I investigate the problem of controlling an off-grid solar inverter when the connected three-phase load becomes unbalanced. In a stand-alone photovoltaic system, the solar inverter must supply stable voltage and frequency to the load. However, unbalanced three-phase loads introduce negative-sequence components in the inverter output voltage, which severely deteriorate the power quality if conventional V/F control is used. In this work, I propose a control method based on decoupled double synchronous reference frames combined with an improved V/F strategy. The method separates the positive- and negative-sequence voltage components, decouples the cross-coupling terms that appear in the double synchronous frames, and then applies individual V/F controllers to each sequence. I developed a complete simulation model in Matlab/Simulink to validate the proposed strategy. The simulation results demonstrate that the proposed solar inverter controller effectively reduces the harmonic distortion of the output voltage, suppresses the negative-sequence component, and supplies high-quality power to unbalanced loads.
1. Introduction and Background
Off-grid power systems are widely used in remote areas, emergency supplies, and isolated loads. In these systems, the solar inverter is the key power conversion device that forms the voltage and frequency reference for all other units. Unlike grid-connected inverters, an off-grid solar inverter has no grid voltage support; therefore, it must maintain the load voltage amplitude and frequency within acceptable limits. The conventional V/F (constant voltage and constant frequency) control method works well under balanced three-phase loads, but in practice the load may become unbalanced due to single-phase loads, sudden load switching, or asymmetrical wiring. When the load is unbalanced, the inverter output voltage contains both positive-sequence and negative-sequence components. The negative-sequence component causes output voltage distortion, increases the total harmonic distortion (THD), and may cause malfunction of sensitive loads.
Many researchers have addressed unbalanced load compensation in standalone inverters. Some methods use instantaneous power theory to derive compensation commands, but the accuracy is limited. Others employ repetitive controllers in the αβ stationary frame, but these methods often suffer from slow dynamic response under sudden load changes. A more elegant approach is to use a decoupled double synchronous reference frame, which is commonly applied in phase-locked loop designs for asymmetric grid faults. I extend this concept to the voltage control loop of the solar inverter. By decomposing the output voltage into positive- and negative-sequence components in two rotating reference frames, and by cancelling the 2ω coupling terms, I can independently control both sequences. Then, an improved V/F controller with two parallel branches is used to regulate the positive-sequence voltage to the rated value and to force the negative-sequence voltage to zero.
Throughout this article, I use the term solar inverter to emphasize the application of the investigated algorithms to stand-alone photovoltaic systems. The proposed control structure can also be used for other types of off-grid inverters, but the focus here is on the solar inverter operating in island mode.
2. System Configuration
The considered off-grid solar inverter system is shown in Figure 1. A DC bus with voltage \(U_{dc}\) supplies a three-phase SVPWM inverter. The inverter output is connected to an LC filter (only the inductance is considered in the main model) and then to a three-phase load. The control system measures the output phase voltages \(U_a, U_b, U_c\) and the load currents \(i_a, i_b, i_c\). These measurements are transformed into the αβ stationary frame and then into positive- and negative-sequence dq reference frames. The decoupled double synchronous algorithm extracts the positive- and negative-sequence components, and the improved V/F controllers generate the voltage commands \(U_d^*\) and \(U_q^*\). Finally, the SVPWM modulator produces the switching signals for the power semiconductor devices.
