The research object of this paper is the parallel system of grid-following and GFM, as illustrated in Figure 1. In this parallel system, the DC-side power of the two converters can be supplied by photovoltaic panels, wind turbines, energy storage devices, or other sources. However, since the focus of this study lies in the characteristics of the converters on the AC side, the structure of the DC-side power sources is disregarded.

The fundamental concept of the HGS system is to combine grid-following and grid-forming control strategies in parallel, thereby integrating the respective characteristics of these two control approaches. The grid-following converter employs a PLL for synchronization control, and its control structure includes sampling, the PLL, inner current control loops, and a PWM generator. In contrast, the grid-forming converter adopts a more mature Virtual Synchronous Generator (VSG) control strategy. Its control circuit primarily consists of active power and reactive power loops, coordinate transformation, dual-loop voltage and current control, and PWM signal modulation. The two converters are connected to the main power grid via PCC.

In Figure 1, V _DC is a constant DC voltage; R _f1, L _f1, R _f2, and L _f2 are the filter resistances and filter inductances for the GFL and GFM converters, respectively; C _f2 is the filter capacitor for the GFM converter; R ₁, L ₁, R ₂, and L ₂ are the line resistances and line inductances for the GFL and GFM converters, respectively; R _g and L _g represent the line resistance and inductance between the PCC and the grid; U _pcc is the voltage at the PCC; i _abc1 and i _abc2 are the converter-side currents for the GFL and GFM converters, respectively; u _abc1 and u _abc2 are the output voltages for the GFL and GFM converters, respectively; V _g is the grid voltage.

The control system of the GFL is shown in Figure 2. The core of the GFL control strategy is the PLL, and its control structure is shown in Figure 3. The PLL synchronizes the generating unit with the grid by estimating and tracking the phase of the grid voltage u _abc1. The outer-loop power control compares the reference values with the actual values to generate an error signal, which serves as the reference for the inner-loop current control. The inner-loop current control generates modulation signals, which are then processed by the PWM stage.

The mathematical model of the PLL can be expressed as follows: d θ pll d t = ω pll d ω pll d t = k i u q 1 + k p d u q 1 d t (1)

In Equation 1: θ _pll is the output phase angle of the PLL; ω _pll is the output angular frequency of the PLL; u _d1 and u _q1 are the d-axis and q-axis voltages of the grid-connected point, respectively; k _i and k _p are the proportional and integral gains of the PI controller. θ _pll is used as the angle for the Park transformation, achieving decoupled control of active and reactive power by aligning u _abc1 with the d-axis, ensuring that u _q1 = 0.

The PLL is used to provide the reference angle for the Park transformation. By aligning u _abc1 to the d-axis, such that u _q1 = 0, decoupled control of active and reactive power is achieved. This ensures precise power regulation and synchronization of the system with the grid. i dref = k p _ p P ref − P GFL + k i _ p P ref − P GFL / s i dref = k p _ u dc U dcref − U dc + k i _ u dc U dcref − U dc / s (2) i qref = k p − uo U oref − U GFL + k i − uo U oref − U GFL / s i qref = k p − q Q ref − Q GFL + k i − q Q ref − Q GFL / s (3)

In Equations 2, 3, i _dref and i _qref represent the d-axis and q-axis reference currents, respectively. k _{p_p} and k _{i_p}, k _{p_udc} and k _{i_udc}, k _{p_uo} and k _{i_uo}, and k _{p_q} and k _{i_q} are the proportional and integral gains of the respective PI controllers. P _ref denotes the active power reference value, and Q _ref represents the reactive power reference value. U _dcref is the reference value for the DC-side voltage, while U _oref is the reference value for the PCC voltage. Finally, U _GFL indicates the output voltage magnitude of the GFL converter.

