The GFVSG grid-connected circuit structure and its control principle are shown in Fig. 1 [11].

In Fig. 1, U_dc denotes DC side voltage and Z_line represents line impedance. L_f and C_f are filter inductance and filter capacitor; θ denotes output phase angle; u_gabc and S are three-phase network voltage and grid-connected switch; u_abc denotes three-phase output voltage and i_abc represents grid-connected current. The rotor equations of motion of the GFVSG and its primary regulating voltage equation can be expressed sequentially as follows: (1)Pref−Pe−Dω0(ω−ω0)=Jω0dωdt=Jω0d(ω−ω0)dt (2)E=E0+kq(Qref−Qe)where P_e and P_ref are GCAP and its reference power, respectively, and J, D are virtual inertia and virtual damping; ω and ω₀ are output angular frequency and the fund-amental; Q_e denotes grid-connected reactive power and Q_ref represents reference reactive power; E and E₀ are output voltage amplitude and nominal voltage, respectively, and k_q is a proportional gain.

It is worth noting that the primary focus of this paper lies in addressing the GCAP dynamic response optimization issue of GFVSG, considering the GCAP and the grid-connected reactive power can be decoupled under the condition that Z_line is inductive and disregarding the impact of the voltage and current bottom double-closed-loop on the power outer-loop with a high control bandwidth. Thus, the relevant content concerning the reactive power control and the bottom double-loop control will not be reiterated [21].

As illustrated in Fig. 1 and combined with the line power transfer theory, the P_e of GFVSG can be approximated by Eq. (3) [25]. (3)Pe=3UgE2Xsin⁡δ≈Kδ=Kω−ωgswhere U_g is the amplitude of u_gabc and ω_g is the angular frequency of u_gabc. X denotes the equivalent reactance of Z_line; the synchronizing voltage coefficient is first formulated as K = 1.5U_gE/X; δ and s are phase angle and laplacian. As shown in Fig. 2, the GCAP closed-loop small-signal control model of GFVSG can be derived from Eqs. (1) and (3). In Fig. 2, “∆” represents the amount of perturbation.

From Fig. 2, it can be seen that while the GFVSG realizes the virtual inertia control by using the first-order low-pass filter 1/(Jω₀s + Dω₀) consisting of J and D, its P_e and ω are affected by ∆P_ref and ∆ω_g disturbances. The transfer functions of ∆P_ref to ∆P_e, ∆ω_g to ∆P_e, ∆P_ref to ∆ω and ∆ω_g to ∆ω for GFVSG are derived as follows: (4){G1(s)=ΔPeΔPref|Δωg=0=KJω0s2+Dω0s+KG2(s)=ΔPeΔωg|ΔPref=0=−(Jω0s+Dω0)KJω0s2+Dω0s+K (5){G3(s)=ΔωΔPref|Δωg=0=sJω0s2+Dω0s+KG4(s)=ΔωΔωg|ΔPref=0=KJω0s2+Dω0s+K

According to Eqs. (4) and (5), it is easy to find that the GCAP closed-loop control system of GFVSG is a second-order oscillation system, and the corresponding natural oscillation angular frequency ω_n and its damping ratio ξ of the system can be expressed as: (6)ωn=KJω0,ξ=Dω021KJω0

It can also be obtained from Eq. (4) that the GCAP steady state deviation ∆P_e0 (∆P_e0 = ∆P_e − ∆P_ref) of GFVSG is shown in Eq. (7). (7)ΔPe0=Lims→0G2(s)Δωg=−Dω0Δωg

