Analysis of Transient Characteristics and Fault Ride-through Control of Hybrid Grid-tied Converters with Grid-following and Grid-forming

1. Introduction

With the implementation of carbon peaking and carbon neutrality policies, renewable energy sources such as wind and photovoltaic power are rapidly developing. The proportion of non-synchronous generators in the power system is increasing significantly [1,2,3]. A high proportion of new energy and power electronic equipment has become a notable characteristic of power systems. However, using grid-following converters as the main grid-connected devices does not provide sufficient inertia and damping for the system, resulting in a further decrease in system inertia, which jeopardizes the safe operation of the power system. Grid-forming converters, which possess the characteristics of a synchronous generator, have garnered widespread attention. Compared to grid-following converters, they can effectively function as a voltage source, allowing them to regulate system voltage and frequency and provide necessary inertia and damping support. Nevertheless, grid-following converters still have advantages that grid-forming converters cannot replace. On the other hand, the current new energy power stations are equipped with grid-following converters, making it impractical to replace them all with grid-forming converters. Therefore, a feasible alternative is to modify some of the already installed converters to grid-forming control. This implies that in future power systems, both grid-following and grid-forming converters will coexist [4,5,6].

In the event of a grid fault, a synchronous generator can inject 6 to 8 times its rated current (pu) to support the grid [7]. However, the overcurrent tolerance of converters is relatively low. Without proper control, this could lead to disconnection of renewable energy generation equipment from the grid or damage to the converters. To address this, IEEE standard 2800-2022 provides detailed technical requirements for renewable energy generation equipment regarding fault ride-through capabilities [8]. Currently, research mainly focuses on the transient processes of single grid-following (GFL) or single grid-forming (GFM) converters during fault ride-through, with an emphasis on the control of fault current and power angle. For GFL (grid-following converters), reference [9] adjusts the active current injection during faults through the feedback of the phase-locked loop (PLL) frequency, but the steady-state operating point and stability are difficult to predict. Reference [10] proposes locking the PLL during faults to maintain the output of the previous moment; however, this approach is challenging to meet the grid code requirements. Reference [11] suggests locking only the integral part of the PLL during faults, but this method fails when there is no equilibrium point after the fault. Reference [12] proposes setting the ratio of active and reactive currents based on the grid impedance after a fault. While this ensures the existence of an equilibrium point, it still cannot fully comply with grid codes. For GFM (grid-forming converters), reference [13] switches the grid-forming control to grid-following control during faults. However, this process requires a backup PLL and may face stability issues in weak grid environments. References [14,15,16] adopt a current limiting method in the current loop of the converter’s double closed-loop control to limit fault currents. However, this approach turns the voltage source into a current source, making power control difficult, reducing stability, and potentially causing instability. Reference [17] proposes a strategy using voltage limiting to control fault current, locking the reactive voltage droop, and calculating the reference voltage based on the maximum allowable current. References [18,19] respectively analyze the transient characteristics of parallel grid-following and grid-forming converters and explore the issue of transient synchronization stability in islanded AC microgrids, considering the dynamic interactions between grid-forming and grid-following converters. However, there is relatively little research on fault ride-through issues under hybrid grid conditions. In fact, during faults, the cooperation between converters with different control strategies can provide better service to the grid.

To overcome the aforementioned limitations, this study introduces a hybrid fault ride-through control strategy aimed at improving the issue of fault ride-through during symmetrical voltage dips when low voltage faults occur in the grid. The main contributions of this strategy are as follows:

• Parallel System Construction: A coupled effect is generated by the parallel connection of Grid Forming (GFM) and Grid Following (GFL) systems, allowing for the analysis of fault ride-through issues and transient stability under these conditions.
• Power Stability: A mechanical power adjustment loop is introduced in GFM, and a grid-structured improved phase-locked loop is employed in GFL to determine the balance point between GFM and GFL.
• Reactive Power Locking: The reactive power output that complies with grid guidelines is supported by replacing the reactive power output with the calculated GFM output voltage U_2F to support the grid containing GFL.
• Transient Fault Current Control: Transient virtual impedance is utilized to suppress transient overcurrent, preventing excessive short-circuit current from damaging power electronic devices in GFM.