The mathematical model of the inner current loop can be expressed as follows: u md = u GFLd + k p _ i i dref − i d + k i _ i i dref − i d / s − ω L f i q u mq = u GFLq + k p _ i i qref − i q + k i _ i i qref − i q / s + ω L f i d (4)

In Equation 4, u _md and u _mq represent the d-axis and q-axis components of the modulation signal, respectively. k _{p_i} and k _{i_i} are the proportional and integral gains of the PI controller, respectively. L _f i _q and L _f i _d denote the decoupling control terms.

The control part of a VSG includes an active power loop, a reactive power loop, dual voltage-current control loops, and PWM signal modulation. The core of the control is the active and reactive power loops. VSG achieves synchronization with the grid by mimicking the rotor characteristic equations of a synchronous generator. Consequently, a grid-forming converter based on virtual synchronous generator control possesses the external characteristics of a synchronous generator. The schematic diagram is shown in Figure 4.

The control system of GFM converter is shown in Figure 5. The voltage control, current control, and PWM generator of the GFM converter are consistent with those of the grid-following converter.

The GFM converter achieves control over its output active and reactive power through the active power loop and reactive power loop. The mathematical model of the active power loop under VSG control can be expressed as follows: d θ vsg d t = ω vsg J d ω vsg d t = P set − P − D p ω vsg − ω n (5)

In Equation 5: θ _vsg is the output phase angle of the VSG; ω _vsg is the derivative of θ _vsg; P _set is the input mechanical power; P is the output active power of the VSG; ω _n is the nominal angular frequency; J is the moment of inertia; and D _p is the damping coefficient.

The active power loop adjusts θ _vsg to regulate the output active power P of the VSG, ultimately ensuring that P = P _set.

The mathematical model of the reactive power loop can be expressed as: U ref = U 0 + k q Q ref − Q (6)

In Equation 6: U _ref is the output voltage of the VSG; U ₀ is the set value of the output voltage; k _q is the droop coefficient of the reactive power loop; Q _ref is the reactive power reference value; and Q is the output reactive power of the VSG.

The reactive power loop adjusts the output voltage reference value based on the output reactive power.

The bandwidth of the current loop in the GFL is significantly larger than the bandwidth of the PLL. By ignoring the dynamic process of the current loop, the GFL can be approximated as a current source. Similarly, in the GFM, the adjustment speed of the inner voltage and current loops is much faster than that of the active and reactive power loops. By neglecting the dynamic process of the inner voltage and current loops, the GFM can be approximated as a voltage source (Li et al., 2021). Circuit impedance is considered negligible, and the phase angle of the grid voltage is taken as the reference angle (Gursoy et al., 2023).The equivalent circuit diagram of Figure 1 is shown in Figure 6.

In this figure, I ₁ is the RMS value of the output current of the current-controlled converter; U ₁ is the RMS value of the output voltage of the GFL; θ _pll is the phase angle difference between the PLL and the grid, corresponding to the power angle of a synchronous generator, and for convenience, it will be referred to as the power angle hereafter; φ ₁ is the phase angle difference between voltage U ₁ and current I ₁; U ₂ is the RMS value of the output voltage of the GFM; δ _vsg is the power angle of the GFM; V _g is the RMS value of the grid voltage.

According to the superposition theorem, the PCC voltage U _pcc∠δ _pcc in Figure 6 can be expressed in the stationary reference frame as: U pcc ∠ δ pcc = j k 1 I 1 ∠ δ pcc + φ 1 + k 2 U 2 ∠ δ vsg + k 3 V g ∠ 0 ° (7)

In Equation 7: k 1 = ω pll L 2 L g L 2 + L g ; k 2 = L g L 2 + L g ; k 3 = L 2 L 2 + L g .

The point where the PLL measures the voltage is U ₁∠δ _pll, which can be expressed as: U 1 ∠ δ pll = j k 4 I 1 ∠ δ pll + φ 1 + k 2 U 2 ∠ δ vsg + k 3 V g ∠ 0 ° (8)

In Equation 8: k 4 = ω pll L 1 + ω pll L 2 L g L 2 + L g .