By applying Eqs. (6) and (7), it can be found that the value of J affects the ξ and ω_n of the GFVSG grid-connected system at the same time, i.e., the larger J is, the smaller the corresponding ξ and ω_n is, and the more pronounced the dynamic oscillation and the longer the dynamic response time of its GCAP under the disturbance of ∆P_ref and ∆ω_g. The value of D also affects ξ and ∆P_e0 of the GFVSG grid-connected system; that is, the larger the value of D, the larger the corresponding ξ and ∆P_e0 are, and the stronger the damping ability of GCAP dynamic oscillation, but the larger the GCAP steady state deviation of P_e when ω_g deviates from ω₀. Consequently, for the GFVSG grid-connected system, the values of J and D dominated the dynamic and steady state response performance of its GCAP. In other words, the values of J and D could only be reasonably determined on the premise of weighing the GCAP dynamic and steady state response performance of GFVSG, resulting in certain limitations in the optimization of its GCAP response performance.

Fig. 3a shows the GCAP closed-loop small-signal control model of LLF-GFVSG. In Fig. 3a, K_p and K_d are the forward compensation coefficient and feedforward compensation coefficient of LLF-GFVSG, respectively. Further, the GCAP closed-loop small-signal equivalent control model of LLF-GFVSG can be obtained by equivalent transformation of the control block diagram in Fig. 3a, as shown in Fig. 3b. Since LLF composed of J, D, K_p and K_d is included in Fig. 3b, the improved strategy is referred to as LLF-GFVSG for short in this text.

According to Fig. 3a,b, the transmission functions of ∆P_ref to ∆P_e, ∆ω_g to ∆P_e, ∆P_ref to ∆ω and ∆ω_g to ∆ω for LLF-GFVSG can be expressed as: (8){G11(s)=K(KdJω0s+Kp)Jω0s2+(Dω0+KKdJω0)s+KKpG21(s)=−(Jω0s+Dω0)KJω0s2+(Dω0+KKdJω0)s+KKp (9){G31(s)=s(KdJω0s+Kp)Jω0s2+(Dω0+KKdJω0)s+KKpG41(s)=K(KdJω0s+Kp)Jω0s2+(Dω0+KKdJω0)s+KKp

It was comparing Eqs. (8) with (4) and Eqs. (9) with (5), respectively, it can be found that, compared with the GFVSG, the LLF-GFVSG, under the premise of ensuring that the order of its GCAP closed-loop control system remains unchanged and is still a second-order system, transforms the natural oscillating angular frequency of its grid-connected system, ω_n1, and its damping ratio, ξ₁, in turn, as follows: (10)ωn1=KKpJω0,ξ1=Dω0+KKdJω021KKpJω0

From Eq. (8), the GCAP steady state deviation ∆P_e01 of LLF-GFVSG is obtained, as shown in Eq. (11). (11)ΔPe01=Lims→0G21(s)Δωg=−(Dω0/Kp)Δωg

By comparing Eqs. (10) with (6) and Eqs. (11) with (7), respectively, it can be found that the value of K_p affects ω_n1, ξ₁ and ∆P_e01 of the LLF-GFVSG grid-connected system at the same time, in order to prioritize to ensure that the GCAP of LLF-GFVSG does not generate steady state deviation under the condition of ω_g deviation from ω₀, i.e., to prioritize to satisfy the condition of ∆P_e01 = ∆P_e0, the paper needs to set K_p = 1, then we have ω_n1 = ω_n, which then ensures that the grid-connected systems of LLF-GFVSG and GFVSG have the same natural oscillation angular frequency, which provides a fair condition for comparing the GCAP dynamic and steady state response performances of LLF-GFVSG and GFVSG, and at the same time simplifies the process of parameter design of LLF-GFVSG.

