In recent years, the increasing depletion of fossil fuels and growing environmental awareness have shifted significant attention toward renewable energy sources. Among the critical components in renewable energy systems, single-phase DC/AC inverters play a pivotal role in applications such as rooftop photovoltaic installations. The performance and efficiency of these single-phase inverters heavily depend on their control strategies. Traditional control methods for single-phase inverters, which are predominantly based on stationary coordinate systems, often face limitations. For instance, conventional PI controllers fail to achieve precise tracking of sinusoidal signals, while proportional resonant (PR) regulators introduce complexity and operational constraints. Furthermore, transitioning between islanded and grid-connected modes poses considerable challenges in stationary frames. To address these issues, this article explores control strategies for single-phase inverters within the dq rotating coordinate system, drawing inspiration from well-established techniques used in three-phase inverter systems.
The primary advantage of employing the dq coordinate system lies in its ability to transform time-varying AC quantities into constant DC values. This transformation simplifies the control design, allowing the use of straightforward PI controllers for accurate regulation without steady-state error. In this work, we develop a comprehensive mathematical model for single-phase inverters in the dq frame, propose control strategies for islanded operation, pre-synchronization, and grid-connected operation, and validate these strategies through simulation and experimental results. The findings demonstrate that the single-phase inverter can output a standard 50 Hz sinusoidal wave in islanded mode, achieve phase synchronization with the grid during pre-synchronization, and seamlessly transition to PQ control when connected to the grid.
Mathematical Modeling of Single-Phase Inverters in the dq Coordinate System
To effectively control a single-phase inverter in the dq rotating coordinate system, it is essential to first establish an accurate mathematical model. The typical topology of a single-phase inverter includes an LC filter, which is suitable for both islanded and grid-connected operations. Using state-space averaging techniques, the average model of the single-phase inverter can be derived. The state-space equations in the stationary frame are given by:
$$ \frac{du_o}{dt} = \frac{1}{C} (i_L – i_o) $$
$$ \frac{di_L}{dt} = \frac{1}{L} (v_o – u_o) $$
where \( u_o \) is the output voltage, \( i_L \) is the inductor current, \( i_o \) is the output current, \( v_o \) is the inverter bridge output voltage, \( L \) is the filter inductance, and \( C \) is the filter capacitance.
However, to facilitate control in the dq frame, it is necessary to generate an orthogonal virtual quantity that complements the actual single-phase variable. This virtual quantity is essential for performing the coordinate transformation. Several methods exist for generating this orthogonal component, including direct delay, Park inverse transformation, and second-order generalized integrator (SOGI) techniques. In this study, the Park inverse transformation method is employed due to its effective filtering of high-frequency harmonics. The transformation from the stationary αβ frame to the rotating dq frame is defined by the matrix:
$$ T_{\alpha\beta\to dq} = \begin{bmatrix} \cos(\omega t) & \sin(\omega t) \\ -\sin(\omega t) & \cos(\omega t) \end{bmatrix} $$
Assuming the actual output voltage is \( u_\alpha = U_m \cos(\omega_0 t + \phi) \), the orthogonal virtual voltage is derived as \( u_\beta = U_m \cos(\omega_0 t + \phi – 90^\circ) \). Applying the transformation, the state equations in the dq frame become:
$$ \frac{d}{dt} \begin{bmatrix} u_{od} \\ u_{oq} \end{bmatrix} = \begin{bmatrix} -\frac{1}{RC} & \omega \\ -\omega & -\frac{1}{RC} \end{bmatrix} \begin{bmatrix} u_{od} \\ u_{oq} \end{bmatrix} + \begin{bmatrix} \frac{1}{C} & 0 \\ 0 & \frac{1}{C} \end{bmatrix} \begin{bmatrix} i_{Ld} \\ i_{Lq} \end{bmatrix} $$
$$ \frac{d}{dt} \begin{bmatrix} i_{Ld} \\ i_{Lq} \end{bmatrix} = \begin{bmatrix} 0 & \omega \\ -\omega & 0 \end{bmatrix} \begin{bmatrix} i_{Ld} \\ i_{Lq} \end{bmatrix} + \begin{bmatrix} -\frac{1}{L} & 0 \\ 0 & -\frac{1}{L} \end{bmatrix} \begin{bmatrix} u_{od} \\ u_{oq} \end{bmatrix} + \begin{bmatrix} \frac{1}{L} & 0 \\ 0 & \frac{1}{L} \end{bmatrix} \begin{bmatrix} v_{od} \\ v_{oq} \end{bmatrix} $$