Transforming Equation 8 to the dq rotating reference frame based on the PLL, the q-axis voltage of U ₁ can be obtained as: u q 1 = k 4 i d 1 + k 2 U 2 ⁡ sin δ vsg − δ pll − k 3 V g ⁡ sin ⁡ δ pll (9)

In Equation 9: typically, δ _vsg, δ _pll∈[0,π/2].

Comparing Equation 9 with the case of a single GFL connected to the grid, it can be observed that the q-axis voltage u _q1 of the GFL has an additional impedance drop coupling term (Paquette and Divan, 2015). The magnitude of this coupling term is related to k ₂, U ₂, δ _vsg, and δ _pll, indicating that the power angle of the PLL is also influenced by the GFM power angle δ _vsg.

From Figure 6, the single-phase output complex power of the GFM can be expressed as: S ˙ 2 = U 2 ∠ δ vsg U 2 ∠ δ vsg − U pcc ∠ δ pcc j X 2 ∗ (10)

In Equation 10, X ₂ = ω _n L ₂, where ω _n is the nominal angular frequency.

Substituting Equation 7 into Equation 10, we obtain: P 2 = k 3 U 2 V g X 2 sin ⁡ δ vsg − k 1 U 2 I 1 X 2 cos δ vsg − δ pll − φ 1 (11) Q 2 = U 2 2 − k 2 U 2 2 X 2 − k 3 U 2 V g X 2 cos ⁡ δ vsg − k 1 U 2 I 1 X 2 sin δ vsg − δ pll − φ 1 (12)

From Equations 11, 12, it can be seen that the GFL affects both the active power loop and the reactive power loop of the GFM. Compared to the single machine case, an additional active power coupling term and a reactive power coupling term are introduced. The magnitude of this impact is related to k ₁、U ₂、I ₁、X ₂、δ _vsg, δ _pll, and φ ₁. The power angle of the GFM is also influenced by the output current I ₁ of the GFL. By calculating the three-phase active power using Equation 11 and substituting it into Equation 5, it can be obtained that the GFM in steady state satisfies Equation 13. P set − 3 k 3 U 2 V g X 2 sin ⁡ δ vsg + 3 k 1 U 2 I 2 X 2 cos δ vsg − δ pll − φ 1 = 0 (13)

In summary, the transient analysis model of the HGS can be obtained, as shown in Figure 7. It can be seen that the integration of the GFL adds a power coupling term to the GFM; similarly, the GFM adds an impedance drop coupling term to the GFL (Zhang et al., 2024).

Setting Equation 9 to 0, the condition for the PLL to have an equilibrium point can be obtained as: k 2 U 2 k 4 2 + k 3 V g k 4 2 + 2 k 2 k 3 U 2 V g k 4 2 cos ⁡ δ vsg ≥ i d 1 (14)

When the GFM operates stably, δ _vsg ∈[0, π/2]. If i d 1 > k 2 U 2 + k 3 V g k 4 , then Equation 14 cannot be satisfied, and there is no equilibrium point for the PLL. If i d 1 ≤ k 2 U 2 + k 3 V g k 4 , then Equation 14 can be satisfied, and an equilibrium point exists for the PLL.

From Equation 13, it can be seen that the presence of I ₁ increases δ _vsg, reducing the stability margin of the GFM. The magnitude of δ _vsg is directly proportional to k ₁, U ₂, and I ₁, and inversely proportional to X ₂. If I ₁ is too large, it may cause the initially stable GFM to lose power angle stability. If the GFM loses power angle stability due to a fault or other reasons, its output angle δ _vsg increases indefinitely. As indicated by Equation 14, when δ _vsg = 2kπ + π, k∈Z, the impact on the PLL is maximized, leading to two possible situations: ① If Equation 14 is not satisfied, the GFL becomes unstable. ② If Equation 14 is satisfied, since δ _vsg increases indefinitely, Equation 9 remains in a state of adjustment and cannot stabilize or converge to 0, indicating that the GFL becomes unstable.