With the theoretical analysis in the previous subsection and the setting of K_p = 1 (∆P_e01 = ∆P_e0 with ω_n1 = ω_n), Eq. (10) can be equated as: (12)ωn1=KJω0,ξ1=Dω0+KKdJω021KJω0

Comparing Eqs. (12) and (6), it is easy to see that LLF-GFVSG has one more K_d control degree of freedom than GFVSG, that is, the former can optimize its GCAP dynamic response performance directly by selecting a reasonable K_d value. It can also be seen from Eq. (8) that G₁₁(s) of LLF-GFVSG is a second-order control system containing negative real zeros under the condition of K_d > 0. With reference to the parameter design method given in [25], G₁₁(s) can be equivalent transformed into: (13)G11(s)=KdJω0sωn12s2+2ξ1ωn1s+ωn12+ωn12s2+2ξ1ωn1s+ωn12

As can be seen from Eq. (13), the GCAP response of LLF-GFVSG mainly includes two parts: differential dynamic response and zero-free typical second-order system response. If K_d is larger, the negative real zero z₀ = −1/(K_dJω₀) of G₁₁(s) will be closer to the origin. The effect of differential dynamic response on the GCAP dynamic response of LLF-GFVSG is more obvious. At the same time, to suppress the GCAP dynamic oscillation of LLF-GFVSG, it is necessary to adjust the value of K_d to make ξ₁ ≥ 1, and when ξ₁ ≥ 1, it can further ensure that there is no power overshot in the dynamic response process of the zero-free typical second-order system in G₁₁(s).

On the one hand, under the condition that ξ₁ is ≥1, it can be obtained by Eq. (12): (14)ξ1≥1⇒Kd≥2KJω0−Dω0KJω0

On the other hand, under the condition that the value of K_d satisfies Eq. (14), regardless of the fact that the two negative real poles s₁ and s₂ of the second-order system with negative real zeros are different or the same, if z₀ is at the right end of s₁ and s₂, i.e., if z₀ is closer to the origin and corresponds to the K_d obtaining a larger value. The differential dynamic response part of G₁₁(s) will have a significant impact on the GCAP dynamic response of LLF-GFVSG and even introduce power overshooting [26]. Given this, it is recommended to select the z₀ of G₁₁(s) in the range s₁ to s₂, that is: (15)(−ξ1−ξ12−1)ωn1⏟s1≤−1KdJω0⏟z0≤(−ξ1+ξ12−1)ωn1⏟s2

According to Eqs. (12), (14) and (15) as well as the main parameters of the GFVSG, the rationalization of the K_d parameters of the additional control degrees of freedom of the LLF-GFVSG can be accomplished.

To verify the correctness and effectiveness of the LLF-GFVSG improvement strategy and its parameter design method, the 100kVA-GFVSG grid-connected system simulation model as shown in Fig. 1, is built by means of MATLAB simulation program. In the simulation process, P_ref = 0 kW, J = 6 kg·m², D = 50.66, K_p = 1 for 100kVA-GFVSG is set. Other main simulation parameters for 100kVA-GFVSG are shown in Table 1 [11].

At the same time, according to the main parameters given in Table 1 can be calculated to get K = 1,452,000, ω_n = ω_n1 = 27.7 rad/s, ξ = 0.15 < 1, and need to set K_d ≥ 3.24 × 10⁻⁵ to ensure that ξ₁ ≥ 1, in this case, K_d = 5.3 × 10⁻⁵ is selected, and thus there is ξ₁ = 1.52 > 1, z₀ = −10, the s₁ = −75, s₂ = −10. It is worth pointing out that in Fig. 3a, K_d is multiplied by ∆P and then feedforward compensated to ∆ω, and for the 100kVA-GFVSG grid-connected system, usually ∆P (order of magnitude 105)>>∆ω (unit of the order of magnitude), so that if the two reach a comparable order of magnitude, it is necessary for K_d to take a magnitude of 10⁻⁵ values.

By inputting the aforementioned parameters into Eqs. (8) and (9), Bode plots comparing the frequency response characteristics of ∆P_e/∆P_ref, ∆P_e/∆ω_g, ∆ω/∆P_ref and ∆ω/∆ω_g for LLF-GFVSG, GFVSG (D = 50.66 J/rad) and GFVSG (D = 335.16 J/rad) are presented in Fig. 4.