These equations reveal coupling between the d and q axes, which necessitates decoupling for independent control. Adopting a feedforward decoupling strategy similar to that used in three-phase systems, the control laws for the current inner loop and voltage outer loop are designed as follows:
Current inner loop control:
$$ v_d^* = K_{Ip} (i_{Ld}^{ref} – i_{Ld}) – \omega L i_{Lq} + u_{od} $$
$$ v_q^* = K_{Ip} (i_{Lq}^{ref} – i_{Lq}) + \omega L i_{Ld} + u_{oq} $$
Voltage outer loop control:
$$ i_{Ld}^{ref} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_d^{ref} – u_{od}) – \omega C u_{oq} $$
$$ i_{Lq}^{ref} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_q^{ref} – u_{oq}) + \omega C u_{od} $$
After decoupling, the system simplifies to:
$$ C \frac{du_{od}}{dt} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_d^{ref} – u_{od}) $$
$$ C \frac{du_{oq}}{dt} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_q^{ref} – u_{oq}) $$
$$ L \frac{di_{Ld}}{dt} = K_{Ip} (i_{Ld}^{ref} – i_{Ld}) $$
$$ L \frac{di_{Lq}}{dt} = K_{Ip} (i_{Lq}^{ref} – i_{Lq}) $$
This decoupled model allows for the use of PI controllers to regulate the d and q components independently, significantly simplifying the control design for the single-phase inverter.
Islanded Operation Control Strategy
In islanded operation, the single-phase inverter must provide stable voltage and frequency support to the local load, typically achieved through V/f control. Traditional methods such as open-loop SPWM, voltage PID control, or voltage-current double-loop control in the stationary frame suffer from poor waveform quality, sensitivity to load changes, and inability to achieve zero steady-state error. By contrast, the dq transformation converts AC quantities into DC values, enabling precise control with PI regulators.
The control structure for islanded operation involves a double-loop strategy with voltage and current controllers in the dq frame. The voltage references are set as \( u_d^{ref} = U_m \) and \( u_q^{ref} = 0 \), corresponding to a sinusoidal output voltage with amplitude \( U_m \) and frequency \( \omega_0 \). The phase angle for the transformation is directly provided by the control system as \( \theta_{ref} = \omega_0 t \). Under steady-state conditions, the d-axis voltage stabilizes at \( U_m \), and the q-axis voltage at zero, ensuring a high-quality sinusoidal output.
To validate this approach, simulations and experiments were conducted. The system parameters are summarized in the table below:
| Parameter | Simulation Value | Experimental Value |
|---|---|---|
| DC Link Voltage \( U_{dc} \) | 400 V | 60 V |
| Filter Inductance \( L \) | 1 mH | 1 mH |
| Filter Capacitance \( C \) | 30 μF | 30 μF |
| Load Resistance \( R \) | 2 Ω | 10 Ω |
| Output Voltage Amplitude \( U_m \) | 311 V | 50 V |
| Switching Frequency | 10 kHz | 10 kHz |
Both simulation and experimental results confirmed that the single-phase inverter outputs a pure sinusoidal voltage at 50 Hz with the specified amplitude. The output current, in phase with the voltage for resistive loads, demonstrated the effectiveness of the V/f control in the dq frame.
Pre-Synchronization Control for Grid Connection
Before connecting the single-phase inverter to the grid, it is crucial to synchronize the inverter’s output voltage with the grid voltage in terms of phase, frequency, and amplitude. In the dq coordinate system, pre-synchronization is achieved by aligning the phase angle of the inverter with that of the grid. The control strategy involves switching from the internal phase reference to the grid phase reference when the pre-synchronization signal is activated.
The phase control method is illustrated in the block diagram below. During islanded operation, the switch is in position “0”, and the inverter operates with its internal reference angle \( \theta_{ref} \). When pre-synchronization is initiated, the switch moves to position “1”, and the inverter’s phase angle tracks the grid phase angle \( \theta_{grid} \). The phase-locked loop (PLL) technique is commonly used to extract \( \theta_{grid} \) from the grid voltage. The dynamics of the phase synchronization can be modeled as:
$$ \frac{d\theta}{dt} = K_p (\theta_{grid} – \theta) + K_i \int (\theta_{grid} – \theta) dt $$
where \( K_p \) and \( K_i \) are the PI controller gains for the PLL.
