An Active Power Dynamic Oscillation Damping Method for the Grid-Forming Virtual Synchronous Generator Based on Energy Reshaping Mechanism

Abstract

The grid-forming virtual synchronous generator (GFVSG) with large virtual inertia can provide a friendly grid-connected operational mode for power electronic converters, but it may also introduce the active power dynamic oscillation problems similar to traditional synchronous generators. In view of this, the dynamic equivalent circuit model of the GFVSG grid-tied active power-angle is established firstly, and, then, the understanding of the GFVSG active power oscillations under variable disturbances is revealed from the perspective of circuit energy flow in this paper. On this basis, an active power dynamic oscillation damping method based on an energy reshaping mechanism for the GFVSG is proposed, and a parameter design method using the second-order equivalent reduced-order control model is given. The MATLAB 2016a simulation as well as experimental test platforms of a 100 kV·A GFVSG grid-connected system are established, then, both the feasibility and effectiveness of the proposed active power dynamic oscillation damping method are verified by using the simulation and experimental comparison results.

1. Introduction

The grid-forming virtual synchronous generator (GFVSG) can provide a certain degree of voltage regulation and inertial support for the grid-connected operation of power electronic converters (PECs) by simulating the primary voltage equation and rotor motion equation of traditional synchronous generators (TSGs), which improves the operation reliability of large-scale renewable energy integration to the power grid, and has attracted wide attention in recent years [1,2]. However, GFVSG with large virtual inertia will inevitably exhibit the grid-connected active power dynamic oscillation problems similar to TSGs when dealing with different disturbances such as the active power command step and the power grid frequency change [3,4]. As the overcurrent tolerance of PECs is usually weak, the large instantaneous current in the GFVSG active power dynamic oscillation process is easy to cause overcurrent protection action or even equipment damage to PECs, thus reducing the GFVSG operation reliability [5].

Damping control methods play a very important role in suppressing the dynamic oscillations of the GFVSG grid-connected active power and its output frequency, which mainly includes three types: the adaptive parameter adjustment method, feedback channel-based compensation method, and feedforward path compensation method [6]. Among them, the adaptive parameter adjustment method improves the ability of GFVSG to suppress its active power dynamic oscillation through the adaptive adjustment of virtual inertia, virtual damping, virtual impedance, or primary frequency modulation parameters. In [7,8], the adaptive adjustment method of the virtual inertia parameter is achieved by detecting the change rule of the GFVSG output frequency. In [9,10], the damping of the GFVSG grid-connected system is further optimized by taking advantage of the fact that both the virtual inertia and the virtual damping parameters of GFVSG can be adjusted adaptively. The damping optimization method of a GFVSG grid-connected system based on an adaptive virtual impedance scheme is proposed in [11]. A GFVSG transient damping control algorithm based on the adaptive adjustment of the GFVSG primary frequency modulation parameter according to different frequency segments is achieved in [12]. It is worth pointing out that the adaptive parameter adjustment method in [7,8,9,10,11,12] has the disadvantages of the parameter adjustment range being difficult to determine and the design process being complicated, and the nonlinear change in adaptive parameters will also affect the operation stability of the GFVSG grid-connected system.

The feedback channel-based compensation method mainly introduces the angular frequency variation or the active power variation into the GFVSG control loop through a feedback channel to suppress the GFVSG active power dynamic oscillation. In [13,14], a first-order low-pass filter (LPF) is used to detect the angular frequency dynamic deviation, which is then fed back to the GFVSG active power command through a proportional link. In [15,16], the dynamic deviation between the GFVSG angular frequency and the grid angular frequency detected using phase locked loop (PLL) is fed back to the GFVSG control loop via a proportional link to simulate the function of the damping windings of TSGs, but the introduction of the PLL will create an undesired negative damping effect on the GFVSG stable operation. In [17,18], both the dynamic variations in the angular frequency and the active power are introduced into the GFVSG control loop through feedback channels to enhance the dynamic damping of the system. It is worth pointing out that, although the above-mentioned damping methods based on feedback channel compensation can effectively suppress the GFVSG active power dynamic oscillation, their damping effects only take effect after the deviation of the controlled variable, which has a certain passivity. Therefore, for a GFVSG grid-tied system with large virtual inertia, if the system disturbance is frequent and large enough, continuous active power dynamic oscillations may be observed in the dynamic response process, which requires a long regulation time to ensure that the GFVSG grid-connected system reaches a new steady state [19].

The damping method based on feedback channel compensation is biased towards passive feedback tracking, while the damping method based on feedforward path compensation is biased towards active feedforward compensation. In [20,21], the GFVSG angular frequency variation is fed forward to its reactive power control loop by referring to the principle of a power system stabilizer, but the dynamic coupling between the active power and reactive power control loops is induced, so that the complex nonlinear characteristics may be introduced into the dynamic response process of the system. In [22], a GFVSG damping optimization method based on the active power command feedforward is proposed, but the control parameter design has the problems of a high dependence on the accuracy of the line impedance parameters and poor applicability to the scenario of the power grid frequency disturbance. In [23,24], a GFVSG damping improved control strategy based on phase feedforward compensation is proposed from the perspective of a reconstructed damping controller, which effectively solves the problem of the high dependence on the system parameters that existed in [22]. However, the open-loop gain of the GFVSG control system has a reduced decline slope in the high frequency band, which weakens the ability of the GFVSG system to resist high-frequency noise. In [25], two GFVSG damping improved control methods based on the active-power differential feedback compensation (ADFC) and the active-power differential feedforward compensation (ADFF) are proposed to enhance the ability to suppress high-frequency noise, but the active power regulation time or the frequency overshoot of the GFVSG system still need to be further improved.

