In the AC microgrid, the VSG control is usually applied to the DC/AC converter to improve the inertia. The active power-frequency P−ω link of the control structure simulates the inertia, damping, and primary frequency modulation characteristics of the synchronous generator, and the original VSG equation is shown in Eq. (1).

(1){Pset−Pe−PD=Jωdωdt≈JωNdωdtPD=Dp(ω−ωN)

where J and Dp are the moments of inertia and damping coefficient, respectively, Pset is the P−ω droop control output power, Pe is the output electromagnetic power, PD is the damping power, ω is the angular frequency, and ωN is the rated value of ω. Based on the control structure of Eq. (1), ωN is introduced into the inertia link, the P−ω droop frequency modulation link, and the damping link. ωc acts as the feedback signal of the frequency modulation link, and the new VSG control equation is shown in Eq. (2).

(2)Jωd(ω−ωN)dt=k(ωN−ωc)+PN−Pe−Dp(ω−ωN)

where PN is the rated power of P−ω droop control, k is the droop coefficient, and ωc is the angular frequency of the DC bus. According to Eq. (2), when the grid frequency changes suddenly, k is used for the primary frequency modulation, Dp is used to suppress the frequency oscillation, and J can make the converter control the active power output under the adjustment of VSG, thereby achieving the effect of suppressing frequency changes. The energy Wr stored in J is shown in Eq. (3).

(3)Wr=12Jω2

Drawing inspiration from the concept of the VSG in the AC microgrid, the AVSG control is implemented in the DC microgrid. Analogous to the P−ω droop control in the AC microgrid, the droop control that utilizes current to suppress the voltage fluctuation is implemented in the DC microgrid. Analogous to Eq. (3), in the energy storage interface DC/DC converter, the energy can be stored by paralleling the virtual capacitance Cv, acting as the moment of inertia J in the AC microgrid. The energy Wc stored in Cv is shown in Eq. (4).

(4)Wc=12Cvudc2

where udc is the DC bus voltage, which is also the output voltage of the DC/DC converter in this paper. The remaining variables associated with the DC/DC converter follow a similar pattern.

The main variable in the AC microgrid is frequency, while in the DC microgrid, the DC bus voltage is the sole indicator to measure the power balance and the stability of the system. Inertia and damping represent the ability to suppress the sudden changes and oscillations in the DC bus voltage. The DC/DC converter lacking inertia features on the DC side is too sensitive to the response of the DC bus voltage, which can cause voltage sudden changes or system instability when responding to external power fluctuations. The introduction of a virtual capacitance with voltage-stabilizing properties slows down voltage changes by utilizing the stored energy. By analogy with the relationship between frequency and power in Eq. (2), the energy stored in J is substituted with the energy stored in Cv, while factoring in the damping effect for voltage oscillation suppression, the control equation for virtual capacitance current is shown in Eq. (5).

(5)Δi=Cvdudc∗dt=iset−idc−id

where udc∗ is the output voltage of the AVSG, and Δi, iset, idc and id are virtual capacitance current, the droop control input current, the DC bus current, and damping current, respectively.

Based on the above analysis, the analogous relationships exist between the DC/AC converter with the VSG in the AC microgrid and the DC/DC converter with the AVSG in the DC microgrid. The corresponding analog parameters of the AC microgrid and the DC microgrid are illustrated in Table 1.

According to the correspondence between the variables between the DC microgrid and the AC microgrid in Table 1, and the analogy of Eqs. (2) and (5), and make adjustments to some variables, the AVSG control equation is shown in Eq. (6).

(6)kdroop(uN−udc)−idc−Dp(udc∗−udc)=Cvd(udc∗−uN)dt

where kdroop is droop coefficient of the AVSG control; uN is the rated voltage of the AVSG.