The structure of this paper is organized as follows: Section 2 constructs the basic model of the grid-following/grid-forming converter hybrid grid-connected system (HGS). Section 3 analyzes the interaction mechanism between GFL and GFM when a low-voltage fault occurs in the power grid. Section 4 proposes a GFM power angle control considering the influence of GFL current injection and a hybrid fault ride-through control strategy based on an improved grid-forming phase-locked loop. During the fault period, this strategy can not only ensure the power angle stability and limited fault current of the GFM but also enable the GFL to inject current in accordance with the grid guidelines. Section 5 verifies the correctness and effectiveness of the proposed control strategy through simulation.

2. Parallel System of Grid-Following and Grid-Forming Converters

2.1. Main Circuit System Topology

The topology of the studied Hybrid Grid System (HGS) is shown in Figure 1. The grid-following converter uses Phase Locked Loop (PLL) control. There are various control methods for the grid-forming converter, such as Virtual Synchronous Generator (VSG) control, droop control, power synchronization control, and virtual oscillation control. This paper focuses on the more mature Virtual Synchronous Generator control as an example for discussion. Both converters are connected to the main grid through a Point of Common Coupling (PCC).

In this figure, 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 Grid-Following (GFL) and Grid-Forming (GFM) converters, respectively; C_f2 is the filter capacitor for the GFM converter; R₁, L₁, R₂, and L₂ 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 Point of Common Coupling (PCC) and the grid; U_pcc is the voltage at the Point of Common Coupling; 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.

2.2. Grid-Following Converter Based on Current Control

The control part of a current-controlled voltage source converter (CCS) includes a phase-locked loop (PLL), power control loop, current control loop, and PWM signal modulation, among others. The core of the control is the PLL, which achieves synchronization of the CCS with the grid by estimating and tracking the phase of the grid-connected point voltage u_abc1. The schematic diagram of the phase-locked loop is shown in Figure 2.

Its mathematical model can be expressed as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\theta }_{\mathrm{p}\mathrm{l}\mathrm{l}}}{dt}}={\omega }_{\mathrm{p}\mathrm{l}\mathrm{l}}\\ {\displaystyle \frac{d{\omega }_{\mathrm{p}\mathrm{l}\mathrm{l}}}{dt}}=k_{\mathrm{i}}{u}_{\mathrm{q}1}+k_{\mathrm{p}}{\displaystyle \frac{d{u}_{\mathrm{q}1}}{dt}}\end{array}\right.$$

(1)

In the equations: θ_pll is the output phase angle of the phase-locked loop; ω_pll is the output angular frequency of the phase-locked loop; 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.

2.3. Grid-Forming Converter Based on VSG Control

The control part of a Virtual Synchronous Generator (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 3.

The mathematical model of the active power loop can be expressed as:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\theta }_{\mathrm{p}\mathrm{l}\mathrm{l}}}{dt}}={\omega }_{\mathrm{p}\mathrm{l}\mathrm{l}}\\ {\displaystyle \frac{d{\omega }_{\mathrm{p}\mathrm{l}\mathrm{l}}}{dt}}=k_{\mathrm{i}}{u}_{\mathrm{q}1}+k_{\mathrm{p}}{\displaystyle \frac{d{u}_{\mathrm{q}1}}{dt}}\end{array}\right.$$

(2)

In the equations: 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:

$$U_{\mathrm{r}\mathrm{e}\mathrm{f}}=U_0+k_{\mathrm{q}}\left(Q_{\mathrm{r}\mathrm{e}\mathrm{f}}-Q\right)$$

(3)

In the equations: U_ref is the output voltage of the VSG; U₀ 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.