3. Mathematical Model and Coordinate Transformations
I begin with the general three-phase voltage vector \(\mathbf{u}=[u_a, u_b, u_c]^T\). Applying the Clarke transformation gives the voltage vector in the stationary αβ frame:
$$
\begin{bmatrix} u_\alpha \\[2mm] u_\beta \end{bmatrix} = C_{32} \begin{bmatrix} u_a \\[2mm] u_b \\[2mm] u_c \end{bmatrix}
$$
where the transformation matrix \(C_{32}\) is:
$$
C_{32} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\[2mm] 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}
$$
Next, the voltage vector in the αβ frame is transformed into a rotating dq frame:
$$
\begin{bmatrix} u_d \\[2mm] u_q \end{bmatrix} = C_{dq} \begin{bmatrix} u_\alpha \\[2mm] u_\beta \end{bmatrix}
$$
For the positive-sequence synchronous frame, the rotation angle is \(\theta_+ = \omega t\), and the transformation matrix is:
$$
C_{dq}^+ = \begin{bmatrix} \cos \theta_+ & \sin \theta_+ \\[2mm] -\sin \theta_+ & \cos \theta_+ \end{bmatrix}
$$
For the negative-sequence synchronous frame, the rotation angle is \(\theta_- = -\omega t\), and the transformation matrix is:
$$
C_{dq}^- = \begin{bmatrix} \cos \theta_- & -\sin \theta_- \\[2mm] \sin \theta_- & \cos \theta_- \end{bmatrix}
$$
Under unbalanced conditions, the voltage vector in stationary coordinates can be decomposed into a positive-sequence component rotating at \(+ω\) and a negative-sequence component rotating at \(-ω\). When viewed from the positive dq reference frame, the positive sequence appears as a DC quantity, while the negative sequence appears as a 2ω alternating component. Similarly, in the negative dq reference frame, the negative sequence appears as DC, while the positive sequence appears as a 2ω alternating component. I denote the actual positive-frame voltages as \(u_{d+}, u_{q+}\), the actual negative-frame voltages as \(u_{d-}, u_{q-}\), and the decoupled DC components as \(\bar{u}_{d+}, \bar{u}_{q+}, \bar{u}_{d-}, \bar{u}_{q-}\). Then the relationship can be written as:
$$
\begin{bmatrix} u_{d+} \\[2mm] u_{q+} \end{bmatrix} = \begin{bmatrix} \bar{u}_{d+} \\[2mm] \bar{u}_{q+} \end{bmatrix} + \mathbf{T}_{2\omega} \begin{bmatrix} \bar{u}_{d-} \\[2mm] \bar{u}_{q-} \end{bmatrix}
$$
$$
\begin{bmatrix} u_{d-} \\[2mm] u_{q-} \end{bmatrix} = \begin{bmatrix} \bar{u}_{d-} \\[2mm] \bar{u}_{q-} \end{bmatrix} + \mathbf{T}_{-2\omega} \begin{bmatrix} \bar{u}_{d+} \\[2mm] \bar{u}_{q+} \end{bmatrix}
$$
where the coupling matrices \(\mathbf{T}_{2\omega}\) and \(\mathbf{T}_{-2\omega}\) are:
$$
\mathbf{T}_{2\omega} = \begin{bmatrix} \cos 2\omega t & \sin 2\omega t \\[2mm] -\sin 2\omega t & \cos 2\omega t \end{bmatrix}, \quad \mathbf{T}_{-2\omega} = \begin{bmatrix} \cos 2\omega t & -\sin 2\omega t \\[2mm] \sin 2\omega t & \cos 2\omega t \end{bmatrix}
$$
These coupling terms must be eliminated before independent control of the positive- and negative-sequence voltages can be achieved. I use a low-pass filter \(F(s)\) to extract the average DC values and a feedforward cancellation network to remove the 2ω ripples. The decoupled estimates are calculated as:
$$
\begin{bmatrix} \hat{u}_{d+} \\[2mm] \hat{u}_{q+} \end{bmatrix} = F(s) \left( \begin{bmatrix} u_{d+} \\[2mm] u_{q+} \end{bmatrix} – \mathbf{T}_{2\omega} \begin{bmatrix} \hat{u}_{d-} \\[2mm] \hat{u}_{q-} \end{bmatrix} \right)
$$
$$
\begin{bmatrix} \hat{u}_{d-} \\[2mm] \hat{u}_{q-} \end{bmatrix} = F(s) \left( \begin{bmatrix} u_{d-} \\[2mm] u_{q-} \end{bmatrix} – \mathbf{T}_{-2\omega} \begin{bmatrix} \hat{u}_{d+} \\[2mm] \hat{u}_{q+} \end{bmatrix} \right)
$$
The low-pass filter has the following transfer function:
$$
LPF(s) = \frac{\omega_f}{s + \omega_f}
$$
In my implementation, I choose \(\omega_f = 222\) rad/s, which corresponds to a cut-off frequency of about 35 Hz. This value is high enough to obtain fast transient response while low enough to attenuate the 2ω (100 Hz) ripple effectively.