To address the above issues, this paper further proposes an active power dynamic oscillation damping method based on energy reshaping mechanism (ERM) for GFVSG (hereafter referred to as ERM-GFVSG). ERM-GFVSG enhances its ability to suppress the active power dynamic oscillation by introducing the feedback channel-based compensation damping method using the dynamic variations in the angular frequency and the active power into the GFVSG control structure. Compared with the existing damping control methods, the proposed ERM-GFVSG significantly obtains a better disturbance rejection ability and dynamic response performance. The main contributions made in this paper as well as their significance are summarized as follows:

(1)

Both the closed-loop small signal model and the dynamic equivalent circuit model of GFVSG are established successively. Therefore, it reflects that the active power command step and the power grid frequency disturbance are regarded as excitation sources to inject dynamic oscillation energy into the GFVSG from the perspective of circuit energy flow.

(2)

Compared with GFVSG, ERM-GFVSG increases the total amount of the consumed energy and the consumption speed of the dynamic oscillation energy on the premise of keeping the total dynamic oscillation energy and the total stored energy unchanged. Accordingly, it is equivalent to increasing the system damping, so as to enhance the ability of suppressing the active power dynamic oscillation.

(3)

The proposed active power dynamic oscillation damping method only works in the dynamic response process, so it does not affect the active power steady-state deviation. Practically, compared with the existing methods, ERM-GFVSG has the smallest frequency overshoot under the active power command step and the shortest active power regulation time under the power grid frequency disturbance.

The rest of the paper is organized as follows. In Section 2, the closed-loop small signal model as well as the dynamic equivalent circuit model of GFVSG are established and the active power dynamic oscillation mechanism is analyzed. In Section 3, the ERM-GFVSG control method is proposed and its parameter design method is presented. In Section 4, the simulation and experimental comparison results are provided. Finally, the conclusions are drawn in Section 5.

2. Control Model and Active Power Dynamic Oscillation Mechanism of GFVSG

2.1. Overview of GFVSG Grid-Connected System

Figure 1 illustrates the circuit topology and the control block diagram of GFVSG grid-connected system [4].

The power stage consists of a DC source, denoted by U_dc, a line equivalent impedance Z_line, a three-phase PEC and an LC filter, where L_f and C_f are the filter inductor and filter capacitor, respectively. u_abc, i_abc, u_gabc, and $u_{\mathrm{abc}}^{*}$ are the output voltage, output current, power grid voltage, and output voltage command, respectively. The GFVSG control principle mainly consists of the power calculation, the stator electrical equation, the rotor motion equation, and the inner control loop. A traditional cascaded voltage and current controller can be used in the inner control loop to achieve a better dynamic response performance [5]. According to Figure 1, the rotor motion equation and the primary voltage equation of GFVSG can be expressed as follows:

$$P_{\mathrm{ref}}-P_{\mathrm{e}}=J{\omega }_0\frac{\mathrm{d}(\omega -{\omega }_0)}{\mathrm{d}t}+D{\omega }_0(\omega -{\omega }_0),$$

(1)

$$E=E_0+k_{\mathrm{q}}\left(Q_{\mathrm{ref}}-Q_{\mathrm{e}}\right),$$

(2)

where P_ref, P_e, Q_ref, and Q_e are the active power command, active power, reactive power command, and reactive power, respectively; J, D, ω₀, and ω are the virtual inertia, virtual damping, rated angular frequency, and output angular frequency, respectively; k_q, E₀, and E are the primary voltage regulation coefficient, rated voltage amplitude, and output voltage amplitude, respectively.

Note that this paper focuses on the GFVSG active power dynamic oscillation derived from the active power control loop rather than the reactive power control loop. Taking into account the AC filter inductor or a transformer on the power grid side, it is assumed that the line equivalent impedance Z_line is inductive, leading to the control loops of the active power and reactive power being decoupled. Meanwhile, the control bandwidth of the inner control loop is commonly designed to be much higher than that of the outer power control loop to maintain the command tracking and stability [6]. To simplify the theoretical analysis, the reactive power control loop and the inner control loop are omitted in the subsequent analysis.

2.2. Small Signal Model of GFVSG and Its Active Power Dynamic Oscillation Analysis

The equivalent power transmission model of GFVSG grid-connected system is illustrated in Figure 2, where X_L, R_L, U_g, and δ are the line reactance, line resistance, power grid voltage amplitude, and power factor angle, respectively.

In this paper, the line equivalent impedance Z_line is assumed to be purely inductive, neglecting the line resistance R_L, to simplify the system model. From Figure 2, the active power P_e of GFVSG can be expressed as

$$P_{\mathrm{e}}=\frac{3U_{\mathrm{g}}E}{2X_{\mathrm{L}}}\delta =K\delta =\frac{3U_{\mathrm{g}}E(\omega -{\omega }_{\mathrm{g}})}{2X_{\mathrm{L}}s},$$

(3)

where K = (3U_gE)/(2X_L) and ω_g are the synchronization voltage coefficient and power grid angular frequency, respectively. By combining Equations (1) and (3), the GFVSG closed-loop small signal control model can be obtained as shown in Figure 3, where “∆” denotes the small signal disturbance amount.

