In my research on grid-connected photovoltaic systems, I have dedicated significant effort to developing an efficient control strategy for a three-phase solar inverter equipped with an LCL filter. The rapid growth of renewable energy sources, particularly solar power, demands high-performance inverters that can inject clean sinusoidal current into the utility grid while maintaining stability and fast dynamic response. The LCL filter, compared to a simple L filter, offers superior attenuation of high-frequency switching harmonics and allows for a reduction in total inductance, which is crucial for minimizing size and cost. In this paper, I present a detailed analysis of the system topology, mathematical modeling in the synchronous rotating d-q frame, and a double-loop current control scheme enhanced by a proportional-resonant (PR) controller. I demonstrate through both simulation and experimental results that the proposed PR-based dual-loop control achieves zero steady-state error in tracking sinusoidal reference currents and effectively suppresses low-order harmonic distortions, making it a robust solution for modern solar inverter applications.
Introduction
With the gradual depletion of fossil fuels and increasing environmental concerns, photovoltaic (PV) generation and wind power have emerged as prominent renewable energy sources. The core component of any grid-connected PV system is the solar inverter, which converts DC power from solar panels into AC power synchronized with the utility grid. Among various inverter topologies, the three-phase voltage source inverter is widely adopted for medium to high power applications. The use of an LCL filter rather than a conventional L filter provides superior harmonic suppression, especially at high frequencies, and reduces the overall inductance requirements. However, the LCL filter introduces resonance issues that must be carefully addressed through proper control design.
In traditional three-phase solar inverter control, a PI controller operating in the stationary reference frame (αβ) cannot achieve zero steady-state error when tracking sinusoidal reference currents. To overcome this limitation, many researchers have implemented control in the synchronous rotating d-q frame, where sinusoidal signals become DC quantities, allowing PI controllers to eliminate steady-state error. However, this approach requires complex coordinate transformations (abc→dq) and introduces cross-coupling between the d and q axes, which complicates the controller design. A more elegant solution is the use of a proportional-resonant (PR) controller in the αβ frame. The PR controller provides infinite gain at the fundamental frequency, thereby achieving zero steady-state error for sinusoidal references without the need for decoupling or coordinate transformation. Moreover, the PR controller can be extended with harmonic compensators (HC) to suppress low-order grid voltage harmonics.
In this paper, I propose a double-loop current control strategy for a three-phase solar inverter with an LCL filter. The inner loop employs capacitor current feedback to actively damp the LCL resonance, while the outer loop uses a PR controller to regulate the grid-side inductor current. I compare the performance of the conventional PI controller versus the PR controller in terms of tracking accuracy, harmonic rejection, and dynamic response. The results confirm that the PR-based dual-loop controller significantly improves the power quality and reliability of the solar inverter.
System Topology and Mathematical Model
The main circuit topology of the three-phase grid-connected solar inverter with an LCL filter is shown in the schematic below. I have inserted a representative image of a solar inverter to illustrate the hardware implementation.
The system consists of a DC bus capacitor C1, six IGBT switches (T1–T6), and an LCL filter composed of inverter-side inductor L1 (with internal resistance R1), filter capacitor C2, and grid-side inductor L2 (with internal resistance R2). The grid is represented by three-phase voltages usa, usb, usc. For a balanced three-phase system, the state equations can be expressed in the stationary αβ frame using the Clarke transformation. The resulting equations are:
$$
\begin{aligned}
\frac{di_{1m}}{dt} &= -\frac{R_1}{L_1} i_{1m} – \frac{1}{L_1} u_{cm} + \frac{1}{L_1} u_m, \\
\frac{di_{2m}}{dt} &= -\frac{1}{L_2} u_{cm} – \frac{1}{L_2} u_{sm}, \\
\frac{du_{cm}}{dt} &= \frac{1}{C_2} i_{1m} – \frac{1}{C_2} i_{2m},
\end{aligned}
$$
where the subscript m denotes α or β component, i1m is the inverter-side inductor current, i2m is the grid-side inductor current, ucm is the capacitor voltage, and um is the inverter output voltage.