Therefore, transient instability in the GFM will trigger a chain reaction, causing transient instability in the GFL.

From Equation 13, the GFM has a steady-state point when the following condition is satisfied: 9 k 3 U 2 V g X 2 2 + 9 k 1 U 2 I 1 X 2 2 − 18 k 1 k 3 U 2 2 V g I 1 X 2 2 sin δ pll + φ 1 ≥ P set (15)

When the GFL operates stably, δ _pll ∈ [0, π/2] and φ ₁ ∈ [-π/2, 0]. If P set > 3 k 3 U 2 V g + k 1 U 2 I 1 X 2 , Equation 15 cannot be satisfied, and there is no equilibrium point for the GFM. If P set ≤ 3 k 3 U 2 V g + k 1 U 2 I 1 X 2 , then Equation 15 can be satisfied, and an equilibrium point exists for the GFM.

From Equation 9, it can be seen that when δ _vsg > δ _pll, δ _pll increases, and the stability margin of the GFL decreases. When δ _vsg < δ _pll, δ _pll decreases, and the stability margin of the GFL increases. When δ _vsg = δ _pll, the coupling term is zero, and δ _pll is not affected by the GFM. If the GFL loses synchronization stability due to a fault or other reasons, the PLL will have no equilibrium point, causing the power angle δ _pll to increase indefinitely, thereby affecting the GFM. There are two possible scenarios: ① If Equation 15 is not satisfied, the GFM becomes unstable. ② If Equation 15 is satisfied, due to δ _pll increasing indefinitely, Equation 13 remains in a state of adjustment and cannot converge to zero, indicating that the GFM becomes unstable.

Therefore, when the GFL experiences transient instability, the increase in its output phase also causes the GFM to experience transient instability.

In summary, it is essential to ensure the simultaneous stability of both converters; instability in either one will affect the stability of the other.

Additionally, during grid voltage sag, the GFM, being a voltage source type, can experience transient overcurrent and steady-state overcurrent. Under fault conditions, the circuit satisfies: L 2 + L g d i 2 F d t = u 2 F − ν gF − L g d i 1 d t (16)

In Equation 16: i _2F is the instantaneous value of the GFM output current during a fault; u _2F is the instantaneous value of the GFM output voltage during a fault; v _gF is the instantaneous value of the grid voltage during a fault; i ₁ is the instantaneous value of the output current of the current-controlled converter.

From Equation 16, it can be seen that the presence of the GFL can reduce the fault current of the GFM.

During a fault, reducing the GFL output current i _d1 not only ensures the existence of the PLL equilibrium point, but also, as seen from Equation 9, the reduction of i _d1 can slow down the acceleration process of the GFL output ω _pll during the fault, thereby reducing the deviation of ω _pll. Since the inertia of the GFM is much larger than the equivalent inertia of the GFL, the increase rate of δ _vsg is slower than that of δ _pll during a fault. Additionally, because the fault duration is short, during the fault, (δ _vsg − δ _pll) ∈ [-π/2, 0] (Cheng et al., 2022). For purely active power output, φ ₁ = 0, and the reduction in i _d1 is much greater than the change in the cosine function, so the effect of the cosine function can be ignored. Therefore, when i _d1 is reduced, the power coupling term in Equation 11 is also reduced. From Equation 5, it can be seen that this also helps to slow down the acceleration of the GFM output ω _vsg during the fault, reducing the deviation of ω _vsg. The phase plane diagram of the HGS under different i _d1 conditions during a fault is shown in Figure 8.

From Figure 8, it can be observed that reducing the active current of the GFL can slow down the acceleration process of both the GFL and GFM during a fault, thereby improving the synchronization stability of the two converters. Moreover, the smaller the i _d1, the better the synchronization stability of the two converters; conversely, the larger the i _d1, the worse the synchronization stability of the two converters.

