The topology of the GFL converter and its control strategy is shown in Figure 1, which consists of a dc bus, a two-level power electronic inverter, an LC filter, a grid impedance, and an ideal voltage source.

Currently, the GFL converter uses a vector control strategy in dq frame. This control samples the output current i _abc and voltage u _gabc at the PCC and then convert them into dq components. Furthermore, the PLL gives the grid phase angle θ in the synchronous reference frame. The control of GFL converters usually uses a dual closed-loop control, where the external voltage loop aims to maintain the dc bus current constant. In contrast, the internal current loop converts the reference current into a voltage reference signal. Then the modulation module generates the modulating waveform m _abc . Finally, the Pulse Width Modulation (PWM) generator drives the converter switching circuit to realize the function of regulating the reactive power on the Alternating Current (AC) side and delivering the active power generated by the RES to the grid. For simplicity, the Direct Current (DC) bus voltage U _dc is assumed to be constant, and the outer loop is ignored for modeling processes in the following sections. The transfer functions of the sampling G _del (s), the inner-loop current controller H _c (s), and the PLL controller H _PLL (s) are given in Eqs 1–3, respectively. G d e l s = G d e l 1 + s T d e l (1) H c s = K p − c + K i − c s (2) H P L L s = K p − p l l + K i − p l l s × 1 s (3) where G _del and T _del are the sampling gain and sampling delay, respectively; K _p-c and K _i-c are the Proportional-Integral (PI) control parameters in the current loop; and K _p-pll and K _i-pll are the PI control parameters in the PLL control loop.

Figure 2 shows the topology of the GFM converter and its control strategy. The GFM control actively forms the voltage output at the PCC point by simulating the synchronous machine power angle characteristics. At the same time, it controls the virtual turn angle according to the active power instead of collecting the voltage and phase at the PCC point through the PLL, and its modulation method is the same as that of the GFL type. The active power control loop of the GFM control simulates the swing equation of a conventional synchronous generator to provide virtual inertia and achieve primary frequency regulation. In contrast, its reactive power control simulates a conventional synchronous generator’s primary voltage regulation characteristics. The virtual rotor equation of motion of the GFM converter is shown below. J d ω d t = P ref − P e − D d ω − ω N (4) θ = 1 S ω (5) where J is the virtual inertia coefficient; D _d is the damping coefficient; ω _N and ω are the nominal and virtual angular frequencies, respectively; and θ is the phase angle generated by the active power control loop; P _ref and P _e are the reference active power and the actual active power output, respectively.

The excitation system of a synchronous generator is controlled by the excitation current, while the equivalent variable in the inverter control is the reference voltage. Therefore, by simulating a synchronous generator’s excitation current control method, the control equation for voltage regulation can be obtained as follows. 2 E = G s 2 U ref − 2 U m (6) where G(s) is the voltage regulator. U _ref and U _m are the effective values of the reference and measured voltages, respectively.

The primary voltage regulation equation of the synchronous generator can be expressed as: D q = Δ Q Δ V = Q ref − Q e 2 U ref − 2 U N (7) where D _q denotes the reactive power regulation coefficient; Q _ref and Q _e are the reference reactive power and real-time reactive power, respectively; and U _N is the effective value of the rated voltage.

Combining Eqs 6, 7, the Q-V equation can be obtained as follows. 2 E = G s D q Q ref − Q e + 2 D q U N − U m (8)

Considering the consistency, the reactive power control loop can be rewritten similarly to the active power loop: 2 E = 1 K s Q ref − Q e + 2 D q U N − U m (9) where K s is the equivalent inertia coefficient of the reactive power loop; E is the effective value of the internal voltage.

The power calculation module in Figure 2 is derived based on the instantaneous power theory. The instantaneous active power P _e and reactive power Q _e output of the GFM converter can be calculated as: P e = 3 2 v α i α + v β i β Q e = 3 2 v β i α − v α i β (10) where i _α , i _β and v _α , v _β are the currents and voltages measured in the αβ reference frame, respectively.