Fig. 2 shows the control block diagram of the energy storage system with the AVSG, which is composed of the BESS (Battery Energy Storage System), the current tracking module, and the AVSG control module. The output voltage udc∗ of the AVSG control module is used as a reference value for the voltage outer loop. In order to eliminate voltage steady-state errors, the outer loop uses the voltage PI controller to make the converter DC side voltage udc track the reference value udc∗ all the time. The output current idc∗ outputs the signals to the switch tubes S1 and S2 of the DC/DC converter through the current tracking module to achieve the control purpose.

As shown in Fig. 3, the AVSG equivalent model is established on the basis of the droop control equivalent model, and the equivalent components of the AVSG control are introduced. The impedance Rd represents the droop coefficient, the inductance LPI represents the voltage PI controller, Cv is the virtual capacitance, and −DpuL is the controllable current source. Compared with the droop control, the AVSG control introduces inertia and damping to the system, thereby improving the voltage quality of the microgrid.

In order to analyze the specific control effect of the energy storage interface DC/DC converter after the introduction of the AVSG control, suppose the duty cycle of the DC/DC converter is d, and the state space equation is shown in Eq. (7).

(7){x˙=C1x+C2ux=[iLudc]Tu=[ubatidc]T

where x and u are state variables, ubat and iL are respectively the input voltage and input current of the energy storage interface DC/DC converter, and udc and idc are respectively the output voltage and output current of the energy storage interface DC/DC converter. The matrices C1and C2 are respectively are shown in Eq. (8).

(8)C1=[RLLb1−dCb1−dbCb0]C2=[1Lb00−1Cb]

The small-signal linearization of the state average equation is carried out, and the disturbance is introduced near the steady-state operating point, and the instantaneous values of the corresponding DC/DC converter variables are shown in Eq. (9).

(9){iL=IL+ΔiLudc=Udc+Δudcubat=Ubat+Δubatd=D+Δdidc=Idc+Δidc

where ΔiL, Δudc, Δubat, Δd and Δibat represent the corresponding small signal disturbance, and IL, Udc, Ubat, D and Idc represent the steady-state values of the corresponding variables. The small signal equation after neglecting the quadratic disturbance is shown in Eq. (10).

(10){sCΔudc=(1−D)ΔiL−ILΔd−ΔidcsCΔiL=Δubat−RLΔiL−(1−D)Δudc+UdcΔd

By sorting out the small signal equation of Eq. (10), the relevant transformation can be deduced. The main transfer function of the DC/DC converter is shown in Eq. (11).

(11){Gii(s)=ΔiL(s)Δidc(s)=1−DCbLbs2+CbRLs+(1−D)2Gid(s)=ΔiL(s)Δd(s)=−CUdcs+(1−D)ILCbLbs2+CbRLs+(1−D)2Gud(s)=Δudc(s)Δd(s)=−LbiLbs−RLiL+(1−D)UdcCbLbs2+CbRLs+(1−D)2Giu(s)=Δidc(s)Δudc(s)=−−Lbs−RLCbLbs2+CbRLs+(1−D)2

where Gii(s) is the transfer function from the output current to the input current, Gid(s) is the transferfunction from the duty cycle to the input current, Gud(s) is the transfer function from the duty cycle to the output voltage and Giu(s) is the the output voltage to the input current.

Neglecting energy losses, the power balance on both sides of the energy storage interface DC-DC converter is depicted in Eq. (12).

(12)ubatiL=udcidc

Linearize the small-signal equation of Eq. (12) and neglect the quadratic disturbance, as shown in Eq. (13).

(13)ILΔubat+UbatΔIL=udc=IdcΔudc+UdcΔidc

According to the superposition theorem, the small signal disturbance Δubat and Δudc in Eq. (13) can be neglected. The resulting simplified transfer function between Δidc and ΔiL is presented in Eq. (14).

(14)GiiL=ΔiL(s)Δidc(s)=UdcUbat

Laplace transform and small-signal linearization are applied to Eq. (6), yielding the small-signal model controlled by the AVSG as shown in Eq. (15).