According to Figure 3, it is not difficult to find that the change in active power command ∆P_ref can be regarded as an internal disturbance, while the change in power grid angular frequency ∆ω_g can be regarded as an external disturbance. Both the ∆P_e and the ∆ω of GFVSG are affected by the two disturbances ∆P_ref and ∆ω_g, and the corresponding closed-loop transfer functions can be obtained as follows:

$$\{\begin{array}{l}{G}_{1\_\mathrm{G}}{(s)=\frac{\Delta P_{\mathrm{e}}(s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}=\frac{K}{J{\omega }_0{s}^2+D{\omega }_{0}s+K}\\ G_{2\_\mathrm{G}}(s)={\frac{\Delta P_{\mathrm{e}}(s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}=-\frac{(J{\omega }_{0}s+D{\omega }_0)K}{J{\omega }_0{s}^2+D{\omega }_{0}s+K}\end{array},$$

(4)

$$\{\begin{array}{l}{G}_{3\_\mathrm{G}}(s){=\frac{\Delta \omega (s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}=\frac{s}{J{\omega }_0{s}^2+D{\omega }_{0}s+K}\\ G_{4\_\mathrm{G}}(s)={\frac{\Delta \omega (s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}=\frac{K}{J{\omega }_0{s}^2+D{\omega }_{0}s+K}\end{array},$$

(5)

where the subscript “G” or the later “E” are applied to declare that the function or variable is definitely used for the GFVSG or ERM-GFVSG. According to Equation (4), the active power steady-state deviation ∆P_e0 (∆P_e0 = ∆P_e − ∆P_ref) of GFVSG can be obtained as follows:

$$\Delta P_{\mathrm{e}0}=\underset{s\to 0}{\mathrm{Lim}}{G}_2(s)\Delta {\omega }_{\mathrm{g}}=-D{\omega }_0\Delta {\omega }_{\mathrm{g}},$$

(6)

According to Equations (4) and (5), it is found that the GFVSG closed-loop control system is a typical second-order oscillation system, and its characteristic function is as follows:

$$J{\omega }_0{s}^2+D{\omega }_{0}s+K=0,$$

(7)

where the natural oscillation angular frequency ω_{n_G} and its damping ratio ξ_G of the GFVSG closed-loop control system are expressed as

$${\omega }_{\mathrm{n}\_\mathrm{G}}=\sqrt{K/\left(J{\omega }_0\right)},{\xi }_{\mathrm{G}}=0.5D{\omega }_0\sqrt{1/\left(KJ{\omega }_0\right)},$$

(8)

It can be seen from Equations (6) and (8) that both ω_{n_G} and ξ_G decrease with the increase in J, which represents the slower dynamic response speed and the more intense dynamic oscillation of P_e or ω under the two disturbances of ∆P_ref and ∆ω_g. In addition, both ξ_G and ∆P_e0 increase with the increase in D, that is, the stronger the ability to inhibit the power active dynamic oscillation, but the larger the steady-state deviation of the active power.

2.3. Equivalent Circuit Model of GFVSG and Its Active Power Dynamic Oscillation Analysis

According to Figure 3, the differential equations can be obtained as follows:

$$\Delta \omega -\Delta {\omega }_{\mathrm{g}}=\frac{1}{K}\frac{\mathrm{d}\Delta P_{\mathrm{e}}}{\mathrm{d}t},$$

(9)

$$\Delta P_{\mathrm{ref}}-\Delta P_{\mathrm{e}}-D{\omega }_0\Delta \omega =J{\omega }_0\frac{\mathrm{d}\Delta \omega }{\mathrm{d}t},$$

(10)

Table 1. The analogy relationships between the electromechanical and electromagnetic variables.

Types	Variables
Electromechanical	1/K	Jω0	ω	Pe	δ
Electromagnetic	L	C	u	i	Ψ

shows the analogy relationships between the electromechanical and electromagnetic variables [26]. In Table 1, L, C, u, i, and Ψ are the inductance, capacitance, voltage, current, and flux linkage, respectively.

According to Table 1, Equations (9) and (10) can be likened to the circuit equations expressed by Equations (11) and (12), respectively.

$$L\frac{\mathrm{d}{i}_{\mathrm{L}}}{\mathrm{d}t}=u_{\mathrm{L}},$$

(11)

$$C\frac{\mathrm{d}{u}_{\mathrm{C}}}{\mathrm{d}t}=i_{\mathrm{ref}}-i_{\mathrm{L}}-\frac{u_{\mathrm{C}}}{R},$$

(12)

where u_L, i_L, u_C, and R = 1/(Dω₀) are the inductance voltage, inductance current, capacitance voltage, and resistance, respectively. According to Equations (11) and (12), the GFVSG closed-loop small-signal control model in Figure 2 can be likened to the dynamic equivalent circuit model shown in Figure 4, where ∆P_ref, Dω₀, sJω₀, s/K, ∆P_e, ∆ω, and ∆ω_g are equal to the current source, conductance, capacitive susceptance, inductive reactance, equivalent current flowing through the inductor 1/K, electromotive force, and voltage source, respectively, and ∆P_ref and ∆ω_g can also be used as two excitation sources in the GFVSG dynamic equivalent circuit.

According to Figure 4, the dynamic oscillation energies W₁ and W₂ injected into the circuit by the current source ∆P_ref and the voltage source ∆ω_g can be expressed as

$$W_1={\displaystyle {\int }_0^t\Delta P_{\mathrm{ref}}\Delta \omega \mathrm{d}t},W_2=-{\displaystyle {\int }_0^t\Delta P_{\mathrm{e}}\Delta {\omega }_{\mathrm{g}}\mathrm{d}t},$$

(13)

It is worth pointing out that, in the GFVSG dynamic response process, ∆P_ref and ∆ω always maintain the same sign, while ∆P_e and ∆ω_g always maintain a different sign. Therefore, the integral calculation results W₁ and W₂ are always positive, which means that the total dynamic oscillation energy (W₁ + W₂) is always injected into the equivalent circuit under the two disturbances of ∆P_ref and ∆ω_g. In Figure 4, W₃, W₄, and W₅ are the energy consumed by the equivalent resistance 1/(Dω₀), the kinetic energy stored by the equivalent capacitance (Jω₀), and the potential energy stored by the equivalent inductor (1/K), respectively. Similarly, W₃, W₄, and W₅ can be expressed as

$$W_3={\displaystyle {\int }_0^{t}D{\omega }_0\Delta {\omega }^2\mathrm{d}t},W_4=\frac{J{\omega }_0\Delta {\omega }^2}{2},W_5=\frac{\Delta P_{\mathrm{e}}^2}{2K},$$

