As a researcher in the field of power electronics and renewable energy systems, I have long been fascinated by the challenges posed by grid integration of photovoltaic (PV) systems. The solar inverter serves as the critical interface between PV arrays and the utility grid, and its performance directly impacts power quality, system stability, and overall energy yield. In practical scenarios, grid voltages are often unbalanced due to asymmetrical loads, faults, or network configurations, leading to degraded performance of conventional inverter control schemes. This article presents my comprehensive study on a novel control strategy designed to enhance the operation of three-phase LCL-filtered solar inverters under such non-ideal grid conditions. The core innovation lies in a double-sequence feedforward decoupling method combined with a compound current controller, which effectively mitigates coupling effects and suppresses harmonic distortion. I will delve into the mathematical foundations, design methodology, and validation through extensive simulations and hardware experiments, aiming to provide a robust solution for modern solar inverter applications.
The proliferation of distributed generation, particularly from PV sources, has necessitated advanced power conversion technologies. A grid-connected solar inverter must not only convert DC to AC power but also ensure synchronization with the grid, maintain high power factor, and inject currents with low harmonic content. The widespread adoption of LCL filters, due to their superior high-frequency attenuation compared to simple L filters, introduces additional complexity in controller design because of inherent resonance peaks. Furthermore, under unbalanced grid voltages, the standard synchronous reference frame (dq-frame) control used in many solar inverters suffers from significant coupling between the d and q axes, leading to unbalanced output currents and increased total harmonic distortion (THD). Existing approaches, such as dual current loop control with active damping or model predictive control, often involve trade-offs between complexity, robustness, and performance. My work addresses these issues by proposing a single-loop control strategy based on inverter-side current feedback, augmented with a precise decoupling mechanism and a harmonic-compensating repetitive controller. This approach simplifies the control structure while achieving excellent steady-state and dynamic performance, making it highly suitable for industrial solar inverter products.
To establish a foundation, I begin with the mathematical model of a three-phase LCL-type grid-connected solar inverter. The main circuit topology consists of a DC-link voltage source \(U_d\) (provided by the PV array via a front-end boost converter), a three-phase voltage source inverter (VSI), an LCL filter, and the grid. The LCL filter includes inverter-side inductors \(L_1\), grid-side inductors \(L_2\), and filter capacitors \(C\). The state variables are the inverter-side currents \(i_{1abc}\), capacitor voltages \(u_{cabc}\), and grid-side currents \(i_{2abc}\). The grid voltages \(u_{gabc}\) may be unbalanced. The inverter is modeled as a gain \(K_{PWM} = U_d / U_{tri}\), where \(U_{tri}\) is the amplitude of the triangular carrier wave. In the synchronous rotating reference frame (dq-frame) rotating at the grid fundamental frequency \(\omega\), the dynamics of the system exhibit cross-coupling terms due to the coordinate transformation. For the d-axis, the differential equations governing the inverter-side current and grid-side current can be derived as:
$$ L_1 \frac{di_{1d}}{dt} = -R_1 i_{1d} + \omega L_1 i_{1q} + u_{id} – u_{cd} $$
$$ L_2 \frac{di_{2d}}{dt} = -R_2 i_{2d} + \omega L_2 i_{2q} + u_{cd} – u_{gd} $$
$$ C \frac{du_{cd}}{dt} = \omega C u_{cq} + i_{1d} – i_{2d} $$
Similarly, for the q-axis:
$$ L_1 \frac{di_{1q}}{dt} = -R_1 i_{1q} – \omega L_1 i_{1d} + u_{iq} – u_{cq} $$
$$ L_2 \frac{di_{2q}}{dt} = -R_2 i_{2q} – \omega L_2 i_{2d} + u_{cq} – u_{gq} $$
$$ C \frac{du_{cq}}{dt} = -\omega C u_{cd} + i_{1q} – i_{2q} $$
Here, \(u_{id}\) and \(u_{iq}\) are the dq components of the inverter output voltage, and \(R_1\), \(R_2\) represent parasitic resistances. The terms \(\omega L_1 i_{1q}\), \(\omega L_2 i_{2q}\), \(\omega C u_{cq}\) in the d-axis equations (and their counterparts in the q-axis) are the coupling components. In a balanced sinusoidal steady state, these are constant, but under unbalanced conditions, they contain both DC and oscillatory components. A conventional PI controller in the dq-frame struggles to regulate these coupled variables, leading to poor performance. Therefore, my strategy focuses on accurately decoupling these terms to achieve independent control of active and reactive power.
