In the realm of renewable energy systems, solar inverters play a critical role in converting DC power from photovoltaic panels into AC power for grid-connected or off-grid applications. Among the various types of solar inverters, off-grid inverters are essential for standalone systems, such as those in remote areas or microgrids, where maintaining output voltage quality is paramount. One significant challenge faced by three-phase off-grid inverters is voltage asymmetry under unbalanced load conditions, which can lead to equipment damage, reduced efficiency, and system instability. This issue is particularly relevant for certain types of solar inverter configurations, including those used in hybrid systems that combine solar power with battery storage. Unbalanced loads, such as uneven distribution of single-phase loads across three phases, cause negative-sequence components in the output voltage, degrading performance. Traditional control methods, like single-sequence double-loop control, often fail to address negative-sequence voltages, resulting in asymmetric outputs. In this article, we explore a dual-sequence control strategy that effectively mitigates voltage asymmetry by incorporating negative-sequence decoupling, enhancing the reliability of off-grid solar inverters. We will derive the necessary mathematical models, present control diagrams, and validate the approach through simulations, while emphasizing the importance of understanding different types of solar inverter topologies for optimal performance.
To begin, let’s consider the main circuit topology of a typical three-phase off-grid inverter, which forms the basis for many types of solar inverter systems. The inverter includes a DC input source, output filter inductors with parasitic resistances, star-connected filter capacitors, and a three-phase load without a neutral connection. The output voltages and currents are governed by differential equations that, under balanced conditions, can be transformed into a synchronous reference frame (dq-axis) for simplified control. However, under unbalanced loads, symmetric component theory reveals that the output comprises positive-sequence, negative-sequence, and zero-sequence components. Since the topology lacks a neutral wire, the zero-sequence voltage is negligible. The positive-sequence components rotate counterclockwise at the fundamental frequency, while the negative-sequence components rotate clockwise. Conventional single-sequence control focuses solely on positive-sequence voltages, using dq-transform with a rotation angle of $\theta^+ = \omega_e t$ (where $\omega_e$ is the positive-sequence angular frequency) to convert AC quantities into DC components for proportional-integral (PI) controller-based regulation. The voltage and current equations in the positive-sequence dq-frame exhibit cross-coupling between the d and q axes, necessitating decoupling strategies for effective control.
For instance, the positive-sequence output voltage equation in the dq-frame is given by:
$$ L_f \frac{d}{dt} \begin{bmatrix} i_{od}^+ \\ i_{oq}^+ \end{bmatrix} = \begin{bmatrix} u_{id}^+ \\ u_{iq}^+ \end{bmatrix} – \begin{bmatrix} u_{od}^+ \\ u_{oq}^+ \end{bmatrix} – r \begin{bmatrix} i_{od}^+ \\ i_{oq}^+ \end{bmatrix} + \omega_e L_f \begin{bmatrix} -i_{oq}^+ \\ i_{od}^+ \end{bmatrix} $$
and the current equation is:
$$ C_f \frac{d}{dt} \begin{bmatrix} u_{od}^+ \\ u_{oq}^+ \end{bmatrix} = \begin{bmatrix} i_{od}^+ \\ i_{oq}^+ \end{bmatrix} – \begin{bmatrix} i_{Ld}^+ \\ i_{Lq}^+ \end{bmatrix} + \omega_e C_f \begin{bmatrix} -u_{oq}^+ \\ u_{od}^+ \end{bmatrix} $$
where $i_{od}^+$ and $i_{oq}^+$ are the d- and q-axis components of the inductor current, $u_{od}^+$ and $u_{oq}^+$ are the output voltage components, $u_{id}^+$ and $u_{iq}^+$ are the inverter bridge voltage components, and $i_{Ld}^+$ and $i_{Lq}^+$ are the load current components. Decoupling is achieved by introducing compensation terms, such as $\omega_e L_f i_{oq}^+$ for the d-axis and $-\omega_e L_f i_{od}^+$ for the q-axis, allowing independent PI control loops for voltage and current regulation. The positive-sequence control block diagram typically includes outer voltage loops and inner current loops with PI controllers, tuned to ensure stability and fast tracking of reference signals. However, this approach ignores negative-sequence voltages, leading to asymmetry when loads are unbalanced.
In contrast, the dual-sequence control method addresses this limitation by incorporating negative-sequence regulation. The negative-sequence components can be transformed into a negative-sequence dq-frame using a rotation angle of $\theta^- = -\omega_e t$, resulting in DC quantities that facilitate control. The negative-sequence voltage equation in the dq-frame is derived as:
$$ L_f \frac{d}{dt} \begin{bmatrix} i_{od}^- \\ i_{oq}^- \end{bmatrix} = \begin{bmatrix} u_{id}^- \\ u_{iq}^- \end{bmatrix} – \begin{bmatrix} u_{od}^- \\ u_{oq}^- \end{bmatrix} – r \begin{bmatrix} i_{od}^- \\ i_{oq}^- \end{bmatrix} – \omega_e L_f \begin{bmatrix} -i_{oq}^- \\ i_{od}^- \end{bmatrix} $$
and the current equation is:
$$ C_f \frac{d}{dt} \begin{bmatrix} u_{od}^- \\ u_{oq}^- \end{bmatrix} = \begin{bmatrix} i_{od}^- \\ i_{oq}^- \end{bmatrix} – \begin{bmatrix} i_{Ld}^- \\ i_{Lq}^- \end{bmatrix} – \omega_e C_f \begin{bmatrix} -u_{oq}^- \\ u_{od}^- \end{bmatrix} $$
Here, the superscript “-” denotes negative-sequence components. Similar to the positive-sequence case, cross-coupling exists and requires decoupling through compensation terms, such as $-\omega_e L_f i_{oq}^-$ for the d-axis and $\omega_e L_f i_{od}^-$ for the q-axis. By adding negative-sequence control branches, the overall system can regulate both positive- and negative-sequence voltages independently. Key to this approach is the use of notch filters to eliminate double-frequency components (e.g., 100 Hz for a 50 Hz system) that arise from the interaction between positive and negative sequences in the dq-transform. The notch filter transfer function is expressed as:
$$ N(s) = \frac{s^2 + \omega_n^2}{s^2 + \frac{\omega_n}{Q}s + \omega_n^2} $$
where $\omega_n$ is the notch frequency (e.g., $2\pi \times 100$ rad/s) and $Q$ is the quality factor, chosen to balance filtering effectiveness and frequency adaptability (e.g., $Q = 0.707$).