(14)

By multiplying Equation (9) by ∆P_e and Equation (10) by ∆ω, and then integrating both sides of the equation over time, the correspondences between Equations (13) and (14) can be acquired, as shown in Equation (15).

$$\underset{Injected energy(W_1+W_2)}{\underbrace{{\displaystyle {\int }_0^t\Delta P_{\mathrm{ref}}\Delta \omega \mathrm{d}t}-{\displaystyle {\int }_0^t\Delta P_{\mathrm{e}}\Delta {\omega }_{\mathrm{g}}\mathrm{d}t}}}=\underset{Stored energy(W_4+W_5)}{\underbrace{\frac{J{\omega }_0\Delta {\omega }^2}{2}+\frac{\Delta P_{\mathrm{e}}^2}{2K}}}+\underset{Consumed energy(W_3)}{\underbrace{{\displaystyle {\int }_0^{t}D{\omega }_0\Delta {\omega }^2\mathrm{d}t}}},$$

(15)

According to Equation (15), it is not difficult to find that the dynamic equivalent circuit of GFVSG can meet the law of the energy conservation, that is, the total dynamic oscillation energy (W₁ + W₂) injected into the circuit by the two disturbances of ∆P_ref and ∆ω_g is equal to the sum of the consumed energy W₃ and the total stored energy (W₄ + W₅), where W₄ and W₅ can be converted into each other. Meanwhile, the value of W₃ represents the capacity to consume the total dynamic oscillation energy, that is, due to the existence of the equivalent resistance 1/(Dω₀), the total dynamic oscillation energy is decreasing continuously and the system eventually reaches a new steady state.

It can also be found from Equation (15) that under a certain disturbance of ∆P_ref or ∆ω_g, the larger the J value of GFVSG, the larger the stored energy W₄, and the longer the system takes to consume W₄, so the regulation time of the system will be longer. The larger the D value of GFVSG, the larger the consumed energy W₃, and the shorter the system takes to consume (W₁ + W₂), the stronger the ability to suppress the system dynamic oscillation. The above results are in one-to-one correspondence with the conclusions from Equation (8).

In summary, for GFVSG grid-connected systems, the active power dynamic oscillation can be suppressed by the following four main methods. First, the method of reducing the stored energy, that is, the adaptive parameter adjustment method using adaptive virtual inertia, adaptive virtual impedance, or other adaptive parameters. Second, the method of reducing the injected energy, that is, the feedforward path compensation method using active power command, grid angular frequency, or other variables. Third, the method of increasing consumed energy, that is, the feedback channel-based compensation method using angular frequency change, active power change, or other changes. Fourth, any combination of the above three methods.

3. The Proposed ERM-GFVSG Control Method

3.1. Control Principle of ERM-GFVSG

To solve the problem of the active power dynamic oscillation existing in the GFVSG grid-connected system, the third method of increasing the consumed energy is used by ERM-GFVSG. In other words, ERM-GFVSG introduces a feedback channel-based compensation method, using the dynamic variations in the angular frequency and the active power in the GFVSG control structure to improve its ability to suppress the active power dynamic oscillation. For ERM-GFVSG, the differential Equations (9) and (10) can be rewritten as follows:

$$\Delta \omega -\Delta {\omega }_{\mathrm{g}}-k_{\mathrm{b}1}\Delta P_{\mathrm{e}}=\frac{1}{K}\frac{\mathrm{d}\Delta P_{\mathrm{e}}}{\mathrm{d}t},$$

(16)

$$\Delta P_{\mathrm{ref}}-\Delta P_{\mathrm{e}}-D{\omega }_0\Delta \omega -k_{\mathrm{b}2}\Delta \omega =J{\omega }_0\frac{\mathrm{d}\Delta \omega }{\mathrm{d}t},$$

(17)

where k_b1 and k_b2 are the active power feedback parameter and angular frequency feedback parameter, respectively. According to Table 1, Equations (16) and (17) can also be likened to the circuit equations expressed by Equations (18) and (19), respectively.

$$L\frac{\mathrm{d}{i}_{\mathrm{L}}}{\mathrm{d}t}=u_{\mathrm{L}}\underset{U_{\mathrm{I}}}{\underbrace{-k_{\mathrm{b}1}{i}_{\mathrm{L}}}},$$

(18)

$$C\frac{\mathrm{d}{u}_{\mathrm{C}}}{\mathrm{d}t}=i_{\mathrm{ref}}-i_{\mathrm{L}}-\frac{u_{\mathrm{C}}}{R}\underset{I_{\mathrm{U}}}{\underbrace{-\frac{u_{\mathrm{C}}}{k_{\mathrm{b}2}}}},$$

(19)

where U_I and I_U are the current controlled voltage source (CCVS) and voltage controlled current source (VCCS), respectively. According to Equations (18) and (19), the dynamic equivalent circuit model of the ERM-GFVSG grid-connected system can also be maintained, as shown in Figure 5, where k_b1 is equal to the resistance, and k_b2 is equal to the conductance. W₆ is the energy consumed by the equivalent resistance k_b1, and W₇ is the energy consumed by the equivalent resistance (1/k_b2). Similarly, W₆ and W₇ can be expressed as

$$W_6={\displaystyle {\int }_0^t{k}_{\mathrm{b}1}\Delta P_{\mathrm{e}}^2\mathrm{d}t},W_7={\displaystyle {\int }_0^t{k}_{\mathrm{b}2}\Delta {\omega }^2\mathrm{d}t},$$

(20)