(15)−kdroopΔudc(s)−Δidc(s)−Dp[Δudc∗(s)−Δudc(s)]=CvsΔudc∗(s)

The AVSG small-signal model of the established energy storage DC/DC converter is shown in Fig. 4. The voltage outer loop and current inner loop adopt the PI control, as shown in Eq. (16).

(16){Gu=kpv+kiv/sGi=kpi+kii/s

where kpu and kiu are the PI parameters of the converter voltage outer loop, and kpi and kii are the PI parameters of the converter current inner loop.

The AVSG small signal model of the energy storage DC/DC converter is shown in Fig. 4, and the AVSG output current disturbance is shown in Eq. (17).

(17)Δidc∗(s)=(kpv+kiv/s)[Δudc∗(s)−Δudc(s)]

Combined with the operating characteristics of the PI controller, Δidc(s) is approximate to Δidc∗(s), and Eq. (17) is substituted into Eq. (15). And the equivalent droop coefficient of AVSG control is shown in Eq. (18).

(18)m=lims→0(Δudc(s)Δidc(s))=−1kdroop

According to Eq. (18), the droop coefficient kdroop determines its equivalent droop coefficient m. Since droop control involves differentiating the voltage, kdroop will impact the bus voltage offset when the power of the microgrid changes. Based on Eqs. (15) and (18), and taking into account the composition of the AVSG control structure along with the derivation of its small signal model, the approximate relationship between the ratio of Δudc(s) and Δidc(s) is depicted in Eq. (19).

(19)Δudc(s)Δidc(s)≈m(Cv/Dp)s+1

This article only requires a brief analysis of the relationship between the inertial parameters of the AVSG and the inertia of the DC microgrid, along with the design of the parameter adaptive equation. Fig. 4 reveals that, beyond the main pathway, the AVSG closed-loop system is also influenced by two additional secondary paths L1 and L2, leading to a relatively intricate analysis. Moreover, the dynamic behavior of the higher-order AVSG system can be directly inferred from the dynamic performance index of the first-order inertial system [20]. As a result, the equivalent first-order transfer function TF(s) of the AVSG closed-loop system is represented in Eq. (20).

(20)TF(s)=Δudc(s)−Δidc(s)≈1/kdroop(Cv/Dp)s+1,τ=Cv/Dp,|m|=1/kdroop

where τ is the inertial time constant. According to Eq. (20), the unit step response curve of the AVSG control parameter variations can be plotted. It can be seen from Fig. 5a that the larger the Cv, the smaller the change of the DC bus voltage perunit time, the longer the time for the DC bus voltage to stabilize, the stronger the inertia of the DC microgrid system. The smaller the Cv, the shorter the time for the DC bus voltage to stabilize, the smaller the inertia of the DC microgrid. In addition, in the equation, τ=Cv/Dp, which reflects the size of the inertia of the system, and Cv is proportional to the inertial time constant τ.

It can be seen from Fig. 5b that a suitable decrease in Dp can extend the time required for voltage stabilization, thereby enhancing the inertia of the DC microgrid system. Conversely, an increase in Dp weakens the inertia. Both Figs. 5a and 5b illustrate that Cv and Dp can impact the inertia, being components of the τ function. Eq. (20) indicates that kdroop is not governed by the τ function and is not correlated with the inertia trend. Nonetheless, as shown in Fig. 5c, it is evident that kdroop influences the voltage offset, showcasing a gradual decrease. Hence, an adaptive adjustments in Cv and Dp values can modify the inertia of the microgrid system.