Applying the Park transformation to the αβ model, I obtain the state equations in the synchronous rotating d-q frame:
$$
\begin{aligned}
\frac{di_{1d}}{dt} &= -\frac{R_1}{L_1} i_{1d} + \omega i_{1q} – \frac{1}{L_1} u_{cd} + \frac{1}{L_1} u_d, \\
\frac{di_{1q}}{dt} &= -\frac{R_1}{L_1} i_{1q} – \omega i_{1d} – \frac{1}{L_1} u_{cq} + \frac{1}{L_1} u_q,
\end{aligned}
$$
$$
\begin{aligned}
\frac{di_{2d}}{dt} &= -\frac{R_2}{L_2} i_{2d} + \omega i_{2q} + \frac{1}{L_2} u_{cd} – \frac{1}{L_2} u_{sd}, \\
\frac{di_{2q}}{dt} &= -\frac{R_2}{L_2} i_{2q} – \omega i_{2d} + \frac{1}{L_2} u_{cq} – \frac{1}{L_2} u_{sq},
\end{aligned}
$$
$$
\begin{aligned}
\frac{du_{cd}}{dt} &= \frac{1}{C_2} i_{1d} – \frac{1}{C_2} i_{2d} + \omega u_{cq}, \\
\frac{du_{cq}}{dt} &= \frac{1}{C_2} i_{1q} – \frac{1}{C_2} i_{2q} – \omega u_{cd}.
\end{aligned}
$$
From these equations, it is evident that the d and q axes are coupled through terms involving ωL1i1q, ωL2i2q, and ωC2ucq etc. Decoupling these axes would require additional feedback signals and increase the dependency on accurate parameter knowledge. Therefore, I prefer to perform the control in the stationary αβ frame, where the coupling is absent, and instead utilize a PR controller to handle the sinusoidal nature of the references.
Current Double-Loop Control
The double-loop control structure for the three-phase solar inverter is illustrated in the block diagram. The inner loop uses capacitor current feedback (with gain Kc) to actively damp the LCL resonance, while the outer loop regulates the grid-side inductor current i2. The open-loop transfer function of the system, when using a conventional PI controller as the outer regulator, is given by:
$$
G_{T3}(s) = \frac{i_{2k}}{i_{2k}^* – i_{2k}} = \frac{G_i(s) G_c(s) G_1(s) G_2(s) G_3(s)}{1 + G_1(s) G_2(s) + G_2(s) G_3(s) + G_1(s) G_c(s)},
$$
where G1(s)=1/(L1s+R1), G2(s)=1/(C2s), G3(s)=1/(L2s+R2), Gc(s)=Kc, and Gi(s)=Kp + Ki/s. The equivalent control block diagram can be simplified to a single-loop form, where the reference I2r = Kc (Kp + Ki/s) I2*.
While the PI controller works well for DC references, it cannot achieve zero steady-state error for sinusoidal references in the αβ frame. To quantify this, consider the closed-loop transfer function of the grid current when a PI controller is used:
$$
I_2 = \frac{K_{PWM} G(s)}{D_{LCL}(s) + K_{PWM} G(s)} I_2^* + \frac{L_1 C_2 s^2 + R_1 C_2 s + 1}{D_{LCL}(s) + K_{PWM} G(s)} U_s,
$$
where DLCL(s) = L1L2C2s3 + (L1R2C2+L2R1C2)s2 + (R1C2+R2C2+L1+L2)s + (R1+R2). When G(s) is a PI controller, its gain at the fundamental frequency ω0 is finite, resulting in a non-zero steady-state tracking error (0 < ε < 1) and a non-zero contribution from the grid voltage disturbance.
PR Resonant Controller Based Double-Loop Control
To eliminate the steady-state error and improve disturbance rejection, I replace the PI controller with a PR (proportional-resonant) controller. The transfer function of an ideal PR controller is:
$$
G_{PR}(s) = K_p + \frac{K_r s}{s^2 + \omega_0^2},
$$
where ω0 is the fundamental angular frequency. At s = jω0, the resonant term becomes infinite, providing theoretically infinite gain at the fundamental frequency. Thus, for a solar inverter operating under normal grid conditions, the PR controller forces the tracking error to zero.
Furthermore, I extend the PR controller to include harmonic compensators (HC) for the 3rd, 5th, and 7th harmonics, which are commonly present in the grid voltage. The combined controller is:
$$
G_{PR+CH}(s) = K_p + \frac{K_r s}{s^2 + \omega_0^2} + \sum_{h=3,5,7} \frac{K_h s}{s^2 + (h\omega_0)^2}.
