Taking a wind farm in Northeast China as the application scenario, the topology structure of wind turbines and ESVG is shown in Fig. 1a, where X_g is the internal impedances of synchronous generator SM. L_f is the grid-connected filter inductor for ESVG. U_g is the grid connected voltage of ESVG. I_g is ESVG compensation current. The synchronous generator is connected to the 220 kV busbar through a transformer and a 5 km transmission line.

The ESVG control structure with direct amplitude and phase control is shown in Fig. 1b. In the ESVG active control loop, f₀ and f_g are the power frequency and the actual system frequency, respectively, and Δf represents the frequency fluctuation of the wind power regional system. KpFS is the ESVG frequency modulation coefficient. P_e represents the active power output of ESVG. When the frequency variation of the wind farm exceeds the dead zone of ESVG frequency regulation, ESVG responds to the system’s active power-frequency demand. In the self-synchronization control loop, P_ref is the active power reference value of the power loop. J and K_D are virtual rotation inertia and virtual damping coefficients. Δω_v, ω₀, ω_v, and θv are the virtual angular velocity adjustment value, rated virtual angular frequency, virtual angular velocity, and virtual phase of ESVG’s virtual swing equation, respectively.

In the AC voltage-reactive power control loop and ESVG electrical topology, U₀, U_ESVG, and ΔU are the ESVG outlet voltage, the rated phase voltage of the PCC node, and the virtual excitation electromotive force adjustment value, respectively. After obtaining the reactive power adjustment value with the voltage adjustment coefficient K_ug, it is summed with the grid-connected reactive power reference Q* and subtracted from the output reactive power of ESVG Q_e. After the obtained difference is integrated by the coefficient K_Q, the PCC phase voltage adjustment value ΔU is obtained and summed with U_g to obtain the virtual excitation electromotive force E_ESVG. The grid-connected power of ESVG is calculated according to the current output and PCC voltage. The virtual phase of ESVG can be obtained by an active power loop, combined with E_ESVG, and the virtual internal potential V_PCC(VPCC=EESVGejθv) is obtained. The amplitude limit E¯ESVG and E_ESVG are usually set to ±1.2 pu~±1.5 pu. Finally, the control pulse is generated by CPS-SPWM modulation to control the ESVG power units. Construct the ESVG swing equation based on Fig. 1b as shown in Eq. (1) (1){dθdt=ωv−ω0Jdωdt=−Te−Td=−Te−KD(ωv−ω0)where T_e is the virtual electromagnetic torque of ESVG, T_d is the virtual damping torque of ESVG, and J is the virtual rotation inertia of ESVG. The virtual internal potential Q-V equation of ESVG is shown in Eq. (2). (2){Qm=Q∗+Kug(U0−Ug)EESVG=KQ∫(Qm−Qe)dt+U0

To simplify the description of the transient stability of ESVG, the dynamic response process of virtual impedance and current inner loop are ignored. The system in Fig. 1 is equivalent to an ESVG single-machine infinite bus system. The output voltage of ESVG is EESVG∠δ, and the PCC voltage is Ug∠0∘. ESVG’s equivalent output impedance Z can be expressed as (3)Z=(Rf+Rg)+jω(Lf+Lg)≈jXwhere X is the impendence of Z. R_f, and L_f are ESVG filter reactance and parasitic resistance. R_g and L_g are the grid impedance. Due to ESVG being directly connected in parallel with the 35 kV high voltage bus, the resistance of the filter in ESVG can be omitted [21,22]. ESVG does not require virtual impedance for current limitations without considering fault conditions. The output power of ESVG is expressed as (4){Pe=3UgEESVGXsin⁡δQe=3UgEESVGXcos⁡δ−3Ug2X

When the wind power system usually operates, ESVG only needs to output reactive power to the wind power system, so ESVG grid-connected active command P_ref is 0. To ensure that Q_e varies with Q_m, the reactive power deviation is integrated to obtain E_ESVG. To respond rapidly to its frequency requirements, its own time constant τ = K_Q/K_D is usually small, and the change of E_ESVG can be ignored [23,24]. Eq. (3) can be reasonably degenerated into a quasi-steady-state equation, according to Eqs. (1)–(4), the second-order differential equation of ESVG control system is obtained. (5){Jd2δdt=PrefX−3EESVGUgsin⁡δω0X−KD(ωv−ω0)Q∗−3E2X−3EESVGUgcos⁡δω0X−Kut(EESVG−U0)=0

Setting the state variable x=[x1,x2]T=[δ,Δω]T, the small disturbance model of ESVG is obtained as follows: (6)[x˙1x˙2]=[x2P∗Jω0−3EESVGUgJω0Xsin⁡x1−KDx2Jω0]

Combined with the analysis of Reference [25], the imbalance between the reference value of the active power output of ESVG and the actual one is the internal cause of VSG transient angle instability. Therefore, in the virtual swing equation, J, K_D, and system frequency fluctuation Δf are primary research variables. According to the second-order differential equations of the system shown in Eq. (6), the phase space trajectory of ESVG is drawn to observe the influence of each sensitive parameter on the electromechanical transient stability of ESVG.