Simulation and experimental tests were performed to verify the pre-synchronization process. Initially, the grid voltage led the inverter output voltage by a phase difference. Upon activating the pre-synchronization command, the inverter’s phase gradually aligned with the grid, achieving zero phase difference within a short transient period. The results demonstrated the robustness and accuracy of the dq-based synchronization method, ensuring smooth transition to grid-connected operation.
Grid-Connected Operation Control Strategy
Once synchronized, the single-phase inverter can be connected to the grid and switched to PQ control mode. In this mode, the inverter acts as a controlled current source, injecting active and reactive power into the grid. The key advantage of the dq transformation is the decoupled control of active and reactive power through the d and q current components, respectively.
Based on instantaneous power theory, the active power \( P \) and reactive power \( Q \) at the inverter terminals are related to the dq components as follows. Let the grid voltage be \( u_s = U_m \cos(\omega t + \phi) \) and the inverter output current be \( i_o = I_m \cos(\omega t) \). The orthogonal virtual quantities are generated, and the dq transformation yields:
$$ i_d = I_m \cos \phi $$
$$ i_q = -I_m \sin \phi $$
The instantaneous active and reactive powers are then:
$$ P = \frac{1}{2} U_m i_d $$
$$ Q = -\frac{1}{2} U_m i_q $$
Thus, the reference currents for the d and q axes are:
$$ i_d^{ref} = \frac{2P^{ref}}{U_m} $$
$$ i_q^{ref} = -\frac{2Q^{ref}}{U_m} $$
The control structure for grid-connected operation includes current loops for \( i_d \) and \( i_q \), with PI regulators ensuring accurate tracking. The decoupling terms compensate for the cross-coupling between axes, as described in the mathematical model. The overall control block diagram for grid-connected mode is shown below, highlighting the seamless transition from voltage control to current control.
To evaluate the PQ control strategy, tests were conducted under different power setpoints. The table below outlines the test conditions:
| Scenario | Load Power | Inverter Active Power \( P \) | Inverter Reactive Power \( Q \) |
|---|---|---|---|
| Simulation 1 | 300 W | 250 W | 0 var |
| Simulation 2 | 300 W | 50 W | 0 var |
| Experiment 1 | 140 W | 105 W | 0 var |
| Experiment 2 | 140 W | 35 W | 0 var |
In all cases, the single-phase inverter accurately tracked the power references, with the grid supplying or absorbing the difference between the load demand and inverter output. The current waveforms remained sinusoidal and in phase with the grid voltage for unity power factor operation, confirming the effectiveness of the dq-based PQ control.
Simulation and Experimental Validation
The proposed control strategies for the single-phase inverter were thoroughly validated through both simulation and experimental studies. The simulation platform was built using PSCAD/EMTDC, while the hardware prototype comprised a DSP TMS320F28335 controller, power switches, LC filters, and grid interface components. The system parameters were consistent with those listed in the previous sections.
For islanded operation, the inverter maintained a stable 50 Hz sinusoidal output voltage under varying load conditions. The transition to pre-synchronization was smooth, with the inverter phase locking to the grid within a few cycles. In grid-connected mode, the inverter successfully regulated active and reactive power as per the references, with fast dynamic response to changes in power setpoints.
The experimental results closely matched the simulations, demonstrating the practicality of the dq-based control approach. The following table summarizes the performance metrics obtained from the tests:
| Operation Mode | Output Voltage THD | Phase Synchronization Time | Power Tracking Error |
|---|---|---|---|
| Islanded | < 3% | N/A | N/A |
| Pre-Synchronization | N/A | < 100 ms | N/A |
| Grid-Connected | N/A | N/A | < 2% |
These results affirm the feasibility and correctness of the proposed methods, highlighting the superiority of the dq coordinate system in simplifying control design for single-phase inverters.
Conclusion
This article has presented a comprehensive study on control strategies for single-phase DC/AC inverters in the dq rotating coordinate system. By leveraging the mathematical model derived in the dq frame, we have developed and validated control schemes for islanded operation, pre-synchronization, and grid-connected operation. The key contribution lies in the simplification of control design through the use of PI regulators for DC quantities, eliminating the need for complex resonant controllers or intricate stationary frame techniques.
The single-phase inverter achieves high-performance output in islanded mode, seamless phase synchronization with the grid, and accurate power control in grid-connected mode. The simulation and experimental results consistently demonstrate the effectiveness of the proposed strategies. However, this study primarily focuses on functional validation; future work will address performance optimization during mode transitions, including minimization of voltage and current overshoot, frequency deviations, and transition times. Overall, the dq-based approach offers a robust and simplified framework for advanced control of single-phase inverters in renewable energy applications.