Then, the modulated waves of the grid-connected converter are generated from the active and reactive power control loops, which can be expressed as follows. m a = = 2 E c o s θ m b = 2 E c o s θ − 2 π / 3 m c = 2 E c o s θ + 2 π / 3 (11)

GFM converter controls its output voltage through the reactive control loop. Assuming that the voltage at the output is a constant value, the grid voltage is U _g ∠0, and the grid resistance is ignored, the active power output of the GFM converter can be expressed as follows: P e = 2 E U g / X g sin ⁡ δ (16)

Eq. 16 shows that the active power output Pe is related to the grid impedance X _g , network side voltage, and virtual power angle. Merging Eqs 4, 16, the active power control loop for the GFM control is: J δ ¨ = P ref − 2 E U g X g sin δ − D d δ ˙ (17)

This equation leads to the GFM control transient model (see Figure 5). Eq. 16 determines the system recovery capability, while the deviation of the power reference value P _ref from the actual active power output drives the virtual rotor motion, the output angular frequency ω, and the angle δ.

From Eq. 17 and the transient model shown in Figure 5, it is clear that the system restoring capability is related to the grid impedance. The Pe-δ curves of the GFM control with different grid strengths and voltage dips are given in Figure 6. The solid line corresponds to the strong grid with short-circuit ratio SCR = 4, while the dashed line corresponds to the weak grid with short-circuit ratio SCR = 2. When in the strong grid condition, assuming that the GFM control initially operates at the equilibrium point a, when the grid voltage drops from 1 pu to 0.6 pu, the power angle characteristic curve will change from the solid blue line to the solid red line. At that time, the acceleration power P _re -P _e is more significant than zero, and the output angular frequency increases. The drive δ red solid line moves to the equilibrium point b and reaches the maximum angular frequency at point b. After passing through point b, the acceleration power becomes negative, so the system output angular frequency decreases. As shown in Figure 6, the deceleration area is more prominent at this point, and δ can re-stabilize at equilibrium point b. However, if the deceleration area is too small so that the angular frequency is still greater than ω _N when passing through point c, the system will lose stability. Therefore, as seen in Figure 6, the weak grid condition indicated by the dashed line leads to a significant decrease in the P _e-δ curve due to the increase in grid impedance. In the same deep voltage dip case, the power angle curve changes from the blue dashed line to the red dashed line after the system contains almost no deceleration area, leading to system instability. The above analysis shows that the reduction in grid strength leads to a reduction in the stability of large disturbances in the GFM control.

GFL control relies on the PLL module to identify terminal voltage U _t to provide synchronous phase θ _pll for synchronous operation. The large disturbance stability of GFL control mainly depends on the closed-loop control composed of a PLL module and current control. The phase-locked loop dynamics can be represented in Figure 7. Under normal operating conditions, the output current vector I, the terminal voltage vector U _t , and the grid voltage vector U _g rotate at the output angular velocity ω _pll of the PLL in a GFL system.

Assuming that the dynamic performance of the GFM control meets the requirement that I is consistent with the reference value I _dqref In this case, there are: I = i dref + j i qref e j θ pll (18)

From Eq. 18, it can be seen that the actual output current I of GFL control is related to I _dqref and the PLL dynamics, and its terminal voltage can be expressed as: U t = U g + Z g ω pll I (19) where Z _g = R _g + jX _g represents the line impedance, and R _g is generally small in a large inductive power grid. According to Eqs 18, 19, the PLL dynamics further influence the terminal voltage by affecting the output current. The terminal voltage is added to the PLL module, which, in turn, affects its output phase.

According to the PLL model, the terminal voltage q-axis component can be expressed as: u t q = R g i qref + ω 0 L g i dref + ω p L g i dref − U g sin δ (20)

As can be seen from Eq. 20, u _tq is determined by grid impedance, grid-side voltage, current reference value, and PLL output phase. Combining Eq. 20 with the PLL dynamics, a transient model with grid control can be obtained (see Figure 8), where δ in the figure is the virtual power angle.