Next, by multiplying Equation (16) by ∆P_e and Equation (17) by ∆ω, and then integrating both sides of the equation over time, the following can be obtained:

$$\underset{Injected energy(W_1+W_2)}{\underbrace{{\displaystyle {\int }_0^t\Delta P_{\mathrm{ref}}\Delta \omega \mathrm{d}t}-{\displaystyle {\int }_0^t\Delta P_{\mathrm{e}}\Delta {\omega }_{\mathrm{g}}\mathrm{d}t}}}=\underset{Stored energy(W_4+W_5)}{\underbrace{\frac{J{\omega }_0\Delta {\omega }^2}{2}+\frac{\Delta P_{\mathrm{e}}^2}{2K}}}+\underset{Consumed energy(W_3+W_6+W_7)>W_3}{\underbrace{{\displaystyle {\int }_0^{t}D{\omega }_0\Delta {\omega }^2\mathrm{d}t+{\displaystyle {\int }_0^t{k}_{\mathrm{b}1}\Delta P_{\mathrm{e}}^2\mathrm{d}t}+{\displaystyle {\int }_0^t{k}_{\mathrm{b}2}\Delta {\omega }^2\mathrm{d}t}}}},$$

(21)

Compared with Equations (15) and (21), it can be seen that ERM-GFVSG can effectively increase the consumption speed of (W₁ + W₂) by increasing the total consumed energy from W₃ to (W₃ + W₆ + W₇) on the premise of keeping (W₁ + W₂) and (W₄ + W₅) unchanged. That is, it is equivalent to increasing the value of D to enhance the damping of the GFVSG grid-connected system and its ability to suppress the active power dynamic oscillation.

In this paper, the two fluctuation variables ∆P_e and ∆ω contained in Equation (21) are extracted by second-order LPFs, which are then fed back to the ERM-GFVSG active power command through k_b1 and k_b2, respectively. Therefore, by combining Equations (16) and (17), the ERM-GFVSG closed-loop small signal control model can be obtained as shown in Figure 6. In Figure 6, ω_c and Q are the cutoff angular frequency and quality factor of the second-order LPF, respectively. The second-order LPF is introduced to effectively filter out the high-frequency harmonics generated by differential operation. It is worth pointing out that the purpose of moving the feedback variable (k_b1∆P_e) forward to the virtual inertia control link is to filter (k_b1∆P_e) again by using the first-order LPF 1/(sJω₀ + Dω₀), which is composed of the virtual inertia link and the virtual damping link on the forward channel in Figure 6.

3.2. Parameter Design Method of ERM-GFVSG

Likewise, according to the closed-loop small signal control model of ERM-GFVSG grid-connected system shown in Figure 6, the closed-loop transfer functions of ERM-GFVSG can be obtained as follows:

$$\{\begin{array}{l}{G}_{1\_\mathrm{E}}(s){=\frac{\Delta P_{\mathrm{e}}(s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}=\frac{K{s}^2+K{\omega }_{\mathrm{c}}/Qs+K{\omega }_{\mathrm{c}}^2}{J{\omega }_0{s}^4+M_3{s}^3+M_2{s}^2+M_{1}s+K{\omega }_{\mathrm{c}}^2}\\ G_{2\_\mathrm{E}}(s)={\frac{\Delta P_{\mathrm{e}}(s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}=-\frac{KJ{\omega }_0{s}^3+N_2{s}^2+N_{1}s+D{\omega }_{0}K{\omega }_{\mathrm{c}}^2}{J{\omega }_0{s}^4+M_3{s}^3+M_2{s}^2+M_{1}s+K{\omega }_{\mathrm{c}}^2}\end{array},$$

(22)

$$\{\begin{array}{l}{G}_{3\_\mathrm{E}}(s){=\frac{\Delta \omega (s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}=\frac{s^3+{\omega }_{\mathrm{c}}/Q{s}^2+{\omega }_{\mathrm{c}}^{2}s}{J{\omega }_0{s}^4+M_3{s}^3+M_2{s}^2+M_{1}s+K{\omega }_{\mathrm{c}}^2}\\ G_{4\_\mathrm{E}}(s)={\frac{\Delta \omega (s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}=\frac{K{s}^2+K({\omega }_{\mathrm{c}}/Q+k_{\mathrm{b}1}{\omega }_{\mathrm{c}}^2)s+K{\omega }_{\mathrm{c}}^2}{J{\omega }_0{s}^4+M_3{s}^3+M_2{s}^2+M_{1}s+K{\omega }_{\mathrm{c}}^2}\end{array},$$

(23)

where M₁ = ω_c(Dω₀ω_c + Kk_b1ω_c + K/Q); M₂ = Jω₀ω_c² + Dω₀ω_c/Q + K+k_b2ω_c²; M₃ = ω₀(Jω_c/Q + D); N₁ = Kω_c(Jω₀ω_c + Dω₀/Q + k_b2ω_c); and N₂ = Kω₀(Jω_c/Q + D). Referring to the parameter selection principle of the second-order LPF in [25], on the one hand, when the filtering time constant τ of the first-order LPF 1/(τs + 1) is set to τ = 1/ω_c, a second-order LPF can be equivalently reduced to a first-order LPF. On the other hand, to weigh the filtering effect, dynamic response speed, and the influence on the system operating stability of the second-order LPF, τ = 0.007 s (ω_c = 1/τ = 142.86 rad /s) and Q = 0.5 are set here to simplify the theoretical analysis process and the system parameter design method. In view of this, Equations (22) and (23) after the order reduction are approximately as follows:

$$\{\begin{array}{l}{G}_{1\_\mathrm{E}}(s){=\frac{\Delta P_{\mathrm{e}}(s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}\approx \frac{K\tau s+K}{J{\omega }_0\tau s^3+M_{21}{s}^2+M_{11}s+K}\\ G_{2\_\mathrm{E}}(s)={\frac{\Delta P_{\mathrm{e}}(s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}\approx -\frac{KJ{\omega }_0\tau s^2+N_{11}s+D{\omega }_{0}K}{J{\omega }_0\tau s^3+M_{21}{s}^2+M_{11}s+K}\end{array},$$