3. Mathematical Model and Transient Characteristics of Parallel System

3.1. Mathematical Modeling of HGS.

The bandwidth of the current loop in the grid-following converter (GFL) is significantly larger than the bandwidth of the phase-locked loop (PLL). By ignoring the dynamic process of the current loop, the GFL can be approximated as a current source. Similarly, in the grid-forming converter (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. Circuit impedance is considered negligible, and the phase angle of the grid voltage is taken as the reference angle. The equivalent circuit diagram of Figure 1 is shown in Figure 4.

In this figure,I₁ is the RMS value of the output current of the current-controlled converter; U₁ 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; φ₁ is the phase angle difference between voltage U₁ and current I₁; U₂ 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 4 can be expressed in the stationary reference frame as:

$$U_{\mathrm{p}\mathrm{c}\mathrm{c}}\mathrm{\angle }{\delta }_{\mathrm{p}\mathrm{c}\mathrm{c}}=j{k}_1{I}_1\mathrm{\angle }\left({\delta }_{\mathrm{p}\mathrm{c}\mathrm{c}}+{\phi }_1\right)+k_2{U}_2\mathrm{\angle }{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}+k_3{V}_{\mathrm{g}}\mathrm{\angle }0°$$

(4)

In the equation:$k_1={\omega }_{\mathrm{pll}}{\displaystyle \frac{L_2{L}_{\mathrm{g}}}{L_2+L_{\mathrm{g}}}};k_2={\displaystyle \frac{L_{\mathrm{g}}}{L_2+L_{\mathrm{g}}}};k_3={\displaystyle \frac{L_2}{L_2+L_{\mathrm{g}}}}$.

The point where the phase-locked loop (PLL) measures the voltage is U₁∠δ_pll, which can be expressed as:

$$U_1\mathrm{\angle }{\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}=j{k}_4{I}_1\mathrm{\angle }\left({\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}+{\phi }_1\right)+k_2{U}_2\mathrm{\angle }{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}+k_3{V}_{\mathrm{g}}\mathrm{\angle }0°$$

(5)

In the equation:${\mathrm{k}}_4={\mathsf{\omega }}_{\mathrm{p}\mathrm{l}\mathrm{l}}{\mathrm{L}}_1+{\mathsf{\omega }}_{\mathrm{p}\mathrm{l}\mathrm{l}}{\displaystyle \frac{{\mathrm{L}}_2{\mathrm{L}}_{\mathrm{g}}}{{\mathrm{L}}_2+{\mathrm{L}}_{\mathrm{g}}}}$.

Transforming equation(5) to the dq rotating reference frame based on the phase-locked loop (PLL), the q-axis voltage of U₁ can be obtained as:

$$u_{\mathrm{q}1}=k_4{i}_{\mathrm{d}1}+k_2{U}_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-{\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}\right)-k_3{V}_{\mathrm{g}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}$$

(6)

In equation (6): typically,δ_vsg,δ_pll∈[0,π/2].

Comparing equation (6) with the case of a single grid-following converter (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. The magnitude of this coupling term is related to k₂, U₂, δ_vsg, and δ_pll, indicating that the power angle of the phase-locked loop (PLL) is also influenced by the GFM power angle δ_vsg.

From Figure 4, the single-phase output complex power of the GFM can be expressed as:

$${\dot{S}}_2=U_2\mathrm{\angle }{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}{\left({\displaystyle \frac{U_2\mathrm{\angle }{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-U_{\mathrm{p}\mathrm{c}\mathrm{c}}\mathrm{\angle }{\delta }_{\mathrm{p}\mathrm{c}\mathrm{c}}}{j{X}_2}}\right)}^{\mathrm{*}}$$

(7)

In the equation, X₂ = ω_n L₂, where ω_n is the nominal angular frequency.