Compared with conventional synchronous generators, large-scale access to new energy power generation devices shows the characteristics of low rotational inertia. According to the grid-forming control structure adopted by the ESVG in the first chapter and the rotational inertia of the synchronous generator shown in Eq. (4), the virtual synchronous control strategy can imitate the rotor motion of the synchronous generator. However, due to the lack of continuous power supply during energy release, ESVG cannot wholly reproduce the continuous output of the prime motor (Steam turbine) during the primary frequency modulation process but mainly imitates the process of converting the mechanical energy from the rotator shaft of the prime motor into electromagnetic energy. (7){J′dΩdt=MT−MEWSG=12J′Ω02WESVG=12CescU2where J^′ represents the rotational inertia of the synchronous generator’s rotor. M_T and M_E represent the mechanical torque of the prime mover and the electromagnetic torque of the synchronous generator, respectively. W_SG is the mechanical energy stored in synchronous generator’s rotor shaft at the rated speed. Ω₀ represents the rated mechanical angular velocity of synchronous generator’s rotor. C_esc represents the equivalent capacitance of all supercapacitors in the chains of ESVG. Unlike the synchronous generator, there is no continuous power injection (no virtual mechanical torque) on the DC side of the ESVG. The equivalent virtual electromagnetic torque is formed only by the electrochemical energy W_ESVG stored in the supercapacitors. Although the energy stored in the above two is similar in the form of expression, ESVG is essentially different from the synchronous machine. Especially in terms of output voltage vector control, ESVG is insufficient.

Taking the frequency reduction condition of the wind power system as an example, the phase change of ESVG and system voltage is shown in Fig. 4. It is assumed that the J of ESVG is infinite. When the active power demand of the wind power system occurs, the α between ESVG and the system increases. However, excessive J results in a constant ESVG phase in a short period.

As shown in Fig. 4, before the frequency of the wind power system decreases, the voltage amplitude and phase of the ESVG are basically identical to the wind power system. After the frequency of the wind power system decreases, excessive J will lead to rapid energy release from the ESVG supercapacitors. Due to the limited energy storage, the voltage of supercapacitors will continue to drop, thus affecting the reactive power compensation performance of ESVG.

Considering the influence of the number of cascaded submodules N on the equivalent capacitance of ESVG, the variation range of ESVG DC cluster voltage U_dc is limited to 0.8~1.1 pu, and the available active power P_av of each phase can be expressed as Eq. (8). (8)Pav=1(∑i=1N1C1+1C2+⋯+1Ci)UdcjdUdcjdt,j=A,B,C

According to Eq. (9), maintaining a constant N, if the capacitance of the supercapacitor in each submodule decreases, the overall equivalent capacitance will decrease. The voltage change rate dU_dcj/dt and the variation of DC voltage will increase, which is not conducive to the stable operation of ESVG in the wind power fluctuation scenario. Based on the wind system model with ESVG, DC cluster voltage simulation results are shown in Fig. 5.

Fig. 5 shows that during wind power fluctuation. However, ESVG can still effectively suppress system frequency changes, and the DC cluster voltage change intensifies with the decrease of individual capacitance, which is consistent with the former analysis in Eq. (9).

According to an actual wind farm in Northeast China, the capacity of an ESVG S_n = 50 MVA, the equivalent inertia time constant of supercapacitors 12 s ≤ H_esc ≤ 20 s, and the capacitance of a single supercapacitor can be expressed as (9){Ci=4NSnHescΔfmaxf0(udcmax2−udcmin2)12Jω02=Hesc⋅SN

According to Eq. (9), it can be seen that the frequency fluctuation increases as is constant. To ensure the active-frequency support ability of ESVG, it is necessary to equip it with a larger supercapacitor capacitance. The increase in the capacitance of the supercapacitor will lead to an increase in H_esc. Although the adjustment time of the frequency is prolonged to some degree, the active-frequency support ability of the ESVG can be enhanced. Here, N = 10 and C_i = 1.5 F. The allowable fluctuation range of DC cluster voltage is 22.86 KV~31.43 KV (0.8~1.1 pu). From Eq. (10), the constraint range of ESVG’s J is 10,860~18,618 kg·m².

The small signal model of the active power control loop of the ESVG in the frequency domain can be obtained by combining Eqs. (1)–(6), as shown in Fig. 6.

