In recent years, the rapid integration of renewable energy sources and distributed generation has significantly increased the penetration of power electronic devices in modern power systems. This shift poses substantial challenges to grid coordination and stability. Among various solutions, the grid-forming grid connected inverter has emerged as a critical technology due to its voltage-source characteristics, enabling it to independently establish voltage and frequency, thereby providing inertial and damping support to the grid. In particular, the single voltage loop grid-forming grid connected inverter offers enhanced small-signal stability and dynamic performance, making it advantageous for renewable energy integration. However, under large disturbances such as grid voltage sags, this type of grid connected inverter exhibits transient responses distinct from synchronous generators, often leading to transient instability characterized by power angle oscillations. This paper aims to analyze the transient stability of single voltage loop grid-forming grid connected inverters, focusing on the transient effects of control loops and controller parameters.
We begin by describing the system configuration of a single voltage loop grid-forming grid connected inverter. The inverter is connected to the grid through an LC filter and a grid-side inductor. The power control loop employs a droop control with a low-pass filter for active power and a droop control for reactive power, generating reference phase and voltage amplitude, respectively. The single voltage loop uses an integral controller to produce the modulated voltage. The DC side is assumed to be a constant voltage source, as front-end converters regulate the DC voltage. Below is a schematic representation of the system.
The mathematical model of the grid connected inverter under large disturbances is derived based on grid voltage sag conditions. The power control loop equations are given by:
$$ \theta^* = \int \left( \omega_0 + \frac{K_p \omega_c}{s + \omega_c} (P_{\text{ref}} – P) \right) dt $$
$$ V_{\text{ref}} = V_0 + K_q (Q_{\text{ref}} – Q) $$
where $\theta^*$ is the reference phase, $V_{\text{ref}}$ is the reference voltage amplitude, $\omega_0$ is the rated angular frequency, $K_p$ and $K_q$ are droop coefficients for active and reactive power, $\omega_c$ is the cutoff angular frequency of the low-pass filter, $P_{\text{ref}}$ and $Q_{\text{ref}}$ are reference active and reactive power, and $P$ and $Q$ are the actual output powers. The single voltage loop integral controller yields:
$$ V^* = \frac{k_{iv}}{s} (V_{\text{ref}} – V_{\text{pcc}}) $$
where $k_{iv}$ is the integral coefficient, and $V_{\text{pcc}}$ is the voltage amplitude at the point of common coupling (PCC). The power angle $\delta$ is defined as the phase difference between the inverter output voltage and the grid voltage, with $\delta \approx \theta^* – \omega_0 t$. The active and reactive powers are expressed as:
$$ P = \frac{3EV}{2(X_{Lf} + X_g)} \sin \delta $$
$$ Q = \frac{3}{2} \cdot \frac{V^2 – EV \cos \delta}{X_{Lf} + X_g} $$
where $E$ is the grid voltage amplitude, $V$ is the inverter output voltage amplitude, $X_{Lf}$ is the filter reactance, and $X_g$ is the grid-side reactance. The relationship between $V$ and $V_{\text{pcc}}$ is derived from the voltage-current phasor diagram:
$$ V_{\text{pcc}} = \frac{\sqrt{E^2 X_{Lf}^2 + V^2 X_g^2 + 2EV X_{Lf} X_g \cos \delta}}{X_{Lf} + X_g} $$
To analyze transient stability, we establish a large-signal model by defining state variables $(x_1, x_2, x_3) = (\delta, \Delta \omega, V)$. The model is represented by the following differential equations:
$$ \dot{x}_1 = x_2 $$
$$ \dot{x}_2 = -\omega_c x_2 + K_p \omega_c \left( P_{\text{ref}} – \frac{3E x_3}{2(X_{Lf} + X_g)} \sin x_1 \right) $$
$$ \dot{x}_3 = k_{iv} \left( V_0 + K_q \left( Q_{\text{ref}} – \frac{3}{2} \cdot \frac{x_3^2 – E x_3 \cos x_1}{X_{Lf} + X_g} \right) – \frac{\sqrt{E^2 X_{Lf}^2 + x_3^2 X_g^2 + 2E x_3 X_{Lf} X_g \cos x_1}}{X_{Lf} + X_g} \right) $$
This nonlinear model captures the dynamics of the grid connected inverter under large disturbances and is solved numerically to obtain transient responses.
The transient stability of the grid connected inverter depends on the power angle response after a disturbance. Two key criteria are identified: first, the existence of a stable equilibrium point post-disturbance, which requires the maximum transmissible power to exceed the reference power; second, the power angle overshoot must not exceed a critical angle, ensuring synchronization with the grid. The phase portrait method is employed to visualize the power angle trajectories and assess stability. For instance, when the grid voltage sags to 0.63 per unit, the system may lose stability if the power angle surpasses the critical angle, as shown in the phase portrait analysis.
