In modern photovoltaic (PV) power generation systems, the grid-connected solar inverter plays a pivotal role in ensuring high-quality power injection into the utility grid. Under unbalanced grid voltage conditions, conventional control strategies based on synchronous reference frame proportional-integral (PI) regulators suffer from degraded performance due to cross-coupling between the d- and q-axes and the presence of negative-sequence components. Moreover, LCL-type filters, commonly used to attenuate switching harmonics, introduce resonance peaks that must be properly damped. In this paper, we propose a comprehensive control strategy for a three-phase LCL-type grid-connected solar inverter operating under unbalanced grid conditions. The method employs a double synchronous reference frame transformation to separate positive- and negative-sequence components, followed by a feed-forward decoupling scheme that completely eliminates the cross-coupling effects. A composite controller combining PI and repetitive control (RC) is utilized to suppress periodic disturbances caused by dead-time effects, thereby reducing the total harmonic distortion (THD) of the grid current. We validate the proposed approach through both simulation and experimental results on a 4.5 kW laboratory prototype. The outcomes demonstrate that the THD of the grid current is reduced from 3.7% to 2.6% under unbalanced voltage, confirming the effectiveness of the proposed method for practical solar inverter applications.
1. Introduction
Grid-connected solar inverters are essential interfaces between photovoltaic arrays and the utility grid. Among various topologies, the three-phase voltage-source inverter with an LCL filter is widely adopted because of its superior harmonic attenuation capability and reduced filter size. However, the LCL filter introduces a resonance peak that must be damped to ensure system stability. Active damping techniques, such as capacitor current feedback, are commonly used but increase system complexity and sensitivity to digital delays. Inverter-side current feedback control, on the other hand, provides a simpler single-loop structure while effectively damping the resonance. Furthermore, under unbalanced grid voltages—caused by asymmetrical loads or faults—the control of a solar inverter becomes more challenging. The conventional synchronous dq-frame PI controller cannot eliminate steady-state errors in both positive- and negative-sequence components, leading to unbalanced and distorted grid currents.
To address these issues, various strategies have been proposed in the literature. Dual synchronous reference frame (DSRF) approaches allow independent control of positive- and negative-sequence currents, but they require complex decoupling networks. Feed-forward decoupling is a well-known technique to cancel the cross-coupling terms between the d- and q-axes in the LCL system. However, under unbalanced conditions, the coupling terms contain both positive- and negative-sequence components, making conventional feed-forward ineffective. In this work, we propose a double sequence feed-forward decoupling method that first separates the positive- and negative-sequence quantities using a dual rotating transformation, then applies feed-forward compensation using only the positive-sequence components. This approach completely eliminates the coupling effects while maintaining a simple control structure.
Additionally, dead-time effects in the pulse-width modulation (PWM) introduce low-order harmonics that deteriorate current quality. Repetitive control, based on the internal model principle, is well suited to suppress periodic disturbances such as those caused by dead-time. By combining a PI controller (for fast dynamic response) with a repetitive controller (for harmonic suppression), we achieve both good steady-state accuracy and low THD. This paper presents a detailed analysis of the proposed control strategy, including mathematical modeling, decoupling coefficient derivation, stability analysis, and experimental validation on a solar inverter prototype.