$$
The block diagram of the PR-based dual-loop control for the three-phase solar inverter is shown in Figure 5 of the original reference. The open-loop and closed-loop transfer functions when using the PR controller derive as:
$$
\frac{I_2}{I_2^* – I_2} = \frac{K_p K_c K_{PWM} s^2 + K_i K_c K_{PWM} s + K_p K_c K_{PWM} \omega_0^2}{w_1 s^5 + w_2 s^4 + w_3 s^3 + (R_1+R_2+w_2 \omega_0^2) s^2 + w_3 \omega_0^2 s + (R_1+R_2)\omega_0^2},
$$
$$
\frac{I_2}{I_2^*} = \frac{K_p K_c K_{PWM} s^2 + K_i K_c K_{PWM} s + K_p K_c K_{PWM} \omega_0^2}{l_1 s^5 + l_2 s^4 + l_3 s^3 + l_4 s^2 + l_5 s + l_6},
$$
where the coefficients are defined as:
| w1 | = L1L2C2 |
| w2 | = L1R2C2 + L2R1C2 + KcKPWML2C2 |
| w3 | = R1R2C2 + L1 + L2 + KcKPWMR2C2 |
| w4 | = KpKcKPWM + R1 + R2 |
| w5 | = KiKcKPWM |
| l1 | = w1 |
| l2 | = w2 |
| l3 | = w3 + w1ω02 |
| l4 | = w4 + w2ω02 |
| l5 | = w5 + w3ω02 |
| l6 | = w4ω02 |
Due to the infinite gain at ω0, the closed-loop transfer function from I2* to I2 approaches unity, while the disturbance transfer function from grid voltage Us to I2 approaches zero. This demonstrates the superior tracking and rejection capabilities of the PR controller in the three-phase solar inverter.
Simulation and Experimental Results
Simulation Setup
To validate the effectiveness of the proposed PR-based dual-loop control, I built a simulation model in MATLAB/Simulink. The system parameters are listed in the table below. The grid voltage was contaminated with 3rd, 5th, and 7th harmonics to test harmonic rejection. Two cases were compared: outer loop using a conventional PI controller and outer loop using the PR controller.
| Parameter | Value |
|---|---|
| DC bus voltage | 400 V |
| Grid voltage (line-to-line RMS) | 380 V |
| Grid frequency | 50 Hz |
| L1 / R1 | 1.8 mH / 0.1 Ω |
| L2 / R2 | 0.9 mH / 0.05 Ω |
| C2 | 10 μF |
| Switching frequency | 10 kHz |
| Controller parameters (PI) | Kp=0.7, Ki=600, Kc=0.49 |
| Controller parameters (PR) | Kp=0.7, Kr=600, Kc=0.49 |
| Harmonic compensators | K3=K5=K7=200 |
Tracking Performance
Figure 7 and Figure 8 of the original literature show the grid current tracking performance when a step change in the reference current is applied. With the PI controller, the actual current lags behind the reference with a noticeable steady-state error. In contrast, the PR controller achieves nearly perfect tracking with zero steady-state error, while the dynamic response time remains similar.
Harmonic Analysis
Figure 9 and Figure 10 display the total harmonic distortion (THD) and harmonic spectra of the grid current under both controllers. When using the PI controller, the grid current contains significant 3rd, 5th, and 7th harmonics due to the grid voltage distortion. With the PR controller augmented by harmonic compensators, these low-order harmonics are dramatically reduced. The THD drops from above 5% with PI to below 2% with PR+HC, meeting the IEEE 519 standard for grid-connected solar inverters.
Experimental Validation
I also conducted a hardware experiment using a 2 kW three-phase solar inverter prototype. The DC bus was supplied by a single-phase rectifier providing 250 V, and the inverter was connected to the 220 V grid through a 1:2 isolation transformer. The switching frequency was 10.5 kHz, and the controller parameters were set to Kp=1.6, Ki=Kr=1600, Kc=1. The grid voltage was distorted with practical harmonics.
The experimental waveforms of phase A voltage and grid current are shown in Figure 11, and the three-phase grid currents are shown in Figure 12. The results demonstrate that even under distorted grid conditions, the grid currents remain sinusoidal and well synchronized with the phase voltages. The current THD measured by a power analyzer was 2.8%, confirming the effectiveness of the PR-based dual-loop control for the solar inverter.
Conclusion
In this paper, I have presented a comprehensive study of the control strategy for a three-phase grid-connected solar inverter employing an LCL filter. By analyzing the mathematical model and the limitations of conventional PI controllers in the stationary frame, I adopted a double-loop control structure with an inner capacitor current loop for active damping and an outer PR resonant controller for grid current regulation. The PR controller provides infinite gain at the fundamental frequency, achieving zero steady-state error for sinusoidal references. Additionally, harmonic compensators were incorporated to suppress low-order grid harmonics. Simulation and experimental results confirm that the proposed method significantly improves the power quality, dynamic response, and robustness of the solar inverter. This work provides a practical and reliable solution for modern grid-connected solar inverter applications, contributing to the efficient integration of photovoltaic systems into the power grid.