The active-inertia disturbance equation and frequency closed-loop transfer function of ESVG can be further obtained from Fig. 6 and can be expressed as (10){P^e(s)ω^g(s)=−3Jω0Un2s+3KDω0Un2Jω0Xs2+KDω0Xs+3Un2ω^v(s)ω^g(s)=3Un2Jω0Xs2+KDXω0s+3Un2/(Xω0)

U_n is the practical value of PCC phase voltage. Considering the demand for primary frequency regulation response of conventional synchronous generators, when the frequency of wind power systems fluctuates, the response time of the ESVG active power loop should be less than 100 ms to exert its fast active power regulation advantage. Combined with the active power control characteristics of ESVG, the natural oscillation angular frequency ω_nf and damping ratio ξ_f of the active inertia response of ESVG can be obtained by Eq. (11). (11){ωnf=3Un2/(Jω0X)ζf=0.5KDω0X/(3JUn2)

When the frequency disturbance occurs in the wind power system, the ESVG control response can accelerate the response speed in the underdamped state in the initial reaction stage. To quickly eliminate ESVG control oscillations and reduce overshoot, choosing critical damping or overdamping is more scientific. Therefore, the value of J should be adapted to the frequent switching between underdamped and overdamped states during disturbance. At the same time, to ensure the stability of ESVG active-inertia closed-loop control, it is necessary to ensure that its characteristic roots are all negative values. Therefore, the boundary condition of ξ_f is written as follows: (12){22≤KD2ω0X3JUn2≤1.2−ωnfξ=−KD4Hesc=−KDPemax2Jω02≤−10

When the ESVG responds to the system’s frequency demand, it cannot exceed its own maximum charge-discharge power, P_emax. (13)2Hescf0max(df(t)dt)PN=JΩN2f0max(df(t)dt)=Pemaxwhere max(df(t)/dt) represents the maximum allowable frequency variation of the wind power system in the unit time set to 1 Hz/s. Based on the above constraints, a feasible region of J-K_D can be drawn, as shown in Fig. 7.

The yellow shaded area in Fig. 7 represents the selection range of J and K_D. After J is tuned, K_D is tuned based on the relationship between the frequency variation of the wind power system and the active power output of ESVG.

The relationship between the frequency variation of the wind power system and the output power of the ESVG can be obtained from Eq. (11). (14)ΔPeΔωg=KDω0

It is assumed that when the wind power system’s frequency change exceeds 0.03 Hz, the ESVG’s output active power equals P_emax. (15)KDω0=30×106×100%2π×0.03≈5×105

Combined with the above J-K_D feasible region, the initial values of J and K_D are taken as 1.3 × 10⁴ kg·m² and 2.5 × 10⁵, respectively.

The essential condition for the stable operation of ESVG is that the output power does not exceed the steady-state power operating range. In the practical application process, ESVG can operate for 1 s with three times the rated active power, and the steady-state overload multiple k_m = 3 is set. Assuming P_e = P_ref, the maximum operating power angle δESVG≈20∘ is obtained from Eq. (16). δESVG<10∘ is taken to analyze the operating boundary. (16)km=mUESVGUgXmUESVGUgXsin⁡δESVG=1sin⁡δESVG=3

The active and reactive power output from ESVG to the wind farm can be expressed as (17){Pe=32⋅XgUESVGUgsin⁡δESVG+(Rf+Rg)(UESVG2−UESVGUgcos⁡δESVG)Xg2Qe=32⋅Xg(UESVG2−UESVGUgcos⁡δESVG)−(Rf+Rg)UESVGUgsin⁡δESVGXg2

Set USSC∠δ=Ud+jUq. According to Kirchhoff voltage law, the ESVG grid connected voltage relationship can be expressed as (18)(Ud+jUq)−Pe+jQeUd+jUq(R+jX)=Ug∠0∘

In addition, further constraints need to be set based on the operating range of ESVG’s DC side voltage mentioned in Section 4.2. (19){0.8pu<Udc<1.1puPESVG_max≤50e6MWQESVG_max≤50e6MW

In summary, the stable output power operating range of ESVG is determined based on the following constraints:

Condition 1: Set the rated capacity of ESVG, combined with Formula (17) to calculate the possible ESVG power operating point. When Pe2+Qe2≤Sn2, the obtained power points need to be further screened according to the following conditions.

Condition 2: According to Eq. (18), determine whether the real and imaginary parts of the equation are equal at both ends. If not, exclude the operating point. On the contrary, ESVG can operate at that power point and further verification is required.

Condition 3: ESVG can operate stably if its DC voltage meets Eq. (19) during ESVG’s active frequency support period.

The power operating points that meet all the above conditions are drawn to form a stable power operating range of ESVG, as shown in Fig. 9.