2. System Modeling
2.1 LCL-Type Grid-Connected Solar Inverter
The main circuit topology of the three-phase LCL-type grid-connected solar inverter is shown in Figure 1 (not cited in text). It consists of a DC-link voltage \(U_d\) (provided by the PV array via a Boost converter for MPPT), a three-phase inverter bridge, and an LCL filter composed of inverter-side inductor \(L_1\), filter capacitor \(C\), and grid-side inductor \(L_2\). The inverter is modulated using sinusoidal PWM (SPWM) and is modeled as a proportional gain \(K_{PWM}\). The mathematical model in the stationary abc frame is given by:
$$
\begin{aligned}
L_1 \frac{di_{1k}}{dt} &= K_{PWM}u_{mk} – u_{ck} – R_1 i_{1k} \\
C \frac{du_{ck}}{dt} &= i_{1k} – i_{2k} \\
L_2 \frac{di_{2k}}{dt} &= u_{ck} – u_{gk} – R_2 i_{2k}
\end{aligned}
$$
where \(k = a,b,c\), \(u_{mk}\) is the modulating signal, \(u_{ck}\) is the capacitor voltage, \(u_{gk}\) is the grid voltage, and \(R_1\), \(R_2\) are parasitic resistances (often neglected in analysis). By applying the synchronous rotating transformation (Park transformation) to the dq-frame rotating at angular frequency \(\omega\), the system equations become coupled due to the rotation. The block diagram for inverter-side current closed-loop control in the dq-frame (without decoupling) is depicted in classical literature. The coupling terms are \(\omega L_1 i_{1q}\), \(\omega L_1 i_{1d}\), \(\omega L_2 i_{2q}\), \(\omega L_2 i_{2d}\), \(\omega C u_{cd}\), and \(\omega C u_{cq}\). These terms must be canceled to achieve independent control of active and reactive power.
2.2 Inverter-Side Current Feedback Control
To damp the LCL resonance without additional sensors, we adopt inverter-side current feedback. The controlled variable is \(i_{1d}\) and \(i_{1q}\), and the grid current \(i_{2d}\) and \(i_{2q}\) are indirectly regulated. The closed-loop transfer function from reference \(i_d^*\) to inverter-side current \(i_{1d}\) is derived using signal flow graph analysis. For a proportional controller \(G_c(s)=k_p\), the characteristic polynomial is:
$$
D(s) = L_1 L_2 C s^3 + k_p K_{PWM} H L_2 C s^2 + (L_1 + L_2)s + k_p K_{PWM} H
$$
The Routh-Hurwitz criterion is applied to verify stability. Table I summarizes the Routh array for this third-order system.
| \(s^3\) | \(L_1 L_2 C\) | \(L_1+L_2\) | 0 |
|---|---|---|---|
| \(s^2\) | \(k_p K_{PWM} H L_2 C\) | \(k_p K_{PWM} H\) | 0 |
| \(s^1\) | \(\frac{(L_1+L_2)(k_p K_{PWM} H L_2 C) – (L_1 L_2 C)(k_p K_{PWM} H)}{k_p K_{PWM} H L_2 C} = L_2\) | 0 | |
| \(s^0\) | \(k_p K_{PWM} H\) | 0 |
Since all coefficients in the first column are positive (assuming \(L_1, L_2, C, k_p, K_{PWM}, H > 0\)), the system is stable. The grid current is given by:
$$
i_{2d}(s) = \frac{G_c(s)K_{PWM}}{D(s)} i_d^*(s) – \frac{L_1 C s^2 + G_c(s) K_{PWM} H C s + 1}{D(s)} u_{gd}(s)
$$
The second term represents disturbance from grid voltage. A full feed-forward of \(u_{gd}\) can eliminate it, with the same coefficients derived in the next section for coupling cancellation.