During a fault, reducing the input mechanical power P _set of the GFM not only ensures the existence of the GFM’s equilibrium point but, as shown in Equation 5, also slows down the acceleration process of the GFM’s output ω _vsg during the fault, reducing the deviation of ω _vsg. A decrease in P _set during a fault causes the δ _vsg of the GFM, which has inertia, to increase more slowly than δ _pll. The smaller the P _set, the slower the increase of δ _vsg, the larger the | δ _vsg − δ _pll |, and the smaller the impedance drop coupling term in Equation 9, resulting in a smaller deviation of the GFL’s output ω _pll. The phase plane diagram of the HGS under different P _set conditions during a fault is shown in Figure 9.

From Figure 9, it can also be seen that reducing the input mechanical power P _set of the GFM can similarly slow down the acceleration process of both the GFM and GFL during a fault, thereby enhancing the synchronization stability of the two converters. Moreover, the smaller the P _set, the better the synchronization stability of the two converters; conversely, the larger the P _set, the worse the synchronization stability of the two converters.

During a grid fault, while maintaining power angle stability, the GFM power angle δ _vsgF is adjusted based on the extent of the grid voltage sag to modify the reactive component of the output current during the fault, thereby supporting the grid. The adjustment rules are as follows: δ vsgF =   V gF V g δ 2 0 , 0.2 < V gF V g < 0.9 0 , V gF V g ≤ 0.2 (17)

In Equation 17, δ 0 2 is the power angle of the GFM before the fault;V _gF is the RMS value of the grid phase voltage after the fault. During the fault, by adjusting P _set, we can control the power angle of the GFM, making it reach δ _vsgF under the corresponding V _gF.

According to Equation 14, the equilibrium point of the PLL is not only related to the output current i _d1 but also affected by the inductances L ₁, L ₂, and L _g. Therefore, by adjusting only the current, especially in a weak grid environment, it is impossible to ensure the existence of the equilibrium point. Hence, the output phase angle θ _vsg of the GFM is transmitted to the PLL. However, under this circumstance, the PLL becomes an open-loop system with a steady-state error, making it impossible to precisely align Ů ₁ with the d-axis.

From Figure 10A, it can be seen that there is a fixed angular difference φ between Ů ₁ and the d-axis d _GFL of the GFL. Therefore, the angle of the GFL needs to be adjusted by adding a fixed angle φ on the basis of θ _vsg. This can be obtained by measuring the deviation of u _q1 and applying PI control. θ pll = θ vsg + ϕ ϕ = k p u q 1 + k i ∫ u q 1 (18)

Using Equation 18, an improved PLL based on the grid-forming model can be obtained, which can also precisely align U ₁ with the d-axis of the two-phase rotating coordinate system of the GFL, as shown in Figure 10B. At this point, the active power and reactive power of the GFL are decoupled.

The P _setF during the fault can be obtained as: P setF = 3 k 3 U 2 V gF X 2 sin ⁡ δ vsgF − 3 k 1 U 2 I 1 X 2 cos ϕ + φ 1 (19)

After introducing the input mechanical power adjustment loop in the GFM and adopting the improved PLL based on the grid-forming model in the GFL, the control block diagrams of the GFM and GFL are shown in Figure 11. In this case, as long as the GFM remains stable, the stability of the GFL can be ensured.

The fault ride-through control process is illustrated in Figure 13. The system operates under normal conditions, monitoring V _g to determine whether a fault has occurred. In the event of a ground fault, the value of V _g decreases.

When the grid voltage drops above a specified threshold, the transient current is evaluated to determine whether it exceeds the current-limiting threshold, thereby deciding whether to activate transient virtual impedance control. If the grid voltage falls below the threshold, transient power angle control is first applied to the hybrid system. Using Equations 17, 21, 22, the fault power angle of the GFM converter and the reference current for the GFL converter are determined. Subsequently, the value of P _setF is calculated according to Equation 19.