Substituting equation (4) into equation (7), we obtain:

$$P_2={\displaystyle \frac{k_3{U}_2{V}_{\mathrm{g}}}{X_2}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-{\displaystyle \frac{k_1{U}_2{I}_1}{X_2}}\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-{\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}-{\phi }_1\right)$$

(8)

$$Q_2={\displaystyle \frac{U_2^2-k_2{U}_2^2}{X_2}}-{\displaystyle \frac{k_3{U}_2{V}_{\mathrm{g}}}{X_2}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-{\displaystyle \frac{k_1{U}_2{I}_1}{X_2}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-{\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}-{\phi }_1\right)$$

(9)

From equations (8) and (9), 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 k₁、U₂、I₁、X₂、δ_vsg, δ_pll, and φ₁. The power angle of the GFM is also influenced by the output current I₁ of the GFL. By calculating the three-phase active power using equation (8) and substituting it into equation (2), it can be obtained that the GFM in steady state satisfies equation (10).

$$P_{\mathrm{s}\mathrm{e}\mathrm{t}}-3{\displaystyle \frac{k_3{U}_2{V}_{\mathrm{g}}}{X_2}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}+3{\displaystyle \frac{k_1{U}_2{I}_2}{X_2}}\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}-{\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}-{\phi }_1\right)=0$$

(10)

In summary, the transient analysis model of the HGS can be obtained, as shown in Figure 5. 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.

3.2. Transient Stability Analysis Considering Coupling Effects

Setting equation (6) to zero, the condition for the PLL to have an equilibrium point can be obtained as:

$$\sqrt{{\left({\displaystyle \frac{k_2{U}_2}{k_4}}\right)}^2+{\left({\displaystyle \frac{k_3{V}_{\mathrm{g}}}{k_4}}\right)}^2+2{\displaystyle \frac{k_2{k}_3{U}_2{V}_{\mathrm{g}}}{k_4^2}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}}}\ge i_{\mathrm{d}1}$$

(11)

When the GFM operates stably,δ_vsg ∈[0, π/2].If $i_{\mathrm{d}1}>{\displaystyle \frac{k_2{U}_2+k_3{V}_{\mathrm{g}}}{k_4}}$,then equation (11) cannot be satisfied, and there is no equilibrium point for the PLL. If $i_{\mathrm{d}1}\le {\displaystyle \frac{k_2{U}_2+k_3{V}_{\mathrm{g}}}{k_4}}$, then equation (11) can be satisfied, and an equilibrium point exists for the PLL.

From equation (10), it can be seen that the presence of I₁ increases δ_vsg, reducing the stability margin of the GFM. The magnitude of δ_vsg is directly proportional to k₁, U₂, and I₁, and inversely proportional to X₂. If I₁ 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 (11), when δ_vsg=2kπ+π, k∈Z, the impact on the PLL is maximized, leading to two possible situations: ① If equation (11) is not satisfied, the GFL becomes unstable.② If equation (11) is satisfied, since δ_vsg increases indefinitely, equation (6) 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 (10), the GFM has a steady-state point when the following condition is satisfied:

$$\begin{array}{rr}& \\ & \sqrt{9{\left({\displaystyle \frac{k_3{U}_2{V}_{\mathrm{g}}}{X_2}}\right)}^2+9{\left({\displaystyle \frac{k_1{U}_2{I}_1}{X_2}}\right)}^2-18{\displaystyle \frac{k_1{k}_3{U}_2^2{V}_{\mathrm{g}}{I}_1}{X_2^2}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}+{\phi }_1\right)}\ge P_{\mathrm{s}\mathrm{e}\mathrm{t}}\end{array}$$

(12)

When the GFL operates stably,δ_pll ∈ [0, π/2] and φ₁ ∈ [-π/2, 0].If ${\mathrm{P}}_{\mathrm{s}\mathrm{e}\mathrm{t}}>3{\displaystyle \frac{{\mathrm{k}}_3{\mathrm{U}}_2{\mathrm{V}}_{\mathrm{g}}+{\mathrm{k}}_1{\mathrm{U}}_2{\mathrm{I}}_1}{{\mathrm{X}}_2}}$, equation (12) cannot be satisfied, and there is no equilibrium point for the GFM.If ${\mathrm{P}}_{\mathrm{s}\mathrm{e}\mathrm{t}}\le 3{\displaystyle \frac{{\mathrm{k}}_3{\mathrm{U}}_2{\mathrm{V}}_{\mathrm{g}}+{\mathrm{k}}_1{\mathrm{U}}_2{\mathrm{I}}_1}{{\mathrm{X}}_2}}$, then equation (12) can be satisfied, and an equilibrium point exists for the GFM.