The active power operation range of ESVG in Fig. 9 determines the active-frequency support capability of ESVG. The threshold limiting can adjust the output active power of ESVG in the range of [P¯max,P_max]∈[−40 MW,40 MW]. Combined with the frequency modulation coefficient adjustment method proposed in the next section, the purpose of stable operation of ESVG can be achieved.

If KpFS is set according to the existing energy storage power station standard, ESVG will meet the response speed as much as possible. However, it may exacerbate the overshoot of its active power output. Therefore, it is necessary to make reasonable restrictions on KpFS. Under the premise of ensuring the inertia response, the position of KpFS is shown in the green block of Fig. 8, it is adaptively adjusted to reduce the ESVG active power fluctuation during frequency modulation [26]. Using the frequency deviation Δf between the real-time frequency of the wind power system and the rated frequency, the active power reference value P_ref is obtained. The improved KpFS and P_ref are represented as follows: (20)KpFS={P¯max0.033,Δf>0.0330,other elseP_max−0.033,Δf<−0.033 (21)Pref={P¯max,Δf>ΔfmaxKpFS⋅Δf,Δfmin<Δf<ΔfmaxP_max,Δf<Δfmin

In Eqs. (20) and (21), Δf_max and Δf_min are the upper and lower critical values of the frequency adjustment range of the ESVG frequency modulator, respectively, and set to ±0.2 Hz. [−0.03, 0.03] Hz is the dead zone of ESVG participating in the frequency modulation of the wind power system. Within this range, the frequency regulator does not work, and there is no active power exchange between ESVG and the wind power system, thereby avoiding frequent charging and discharging of supercapacitors. If Δf ∈ [−0.2, −0.03] ∪ [0.03, 0.2] Hz, the frequency modulator calculates P_ref. When P_ref exceeds the critical values of the frequency adjustment range, it is limited to the active power output threshold P¯max or P_max.

From the analysis of Section 2, it can be seen that ESVG’s Δω_v characterizes the fluctuation degree of active power output during the frequency support process, which can be expressed as (22)Δωv=(PΔf+Pref−Pe)⋅ωv−1−Jdωv/dtKD

From Eq. (22), it can be seen that J and K_D are negatively correlated with the ESVG virtual angular velocity change rate dω_v/dt and the angular velocity change amount Δω_v, respectively. Active power overshooting can be suppressed by simultaneously reducing J and increasing K_D. When dω_v/dt > 0, a larger J can reduce the system overshoot to a certain extent, but it will prolong the adjustment time. When dω_v/dt < 0, setting a smaller J can quickly stabilize the system, but the overshoot tends to increase. To alleviate this contradiction, in the process of adjusting J, when the change of ω_v exceeds the adjustment threshold, K_D should be increased to suppress the excessive |Δω_v|. When dω_v/dt > 0, ALP increases J to suppress overshoot and simultaneously increases K_D, so the output power steadily reaches the command value. Based on the above analysis, as shown in the blue block of Fig. 8, the ALP control strategy of J and K_D is designed as follows: (23)J={J0,|dωvdt|≤0.16J0−Kj1|(dωvdt)⋅Δωv|,(dωvdt)⋅Δωv<0∩|dωvdt|>0.16J0+Kj2|(dωvdt)⋅Δωv|,(dωvdt)⋅Δωv>0∩|dωvdt|>0.16 (24)KD={KD0,|Δωv|≤0.19KD0+Kd|Δωv|,|Δωv|>0.19

In Eqs. (23) and (24), J₀ and K_D0 are the initial values of ESVG’s J and K_D (which can be obtained by combining with the plane diagram of J-K_D feasible region in Fig. 6). K_j1, K_j2, and K_d are the virtual rotation inertia and damping adjustment coefficient, respectively. The frequency modulation range of ESVG is set to [−0.2, −0.03] Hz and [0.03, 0.2] Hz. The transfer function of the ESVG active control loop can be regarded as a type-II system. According to the system simulation parameters provided in Table 1, the virtual angular velocity oscillation curve of ESVG after a small disturbance is shown in Fig. 10.

According to the analysis of Eqs. (23) and (24) and Fig. 10, it can be seen that in the process of ESVG starting to respond to the active power demand of the system in stage I, its virtual phase is no longer consistent with the phase of the wind power system. At this point, |Δω_v | increases and exceeds the dead zone, and K_D will increase while J increases. After ω_v reaches the peak, its change enters the second stage. At this time, |Δω_v | will re-exceed the dead zone. The increase of K_D and the decrease of J can shorten the oscillation time of virtual angular velocity and effectively suppress the active power fluctuation. The third and fourth stages are similar to the first and second stages.

To verify the effectiveness of the proposed ESVG adjustment control strategy using ALP in the wind farm, two conditions of wind turbine tripping and load cutting off are set up. The wind field regional interconnection system model with ESVG and synchronous generator is built, as shown in Fig. 1a. The relevant parameters are shown in Table 1.