The dual-sequence control strategy involves transforming measured output voltages and currents into both positive- and negative-sequence dq-frames, applying notch filters, and then using separate PI controllers for each sequence. The outputs are combined through inverse transforms to generate modulation signals for pulse-width modulation (PWM). This method ensures that negative-sequence voltages are suppressed, maintaining symmetric output voltages even under severe load imbalances. It is particularly beneficial for various types of solar inverter systems, including hybrid inverters that integrate battery storage, as they often face dynamic load variations.
To illustrate the performance of different control methods, we conducted simulations using MATLAB/Simulink, comparing conventional single-sequence control, proportional-resonant (PR) control in the stationary frame, and the proposed dual-sequence control. The system parameters are summarized in the table below, which highlights key aspects relevant to various types of solar inverter designs, such as off-grid and hybrid configurations.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rated Power ($P_N$) | 15 kW | Switching Frequency ($f_s$) | 10 kHz |
| DC Input Voltage ($U_{dc}$) | 700 V | Filter Inductance ($L_f$) | 4 mH |
| Filter Capacitance ($C_f$) | 10 μF | Parasitic Resistance ($r$) | 0.01 Ω |
| Voltage PI Gains ($K_{vp}$, $K_{vi}$) | 0.13, 1.5 | Current PI Gains ($K_{ip}$, $K_{ii}$) | 0.0028, 0.150 |
| Fundamental Frequency ($\omega_e$) | 100π rad/s | Notch Filter $Q$ | 0.707 |
In the simulations, the inverter was subjected to an unbalanced load with phase A at rated resistance (9.67 Ω), phase B at light load (13 Ω), and phase C at heavy load (6 Ω). Under conventional single-sequence control, the output voltages exhibited significant asymmetry, with a negative-sequence voltage RMS value of approximately 9 V, resulting in a voltage unbalance factor of 3%, which exceeds the IEEE standard of 2%. PR control, while better, still showed a negative-sequence voltage of 3.5 V and an unbalance factor of 1.9%. In contrast, the dual-sequence control reduced the negative-sequence voltage to nearly 0 V, achieving an unbalance factor of 0.1%, demonstrating superior performance. Additionally, during transient events such as load switching or fault conditions (e.g., single-phase short-circuit), the dual-sequence control maintained voltage symmetry with minimal disturbance, whereas PR control struggled with higher unbalance factors up to 5%.
The effectiveness of the dual-sequence control can be further analyzed through the closed-loop transfer functions. For the positive-sequence current loop, after decoupling, the control block diagram simplifies to a first-order system with a transfer function $G_i(s) = \frac{K_{PWM}}{1 + s T_i}$, where $K_{PWM}$ is the PWM gain and $T_i$ is the time constant. Similarly, the voltage loop ensures rapid tracking of reference signals. The negative-sequence loops mirror this structure, with appropriate adjustments for the reversed rotation direction. The overall system stability is enhanced by the independent regulation of sequences, which is crucial for the diverse types of solar inverter applications, including those in harsh environments where load imbalances are common.
In terms of implementation, the dual-sequence control requires additional computational resources for the dual dq-transforms and notch filters, but it offers significant advantages in power quality. For instance, in hybrid solar inverters that combine PV generation with battery storage, such as the one depicted in the image, maintaining voltage symmetry under fluctuating loads is essential for prolonging battery life and ensuring reliable operation. The dual-sequence method also outperforms PR control in fault ride-through scenarios, as it does not rely solely on resonant controllers that may fail under severe asymmetries.
To summarize, the dual-sequence control strategy represents a robust solution for off-grid solar inverters facing unbalanced loads. By deriving the negative-sequence coupling equations and implementing decoupled double-loop control, this approach effectively suppresses voltage asymmetry, aligning with the requirements for high-performance types of solar inverter systems. Future work could explore adaptive tuning of PI gains or integration with advanced topologies, such as three-phase four-leg inverters, to further enhance resilience. As solar energy adoption grows, innovations in inverter control will continue to play a vital role in ensuring grid stability and power quality.
In conclusion, we have presented a comprehensive analysis of dual-sequence control for off-grid solar inverters, highlighting its superiority over conventional methods through mathematical models, control diagrams, and simulation results. This discussion underscores the importance of selecting appropriate control strategies for different types of solar inverter setups, particularly in off-grid and hybrid configurations where load imbalances are prevalent. By embracing such advanced techniques, we can improve the reliability and efficiency of renewable energy systems, contributing to a sustainable energy future.