In the operation of off-grid solar inverters, unbalanced three-phase loads are common due to asymmetric load switching, single-phase faults, or varying local demand. When such imbalance occurs, the output voltage of a solar inverter contains both positive-sequence and negative-sequence components. Traditional V/F (constant voltage constant frequency) control strategies, which only regulate the positive-sequence component, lead to severe voltage distortion, high total harmonic distortion (THD), and degraded power quality. To address this challenge, I propose a control algorithm based on decoupled double synchronous reference frames (DDSRF) combined with an improved dual V/F structure. This method effectively separates the positive- and negative-sequence components, decouples the cross-coupling terms, and independently regulates each sequence using dedicated V/F controllers. The performance is verified through Matlab/Simulink simulations, demonstrating that the proposed strategy significantly reduces THD and maintains stable voltage output even under severe load imbalance. The work focuses on enhancing the reliability and power quality of solar inverter systems in autonomous microgrids or stand-alone power supplies.
Solar inverters play a critical role in converting DC power from photovoltaic panels into AC power for local loads. In off-grid applications, the solar inverter must act as a voltage source, providing both voltage magnitude and frequency reference. However, unbalanced loads introduce negative-sequence currents that distort the output voltage. Traditional synchronous reference frame control fails to suppress these distortions because the negative-sequence components appear as 100 Hz oscillations in the dq rotating frame. I therefore adopt the decoupled double synchronous reference frame technique, which uses two rotating frames rotating at opposite angular speeds to separate positive- and negative-sequence quantities. After separation, a feed-forward decoupling network eliminates the 2ω cross-coupling terms, enabling independent regulation. The improved V/F control then sets the positive-sequence d-axis voltage reference to 310 V (peak phase voltage) and q-axis reference to zero, while the negative-sequence d- and q-axis references are set to zero to actively cancel negative-sequence voltage. This dual V/F approach ensures that the solar inverter output remains symmetrical and free from harmonic pollution.
The mathematical foundation of the DDSRF is crucial. Consider the unbalanced grid voltage (or inverter output voltage) represented in the stationary αβ frame. After applying the Clarke transformation, the voltage vector can be expressed as the sum of positive- and negative-sequence phasors rotating at angular speed ω and –ω respectively. The Park transformation with angle θ = ωt projects the positive-sequence component to DC quantities in the positive dq+ frame and transforms the negative-sequence component to double-frequency AC. Similarly, the negative dq– frame yields DC for negative sequence and 2ω AC for positive sequence. This coupling is described by:
$$
\mathbf{u}_{dq}^+ = \mathbf{u}_{dq}^{+’} + \mathbf{T}_{2\omega} \mathbf{u}_{dq}^{-‘}
$$
$$
\mathbf{u}_{dq}^- = \mathbf{u}_{dq}^{-‘} + \mathbf{T}_{-2\omega} \mathbf{u}_{dq}^{+’}
$$
where \(\mathbf{T}_{2\omega}\) is the rotation matrix at 2ω. By applying a low-pass filter and a feed-forward subtraction of the estimated opposite-sequence component, the decoupled outputs become pure DC quantities. The decoupling network structure for voltage (and similarly for current) is designed as shown in Figure 2 of the reference (not reproduced here). This decoupling is essential for the solar inverter to accurately track positive- and negative-sequence references without oscillation.
I now present the overall system configuration of the off-grid solar inverter. The DC bus voltage is 750 V, provided by a photovoltaic array or battery. A three-phase two-level inverter uses space vector pulse width modulation (SVPWM) with a switching frequency of 12 kHz. The output filter inductor is 8 mH per phase, and the load is a star-connected resistor bank. The control system samples three-phase voltages and currents, transforms them into the αβ frame, then applies the decoupled double synchronous reference frame to obtain positive- and negative-sequence dq components. Two separate V/F controllers generate the required dq voltage references for each sequence. These references are summed to produce the total dq voltage commands, which are transformed back to αβ and fed to the SVPWM modulator.
The improved dual V/F control structure is illustrated conceptually in Figure 6 of the reference. The positive-sequence V/F controller regulates Ud+ to 310 V and Uq+ to 0 V, while the negative-sequence controller forces Ud- and Uq- to 0 V. A first-order low-pass filter with cutoff frequency 22 Hz is used for decoupling. The PI parameters for the voltage loops are tuned to ensure fast dynamic response and zero steady-state error. The mathematical model of the solar inverter in the dq frames including the cross-coupling terms is given by:
$$
U_d^+ = L \frac{di_d^+}{dt} + R_g i_d^+ – \omega L i_q^+
$$
$$
U_q^+ = L \frac{di_q^+}{dt} + R_g i_q^+ + \omega L i_d^+
$$
Similar equations hold for negative-sequence variables with ω replaced by –ω. The controller compensates these terms using feed-forward decoupling, and the PI regulators handle the remaining dynamics.
Simulation setup is implemented in Matlab/Simulink. The nominal line-to-line voltage is 380 V (RMS), peak phase voltage 310 V, frequency 50 Hz. Initially, all three phase resistors are 50 Ω, balanced. At t = 0.1 s, a step change is applied: phase A becomes 40 Ω, phase B remains 50 Ω, and phase C becomes 60 Ω, creating a severe unbalanced condition. I compare the proposed DDSRF dual V/F control with the conventional single V/F control. The simulation results clearly show that under balanced conditions before 0.1 s, both control strategies produce symmetrical sinusoidal voltage waveforms. After the imbalance, the conventional control output becomes heavily distorted, with visible amplitude modulation and harmonics, while the proposed control restores a clean sinusoidal waveform within about 0.03 s (i.e., 1.5 cycles).