We now analyze the transient effects of each control loop in the grid connected inverter. The active power control loop influences the damping characteristics of the power angle response. The transfer function between power angle and reference active power is approximated as:
$$ \delta = \frac{K_p \omega_c}{s(s + \omega_c) + K_p \omega_c G_1} P_{\text{ref}} $$
where $G_1 = \frac{3E V}{2(X_{Lf} + X_g)}$ is a gain from circuit parameters. The damping ratio $\xi$ is given by:
$$ \xi = \frac{\omega_c}{2 \sqrt{K_p G_1}} $$
A higher damping ratio reduces power angle overshoot. Thus, decreasing $K_p$ or increasing $\omega_c$ enhances transient stability of the grid connected inverter. This is summarized in the table below for different parameter sets.
| Parameter | Value (p.u.) | Effect on Damping Ratio | Transient Stability |
|---|---|---|---|
| $K_p = 0.28$ | $\omega_c = 1.5$ | Low | Unstable |
| $K_p = 0.18$ | $\omega_c = 1.5$ | Moderate | Stable |
| $K_p = 0.10$ | $\omega_c = 1.5$ | High | Stable (no overshoot) |
| $K_p = 0.18$ | $\omega_c = 0.5$ | Low | Oscillatory |
The reactive power control loop affects the inverter output voltage amplitude during transients. By reducing the droop coefficient $K_q$, the voltage amplitude $V$ increases after a disturbance, as described by:
$$ V = V_0 + K_q (Q_{\text{ref}} – Q) – \frac{k_{iv}}{s} (V_{\text{ref}} – V_{\text{pcc}}) $$
This voltage support elevates the maximum transmissible power, helping maintain a stable equilibrium point. The relationship between $V$ and $\delta$ is implicit, but numerical solutions show that smaller $K_q$ values lead to higher $V$ and improved stability for the grid connected inverter. The following table illustrates this effect for a grid voltage sag to 0.63 p.u.
| $K_q$ (p.u.) | Steady-State $V$ (p.u.) | Steady-State $\delta$ (rad) | Stability Outcome |
|---|---|---|---|
| 0.018 | 1.05 | 1.64 | Stable |
| 0.012 | 1.08 | 1.59 | Stable |
| 0.002 | 1.12 | 1.55 | Stable |
The single voltage loop, with its integral control, governs the rate of change of the inverter output voltage. The dynamics are captured by:
$$ \dot{V} = k_{iv} \left( V_{\text{ref}} – V_{\text{pcc}} \right) $$
A larger integral coefficient $k_{iv}$ accelerates the voltage response, increasing $\dot{V}$ during transients. This rapid voltage adjustment promotes faster power output, reducing the power angle rate $\dot{\delta}$ and suppressing overshoot. Importantly, $k_{iv}$ does not alter the steady-state voltage or power angle, but it critically impacts transient stability. The table below shows how varying $k_{iv}$ influences the grid connected inverter’s response.
| $k_{iv}$ | Voltage Rate $\dot{V}$ (p.u./s) | Power Angle Overshoot | Transient Stability |
|---|---|---|---|
| 10 | 0.5 | Large | Unstable |
| 20 | 1.0 | Moderate | Stable |
| 30 | 1.5 | Small | Stable |
| 40 | 2.0 | None | Stable |
To validate the analysis, we conduct simulations using MATLAB/Simulink for the grid connected inverter. The system parameters are based on typical values, as listed below.
| Parameter | Value | Per Unit (p.u.) |
|---|---|---|
| Rated Power $S_N$ | 60 kW | 1.0 |
| Reference Active Power $P_{\text{ref}}$ | 30 kW | 0.5 |
| Reference Reactive Power $Q_{\text{ref}}$ | 0 kvar | 0 |
| Base Voltage $V_0$ | 311 V | 1.0 |
| Grid Voltage $E$ | 311 V | 1.0 |
| Grid Angular Frequency $\omega_0$ | 314 rad/s | 1.0 |
| Grid-Side Inductance $L_g$ | 9.5 mH | 1.23 |
| Filter Inductance $L_f$ | 1.35 mH | 0.18 |
| Active Droop Coefficient $K_p$ | 0.28 $\omega_0/S_N$ | 0.28 |
| Low-Pass Filter Cutoff $\omega_c$ | 1.5 $\omega_0$ | 1.5 |
| Reactive Droop Coefficient $K_q$ | 0.018 $V_0/S_N$ | 0.018 |
| Voltage Integral Coefficient $k_{iv}$ | 10 | — |
Four case studies are performed, with a grid voltage sag to 0.63 p.u. at 5 seconds. In Case 1, with initial parameters, the grid connected inverter becomes unstable, exhibiting diverging power angle oscillations. In Case 2, reducing $K_p$ to 0.18 p.u. and adjusting $\omega_c$ stabilizes the system, confirming that active power control loop optimization enhances damping. In Case 3, decreasing $K_q$ to 0.012 p.u. increases the inverter output voltage, providing better voltage support and stability. In Case 4, increasing $k_{iv}$ to 20 accelerates voltage response, suppressing power angle overshoot and restoring stability. These results align with the theoretical analysis, demonstrating that controller parameter tuning is crucial for the transient stability of single voltage loop grid-forming grid connected inverters.
The simulation outcomes underscore the importance of each control loop. The active power control loop primarily affects the power angle damping, where lower $K_p$ and higher $\omega_c$ values improve stability. The reactive power control loop influences the voltage amplitude, with lower $K_q$ offering stronger voltage support during faults. The single voltage loop controls the voltage rate, where higher $k_{iv}$ speeds up transient response, mitigating power angle deviations. Collectively, these insights provide a framework for designing robust grid connected inverters in renewable-rich grids.
In conclusion, the transient stability of single voltage loop grid-forming grid connected inverters is governed by power angle dynamics under large disturbances. Through mathematical modeling, phase portrait analysis, and parameter studies, we have shown that the active power control loop, reactive power control loop, and single voltage loop each play distinct roles in transient behavior. Optimizing controller parameters—specifically, reducing active and reactive droop coefficients, increasing the low-pass filter cutoff frequency, and raising the voltage integral coefficient—can significantly enhance the transient stability of grid connected inverters. This work contributes to the reliable integration of renewable energy sources, ensuring that grid connected inverters maintain synchronization and support grid stability during faults. Future research may explore coordinated parameter optimization strategies and the impact of current limiting measures on transient stability in grid connected inverter systems.