Next, the fault current of the GFM converter is monitored. If the current exceeds the transient current-limiting threshold, both the reactive power loop and transient virtual impedance control for the GFM converter are activated. If the current only exceeds the steady-state current-limiting threshold of the GFM converter, only the reactive power loop control is applied.

Once the reactive power loop control is engaged in the GFM converter, the previously determined GFM fault power angle and GFL reference current are used to calculate P _setF and U _2F according to Equation 23. When V _g recovers above the threshold value, the hybrid system transitions back to the normal control strategy.

When the three-phase grid voltage drops to 0.5, the output currents and power angles of the GFM and GFL are shown in Figure 14. The simulation duration is set to 3 s, with a fault occurring at 0.5 s and clearing at 2 s. As seen in Figure 14, during the fault, the power angles of both the GFM and GFL increase indefinitely. The output currents of the two converters are affected by the power angles, resulting in oscillations. Since the GFM is a voltage source type, its maximum current amplitude reaches three times its rated operating value. The output active and reactive power of both converters also experience oscillations.

Figure 15A shows the power angle curves of both converters when the d-axis current reference value I *d1 of the grid-following converter is changed from 1 pu to 2.5 pu at 0.5 s and restored to 1 pu at 2 s. Figure 15B shows the power angle curves of both converters when the input mechanical power P _set of the grid-forming converter is changed from 1 pu to 3 pu at 0.5 s and restored to 1 pu at 2 s. Based on the output power angles, it can be seen that the occurrence of power angle instability in the GFL immediately causes power angle instability in the GFM. Similarly, the occurrence of power angle instability in the GFM immediately causes power angle instability in the GFL. This verifies the analysis of the mutual influence between the two converters mentioned earlier.

Figure 16 shows the output waveforms when the grid voltage symmetrically drops to 0.4 pu at 0.5 s and rises back to 1 pu at 2 s. During the fault, transient power angle control is applied to the HGS, and both the GFL and GFM remain stable after a brief transient process. As seen in Figure 16B, during the fault, by adjusting the input mechanical power P _set of the GFM and using the improved PLL based on the grid-forming model for the GFL, the power angles of both converters can be kept stable. However, as shown in Figure 16D, the transient current output by the GFM is too large, and the steady-state current also exceeds 1.5 pu, approaching 2 pu. From Figure 16F, it can be seen that this is due to the excessive reactive current output by the GFM. Although reactive current can support the grid, there is still a risk of damaging transistors. Therefore, it is necessary to suppress transient and steady-state overcurrents through fault current control.

Figure 17 shows the waveforms of the HGS when transient power angle and fault current control are implemented as the grid voltage symmetrically drops to 0.4 pu. Similarly, the GFL and GFM remain stable after a brief transient process. As shown in Figure 17D, during the fault, by reducing the output voltage of the GFM, the GFM’s output current is precisely controlled to 1.5 pu. Thanks to the virtual impedance, the transient current is also effectively suppressed. As depicted in Figure 17E, the GFL, while remaining stable, is able to provide reactive current in accordance with grid codes, supporting the grid. After the fault is cleared, both converters can return to their pre-fault operating state after a brief transient process.

Figure 18 presents the waveforms of transient power angle and fault current control for the HGS system during a symmetrical grid voltage dip to 0.6 pu. Similarly, both the GFL and GFM converters remain stable after a brief transient process. As shown in Figure 17C, during the fault, the output current of the GFM converter is below 1.5 pu, in contrast to the scenario where the grid voltage symmetrically drops to 0.4 pu. This reduction is attributed to the implementation of virtual impedance, which effectively suppresses transient currents. As depicted in Figure 17D, the GFL converter, while maintaining stability, provides reactive current in compliance with grid codes, thereby supporting the grid. After the fault is cleared, both converters recover to their normal operating conditions following a brief transient period.