From equation (6), 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 (12) is not satisfied, the GFM becomes unstable. ② If equation (12) is satisfied, due to δ_pll increasing indefinitely, equation (10) 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:

$$\begin{array}{rr}& \\ & \sqrt{9{\left({\displaystyle \frac{k_3{U}_2{V}_{\mathrm{g}}}{X_2}}\right)}^2+9{\left({\displaystyle \frac{k_1{U}_2{I}_1}{X_2}}\right)}^2-18{\displaystyle \frac{k_1{k}_3{U}_2^2{V}_{\mathrm{g}}{I}_1}{X_2^2}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_{\mathrm{p}\mathrm{l}\mathrm{l}}+{\phi }_1\right)}\ge P_{\mathrm{s}\mathrm{e}\mathrm{t}}\end{array}$$

(13)

In the equation: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;i₁ is the instantaneous value of the output current of the current-controlled converter.

From equation (13), 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 (6), 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] [18]. For purely active power output, φ₁ = 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 (8) is also reduced. From equation (2), 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 6.

From Figure 6, 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 (2), 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 (6), 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 7.

From Figure 7, 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.

4. Fault Ride-through Control of the HGS

During a low voltage sag fault in the grid, the HGS faces three issues: power angle instability caused by the power imbalance between the GFM and GFL; outputting reactive current in compliance with grid codes to support the grid; and the risk of overcurrent due to the large short-circuit current of the GFM, which threatens power electronic equipment.

From the previous analysis, it is known that reducing i_d1 and P_set during a grid fault helps ensure the existence of the equilibrium points of the GFL and GFM, and simultaneously enhances the transient stability of both converters. However, the conditions for the existence of the equilibrium points of both converters are coupled through the GFL power angle δ_pll, the GFL output current I₁, the GFM power angle δ_vsg, and the GFM output voltage U₂. Directly controlling the power angles by adjusting i_d1 and P_set is quite challenging. Therefore, this paper first determines the values of the GFM power angle δ_vsgF, the fault currents I_1F and I_2F during the fault, and then calculates the required VSG output voltage U_2F and mechanical power P_setF based on the power angles.

4.1. Transient Power Angle Control

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:

$$\begin{array}{rr}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}& =\left\{\begin{array}{ll}{\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}{\delta }_2^0& ,0.2<{\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}<0.9\\ & \\ & \\ 0& ,{\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}\le 0.2\\ & \end{array}\right.\end{array}$$

(14)

In the equation, 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 (11), the equilibrium point of the PLL is not only related to the output current i_d1 but also affected by the inductances L₁, L₂, 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 $U(。)$₁ with the d-axis.

From Figure 8(a), it can be seen that there is a fixed angular differenceφ between $U(。)$₁ and the d-axis d_GFL of the GFL. Therefore, the angle of the GFL needs to be adjusted by adding a fixed angleφon the basis of θ_vsg. This can be obtained by measuring the deviation of u_q1 and applying PI control.

$$\begin{array}{rr}& \\ & \left\{\begin{array}{ll}{\theta }_{\mathrm{p}\mathrm{l}\mathrm{l}}={\theta }_{\mathrm{v}\mathrm{s}\mathrm{g}}+\varphi & \\ \varphi =k_{\mathrm{p}}{u}_{\mathrm{q}1}+k_{\mathrm{i}}\int u_{\mathrm{q}1}& \end{array}\right.\end{array}$$

(15)

Using equation (15), an improved PLL based on the grid-forming model can be obtained, which can also precisely align U₁ with the d-axis of the two-phase rotating coordinate system of the GFL, as shown in Figure 8(b). At this point, the active power and reactive power of the GFL are decoupled.