To enable the AVSG control to dynamically adjust Cv and Dp in response to changes in the DC bus voltage, Cv and Dp should be formulated as functions of the rate of voltage change. Therefore, the voltage change rate tracking coefficients A and B are introduced, and |dudc/dt|>kv is used as a limiting condition to determine whether the DC bus voltage fluctuates. The flexible adjustment of Cv and Dp can improve the inertia of the DC microgrid system and effectively address the voltage fluctuations generated by the distributed power sources and load power changes. Based on the aforementioned impact of Cv and Dp trends on the inertia of the DC microgrid, the adaptive parameter conditions are designed, as depicted in Eqs. (21) and (22).

(21){Cv=Cv0(|udc−uN|≤ku)∩(|dudc/dt|≤kv)Cv=Cv0+A|dudc/dt||dudc/dt|>kv

(22){Dp=Dp0(|udc−uN|≤ku)∩(|dudc/dt|≤kv)Dp=Dp0−B|dudc/dt||dudc/dt|>kv

where Cv0 and Dp0 are the initial virtual capacitance and damping coefficient, respectively, and A and B are the voltage conversion rate tracking coefficients, which can be selected according to the actual situation, and ku and kv are the minimum voltage change and voltage change rate that are allowed to fluctuate. This article sets the value of ku and kv to 0 for simplifying.

When performing parameter adaptive optimization, a high-pass filter (HPF) is used to mitigate the disturbance caused by introducing the differential link to obtain the voltage change rate within the system. This approach facilitates the acquisition of the voltage change rate. Simultaneously, to prevent output voltage oscillations resulting from sudden changes in Cv and Dp, a low-pass filter (LPF) has been added to the output terminals of Cv and Dp, respectively. The block diagram illustrating the adaptive control of the AVSG is shown in Fig. 6.

To simplify the analysis, ignoring the influence caused by two secondary paths L1 and L2 in Fig. 4, the closed-loop transfer function TFff(s) between Δudc(s) and −Δidc(s) is shown in Eq. (23).

(23)TFff=Δudc(s)/(−Δidc(s))=(GudGiGuGii)/[(Cvs+Dp)(1+GidGi)+(kdroop+Cvs)GudGiGuGii]

By analyzing the closed loop pole distribution trends of Cv, Dp and kdroop changes, a rough range of the AVSG control parameters that ensure system stability can be determined.

As shown in Fig. 7a, in addition to the fixed poles that have satisfied stability, it is also necessary to analyze the influence of changing poles on the system stability. With the increase of Cv, the poles labeled as P11, P12 and P13 are from left to right. P11 gradually moves toward the imaginary axis, weakening the system stability. P12 gradually moves away from the imaginary axis, enhancing the system stability. As the dominant pole closest to the imaginary axis, P13 moves toward the imaginary axis, and the system stability tends to weaken, but it does not enter the right half-plane, and the entire system remains stable. Similarly, in Figs. 7b and 7c, the dominant poles P23, P32, and P33 closest to the imaginary axis determine the trend of system stability. The larger the values of Dp and kdroop, the closer their poles are to the intersection of the left half-plane and the right half-plane, and the worse the overall system stability will be.

To validate the efficacy of the proposed control strategy, a simulation platform is constructed using MATLAB/Simulink, as depicted in Fig. 1. The PV unit operates at a rated temperature of 25°C, with an initial solar irradiance of 600 W/m². The parameters of the DC microgrid system are provided in Table 2.

The effects of the voltage change rate tracking coefficients A and B on the AVSG are still examined in Scenario 2. The roles of A and B in Scenario 3 are similar to those in Scenario 2, eliminating the need for further explanation. Whenever there is a change in the power of the DC microgrid, the AVSG adaptive control can recalibrate the values of Cv and Dp by adjusting A and B. As illustrated in Figs. 14a and 14b, the waveform variations of Cv and Dp can be observed with different values of A and B.

In Fig. 14c, with the increase of A and B, Cv becomes larger and Dp becomes smaller. By adjusting the values of A and B, the values of Cv and Dp can be modified, enabling the implementation of the AVSG adaptive control. This improvement enables the AVSG controller to detect changes in the DC bus voltage with greater sensitivity.