The proposed decoupling method is based on a double synchronous reference frame (DSRF) theory. Under unbalanced grid voltages, the three-phase quantities contain both positive-sequence (fundamental) and negative-sequence components. Transforming these quantities into a single dq-frame rotating at \(\omega\) results in DC components for the positive sequence but \(2\omega\) oscillatory components for the negative sequence. To separate them, I employ two rotating frames: the positive-sequence frame (dq+) rotating counterclockwise at \(\omega\), and the negative-sequence frame (dq-) rotating clockwise at \(\omega\). Any vector \(\mathbf{V}\) in the stationary \(\alpha\beta\)-frame can be expressed as:
$$ \mathbf{V} = \mathbf{V}^+ e^{j\omega t} + \mathbf{V}^- e^{-j\omega t} $$
In the dq+ frame, \(\mathbf{V}^+\) appears as a constant complex number \(V_d^+ + jV_q^+\), while \(\mathbf{V}^-\) appears as a complex oscillation at \(2\omega\). The transformation from \(\alpha\beta\) to dq+ and dq- is given by:
$$ \begin{bmatrix} V_d^+ \\ V_q^+ \end{bmatrix} = \begin{bmatrix} \cos(\omega t) & \sin(\omega t) \\ -\sin(\omega t) & \cos(\omega t) \end{bmatrix} \begin{bmatrix} V_\alpha \\ V_\beta \end{bmatrix} $$
$$ \begin{bmatrix} V_d^- \\ V_q^- \end{bmatrix} = \begin{bmatrix} \cos(\omega t) & -\sin(\omega t) \\ \sin(\omega t) & \cos(\omega t) \end{bmatrix} \begin{bmatrix} V_\alpha \\ V_\beta \end{bmatrix} $$
However, due to the mutual interference between sequences, the measured \(V_d^+\) actually contains a contribution from the negative sequence: \(V_d^+ = D^+ + D^- \cos(2\omega t) + Q^- \sin(2\omega t)\), where \(D^+\) is the true positive-sequence d-component, and \(D^-\), \(Q^-\) are the negative-sequence components. Using the DSRF decoupling network, I extract the pure DC components \(D^+\), \(Q^+\), \(D^-\), \(Q^-\) through a structure involving low-pass filters (LPFs) and cross-feedback of the estimated components. This allows me to obtain accurate positive-sequence quantities for control purposes while simultaneously regulating the negative-sequence components to zero to mitigate unbalance.
With the positive-sequence components isolated, I design a feedforward decoupling scheme for the solar inverter current control loop. The control objective is to regulate the inverter-side current \(i_1\) as it offers inherent active damping of the LCL resonance without additional feedback paths. The block diagram of the decoupled system in the dq+ frame is shown below, where the coupling terms \(\omega L_1 i_{1q}\), \(\omega L_2 i_{2q}\), and \(\omega C u_{cq}\) are compensated by feedforward signals with gains A, B, E, F, M, N derived via signal flow graph analysis and Mason’s gain formula.
Let \(G_c(s)\) be the current controller, \(H\) the inverter-side current feedback gain. The transfer function from the coupling disturbance \(\omega L_2 i_{2q}\) to the output \(i_{2d}\) can be nullified by injecting a feedforward signal with gain \(A = H C s\) and \(B = (L_1 C s^2 + 1)/K_{PWM}\). Similarly, for \(\omega C u_{cq}\), the gains are \(E = H\), \(F = L_1 s / K_{PWM}\), and for \(\omega L_1 i_{1q}\), \(M=0\), \(N=1/K_{PWM}\). These derivations ensure that the coupling effects are perfectly canceled in the ideal case, allowing the controller \(G_c(s)\) to treat the plant as a simple decoupled first-order system. The comprehensive feedforward action transforms the system into:
$$ i_{2d}(s) = \frac{G_c(s) K_{PWM}}{D(s)} i_{d}^*(s) $$
where \(D(s) = L_1 L_2 C s^3 + G_c(s) K_{PWM} H L_2 C s^2 + (L_1 + L_2)s + G_c(s) K_{PWM} H\). This simplification is crucial for designing a robust controller for the solar inverter.
The current controller \(G_c(s)\) itself is a compound structure combining a PI regulator and a repetitive controller (RC). The PI controller provides fast dynamic response and zero steady-state error for DC references, while the RC suppresses periodic disturbances caused by factors like dead-time effects and grid voltage harmonics. The dead-time effect in PWM inverters introduces low-order odd harmonics (e.g., 5th, 7th) in the output current, which significantly degrades the THD. The RC, based on the internal model principle, incorporates a delay line of one fundamental period \(N = f_s / f_1\) (where \(f_s\) is the sampling frequency and \(f_1\) is the grid frequency) to generate high gain at harmonic frequencies. The discrete-time RC transfer function is:
$$ G_{RC}(z) = \frac{Q(z) z^{-N}}{1 – Q(z) z^{-N}} C(z) $$
Here, \(Q(z)\) is a filter typically chosen as a constant slightly less than 1 (e.g., 0.95) for stability, and \(C(z) = K_r z^k S(z)\) is a compensator. \(K_r\) is the gain, \(k\) is a phase lead step to compensate system delay, and \(S(z)\) is a low-pass filter to attenuate high-frequency noise. For my solar inverter design, I selected \(S(z)\) as a second-order Butterworth filter with a cut-off frequency of 800 Hz:
$$ S(z) = \frac{0.0902z + 0.06461}{z^2 – 1.213z + 0.3679} $$
The compound controller is then \(G_c(z) = G_{PI}(z) + G_{RC}(z)\). The PI part is tuned for a bandwidth of about 1 kHz, ensuring good tracking of current references. The parameters used in my simulation and experimental studies are summarized in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link Voltage | \(U_d\) | 700 V |
| Grid Voltage (line-to-line) | \(U_{g,ll}\) | 380 V RMS |
| Grid Frequency | \(f_1\) | 50 Hz |
| Switching Frequency | \(f_{sw}\) | 10 kHz |
| Inverter-side Inductance | \(L_1\) | 3 mH |