4. Decoupled Double Synchronous Frame Implementation
The double synchronous reference frame contains two rotating coordinate systems. The positive-system dq\(^+\) frame rotates counter-clockwise with angular velocity \(+ω\), and the negative-sequence dq\(^-\) frame rotates clockwise with angular velocity \(-ω\). Both frames rotate at the same absolute speed, but in opposite directions. By using the decoupling network described above, the positive- and negative-sequence components can be independently extracted from the asymmetric voltage vector.
I implement the decoupling network in both the voltage measurement path and the current measurement path. The voltage decoupling is used for the improved V/F controller, while the current decoupling helps to identify the positive- and negative-sequence load currents. The decoupled current components are also used for monitoring and protection purposes. The block diagram of the decoupling network is given in Figure 2 conceptually; it uses the estimated opposite-sequence value multiplied by the coupling matrix to subtract the ripple from the direct measurement.
For the positive-sequence frame, the output voltage model of the solar inverter with the filter inductance \(L\) and load resistance \(R_g\) is:
$$
L \frac{d}{dt} \begin{bmatrix} i_{d+} \\[2mm] i_{q+} \end{bmatrix} = \begin{bmatrix} U_{d+} \\[2mm] U_{q+} \end{bmatrix} – R_g \begin{bmatrix} i_{d+} \\[2mm] i_{q+} \end{bmatrix}
$$
Similarly, for the negative-sequence frame, I have:
$$
L \frac{d}{dt} \begin{bmatrix} i_{d-} \\[2mm] i_{q-} \end{bmatrix} = \begin{bmatrix} U_{d-} \\[2mm] U_{q-} \end{bmatrix} – R_g \begin{bmatrix} i_{d-} \\[2mm] i_{q-} \end{bmatrix}
$$
These equations are valid after the positive- and negative-sequence separation, because the coupling between sequences has been removed by the decoupling network. The cross-coupling between the d and q axes due to the rotating frames can be treated as disturbance and is compensated by the PI controllers in the V/F loops. In practice, I add feedforward compensation terms \(ω L i_{q}\) and \(-ω L i_{d}\) to improve the transient response. The full positive-sequence model then becomes:
$$
L \frac{d i_{d+}}{dt} = -R_g i_{d+} + \omega L i_{q+} + U_{d+}
$$
$$
L \frac{d i_{q+}}{dt} = -R_g i_{q+} – \omega L i_{d+} + U_{q+}
$$
and the full negative-sequence model is:
$$
L \frac{d i_{d-}}{dt} = -R_g i_{d-} – \omega L i_{q-} + U_{d-}
$$
$$
L \frac{d i_{q-}}{dt} = -R_g i_{q-} + \omega L i_{d-} + U_{q-}
$$
These models are used as the basis for designing the improved V/F controllers.
5. Improved V/F Control Strategy
The conventional V/F controller for an off-grid solar inverter regulates only the positive-sequence voltage. It sets the reference voltage \(U_{d}^{ref}=310\) V (phase amplitude) and \(U_q^{ref}=0\) V, and treats the measured \(U_d\) and \(U_q\) as if they were pure DC quantities. Under balanced loads, this is sufficient because the negative-sequence component is zero. However, under unbalanced loads, the measured \(U_d\) and \(U_q\) contain large 2ω ripples, and the negative-sequence component remains uncontrolled, leading to severe output voltage distortion.
