In the context of achieving carbon neutrality, renewable energy sources are increasingly integrated into power systems through inverter-based resources. Among various types of solar inverters, grid-forming (GFM) inverters have attracted significant attention due to their ability to emulate synchronous generator behavior and provide grid support. However, traditional power control methods for GFM inverters suffer from limited dynamic performance and low control flexibility. In this work, I propose a flexible grid-connected power control method for GFM inverters that addresses these issues by introducing proportional-integral (PI) droop control with configurable zeros, enabling first-order dynamic response with customizable settling time and zero overshoot.
1. Introduction
The increasing penetration of renewable energy, particularly through types of solar inverters, has led to challenges in power system stability, such as reduced inertia and frequency regulation capability. Grid-following (GFL) inverters behave as current sources and rely on phase-locked loops, while grid-forming (GFM) inverters act as voltage sources and can autonomously respond to grid frequency and voltage variations. Among GFM techniques, virtual synchronous generator (VSG) control, droop control, and virtual oscillator control are widely used. However, conventional droop control often exhibits second-order dynamics, leading to overshoot and oscillation, and suffers from coupling between droop coefficients and damping. To overcome these limitations, I propose a power control strategy that leverages feedforward paths to introduce configurable zeros, effectively reducing the system order to first-order while allowing independent adjustment of settling time and damping.
2. Proposed Power Control Architecture
The proposed control scheme is illustrated conceptually in Figure 1 (see the image link below). It consists of an outer power loop and an inner voltage loop. For active power control, I employ a PI controller with feedforward coefficients \(k_{set}\) and \(k_D\), as shown in the small-signal model. The key idea is to configure the zeros of the closed-loop transfer function to cancel one pole, yielding a first-order response. The active power control block diagram includes the PI gains \(k_{ip}\) and \(k_{i\omega}\), and the feedforward terms enable independent tuning of the droop coefficient \(D_p\) and the damping characteristic. Similarly, the reactive power control uses an integral controller with gain \(k_{iq}\) and a droop coefficient \(D_q\) to regulate voltage support.
In grid-connected operation, the inverter output power \(S = p + jq\) is related to the line impedance and voltage angles. For an inductive grid (\(X \gg R_g\)), the active and reactive power can be approximated as:
$$
p \approx \frac{U_C U_g \sin(\Delta\delta)}{X} \approx C_p \Delta\delta
$$
$$
q \approx \frac{U_C (U_C – U_g \cos(\Delta\delta))}{X} \approx C_q \Delta U
$$
where \(C_p = U_C U_g / X\) and \(C_q = U_C / X\) are local constants under nominal conditions.
3. Active Power Control Design
From the small-signal model, the active power transfer function from the power reference \(p_{set}\) and grid frequency deviation \(\Delta\omega_g\) to the output active power \(p\) is derived as:
$$
p = G_{rp}(s) p_{set} + G_{\omega p}(s) \Delta\omega_g
$$
$$
G_{rp}(s) = \frac{C_p (k_{ip} + k_{set}) s + C_p k_{i\omega}}{s^2 + C_p k_{ip} s + C_p k_{i\omega}}
$$
$$
G_{\omega p}(s) = \frac{C_p (D_p k_D + D_p k_{ip} + 1) s + C_p D_p k_{i\omega}}{s^2 + C_p k_{ip} s + C_p k_{i\omega}}
$$
At steady state, \(G_{rp}(0)=1\) and \(G_{\omega p}(0)=D_p\), confirming accurate tracking and droop response. To achieve first-order dynamics, I configure the zero of \(G_{rp}(s)\) to cancel one pole. Let the denominator poles be \(p_1\) and \(p_2\). The zero is \(z_1 = k_{i\omega} / (k_{ip} + k_{set})\). By setting \(z_1 = p_2\), the transfer function reduces to:
$$
G_{rp}(s) = \frac{p_1}{s + p_1}
$$
where \(p_1\) is the desired bandwidth, and the settling time \(T_{Pset} = 4.6 / p_1\). The controller parameters are then uniquely determined:
$$
k_{ip} = \frac{p_1 + p_2}{C_p}, \quad k_{i\omega} = \frac{p_1 p_2}{C_p}, \quad k_{set} = \frac{p_1}{C_p} – k_{ip}
$$
Similarly, for the frequency response, I set the zero \(z_2 = \frac{D_p k_{i\omega}}{D_p k_D + D_p k_{ip} + 1}\) equal to \(p_2\), yielding:
$$
k_D = \frac{k_{i\omega}}{p_2} – k_{ip} – \frac{1}{D_p}
$$
Thus, the frequency-to-power transfer function also becomes first-order:
$$
G_{\omega p}(s) = \frac{D_p}{s + p_1}
$$
This design decouples the droop coefficient \(D_p\) from the dynamic response, allowing independent adjustment of steady-state frequency support and transient performance.