| Grid-side Inductance | \(L_2\) | 1.1 mH |
| Filter Capacitance | \(C\) | 4.7 μF |
| Inverter-side Current Feedback Gain | \(H\) | 0.5 |
| PI Controller Proportional Gain | \(K_p\) | 0.8 |
| PI Controller Integral Gain | \(K_i\) | 100 |
| Repetitive Control Gain | \(K_r\) | 0.8 |
| Phase Lead Steps | \(k\) | 2 |
| Internal Model Filter | \(Q\) | 0.95 |
Stability analysis of the overall system is performed using the Routh-Hurwitz criterion on the characteristic polynomial \(D(s)\). With \(G_c(s)\) simplified to a proportional gain \(K_p\) for analysis, the polynomial becomes:
$$ D(s) = L_1 L_2 C s^3 + K_p K_{PWM} H L_2 C s^2 + (L_1 + L_2)s + K_p K_{PWM} H $$
Constructing the Routh array confirms that all coefficients are positive and the first column has no sign changes, guaranteeing stability for properly chosen gains. This robust stability margin is essential for a commercial solar inverter operating under varying grid conditions.
To validate the proposed strategy, I conducted detailed simulations in MATLAB/Simulink. The test scenario involved a severely unbalanced grid voltage: phase A and B at 311 V peak (220 V RMS), but phase C at 291 V peak (206 V RMS). Without the double-sequence feedforward decoupling, the conventional PI controller produced unbalanced output currents with a significant negative-sequence component, as expected. The waveform distortion was evident, particularly during zero-crossings due to dead-time effects. However, with the full decoupling and compound control enabled, the three-phase currents became balanced and sinusoidal, closely tracking their references. The total harmonic distortion (THD) of the grid current was measured. Without the repetitive controller, the THD was 1.32% at a fundamental current of 10 A. After activating the repetitive controller, the THD dropped to 0.62%, demonstrating the effective suppression of dead-time harmonics. These simulation results unequivocally prove the efficacy of my approach for high-performance solar inverters.
Following simulations, I developed a 4.5 kW three-phase solar inverter prototype to experimentally verify the control strategy. The platform included a programmable AC source to emulate unbalanced grid voltages, a DC power supply simulating the PV array, and a dSPACE DS1103 controller for real-time implementation. The control algorithm, including the DSRF decoupling, feedforward compensation, and compound controller, was executed at a 10 kHz sampling rate. The experimental waveforms captured under the same unbalanced voltage conditions showed a remarkable improvement. With only PI control and no sequence decoupling, the phase currents were unbalanced and exhibited noticeable distortion, with a THD of 3.7% as measured by a power quality analyzer. Enabling the double-sequence feedforward decoupling immediately balanced the currents. Further enabling the repetitive controller reduced the THD to 2.6%. The current waveforms became smooth and sinusoidal, with perfect synchronization to the grid voltage. The dynamic performance was also tested by step-changing the current reference from 5 A (half-load) to 10 A (full-load) and vice versa. The system responded within one fundamental cycle (20 ms) without overshoot or instability, confirming the fast dynamic response preserved by the PI controller. These experimental results align perfectly with the simulations and underscore the practical viability of my proposed strategy for industrial solar inverter products.
In conclusion, I have presented a comprehensive and effective control strategy for three-phase LCL-filtered grid-connected solar inverters operating under unbalanced voltage conditions. The key contributions are threefold. First, the inverter-side current feedback loop provides inherent active damping of the LCL resonance, simplifying the control structure and enhancing robustness. Second, the double-sequence feedforward decoupling method accurately cancels the cross-coupling terms in the dq-frame by extracting and utilizing only the positive-sequence components, thereby enabling independent control of active and reactive power and ensuring balanced output currents even on an unbalanced grid. Third, the compound controller integrating a PI regulator and a repetitive controller achieves excellent reference tracking while significantly suppressing periodic disturbances like dead-time harmonics, leading to a lower THD. Both simulation and experimental results on a 4.5 kW prototype validate the superiority of this approach, showing THD reduction from 3.7% to 2.6% under severe unbalance. This strategy offers a balanced trade-off between performance, complexity, and cost, making it highly attractive for next-generation solar inverters aiming for superior power quality and grid support capabilities. Future work may focus on extending this method to single-phase systems or integrating adaptive mechanisms for time-varying grid impedances.
The mathematical rigor and practical validation detailed in this article provide a solid foundation for engineers and researchers working on advanced grid-tied solar inverter technologies. As renewable penetration continues to grow, such control advancements will be pivotal in maintaining grid stability and power quality. I believe that the principles outlined here can be further adapted and optimized for various power ratings and grid standards, contributing to the global transition towards sustainable energy systems.