(24)

$$\{\begin{array}{l}{G}_{3\_\mathrm{E}}(s){=\frac{\Delta \omega (s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}\approx \frac{\tau s^2+s}{J{\omega }_0\tau s^3+M_{21}{s}^2+M_{11}s+K}\\ G_{4\_\mathrm{E}}(s)={\frac{\Delta \omega (s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}\approx \frac{K(\tau +k_{\mathrm{b}1})s+K}{J{\omega }_0\tau s^3+M_{21}{s}^2+M_{11}s+K}\end{array},$$

(25)

where M₂₁ = Jω₀ + Dω₀τ + k_b2; M₁₁ = Dω₀ + Kk_b1 + Kτ; and N₁₁ = K(Jω₀ + Dω₀τ + k_b2). By comparing Equations (4) and (24), or Equations (5) and (25), it is not difficult to find that compared with GFVSG, the closed-loop control system of ERM-GFVSG upgrades the system order of the former to the third order, and adds a zero as well as a pole, so the former can effectively improve the ability to suppress the active power dynamic oscillation by optimizing the configurations of an additional zero and an additional pole.

Meanwhile, combined with the parameter design method based on the second-order equivalent reduced-order control model proposed in [25], that is, the influence of the control items containing variable τ in Equations (24) and (25), the order of which is larger than that of one order is ignored. Based on the above order reduction principle, Equations (24) and (25) are further reduced to an equivalent second-order control system as shown in Equations (26) and (27).

$$\{\begin{array}{l}{G}_{1\_\mathrm{E}}(s){=\frac{\Delta P_{\mathrm{e}}(s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}\approx \frac{K\tau s+K}{(J{\omega }_0+k_{\mathrm{b}2})s^2+M_{11}s+K}\\ G_{2\_\mathrm{E}}(s)={\frac{\Delta P_{\mathrm{e}}(s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}\approx -\frac{N_{11}s+D{\omega }_{0}K}{(J{\omega }_0+k_{\mathrm{b}2})s^2+M_{11}s+K}\end{array},$$

(26)

$$\{\begin{array}{l}{G}_{3\_\mathrm{E}}(s){=\frac{\Delta \omega (s)}{\Delta P_{\mathrm{ref}}(s)}|}_{\Delta {\omega }_{\mathrm{g}}=0}\approx \frac{s}{(J{\omega }_0+k_{\mathrm{b}2})s^2+M_{11}s+K}\\ G_{4\_\mathrm{E}}(s)={\frac{\Delta \omega (s)}{\Delta {\omega }_{\mathrm{g}}(s)}|}_{\Delta P_{\mathrm{ref}}=0}\approx \frac{K(\tau +k_{\mathrm{b}1})s+K}{(J{\omega }_0+k_{\mathrm{b}2})s^2+M_{11}s+K}\end{array},$$

(27)

Therefore, the parameters k_b1 and k_b2 of ERM-GFVSG can be designed directly using the second-order control system described by Equation (26) or (27). The characteristic function of the ERM-GFVSG grid-connected system can be written as

$$(J{\omega }_0+k_{\mathrm{b}2})s^2+(D{\omega }_0+K{k}_{b1}+K\tau )s+K=0,$$

(28)

where the natural oscillation angular frequency ω_{n_E}, the damping ratio ξ_E, the phase angle margin γ_E, and the cut-off angular frequency ω_{c_E} of the ERM-GFVSG closed-loop control system can be expressed as

$${\omega }_{\mathrm{n}\_\mathrm{E}}=\sqrt{\frac{K}{J{\omega }_0+k_{\mathrm{b}2}}}, {\xi }_{\mathrm{E}}=\frac{D{\omega }_0+K{k}_{b1}+K\tau }{2\sqrt{(J{\omega }_0+k_{\mathrm{b}2})K}},$$

(29)

$${\gamma }_{\mathrm{E}}=\arctan \frac{2{\xi }_{\mathrm{E}}}{\sqrt{\sqrt{1+4{\xi }_{\mathrm{E}}^4}-2{\xi }_{\mathrm{E}}^2}}, {\omega }_{\mathrm{c}\_\mathrm{E}}={\omega }_{\mathrm{n}\_\mathrm{E}}\sqrt{\sqrt{1+4{\xi }_{\mathrm{E}}^4}-2{\xi }_{\mathrm{E}}^2}$$

(30)

In order to ensure that ERM-GFVSG has good operating stability and to eliminate its active power dynamic oscillation, it is necessary to set ξ_E ≥ 1 and γ_E ≥ 45°. Therefore, by substituting the set characteristic values ξ_E and γ_E into Equations (29) and (30), the optimal designs of k_b1 and k_b2 for ERM-GFVSG can be achieved. To sum up, according to the main parameters of 100 kV·A GFVSG included in

Table 2. The main parameters of a 100 kV·A GFVSG.

Symbol	Parameter	Value
E0	Rated voltage amplitude	311 V
Udc	DC bus voltage	700 V
ω0	Rated angular frequency	314.15 rad/s
J	Virtual inertia	8 kg·m2
kq	Primary voltage regulation coefficient	1.4 × 10−4 V/var
D	Virtual damping	50.66 J/rad
Lf	Filter inductance	50.6 mH
Cf	Filter capacitor	270 uF
fs	Sampling frequency	5 kHz
XL	Line reactance	0.15 Ω

, the ω_{n_G} = 19.62 rad /s, ξ_G = 0.16, τ = 0.007 s, ω_c = 1/τ = 142.86 rad/s, Q = 0.5, k_b1 = 0.12, k_b2 = 2000, ω_{n_E}= 14.64 rad /s, and ξ_E = 1.05 > 1 and γ_E = 77.6° > 45° can be calculated.