Fig. 5 shows the simulation comparison results of P_e vs. output frequency f for LLF-GFVSG, GFVSG (D = 50.66) and GFVSG (D = 335.16), respectively, in response to the dynamic process of the P_ref stepping from 20 to 60 kW and the grid frequency f_g stepping from 50 to 49.95 Hz.

It is not difficult to see from Figs. 4 and 5 that when D = 50.66, GFVSG is ξ = 0.15 of the underdamped system. The four bode diagrams representing GFVSG (D = 50.66) shown in Fig. 4 all have a resonant peak before the cutoff frequency. Therefore, its P_e and f under both the P_ref and f_g disturbances appear dynamic oscillations. When D = 335.16 is increased, GFVSG is an overdamped system ξ = 1.006, and the four bode diagrams representing GFVSG (D = 335.16) shown in Fig. 4 do not have a resonance peak before the cutoff frequency, so P_e and f do not oscillate dynamically under P_ref and f_g disturbances. However, its P_e introduces active steady state deviation at f_g = 49.95 Hz (∆P_e0 = 28.1 kW); while the P_e and f of LLF-GFVSG do not show dynamic oscillations under the two perturbations of P_ref and f_g, and its P_e does not have active steady state deviation at f_g = 49.95 Hz (∆P_e01 = 0) as shown in Fig. 5.

Fig. 6 presents the simulation results of the P_e and f during the implementation of the FOVSG control method in reference [27], as well as the active reference command P_ref of LLF-GFVSG transitioning from 20 to 60 kW and power grid frequency f_g varying from 50 to 49.95 Hz under the identical conditions. As illustrated in Fig. 6, the LLF-GFVSG and FOVSG exhibit comparable dynamic performance in terms of active power P_e under disturbances in P_ref and f_g. However, the output frequency overshoot observed in FOVSG is significantly higher than that in LLF-GFVSG. During the f_g step change, the output frequency response f of LLF-GFVSG is observed to be slower than that of FOVSG, suggesting a superior inertia response in LLF-GFVSG compared to FOVSG.

In summary, the LLF-GFVSG can effectively solve the problem that it is challenging to balance the GCAP dynamic response performance and its steady state response performance of GFVSG under both the disturbances of P_ref and f_g, i.e., the LLF-GFVSG can ensure that its P_e has both good dynamic and steady state response performance under the two disturbances of P_ref and f_g. It should be noted that since the LLF-GFVSG directly utilizes ∆P to feedforward compensate ∆ω, and there is a large deviation between P_ref and P_e at the beginning of the disturbance, i.e., the dynamic feedforward compensation to ∆ω is larger, so while improving the dynamic response speed of P_e, it is also easy to cause a large overshoot amplitude of f under the disturbance of P_ref.

In order to further verify the efficacy and superiority of the described LLF-GFVSG over the GFVSG in optimizing its GCAP dynamic response performance, experimental comparative verification is carried out on the energy storage microgrid system test platform presented in Fig. 7. The testing platform mainly includes two 100kVA-GFVSGs, two 100kVA bi-directional controllable rectifiers that provide a stable DC voltage and bi-directional energy supply for the 100kVA-GFVSGs (which can be used as a storage battery simulator), and a set of 250 kW resistive controllable loads and the distribution network [25].

During the experimental test, P_ref = 20 kW, J = 6 kg·m², D = 50.66, K_p = 1, K_d = 5.3 × 10⁻⁵ were set for 100kVA-GFVSG. Other main experimental parameters were consistent with Table 1 and the simulation parameters. Fig. 8a shows the experimental results under a power grid voltage step change of −1%. Fig. 8b shows the experimental outcomes for Q_ref transitioning from 0 to 30 kvar. As inferred from Eq. (2), the primary voltage regulation equation represents a first-order system. Regardless of whether the voltage step is −1% or the Q_ref steps from 0 to 30 kvar, the change in Q_e remains non-oscillatory. Nevertheless, Fig. 8 demonstrates that the LLF-GFVSG can effectively suppress dynamic oscillations in active power caused by grid voltage and Q_ref disturbances. Consequently, this study focuses exclusively on the changes in the grid-connected active power P_e and output frequency f.