To overcome this limitation, I propose an improved V/F controller with two independent branches: one for the positive-sequence component and one for the negative-sequence component. The positive-sequence V/F controller uses the same reference values as the conventional one:
$$
U_{d+}^{ref} = 310 \text{ V}, \quad U_{q+}^{ref} = 0 \text{ V}
$$
The negative-sequence V/F controller sets both reference values to zero because the goal is to eliminate the negative-sequence voltage:
$$
U_{d-}^{ref} = 0 \text{ V}, \quad U_{q-}^{ref} = 0 \text{ V}
$$
Each controller is a standard proportional-integral (PI) regulator. For example, the positive-sequence d-axis controller gives:
$$
U_{d+}^* = K_{p+} \left( U_{d+}^{ref} – \hat{u}_{d+} \right) + K_{i+} \int \left( U_{d+}^{ref} – \hat{u}_{d+} \right) dt
$$
and the negative-sequence d-axis controller gives:
$$
U_{d-}^* = K_{p-} \left( U_{d-}^{ref} – \hat{u}_{d-} \right) + K_{i-} \int \left( U_{d-}^{ref} – \hat{u}_{d-} \right) dt
$$
The same structure is used for the q-axis components. The outputs of the positive- and negative-sequence controllers are summed to form the final voltage commands:
$$
U_d^* = U_{d+}^* + U_{d-}^*
$$
$$
U_q^* = U_{q+}^* + U_{q-}^*
$$
These two signals are then transformed back to the αβ frame and fed into the SVPWM modulator. The SVPWM module calculates the switching states of the solar inverter power bridge. Because the controller explicitly drives the negative-sequence voltage to zero, the inverter output voltage is forced to be increasingly symmetric even when the load is unbalanced.
I summarize the control strategy in the following steps:
- Measure the instantaneous output phase voltages and load currents.
- Transform the measured quantities to the αβ stationary frame using the Clarke transform.
- Apply the positive- and negative-sequence Park transforms to obtain \(u_{d+}, u_{q+}, u_{d-}, u_{q-}\).
- Use the decoupled double synchronous network to obtain the filtered estimates \(\hat{u}_{d+}, \hat{u}_{q+}, \hat{u}_{d-}, \hat{u}_{q-}\).
- Compare the estimates with their references in the improved V/F controllers and compute \(U_{d+}^*, U_{q+}^*, U_{d-}^*, U_{q-}^*\).
- Add the positive- and negative-sequence controller outputs to obtain the total command voltage \(U_d^*, U_q^*\).
- Generate the SVPWM pulses.
This approach enables the solar inverter to operate robustly under unbalanced load conditions while maintaining the output voltage at the rated amplitude and frequency.
6. Simulation Setup and Parameters
I carried out the simulation in Matlab/Simulink using the SimPowerSystems toolbox. The complete model consists of a DC voltage source representing the PV side (through a DC-DC converter or battery), a three-phase two-level SVPWM inverter, an LC filter, and a dynamically switchable three-phase load. The control algorithms are implemented with discrete-time blocks at a sampling frequency of 12 kHz. The simulation time is set to 0.2 s. The main parameters of the solar inverter system are given in Table 1.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| DC bus voltage | \(U_{dc}\) | 750 | V |
| Filter inductance | \(L\) | 8 | mH |
| DC-link capacitance | \(C_{dc}\) | 2100 | μF |
| Switching frequency | \(f_{sw}\) | 12 | kHz |
| Sampling frequency | \(f_s\) | 12 | kHz |
| Rated phase voltage amplitude | \(U_{ph}\) | 310 | V |
| Rated frequency | \(f\) | 50 | Hz |
| Angular frequency | \(\omega\) | 314 | rad/s |
| LPF cut-off frequency | \(\omega_f\) | 222 | rad/s |
I define three load scenarios to investigate the behavior of the solar inverter:
| Time interval | Phase A resistance | Phase B resistance | Phase C resistance | Load condition |
|---|---|---|---|---|
| 0 s – 0.1 s | 50 Ω | 50 Ω | 50 Ω | Balanced |
| 0.1 s – 0.2 s | 40 Ω | 50 Ω | 60 Ω | Unbalanced |
At \(t=0.1\) s, a circuit breaker changes the phase resistances from the balanced values to the unbalanced values. This simulates the sudden connection of unsymmetrical single-phase loads in an off-grid solar inverter system.