The model in Figure 8 can be organized as: u i n = a 1 δ ¨ + a 2 δ ˙ + U g sin δ (21)

Where, u i n = R g i qref + ω 0 L g i dref (22) a 1 = 1 − k p − pll L g i dref / k i − pll (23) a 2 = − L g i dref + k p − pll U g cos δ / k i − pll (24)

According to the GFL control transient model and Eq. 21, the PI parameter (k _p-pll + k _i-pll/s ) of PLL can be regarded as virtual inertia; L _gidref can be considered as system damping. The deviation between the input u _in and U _g sin(δ) drives the PLL action, and outputs angular frequency ω _pll and virtual power angle δ. Unlike the configuration control adjustable damping coefficient, there are: − L g i dref + k p − pll U g cos δ / k i − pll > 0 (25) δ < arccos L g l dref k i − pll U g k p − pll , δ ∈ 0 , π (26)

In other words, when Eqs 25, 26 are satisfied, there is positive damping with GFL control, which is conducive to the recovery of the converter in the transient process. Otherwise, GFL control will appear as negative damping and thus, destroying its large disturbance stability.

Figure 9 shows the u _tq -δ curve of GFL control under transient conditions. Assuming that the GFL control initially runs at the equilibrium point b, when the voltage input increases from u _ino to u _in , the accelerating voltage (u _in -u _tq ) drives u _tq to point a. At this point, the accelerating voltage is zero, but the angular frequency ω _pll is still higher than the synchronous angular frequency ω _o , causing δ increase. When the δ is greater than δ _a , the accelerating voltage becomes negative, and the driving ω _pll decelerates to ω _o and reaches point c. However, since the accelerating voltage is still negative, the ω _pll continues to decelerate below ω _o while moving toward point a. If there is positive damping in GFL control, the δ will stabilize at the new equilibrium point. Under the condition of no damping or negative damping, the δ will constantly oscillate between point b and point c, or even away from the original equilibrium point. Since the damping coefficient is negatively correlated with L _g , a large L _g will cause negative damping of the system, resulting in system instability under a weak grid. In addition, increasing L _g will also increase u _in , thereby increasing the acceleration area in the transient process, which is not conducive to system stability.

In summary, the impedance of the power grid affects both GFL control and GFM control. Strong grid conditions can improve the large disturbance stability of both controls. However, under a weak power grid, the damping coefficient of the GFM control converter is only determined by the control parameter D _d , which can easily maintain the positive damping effect. However, due to the damping coefficient of the GFL control converter, the parameter L _g is affected. When the grid impedance is high under the weak grid, the GFL control converter will show negative damping and reduce its large disturbance stability.

In this section, different short-circuit ratios are used to simulate the power grid strength to verify the small disturbance stability of GFL and GFM control under the weak power grid. In the simulation experiment, these controls are first utilized in the strong grid condition (SCR = 4). The impedance switch between GFL and GFM control in 1.5 s, respectively, to make them work in the weak grid condition (SCR = 2). Figure 10 presents the results of the three-phase current simulation experiment under the simulated grid strength of GFL control under different short-circuit ratios and the analysis results of the Total Harmonic Distortion (THD). It can be seen that in a strong power grid, the GFL control can operate stably, and the harmonic components of the output current are nearly zero. The three-phase current remains balanced, but when switching to the weak grid for SCR = 2, the three-phase current waveform has equal amplitude oscillations and loses stability, THD is as high as 20%. In contrast, the three-phase current with GFM control in Figure 11 has amplitude reduction oscillations under the condition of a strong power grid. The oscillation frequency and amplitude comparison performance are smaller than that of the weak grid type and the THD of GFM control under strong power grid conditions is approximately 9%. At the same time, GFM control can operate stably under weak grid conditions, and the three-phase current will be balanced quickly, and THD is only 1%. The simulation results are consistent with the analysis results of the small disturbance stability of the two types of control in a weak power grid in this paper. The small disturbance stability of GFM control under the weak network is significantly better than that of GFL control.