The positive- and negative-sequence dq voltages are extracted and plotted. Under balance, only positive-sequence exists: Ud+ = 310 V, Uq+ = 0 V, and both Ud- and Uq- are zero. After the load change, the negative-sequence components appear transiently but are forced back to zero within 0.03 s by the dual V/F control. This demonstrates the effective cancellation of negative-sequence voltage. Meanwhile, the positive-sequence Ud+ remains at 310 V with minimal fluctuation, and Uq+ stays near zero. The load currents show positive-sequence components that adjust to the new unbalanced load, while negative-sequence currents persist because the load itself is unbalanced; the controller only regulates the voltage.
FFT analysis of phase A voltage is performed in Matlab. Under balanced load, the THD is 1.10%, well below the IEEE 3% limit. Under unbalanced load with the proposed control, THD increases slightly to 2.07%, still meeting the standard. In contrast, the conventional control yields THD exceeding 10% under the same imbalance, confirming the necessity of the proposed method. The output frequency remains locked at 50 Hz except for a short transient of ±0.1 Hz during the load step, which settles within 0.02 s.
To summarize the key parameters and results, the following tables are provided.
| Parameter | Symbol | Value |
|---|---|---|
| DC bus voltage | Udc | 750 V |
| Filter inductance | L | 8 mH |
| DC link capacitor | C | 2100 μF |
| Switching frequency | fsw | 12 kHz |
| Nominal peak phase voltage | Um | 310 V |
| Nominal frequency | f | 50 Hz |
| Load resistors (balanced) | Ra=Rb=Rc | 50 Ω |
| Unbalanced load at t=0.1s | Ra, Rb, Rc | 40 Ω, 50 Ω, 60 Ω |
| LPF cutoff frequency | ωf | 22 Hz |
| PI gains (voltage loop) | Kp, Ki | 0.5, 20 |
| Load Condition | Control Strategy | THD (%) | Voltage Symmetry |
|---|---|---|---|
| Balanced (0–0.1 s) | Conventional V/F | 1.10 | Symmetrical |
| Balanced (0–0.1 s) | Proposed DDSRF+V/F | 1.10 | Symmetrical |
| Unbalanced (0.1–0.2 s) | Conventional V/F | >10 | Severely distorted |
| Unbalanced (0.1–0.2 s) | Proposed DDSRF+V/F | 2.07 | Symmetrical after 0.03 s |
| Time Segment | Ud+ (V) | Uq+ (V) | Ud– (V) | Uq– (V) |
|---|---|---|---|---|
| Before 0.1 s (balanced) | 310 | 0 | 0 | 0 |
| After 0.13 s (unbalanced, steady) | 310 | 0 | 0 | 0 |
From the simulation results, it is evident that the decoupled double synchronous reference frame combined with improved dual V/F control provides excellent voltage regulation for off-grid solar inverters under unbalanced loads. The negative-sequence voltage is effectively eliminated, the THD remains within acceptable limits, and the dynamic response is fast (3/4 of a fundamental cycle). This makes the solar inverter suitable for sensitive loads in remote areas or microgrids where grid support is unavailable.
Further improvement could be achieved by incorporating a repetitive controller in parallel with the PI to achieve zero steady-state error for the 2ω oscillation residuals. However, the current approach already satisfies typical power quality standards. The control strategy is also applicable to other power converters in renewable energy systems, such as wind inverters or battery energy storage inverters, when operating in islanded mode.
In conclusion, I have presented a robust control scheme for off-grid solar inverters that addresses the critical issue of unbalanced loads. The method uses a decoupled double synchronous reference frame to separate positive- and negative-sequence voltages, a feed-forward decoupling network to remove cross-coupling, and dual V/F regulators to independently control each sequence. Simulation verification confirms that the solar inverter output voltage maintains symmetry, low THD, and stable frequency. This work contributes to the reliable operation of solar inverters in autonomous power systems, ensuring high-quality power supply even under adverse load conditions.
The key formulas used in the control design are summarized below for clarity:
Clarke transformation:
$$
\begin{bmatrix} u_\alpha \\ u_\beta \end{bmatrix} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} u_a \\ u_b \\ u_c \end{bmatrix}
$$
Park transformation for positive-sequence:
$$
\begin{bmatrix} u_d^+ \\ u_q^+ \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} u_\alpha \\ u_\beta \end{bmatrix}
$$
Decoupling equations in DDSRF:
$$
\mathbf{u}_{dq}^{+*} = \mathbf{LPF}(s) \left( \mathbf{u}_{dq}^+ – \mathbf{T}_{2\omega} \mathbf{u}_{dq}^{-*} \right)
$$
$$
\mathbf{u}_{dq}^{-*} = \mathbf{LPF}(s) \left( \mathbf{u}_{dq}^- – \mathbf{T}_{-2\omega} \mathbf{u}_{dq}^{+*} \right)
$$
Voltage loop plant model (positive-sequence):
$$
\begin{bmatrix} U_d^+ \\ U_q^+ \end{bmatrix} = \begin{bmatrix} R_g + sL & -\omega L \\ \omega L & R_g + sL \end{bmatrix} \begin{bmatrix} i_d^+ \\ i_q^+ \end{bmatrix}
$$
These mathematical relations form the backbone of the proposed control for the solar inverter.