7. Simulation Results and Analysis
7.1 Output Voltage Waveforms
First, I compare the output phase voltage waveforms of the proposed control strategy with those of the conventional V/F control. In the balanced interval from 0 to 0.1 s, both controllers produce stable, symmetric phase voltages with an amplitude of 310 V and a frequency of 50 Hz. The waveforms are nearly identical. At \(t=0.1\) s, when the load becomes unbalanced, the conventional controller produces a severely distorted voltage waveform. The amplitude is not maintained and the waveform contains visible notches. In contrast, the proposed decoupled double synchronization plus improved V/F controller restores the output voltage to a balanced, sinusoidal waveform within about 30 ms. After the transient, the phase voltage amplitude returns to 310 V and the waveform is highly sinusoidal.
7.2 Positive- and Negative-Sequence Voltage Components
I also recorded the dq-axis voltage components in both synchronous frames. Before the load change, the positive-sequence d-axis voltage is 310 V, the positive-sequence q-axis voltage is 0 V, and both negative-sequence dq voltages are 0 V. This indicates that the output voltage contains only the positive-sequence component. After the load becomes unbalanced, the negative-sequence dq voltages briefly jump to non-zero values because of the sudden asymmetry. The proposed controller acts quickly to cancel these components. After about 0.03 s, the negative-sequence dq voltages return to zero, and the positive-sequence dq voltages settle back to their references. This observation confirms that the improved V/F controller successfully forces the negative-sequence voltage to zero.
7.3 Load Current Behavior
The load currents are also decomposed into positive- and negative-sequence components. In the balanced interval, the negative-sequence dq currents are zero. After the load becomes unbalanced, the phase resistances are unequal, so the load currents are inherently unbalanced. The negative-sequence dq currents become non-zero and remain non-zero throughout the unbalanced interval because the load itself is unbalanced. The positive-sequence currents are still regulated so that the voltage remains constant. This behavior is expected: the solar inverter provides a balanced voltage, but the load current is determined by the load impedances.
7.4 Harmonic Analysis
I used the FFT analysis tool in Matlab/Simulink to compute the total harmonic distortion of the A-phase voltage. In the balanced load interval, the THD is as low as 1.10%. After switching to the unbalanced load and after the transient settles, the THD of the proposed controller is 2.07%. This value remains below the common design limit of 3%. The conventional V/F controller, on the other hand, produces a THD that is significantly higher than 3%, making the output unacceptable for many loads. Therefore, the proposed control strategy effectively reduces harmonic distortion caused by unbalanced loads in the solar inverter.
Table 3 summarizes the key performance indices of the proposed control strategy under unbalanced load conditions.
| Quantity | Balanced load (0–0.1 s) | Unbalanced load (after 0.13 s) |
|---|---|---|
| A-phase voltage THD | 1.10% | 2.07% |
| Positive-sequence d-axis voltage | 310 V | 310 V |
| Positive-sequence q-axis voltage | 0 V | 0 V |
| Negative-sequence d-axis voltage | 0 V | 0 V |
| Negative-sequence q-axis voltage | 0 V | 0 V |
| Output frequency | 50 Hz | 50 Hz |
| Recovery time after load change | — | ≈30 ms |
7.5 Frequency Stability
The output voltage frequency is another important metric for an off-grid solar inverter. I measured the instantaneous frequency of the output voltage. During the balanced interval, the frequency is exactly 50 Hz. At the moment of the load change, the frequency experiences a small transient deviation because the sudden unbalance causes a temporary asymmetry in the voltage vector. However, the controller damps this deviation quickly, and the frequency returns to 50 Hz after about 0.03 s. This result demonstrates that the proposed control strategy maintains frequency stability even under severe load imbalance.
7.6 Dynamic Response
To evaluate the dynamic behavior of the proposed solar inverter controller, I examined the transient response of the voltage envelope during the load switching event. The voltage envelope shows a small overshoot and a brief oscillatory transient that disappears within two to three fundamental cycles. The settling time is approximately 30 ms, which corresponds to about 1.5 cycles at 50 Hz. This is a satisfactory dynamic response for a solar inverter in island mode. The decoupled double synchronous filter with \(\omega_f=222\) rad/s provides fast enough tracking without introducing excessive ripple.