By bringing the above parameters into the G_{1_E}(s) in Equations (22), (24), and (26) successively, the comparison results of the step response of ERM-GFVSG under different order control models can be obtained as shown in Figure 7. As can be seen from Figure 7, the step response curves of the three are almost identical, that is, the three have a very similar dynamic response performance, which indicates that it is reasonable and feasible to simplify the design of the system parameters by using the ERM-GFVSG second-order equivalent reduced-order control model as described in Equation (26).

Moreover, based on the proposed ERM-GFVSG control method, the closed-loop Bode diagrams of ΔP_e(s)/ΔP_ref(s), ΔP_e(s)/Δω_g(s), Δω(s)/ΔP_ref(s), and Δω(s)/Δω_g(s) in the GFVSG grid-connected system shown in Figure 1 are compared in Figure 8a, Figure 8b, Figure 8c and Figure 8d, respectively.

As explained before, the GFVSG closed-loop control system is a typical second-order oscillation system. Hence, when the value of ξ_G (ξ_G = 0.16 < 1) is too small, all the Bode diagrams of GFVSG have a resonance peak before the cutting frequency, as shown in Figure 8, which means that a dynamic oscillation exists at the resonance frequency. Thus, both the P_e and the ω of GFVSG with ξ_G = 0.16 inevitably have the dynamic oscillations under both the disturbances of the P_ref and the ω_g. And, unlike GFVSG, it can be seen from Figure 8 that the resonance peak in all the Bode diagrams of ERM-GFVSG is eliminated effectively by increasing the value of ξ_E (ξ_E = 1.05 > 1), which illustrates that a dynamic oscillation does not occur at the resonance frequency. Therefore, both the P_e and the ω of ERM-GFVSG with ξ_E = 1.05 do not have any dynamic oscillations under both the disturbances of the P_ref and the ω_g. According to the above theoretical analysis, the proposed ERM-GFVSG can solve the dynamic oscillation problems existing in the P_e and the ω of GFVSG under different disturbances.

4. Simulation Comparison Results

The simulation test platform of a 100 kV·A GFVSG grid-connected system shown in Figure 1 is established by using the Matlab2016a/Simulink simulation software to verify the effectiveness of the proposed ERM-GFVSG in damping the active power dynamic oscillation. Meanwhile, comparison studies between the other damping control methods and the proposed ERM-GFVSG have also been implemented on the simulation test platform, including the GFVSG₁ (D = 50.66 J/rad), the GFVSG₂ (D = 335.16 J/rad), the active-power fractional differential correction (AFDC) [3], the ADFC [25], and the ADFF [25]. In order to achieve a fair comparison, a general rule is used to make each damping control method obtain the optimal dynamic performance in consideration of both the disturbances of the P_ref and the ω_g.

In this simulation, τ = 0.007 s, ω_c = 142.86 rad/s, and Q = 0.5 are selected. The main parameters used in each damping control method are summarized in

Table 3. The main parameters of each damping control method.

Method	Symbol	Parameter	Value
AFDC-GFVSG	μ	Fractional differential order	0.8
ADFC-GFVSG	KB	Differential feedback parameter	0.08
	ξB	Damping ratio	1.02
ADFF-GFVSG	KF	Differential feedforward parameter	0.08
	ξF	Damping ratio	1.02
GFVSG1	D	Virtual damping	50.66 J/rad
	ξG	Damping ratio	0.16
GFVSG2	D	Virtual damping	335.16 J/rad
	ξG	Damping ratio	1.07
ERM-GFVSG	kb1	Active power feedback parameter	0.12
	Kb2	Angular frequency feedback parameter	200
	ξE	Damping ratio	1.05

and the other parameters are shown in Table 1.

Figure 9 shows the response waveforms of the P_e and the output frequency f for various damping control methods when the P_ref steps from 20 kW to 60 kW at 4 s. Figure 9a illustrates the P_e response waveforms, whereas Figure 9b illustrates the f response waveforms. In Figure 9a, as the damping is only provided by the D (D = 50.66 J/rad) in the GFVSG₁ (ξ_G = 0.16), the damping effect is limited and not enough to suppress the active power dynamic oscillation. On the other hand, there is no oscillation in the GFVSG₂ (D = 335.16 J/rad), the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the ERM-GFVSG. In terms of the dynamic response speed, the regulation time of the ERM-GFVSG and the ADFC-GFVSG is very similar, which is much longer than the other three damping control methods, and the regulation time of the GFVSG₂ is slightly longer than that of the AFDC-GFVSG and the ADFF-GFVSG.

In Figure 9b, the response waveform of f in the GFVSG₁ (ξ_G = 0.16) is obviously oscillatory when the P_ref steps from 20 kW to 60 kW. Although the frequency overshoot exists in all damping control methods, the frequency overshoot of the ERM-GFVSG is smallest with a soft change slope at 4 s. Meanwhile, the frequency overshoot of the AFDC-GFVSG and the ADFF-GFVSG is similar with a severe change slope at 4 s, which is much larger than the other four damping control methods.

Figure 10 shows the response waveforms of the P_e and the f for various methods when the power grid frequency f_g changes from 50 Hz to 49.95 Hz at 7 s.

Figure 10a illustrates the P_e response waveforms, whereas Figure 10b illustrates the f response waveforms. In Figure 10a, the damping ratio of the GFVSG₁ (D = 50.66 J/rad) is small and not large enough to suppress the active power dynamic oscillation. Although the damping effect of the GFVSG₂ (ξ_G = 1.07) is large enough to suppress the active power dynamic oscillation by increasing the D (D = 335.16 J/rad), a large active power steady state deviation ∆P_e0 (∆P_e0 = 28.1 kW) can be observed under the condition that the f_g deviates from 50 Hz. On the other hand, there is no oscillation in the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the ERM-GFVSG when the f_g drops from 50 Hz to 49.95 Hz. It should be noted that in terms of the dynamic response speed, ERM-GFVSG has a relatively small active power overshoot, so its active power regulation time is slightly shorter than in the other damping control methods.