3. Proposed Control Strategy
3.1 Double Sequence Feed-Forward Decoupling
Under unbalanced grid voltage, the three-phase quantities contain both positive- and negative-sequence components. The conventional single synchronous reference frame transformation mixes these components, leading to oscillation at twice the fundamental frequency. To achieve decoupling, we employ a dual synchronous reference frame (DSRF) structure shown conceptually. The positive-sequence frame (rotating counterclockwise at \(\omega\)) and the negative-sequence frame (rotating clockwise at \(-\omega\)) are used. The transformation matrices are:
$$
\begin{bmatrix} d \\ q \end{bmatrix} = \begin{bmatrix} \sin\omega t & -\cos\omega t \\ \cos\omega t & \sin\omega t \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix}
$$
$$
\begin{bmatrix} d^- \\ q^- \end{bmatrix} = \begin{bmatrix} -\sin\omega t & -\cos\omega t \\ \cos\omega t & -\sin\omega t \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix}
$$
Let \(V^+\) and \(V^-\) represent the positive- and negative-sequence vectors. In the positive dq-frame, the projection of \(V^-\) appears as an alternating component at \(2\omega\). Specifically, for the d-axis:
$$
V_d = D + D^- \cos 2\omega t + Q^- \sin 2\omega t
$$
where \(D\) and \(Q\) are DC components of positive sequence, and \(D^-\), \(Q^-\) are DC components of negative sequence. Similar expressions hold for other axes. To extract the DC components, low-pass filters (LPF) are used, and the decoupling network in the DSRF-PLL (phase-locked loop) eliminates the cross-coupling between frames. The decoupling equations are:
$$
\begin{aligned}
D &= V_d – D^- \cos 2\omega t – Q^- \sin 2\omega t \\
Q &= V_q + D^- \sin 2\omega t – Q^- \cos 2\omega t \\
D^- &= V_{d^-} – D \cos 2\omega t + Q \sin 2\omega t \\
Q^- &= V_{q^-} – D \sin 2\omega t – Q \cos 2\omega t
\end{aligned}
$$
Once the positive-sequence components are isolated, we apply feed-forward decoupling to cancel the cross-coupling terms in the LCL model. Using the signal flow graph method, we compute the feed-forward coefficients for each coupling path. For example, the transfer function from the coupling term \(\omega L_2 i_{2q}\) to the output \(i_{2d}\) is derived, and the required feed-forward gain is found to be:
$$
A = H C s, \quad B = \frac{L_1 C s^2 + 1}{K_{PWM}}
$$
Similarly, for other coupling terms, the coefficients are summarized in Table II.
| Coupling Term | Feed-Forward Coefficient from Controller Output | Feed-Forward Coefficient from Disturbance |
|---|---|---|
| \(\omega L_2 i_{2q}\) to \(i_{2d}\) | \(A = H C s\) | \(B = (L_1 C s^2+1)/K_{PWM}\) |
| \(\omega C u_{cq}\) to \(i_{2d}\) | \(E = H\) | \(F = L_1 s / K_{PWM}\) |
| \(\omega L_1 i_{1q}\) to \(i_{2d}\) | \(M = 0\) | \(N = 1/K_{PWM}\) |
These coefficients are applied as feed-forward signals added to the controller output in the d- and q-axes. Because the positive-sequence components are used, the decoupling is effective even under unbalanced conditions. The negative-sequence components are independently regulated by PI controllers with zero reference to eliminate negative-sequence currents.
3.2 Composite PI and Repetitive Controller
Dead-time in the inverter bridge introduces low-order harmonics (5th, 7th, etc.) that degrade current quality. To suppress these periodic disturbances, we incorporate a repetitive controller in parallel with the PI regulator. The discrete transfer function of the repetitive controller is:
$$
G_{RC}(z) = \frac{z^{-N}}{1 – Q(z) z^{-N}} C(z)
$$
where \(N = f_s / f_0 = 200\) (sampling frequency 10 kHz, fundamental 50 Hz), \(Q(z)\) is a low-pass filter or constant (we choose \(Q=0.95\) for good stability margin), and \(C(z)\) is a compensator designed to invert the plant dynamics at low frequencies. Specifically, \(C(z) = K_r z^k S(z)\), with \(K_r = 0.8\), phase lead \(k=2\), and \(S(z)\) a second-order low-pass filter with cutoff 800 Hz (damping ratio 1). The filter is:
$$
S(z) = \frac{0.0902 z + 0.06461}{z^2 – 1.213 z + 0.3679}
$$
The overall controller in the z-domain is:
$$
G_c(z) = G_{PI}(z) + 0.8 z^{-198} \frac{0.0902 z + 0.06461}{(1 – 0.95 z^{-200})(z^2 – 1.213 z + 0.3679)}
$$
This composite controller achieves high gain at multiples of the fundamental frequency, thereby effectively canceling dead-time harmonics. The PI part guarantees fast transient response.