8. Discussion
The simulation results reveal the effectiveness of the proposed control strategy. The key improvement is the addition of a negative-sequence V/F controller. In the conventional V/F method, the negative-sequence voltage component is completely ignored. The measured dq voltages contain a 100 Hz ripple caused by the negative sequence. The PI controller cannot eliminate this ripple, so the output voltage remains unbalanced and distorted. In my proposed method, the decoupling network splits the positive and negative sequences and removes the 2ω coupling. Then, the negative-sequence controller actively drives the negative-sequence voltage to zero. This is analogous to a feedforward compensation of the unbalance disturbance.
One important practical consideration is the performance of the low-pass filter in the decoupling network. If \(\omega_f\) is too low, the dynamic response becomes sluggish. If \(\omega_f\) is too high, the 2ω ripple is not sufficiently attenuated. My choice of \(\omega_f=222\) rad/s is a reasonable trade-off. In a digital implementation, the filter is discretized with the Tustin method or another suitable approximation.
Another consideration is the interaction between the positive- and negative-sequence controllers. Because the decoupling network cancels the cross-coupling, the two controllers can be designed independently. The positive-sequence controller mainly determines the voltage amplitude and frequency, while the negative-sequence controller is responsible for unbalance mitigation. This decoupled design greatly simplifies the tuning process.
I also note that the proposed control strategy does not require any additional hardware. It only modifies the control algorithm of the solar inverter, making it an attractive solution for existing systems. The computational burden is modest: the coordinate transformations, decoupling calculations, and two additional PI controllers are easily executed by a modern DSP or microcontroller.
9. Comparison with Conventional Control
For completeness, I present a comparison between the conventional V/F control and the proposed improved V/F control. The comparison is qualitative because the conventional controller produces highly distorted waveforms under unbalanced load.
| Aspect | Conventional V/F | Proposed decoupled double synchronization + improved V/F |
|---|---|---|
| Positive-sequence control | Yes | Yes |
| Negative-sequence control | No | Yes |
| Handling of unbalanced load | Poor | Good |
| THD under unbalanced load | High (above 3%) | 2.07% |
| Voltage amplitude regulation | Lost | Maintained at 310 V |
| Frequency regulation | Distorted | Maintained at 50 Hz |
| Control complexity | Low | Moderate |
| Computation requirement | Low | Acceptable |
From a practical standpoint, the increase in complexity is justified by the significant improvement in power quality. The solar inverter can continue to feed sensitive loads even when the unbalanced condition persists for a long time, which is a major advantage for off-grid photovoltaic applications.
10. Conclusion
In this work, I presented a control strategy for an off-grid solar inverter based on decoupled double synchronous reference frames and an improved V/F method. The proposed strategy separates the positive- and negative-sequence components of the inverter output voltage, decouples the 2ω coupling terms, and applies separate V/F controllers to each sequence. The positive-sequence controller regulates the output voltage to the rated amplitude and frequency, while the negative-sequence controller forces the negative-sequence voltage to zero, thereby eliminating the harmful effects of unbalanced loads.
Simulation results in Matlab/Simulink confirm the feasibility and effectiveness of the proposed method. Under balanced loads, the output voltage has a THD of 1.10%. When the load suddenly becomes unbalanced, the proposed solar inverter controller restores a balanced sinusoidal voltage within about 30 ms, and the THD remains under 2.07%, which is below the standard 3% limit. The output frequency stays at 50 Hz, and the negative-sequence voltage is almost completely suppressed. The conventional V/F controller cannot achieve these results under the same conditions.
Future work will focus on improving the steady-state tracking accuracy of the PI controllers. The PI controller cannot perfectly track sinusoidal disturbances without a resonant term. I plan to investigate a composite controller that combines a repetitive controller with the PI controller to achieve zero steady-state error. In addition, I will extend the proposed method to a solar inverter with LCL filters and examine its performance under nonlinear loads. The ultimate goal is to develop a robust, high-performance control solution for off-grid solar inverters that can handle a wide range of load conditions with excellent power quality.