.

3.3. Influencing Factor

When there is no reactive-voltage outer loop, the expression for G_θsI(s) and G_θsθdq(s) is relatively simple, and their expressions are as follows:

$$\begin{array}{cc}& \end{array}\left\{\begin{array}{l}{G}_{\theta sI}(s)={\displaystyle \frac{V_g{G}_{PI}(s)(X_g\cos {\theta }_{I0}{G}_{PI}(s)+\sin {\theta }_{I0})}{1+X_g^2{G}_{PI}^2(s)}}\\ G_{\theta s\theta dq}(s)={\displaystyle \frac{V_g{G}_{PI}(s)(\cos {\theta }_{I0}-X_g\sin {\theta }_{I0}{G}_{PI}(s))}{I_0(1+X_g^2{G}_{PI}^2(s))}}\end{array}\right.\begin{array}{cc}& \end{array}$$

(12)

where G_PI(s) is equal to K_p1+K_i1/s.

When the initial state of the system is determined, the amplitude and phase motion of the VSG’s current vector are jointly determined by grid strength and the PI controller’s parameters of the terminal voltage control. When the reactive-voltage outer loop is taken into account, K_q and K will be introduced into the expression of G_θsI(s) and G_θsθdq(s). Next, the influence of these five parameters on the current vector of a VSG will be analyzed in turn. Based on the previous analysis, the following paper focuses on the influence of these parameters on the amplitude motion of the current vector. Except for the parameters analyzed in each case, the values in the following

Table 1. The values of simulation parameters.

Parameter	Value	Parameter	Value
Sbase	600 MVA	Vbase	800 V
D	60	J	8
Pref	0.7	Vg	1.05
K	10	Kq	100
Kp1	3.5	Ki1	5
Kp2	0.5	Ki2	20
Lf1	0.01 pu	Lf2	0.001 pu
Cf	0.05 pu	Ltg	0.289 pu
LT	0.1 pu	Kfv	1

are used for the parameters.

When these parameters are within the range selected in the following paper, the frequency of low-frequency oscillation is maintained between 1.2–1.7 Hz, so this paper only focuses on the Bode diagram of G_θsI(s) in this frequency band. Figure 12a shows the change of G_θsI(s)’s Bode diagram with K_p1. With the increase in K_p1, the gain of the current vector’s amplitude motion increases, and phase lag decreases, but the trend of this increase is slowing down. At this time, the change of damping of low-frequency oscillation cannot be qualitatively judged according to Equation (12), so the root locus shown in Figure 12b can be considered for quantitative analysis. Figure 12b shows that the damping of low-frequency oscillation increases with the increase in K_p1. When K_p1 increases from 2 to 5, the damping ratio increases from −0.0371 to 0.2669, and the oscillation frequency increases from 1.2760 Hz to 1.4143 Hz.

Figure 13a shows the change of G_θsI(s)’s Bode diagram with K_i1. With the increase in K_i1, the gain of the current vector’s amplitude motion increases, but the phase lag decreases. Therefore, the effect of K_i1 on negative damping cannot be explained simply from the angle of the current vector’s amplitude motion. This paper uses the root locus diagram shown in Figure 13b for analysis. Figure 13b shows that the damping of low-frequency oscillation decreases with the increase in K_i1. When K_i1 increases from 4 to 10, the damping ratio increases from 0.1827 to 0.1534, and the oscillation frequency increases from 1.3690 Hz to 1.4578 Hz. The damping of low-frequency oscillation is not significantly affected by K_i1.

Figure 14a shows the change of G_θsI(s)’s Bode diagram with X_g. With the increase in grid strength, the gain and phase lag of the current vector’s amplitude motion significantly increase, and the negative damping provided by the amplitude motion of the VSG’s current vector increases. Therefore, the VSG is more prone to low-frequency oscillation under strong grid conditions. Figure 14b also illustrates this. When X_g is 0.2 pu, the system is at risk of destabilizing with low-frequency oscillation. When X_g increases from 0.2 pu to 0.5 pu, the damping ratio increases from −0.0661 to 0.2531, and the oscillation frequency decreases from 1.5033 Hz to 1.2726 Hz.

Figure 15a and Figure 16a, respectively, show the change of G_θsI(s)’s Bode diagram with K_q and K. With the increase in K_q, the gain of the current vector’s amplitude motion increases; with the increase in K, the gain of the current vector’s amplitude motion decreases. In both cases, the change of the phase lag is not monotonic, and there is a crossover situation. Therefore, the root locus shown in Figure 15b and Figure 16b are needed to quantitatively analyze the influence of K_q and K on the damping of low-frequency oscillation. From Figure 15b and Figure 16b, it can be seen that the damping of low-frequency oscillation decreases first and then increases with the increase in K_q and K. The difference is that the frequency of low-frequency oscillation increases as K_q increases, and the frequency of low-frequency oscillation decreases as K increases. When K_q increases from 50 to 200, the damping ratio decreases from 0.2042 to 0.1760 and then increases to 0.2051, and the oscillation frequency increases from 1.2093 Hz to 1.5531 Hz. When K increases from 1 to 20, the damping ratio decreases from 0.3373 to 0.2040, and the oscillation frequency decreases from 1.6449 Hz to 1.2051 Hz.

To sum up, by combining the vector motion and damping torque analysis, it can be seen that the increase in the proportional coefficient of the TVC is beneficial to system stability, while the influence of other parameters are not apparent or non-monotonic.

4. Simulation Verification

In order to verify whether the influence rules of different parameters on the low-frequency oscillation of a VSG obtained in the previous section are correct, the corresponding model is built in the MATLAB 2022a/Simulink platform for verification. The perturbation is a step signal with a duration of 0.1 s and an amplitude of 0.03 pu applied to the P_ref at the corresponding time. The time domain simulation results are shown in Figure 17.

Figure 17a illustrates that the decrease in K_p1 deteriorates system stability, and Figure 17b shows that the increase in K_i1 slightly destabilizes the system. When K_q and K increase, the system stability first weakens, and then strengthens, as shown in Figure 17c,d. Finally, the impact of grid strength is shown in Figure 17e. The system becomes more stable when the grid strength decreases.

The time domain simulation results are consistent with the analysis results in the previous section, which proves that the influence rules of the five parameters (K_p1, K_i1, K_q, K and X_g) on the low-frequency oscillation of a VSG are correct.

5. Conclusions

The low-frequency oscillation of a VSG grid-connected system is studied in this paper. The contributions of this paper can be summarized as follows:

The negative damping of the amplitude motion of the current vector is caused by the phase lag of the amplitude of the current vector caused by the terminal voltage control loop.

The increase in grid strength significantly increases the negative damping provided by the amplitude motion of the current vector of the VSG, which makes the VSG more prone to low-frequency oscillation under strong grid conditions.

The influence of key control parameters is analyzed, illustrating that increasing the proportional parameter of the TVC is very beneficial to enhancing system stability.

The analysis in this paper proposes a clear and intuitive method to study the control effect on stability and clarifies the interaction between voltage and current vectors. According to the analysis, the control parameters can be tuned to pursue better stability performance. Further studies include expanding the study into multi-machine systems and considering more realistic operating conditions.