The P_setF during the fault can be obtained as:

$$P_{\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{F}}=3{\displaystyle \frac{k_3{U}_2{V}_{\mathrm{g}\mathrm{F}}}{X_2}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}-3{\displaystyle \frac{k_1{U}_2{I}_1}{X_2}}\mathrm{c}\mathrm{o}\mathrm{s}\left(\varphi +{\phi }_1\right)$$

(16)

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 9. In this case, as long as the GFM remains stable, the stability of the GFL can be ensured.

4.2. Fault Current Control

4.2.1. Steady-State Current Control

Although the GFM is stabilized by adjusting the input mechanical power, this does not effectively suppress the overcurrent phenomenon. Even though the presence of the GFL can reduce the fault current of the GFM, the GFM may still generate a significant fault current when there is a substantial drop in grid voltage. Therefore, when the fault current I_2F exceeds 1.5 times the rated current, the control of the GFM fault current I_2F is as follows:

$$I_{2\mathrm{F}}=1.5I_{2\mathrm{n}}$$

(17)

In equation (17): I_2n is the rated value of the GFM output current.

Since the GFL is a current source type, it will not generate steady-state overcurrent during a fault. However, to support the grid, it needs to output reactive current. The adjustment rules for the output current of the grid-following type are as follows:

$$\begin{array}{rr}{I}_{\mathrm{q}1\mathrm{F}}& =\left\{\begin{array}{ll}0& , {\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}\ge 0.9\\ & \\ -1.5\left(0.9-{\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}\right)I_{1\mathrm{n}}& , 0.2\le {\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}<0.9\\ & \\ -1.05I_{1\mathrm{n}}& , {\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}<0.2\end{array}\right.\end{array}$$

(18)

$$I_{\mathrm{d}1\mathrm{F}}=\left\{\begin{array}{c}\sqrt{I_{1\mathrm{n}}^2-I_{\mathrm{q}1\mathrm{F}}^2} ,{\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}\ge 0.2\\ 0 ,{\displaystyle \frac{V_{\mathrm{g}\mathrm{F}}}{V_{\mathrm{g}}}}<0.2\end{array}\right.$$

(19)

In the equation:I_1n is the rated value of the GFL output current;I_d1F is the d-axis output current of the GFL during a fault;I_q1F is the q-axis output current of the GFL during a fault.

Based on equations (14), (18), and (19), the values of δ_vsgF, I_q1F, and I_d1F are obtained, respectively. From equations (4), (6), (8), and (17), the following system of equations can be derived:

$$\left\{\begin{array}{rr}{\left(A-k_3{V}_{\mathrm{g}}\right)}^2+B^2-2{\left(1.5X_2{I}_{2\mathrm{n}}\right)}^2& =0\\ k_4{i}_{\mathrm{d}1\mathrm{F}}-k_2{U}_{2\mathrm{F}}\mathrm{s}\mathrm{i}\mathrm{n}\varphi -k_3{V}_{\mathrm{g}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}+\varphi \right)& =0\\ P_{\mathrm{s}\mathrm{e}\mathrm{f}\mathrm{F}}=C\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}-D\mathrm{c}\mathrm{o}\mathrm{s}\left(\varphi +{\phi }_{\mathrm{l}\mathrm{F}}\right)& \end{array}\right.$$

(20)

In equation (20):