In Figure 10b, the response waveform of f in the GFVSG₁ (ξ_G = 0.16) is obviously oscillatory when the f_g changes from 50 Hz to 49.95 Hz. For the other five damping control methods, the slopes of the frequency changes at 7 s of the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the ERM-GFVSG are all steeper than that of the GFVSG₂. However, their frequency response time is much shorter than that of the GFVSG₂. Moreover, the damping performance of the ERM-GFVSG is satisfactory under the disturbance of the f_g change, as it has a slightly dynamic smaller frequency drop than that of the AFDC-GFVSG, the ADFC-GFVSG, or the ADFF-GFVSG, and the steady-state frequency drop of each damping method described above, including the GFVSG with D = 335.16, is 49.95 Hz.

5. Experimental Comparison Results

In order to further validate the feasibility of the proposed ERM-GFVSG control method, experimental tests were carried out on a microgrid experiment platform, as shown in Figure 11 [25]. A structure diagram and a photograph of the microgrid platform are shown in Figure 11a and 11b, respectively. Figure 11c presents the test process of a 100 kV·A GFVSG, the main parameters of which are given in Table 1 and Table 3. The 100 kV·A GFVSG was controlled by an independent DSP TMS320F28335, which implemented the various damping control methods, as discussed in the above sections.

Figure 12 shows the response waveforms of the P_e and the f for various damping control methods when the P_ref steps from 20 kW to 60 kW. Figure 12a illustrates the P_e response waveforms, whereas Figure 12b illustrates the f response waveforms. In Figure 12a, the damping effect of the GFVSG₁ (D = 50.66 J/rad) is limited and not enough to suppress the active power dynamic oscillation. However, there is no oscillation in the GFVSG₂ (D = 335.16 J/rad), the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the ERM-GFVSG, and the regulation time of the ERM-GFVSG and the ADFC-GFVSG is very similar, which is much longer than the other three damping control methods. The regulation time of the GFVSG₂ is slightly longer than that of the AFDC-GFVSG and the ADFF-GFVSG, which is consistent with Figure 9a.

In Figure 12b, the response waveform of f in the GFVSG₁ (ξ_G = 0.16) is obviously oscillatory when the P_ref steps from 20 kW to 60 kW. Although the frequency overshoot exists in all damping control methods, the frequency overshoot of the ERM-GFVSG is smallest with a soft change slope, which can correspond to Figure 9b. However, the frequency overshoot of the AFDC-GFVSG and the ADFF-GFVSG is similar with a severe change slope, which is much larger than the other four damping control methods.

Figure 13 shows the response waveforms of the P_e and the f for various damping control methods when the f_g drops from 50 Hz to 49.95 Hz. Figure 13a illustrates the P_e response waveforms, whereas Figure 13b illustrates the f response waveforms. In Figure 13a, the damping ratio of the GFVSG₁ (D = 50.66 J/rad) is not large enough to suppress the active power dynamic oscillation. Although the GFVSG₂ (ξ_G = 1.07) is able to suppress the active power dynamic oscillation by increasing the D (D = 335.16 J/rad), the ∆P_e0 = 28.3 kW can be observed when the f_g deviates from 50 Hz in the steady state. And there is no oscillation in the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the ERM-GFVSG when the f_g changes from 50 Hz to 49.95 Hz. In terms of the dynamic response speed, the regulation time of the ERM-GFVSG is slightly smaller than the other five damping control methods, which is shown in Figure 10a.

In Figure 13b, the response waveform of f in the GFVSG₁ (ξ_G = 0.16) is obviously oscillatory when the f_g drops from 50 Hz to 49.95 Hz. The slopes of the frequency changes in the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the ERM-GFVSG are all steeper than the GFVSG₁ or the GFVSG₂. However, their frequency response speed is much faster than the GFVSG₁ or the GFVSG₂. And the damping performance of the ERM-GFVSG is satisfactory considering the disturbance of the f_g change, as it has a smaller frequency drop than that of the AFDC-GFVSG, the ADFC-GFVSG, or the ADFF-GFVSG, which is consistent with Figure 10b.

6. Conclusions

To solve the dynamic oscillation problems existing in the P_e and the ω of GFVSG under the two disturbances of the P_ref step and the f_g change, an active power dynamic oscillation damping method based on ERM for the GFVSG is proposed in this paper. Through the theoretical analysis, mathematical modeling, energy reshaping, parameter design, simulation, and experimental validations, the following conclusions are drawn:

(1) From the perspective of energy flow, the larger the J value of GFVSG, the larger the stored energy and the longer the system takes to consume the stored energy, so the regulation time of the system will be longer. Meanwhile, the larger the D value, the larger the consumed energy and the shorter the system takes to consume the dynamic oscillation energy, the stronger the ability to suppress the active power dynamic oscillation.

(2) Although the P_e and the ω of the AFDC-GFVSG, the ADFC-GFVSG, the ADFF-GFVSG, and the proposed ERM-GFVSG all have neither dynamic oscillation nor active power steady-state deviation under the two disturbances, the frequency overshoot of the ERM-GFVSG under the P_ref step disturbance is the smallest at the expense of the P_e regulation time, and the P_e regulation time of the ERM-GFVSG under the f_g step disturbance is the shortest. Therefore, in the practical application of the ERM-GFVSG, the parameters k_b1 and k_b2 of ERM-GFVSG need to be selected in a compromise to balance the relationship between the regulation time and the frequency overshoot.

It is worth pointing out that the ERM-GFVSG control method is proposed based on the closed-loop small signal control model of a GFVSG grid-connected system, so the applicability and effectiveness of the active power oscillation damping method based on the ERM in islanded microgrids with multiple GFVSGs still need to be further studied.