3.3 Overall Control Architecture
The complete control block diagram for the solar inverter is illustrated conceptually in standard format. The system uses double synchronous reference frame transformations to separate positive and negative sequences. The positive-sequence currents \(i_{1d}^+\) and \(i_{1q}^+\) are regulated by the composite PI+RC controller. The feed-forward decoupling terms (using coefficients from Table II) are added to cancel the cross-coupling. The negative-sequence currents \(i_{1d}^-\) and \(i_{1q}^-\) are controlled to zero by dedicated PI regulators. The grid voltage feed-forward is also applied to eliminate its disturbance. The outputs are transformed back to abc and passed to the SPWM modulator.
4. Simulation and Experimental Validation
4.1 Simulation Results
We built a simulation model in MATLAB/Simulink to verify the proposed strategy. The system parameters are: DC-link voltage \(U_d = 650\) V, grid voltage 311 V peak per phase (with unbalanced condition: two phases at 311 V, one phase at 291 V), LCL filter: \(L_1 = 3\) mH, \(L_2 = 1.1\) mH, \(C = 4.7\) μF, switching frequency 10 kHz, dead-time 5 μs. The controller parameters: \(k_p = 0.5\), \(k_i = 100\), repetitive control parameters as above.
Figure 2 shows the simulation results. Without the double-sequence feed-forward decoupling, the three-phase currents are unbalanced (different amplitudes). With the proposed decoupling, the currents become balanced. The THD of the grid current without RC is 1.32%; after adding RC, the THD drops to 0.62%, demonstrating effective dead-time harmonic suppression.
4.2 Experimental Results
A 4.5 kW laboratory prototype of the solar inverter was built using a TI TMS320F28335 DSP. The grid was emulated by a programmable AC source with unbalanced voltages: a-phase 311 V, b-phase 311 V, c-phase 291 V. The LCL filter parameters were identical to simulation. The switching frequency was 10 kHz with 5 μs dead-time. Grid currents were measured via current probes and a power quality analyzer.
Experimental waveforms are shown in Figure 3. With conventional PI control (without double-sequence decoupling), the three-phase currents are visibly unbalanced and distorted, with THD = 3.7%. After applying the proposed double-sequence feed-forward decoupling and composite controller, the currents become nearly balanced (slight asymmetry due to filter parameter tolerances) and the THD reduces to 2.6%. The dynamic response test (Figure 4) shows smooth transition between half-load (5 A) and full-load (10 A) within about one fundamental cycle, confirming that the repetitive controller does not degrade transient performance.
| Control Scheme | THD (%) |
|---|---|
| Conventional PI without decoupling | 3.7 |
| Proposed double-sequence decoupling + PI | 3.0 |
| Proposed double-sequence decoupling + PI+RC | 2.6 |
These results confirm that the proposed method effectively handles unbalanced grid conditions and suppresses dead-time harmonics, making it highly suitable for high-performance solar inverter applications.
5. Conclusion
In this paper, we have presented a robust control strategy for a three-phase LCL-type grid-connected solar inverter operating under unbalanced grid voltages. The key contributions are:
- Inverter-side current feedback is used to damp LCL resonance, enabling a simple single-loop control structure.
- A double synchronous reference frame transformation isolates positive- and negative-sequence components. Feed-forward decoupling using only positive-sequence components eliminates cross-coupling effects, achieving independent control of active and reactive power even under unbalanced conditions.
- A composite PI and repetitive controller effectively suppresses dead-time-induced harmonics, reducing the grid current THD significantly.
Simulation and experimental results on a 4.5 kW prototype demonstrate that the THD is reduced from 3.7% to 2.6%, while maintaining good dynamic response. The proposed method provides a practical and effective solution for modern solar inverter systems operating in non-ideal grid environments.
Acknowledgment
The authors sincerely thank the support from the National Natural Science Foundation of China and the Guangxi Key Research and Development Program (project numbers omitted per policy). The prototype used was assembled with components from industry partners.
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