In order to verify the large disturbance stability of GFL and GFM control, the grid voltage drop of different degrees is simulated for these controls based on different power grid strengths. The grid voltage is reduced for a small voltage drop from 1 pu to 0.8 pu at 2.5 s. Figures 12A, B are the three-phase current simulation results of GFM and GFL control under the strong grid condition with SCR = 4, respectively. It can be seen that under strong grid conditions when the voltage drops slightly, both controls can maintain stability after the voltage drops. GFM control in Figure 12A has enough deceleration area to remain stable after a voltage drop. However, its transient process is slightly longer than the GFL control, as shown in Figure 12B. The transient and steady-state currents after returning to stability are more significant than the GFL control. It can be seen that under a strong power grid condition, GFL control can stabilize faster than GFM control after a small voltage drop. The current amplitude of GFL control after stabilization is also lower than that of GFM control, which makes it difficult to exceed the current protection limit after the voltage drop.

Figures 13A, B are the three-phase current simulation results of the two controls under weak grid conditions with SCR = 2. Under weak grid conditions, although the increase of grid impedance reduces the power angle characteristic curve of GFM control due to the low voltage drop amplitude, GFM control still has a sufficient deceleration area (see Figure 13A). GFM control can remain stable, and the transient and steady-state currents are more substantial and lower in the power grid state. For GFL control shown in Figure 13B, the three-phase current waveform is unstable be-fore and after the voltage drop. The oscillation amplitude increases after the voltage drop because the voltage is initially unstable before and after. The weak grid makes GFL control appear negative damping; thus, the three-phase current waveform is un-stable before and after the voltage drop. The oscillation amplitude further increases after the voltage drop.

Considering the significant voltage drop, we reduce the grid voltage from 1 pu to 0.6 pu at 2.5 s about these two controls. Figure 14A is the three-phase current simulation results of GFM control under the condition of a strong power grid. According to the analysis of Section 4.1, at this time, due to the acceleration area still tiny than the deceleration area, after a transient process for about 0.4 s, the GFM control under the strong grid is stable after the voltage drop and the three-phase cur-rent is balanced. However, the transient process is more extended, and the steady-state current is high. With the same condition, the GFL control shown in Figure 14B has significantly superior performance after voltage drop, the transient process is shorter, and the current amplitude is unchanged after stabilization, which can better avoid the over-current.

In the case of a weak grid, the GFM control shown in Figure 15A becomes unstable after the voltage drop. According to the power angle characteristic curve of GFM control shown in Figure 6, due to the large impedance of the power grid, at this time, the power angle curve is significantly reduced. Moreover, the deceleration area is insufficient after the voltage drop, resulting in the instability of GFM control. For Grid-following control under the same condition, as shown in Figure 15B, the three-phase current waveform is similar to the case of a slight voltage drop. The more significant grid impedance makes GFL control unstable before the voltage drop. The voltage drop further aggravates the oscillation due to the negative damping effect. The above simulation results are consistent with the analysis of enormous disturbance stability in Section 4.

In summary, according to the analysis of the stability of small disturbance and large disturbance of the GFL control and GFM control, as well as the simulation experiment results, the following conclusions can be further drawn: GFL control is more suitable for strong power grids with SCR >4, while instability is prone to occur under weak power grid conditions with SCR <2. On the other hand, for GFM control, under strong power grid conditions with SCR >4, its stability under small disturbances is worse than that of GFL control. However, as SCR decreases, indicating a decrease in grid strength, GFM control gradually outperforms GFL control in terms of stability. Under weak power grid conditions with SCR <2, significant large disturbance stability issues only exist when the grid voltage decreases significantly. In other cases, GFM control demonstrates significant advantages in stability compared to GFL control.