4. Reactive Power Control Design
The reactive power loop is inherently first-order due to the integral controller. The transfer function from the reactive power reference and grid voltage deviation is:
$$
q = G_{rq}(s) q_{set} + G_{Vq}(s) \Delta U_g
$$
$$
G_{rq}(s) = \frac{C_q k_{iq}}{s + C_q k_{iq}}, \quad G_{Vq}(s) = \frac{C_q s + D_q C_q k_{iq}}{s + C_q k_{iq}}
$$
At steady state, \(G_{rq}(0)=1\) and \(G_{Vq}(0)=D_q\). The settling time \(T_{Qset}\) is chosen, and the gain is:
$$
k_{iq} = \frac{4.6}{C_q T_{Qset}}
$$
The voltage droop coefficient \(D_q\) is selected based on the inverter’s remaining capacity and allowable voltage range.
5. Simulation Results and Analysis
I validated the proposed method using MATLAB/Simulink. The system parameters are listed in Table 1. Control parameters for three different settling times are summarized in Table 2.
| Symbol | Description | Value |
|---|---|---|
| \(u_{dc}\) | DC-link voltage | 700 V |
| \(f_N\) | Nominal frequency | 50 Hz |
| \(f_{sw}\) | Switching frequency | 20 kHz |
| \(L_f\) | Filter inductor | 2 mH |
| \(R_f\) | Filter resistor | 0.3 Ω |
| \(C_f\) | Filter capacitor | 20 μF |
| \(R_C\) | Damping resistor | 0.1 Ω |
| \(R_g\) | Grid resistor | 0.1 Ω |
| \(L_g\) | Grid inductor | 2 mH |
| \(T_{Pset}=T_{Qset}\) | \(D_p=1000,\; D_q=100\) | \(k_{ip}\) | \(k_{i\omega}\) | \(k_{set}\) | \(k_D\) | \(k_{iq}\) |
|---|---|---|---|---|---|---|
| 1.0 s | 5.9765×10-5 | 1.3746×10-4 | -2.9882×10-5 | -0.1 | 0.0093 | |
| 0.2 s | 2.9882×10-4 | 0.0034 | -1.4941×10-4 | -0.1001 | 0.0465 | |
| 0.5 s | 1.1953×10-4 | 5.4984×10-4 | -5.9763×10-5 | -0.10006 | 0.0186 |
Figure 5 shows the active and reactive power responses for a step change of +5 kW (+5 kvar) at t=2.0 s and recovery at t=3.5 s, with a designed settling time of 1.0 s. The proposed method exhibits a first-order response without overshoot, matching the theory. The traditional droop control also shows similar dynamics. When the settling time is reduced to 0.2 s (Figure 6), the proposed method still tracks cleanly, while traditional droop begins to exhibit overshoot. At 0.1 s (Figure 7), the traditional method becomes highly oscillatory, whereas the proposed method maintains near-first-order behavior with minimal overshoot.
For frequency support, a 1 Hz grid frequency step was applied (Figure 8). The inverter autonomously responds by injecting \(D_p \times 1\) Hz = 1000 units of active power (scaled appropriately), with a first-order transient. Similarly, a 5 V grid voltage step (Figure 9) causes the inverter to supply \(D_q \times 5\) = 500 vars of reactive power. Both demonstrate the desired droop characteristics.
Robustness tests under a voltage sag to 150 V (Figure 10) show that the active power response remains nearly first-order with only slight overshoot, and the reactive power retains its first-order shape. Under unbalanced grid voltage (Figure 11), the power waveforms contain second-order harmonic oscillations, but the overall dynamics still exhibit first-order characteristics, confirming the controller’s suitability for further fault-ride-through designs.
Sensitivity to FLL noise was analyzed via the frequency-domain transfer function \(G_{\omega p}(s)\) shown in Figure 12. At high frequencies, the filter roll-off attenuates noise. Adding white noise to the FLL output (Figure 13) introduces small steady-state ripple but does not affect the settling time (Figure 14), confirming robustness.
6. Experimental Verification
I built a 10 kVA inverter prototype based on TMS320F28379, sampling at 20 kHz with 32-bit floating-point precision. Figure 15 shows the active and reactive power step response (5 kW/5 kvar) with a designed settling time of 2.0 s. The measured settling time is approximately 2.0 s, matching theory. Figure 16 shows a 5.0 s settling time case, also consistent. Figure 17 presents the response to a 0.2 Hz grid frequency drop: the inverter increases active power by 0.2 kW (linear with \(D_p\)), and the response time is about 2.0 s. Figure 18 shows the response to a 5 V grid voltage increase: the inverter reduces reactive power by 0.5 kvar (linear with \(D_q\)), confirming the droop control functionality.
7. Conclusion
I have presented a flexible grid-connected power control method for GFM inverters, one of the essential types of solar inverters for future power systems. By introducing feedforward terms in the active power loop, the system is reduced to a first-order model with customizable settling time and zero overshoot. The reactive power loop also achieves first-order dynamics. The droop coefficients are decoupled from transient behavior, allowing independent design. Simulation and experimental results validate the theoretical analysis, demonstrating excellent dynamic performance, robustness, and adaptability. This control strategy is particularly suitable for grid-forming applications that demand high-quality dynamic response and stable operation under various grid conditions.