In equation (20):${\phi }_{1\mathrm{F}}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{I_{\mathrm{q}1\mathrm{F}}}{I_{\mathrm{d}1\mathrm{F}}}}\right);I_{1\mathrm{F}}=\sqrt{I_{\mathrm{d}1\mathrm{F}}^2+I_{\mathrm{q}1\mathrm{F}}^2};$

$$\begin{gathered} A=U_{2\mathrm{F}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}+k_1{I}_{1\mathrm{F}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}+\varphi +{\phi }_{1\mathrm{F}}\right)-k_2{U}_{2\mathrm{F}}\mathrm{c}\mathrm{o}\mathrm{s}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}; \\ B=U_{2\mathrm{F}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}-k_1{I}_{1\mathrm{F}}\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}}+\varphi +{\phi }_{1\mathrm{F}}\right)-k_2{U}_{2\mathrm{F}}\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{\mathrm{v}\mathrm{s}\mathrm{g}\mathrm{F}};C=3{\displaystyle \frac{k_3{U}_{2\mathrm{F}}{V}_{\mathrm{g}}}{X_2}}. \end{gathered}$$

Equation (20) can be solved using numerical methods to obtain the output voltage U_2F of the GFM and the input mechanical power P_setF during a fault. Since the reactive power loop adjusts the output voltage command, this may lead to inaccurate fault current control. Therefore, during a fault, the reactive power loop is locked, and the calculated U_2F is used to replace the output of the reactive power loop.

4.2.2. Steady-State Current Control

The transient current of the GFL is small and has a minimal impact on the system, so it can be neglected. However, the transient current of the GFM is relatively large, and adjusting the reference voltage of the reactive power loop alone is not effective. If left unchecked, it can damage transistors.

Therefore, transient virtual impedance [20] is used to suppress transient overcurrent. The threshold current Ith for the virtual impedance is set slightly above 1.5 times I_2n. When the detected fault current exceeds Ith, the virtual impedance is activated; otherwise, the virtual impedance remains inactive. By combining the virtual impedance with the adjustment of the reactive power loop, the output voltage of the GFM can be expressed as:

$$U_{\mathrm{r}\mathrm{e}\mathrm{f}}=U_{2\mathrm{F}}-I_{2\mathrm{F}}{Z}_{\mathrm{v}}$$

(21)

In equation (21): Z_v is the virtual impedance.

The overall current control block diagram is shown in Figure 10:

5. Case Study

To verify the theoretical analysis and the proposed fault ride-through control strategy, a hybrid grid-connected model of grid-following and grid-forming converters, as shown in Figure 1, was built in Matlab/Simulink. The system parameters are listed in Table 1.

5.1. Output Characteristics of GFM and GFL during Fault Conditions

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 11. The simulation duration is set to 3 seconds, with a fault occurring at 0.5 seconds and clearing at 2 seconds. As seen in Figure 11, 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 12. a) 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 12(b) 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.

5.2. Verification of the Proposed Fault Ride-through Control Strategy

Figure 13 shows the output waveforms when the grid voltage symmetrically drops to 0.4 pu at 0.5 seconds and rises back to 1 pu at 2 seconds. 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 13(b), 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 13(d), 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 13(f), 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 14 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 14(d), 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 14(e), 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.

6. Conclusion

Focusing on the parallel hybrid grid connection of GFL and GFM converters, a mathematical model is established to study the challenges of achieving fault ride-through during symmetric grid voltage sags. The following conclusions are drawn.

1) Due to the coupling effect, the instability of one converter will cause the instability of the other converter. During a fault, reducing the i_d1 of the GFL or the P_set of the GFM can simultaneously improve the transient stability of both converters.

2) During a fault, the GFM adjusts the power angle according to the extent of the grid voltage sag and modifies the output voltage while considering the impact of the current injected by the GFL when limiting current. This approach can effectively suppress short-circuit overcurrent and maintain power angle stability. The GFL uses an improved PLL based on grid-forming control, which can effectively prevent GFL instability and inject current into the grid in accordance with grid codes.

3) The study preliminarily explores the mutual influence mechanism and fault ride-through control of GFL and GFM parallel grid connection under symmetrical faults. The transient process and fault control of both converters under asymmetrical faults are more complex. The next step will be to study control strategies under asymmetrical faults.