Figs. 9a and 10a illustrate the experimental results for the P_e and f under the condition of X = 0.1 Ω, respectively. The comparison includes LLF-GFVSG, GFVSG (D = 50.66), and GFVSG (D = 335.16) during the transitions of P_ref from 20 to 60 kW and f_g from 50 to 49.95 Hz. Conversely, Figs. 9b and 10b illustrate the experimental results for the P_e and f under the condition of X = 0.05 Ω, respectively. The comparison includes LLF-GFVSG, GFVSG (D = 50.66), and GFVSG (D = 335.16) during the transitions of P_ref from 20 to 60 kW and f_g from 50 to 49.95 Hz.

According to the experimental comparison results shown in Figs. 9a and 10a, it is evident that the experimental outcomes during the transitions of P_ref from 20 to 60 kW and f_g from 50 to 49.95 Hz maintain a one-to-one correspondence with the simulation results presented in Fig. 5. According to the experimental comparison results shown in Figs. 9b and 10b, it is evident that the system’s damping ratio decreases as the line impedance decreases, resulting in more pronounced oscillations for GFVSG (D = 50.66). In contrast, the effectiveness and adaptability of LLF-GFVSG under the same conditions of reduced line impedance are markedly superior to those of GFVSG. Specifically, when the value of D increases from 50.66 to 335.16, the P_e and f for the GFVSG grid-connected system do not vibrate under the P_ref and f_g disturbances. However, the P_e has an active steady state deviation of ∆P_e0 = 28.3 kW at f_g = 49.95 Hz, while the P_e and f of LLF-GFVSG do not have dynamic oscillation when responding to two disturbances of P_ref and f_g, and the P_e can eliminate the active steady state deviation at f_g = 49.95 Hz. So ∆P_e01 is equal to 0.

In order to verify the effectiveness of LLF-GFVSG, we conducted parallel networking verification on the two GFVSGs shown in Fig. 7. The main system parameters for the parallel network system were configured as follows: P_ref1 = 2P_ref2 = 40 kW, D₁ = 2D₂ = 200, J₁ = 2J₂ = 6 kg·m², K_d1 = 2K_d2 = 5.3 × 10⁻⁵, X₁ = 2X₂ = 0.1 Ω, k_q1 = 2k_q2 = 1.4 × 10⁻⁴. Fig. 11 shows the comparison of experimental results between LLF-GFVSG and traditional GFVSG during the running of two GFVSGs parallel networking. At the initial time, the GFVSG1 and GFVSG2 jointly support a 60 kW resistive load to maintain stable operation. The 60 kW resistive step load is applied at 1 s and removed at 1.6 s. As shown in Fig. 11a, the LLF-GFVSG parallel networking control method can effectively mitigate the dynamic oscillation of the output active power P_e in the parallel networking system under step disturbances of the system load, compared to the existing traditional GFVSG parallel networking control method. Similarly, as shown in Fig. 11b, the LLF-GFVSG parallel networking control method can also effectively address the dynamic oscillation of the output frequency f_s in the parallel networking system under step disturbances of the system load, compared to the existing GFVSG parallel networking control method.

It is worth noting that because the proposed LLF-GFVSG directly employs ∆P for dynamic feed-forward compensation of ∆ω, the harmonic components present in P_e are introduced into f, resulting in the experimental test waveform of f containing more harmonics compared to that of the GFVSG. Consequently, the dynamic response waveform of f in the LLF-GFVSG exhibits a coarser envelope shape. Therefore, optimizing the dynamic response performance and waveform quality of the f of LLF-GFVSG when dealing with P_ref and f_g disturbances is one of the subsequent research works.