In recent years, the increasing depletion of fossil fuels and growing environmental awareness have shifted focus towards renewable energy sources. Among these, the single phase inverter plays a critical role in applications such as photovoltaic and wind power systems. Traditional control strategies for single phase inverters are often based on stationary coordinate systems, which present limitations in achieving precise control. For instance, conventional PI controllers fail to eliminate steady-state errors for sinusoidal signals, while proportional resonant (PR) regulators introduce complexity and operational constraints. Moreover, transitioning between islanded and grid-connected operations in stationary coordinates is challenging. To address these issues, this paper explores the control of single phase inverters in the rotating dq coordinate system, drawing inspiration from three-phase inverter systems. By transforming time-varying AC quantities into constant DC components through coordinate transformation, the single phase inverter can utilize simple PI controllers for accurate tracking without the need for intricate designs. This approach significantly simplifies control strategies for various operational modes, including islanded operation, pre-synchronization, and grid-connected operation.
The foundation of this study lies in establishing a mathematical model for the single phase inverter in the dq coordinate system. The average model of the single phase inverter, considering an LC filter topology, is derived using state-space averaging techniques. The dynamics are described by the following equations:
$$ C \frac{du_o}{dt} = i_L – i_o $$
$$ L \frac{di_L}{dt} = v_o – u_o $$
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, \( C \) is the filter capacitance, and \( L \) is the filter inductance. To facilitate transformation into the dq coordinate system, an orthogonal virtual quantity is generated, often using methods such as direct delay, Park inverse transformation, or second-order generalized integrators. The Park inverse transformation method is employed here for its filtering capabilities, which attenuate high-frequency harmonics. The transformation from the stationary αβ coordinate system to the rotating dq coordinate system is given by:
$$ T_{\alpha\beta\to dq} = \begin{pmatrix} \cos \omega t & \sin \omega t \\ -\sin \omega t & \cos \omega t \end{pmatrix} $$
This transformation converts the sinusoidal quantities into DC components, enabling the use of PI controllers for zero steady-state error. The state equations in the dq coordinate system are derived as follows:
$$ \frac{d}{dt} \begin{pmatrix} u_{od} \\ u_{oq} \end{pmatrix} = \begin{pmatrix} -\frac{1}{RC} & \omega & \frac{1}{C} & 0 \\ -\omega & -\frac{1}{RC} & 0 & \frac{1}{C} \end{pmatrix} \begin{pmatrix} u_{od} \\ u_{oq} \\ i_{Ld} \\ i_{Lq} \end{pmatrix} $$
$$ \frac{d}{dt} \begin{pmatrix} i_{Ld} \\ i_{Lq} \end{pmatrix} = \begin{pmatrix} -\frac{1}{L} & 0 & 0 & \omega \\ 0 & -\frac{1}{L} & -\omega & 0 \end{pmatrix} \begin{pmatrix} u_{od} \\ u_{oq} \\ i_{Ld} \\ i_{Lq} \end{pmatrix} + \begin{pmatrix} v_{od} \\ v_{oq} \end{pmatrix} $$
These equations reveal coupling between the d-axis and q-axis components, necessitating decoupling control. By employing feedforward decoupling, the system can be simplified, allowing independent control of d and q axes. The current inner loop and voltage outer loop controllers are designed as:
$$ v_d^* = K_{Ip} (i_{Ld}^{ref} – i_{Ld}) – \omega L i_{Lq} + u_{od} $$
$$ v_q^* = K_{Ip} (i_{Lq}^{ref} – i_{Lq}) + \omega L i_{Ld} + u_{oq} $$
and
$$ i_{Ld}^{ref} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_d^{ref} – u_{od}) – \omega C u_{oq} $$
$$ i_{Lq}^{ref} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_q^{ref} – u_{oq}) + \omega C u_{od} $$
After decoupling, the system dynamics become:
$$ C \frac{du_{od}}{dt} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_d^{ref} – u_{od}) $$
$$ C \frac{du_{oq}}{dt} = \left(K_{Up} + \frac{K_{Ui}}{s}\right) (u_q^{ref} – u_{oq}) $$
$$ L \frac{di_{Ld}}{dt} = K_{Ip} (i_{Ld}^{ref} – i_{Ld}) $$
$$ L \frac{di_{Lq}}{dt} = K_{Ip} (i_{Lq}^{ref} – i_{Lq}) $$
This decoupled model forms the basis for implementing various control strategies for the single phase inverter.
In islanded operation, the single phase inverter employs V/f control to provide voltage and frequency support. The control strategy involves setting the d-axis voltage reference to the desired amplitude and the q-axis reference to zero, ensuring a standard sinusoidal output. For example, with \( u_d^{ref} = 311 \, \text{V} \) and \( u_q^{ref} = 0 \, \text{V} \), the inverter outputs a 220 V RMS voltage at 50 Hz. The effectiveness of this approach is validated through simulations and experimental tests on a hardware platform using a DSP controller. The results demonstrate that the single phase inverter maintains stable output under resistive loads, with voltage and current waveforms aligning as expected.
For grid connection, pre-synchronization is essential to match the phase of the inverter output voltage with the grid voltage. The control system switches from V/f control to phase tracking mode, where the inverter’s output phase locks onto the grid phase. This is achieved by adjusting the reference angle \( \theta_{ref} \) in the coordinate transformation. Simulations and experiments show that initially, a phase difference exists between the inverter output and grid voltage, but upon activation of the pre-synchronization signal, the phases synchronize rapidly, enabling seamless grid integration.
Once synchronized, the single phase inverter transitions to grid-connected operation using PQ control. In this mode, the inverter acts as a controlled current source, with active and reactive power regulated independently via d-axis and q-axis currents. Based on instantaneous power theory, the relationship between power and current components is derived. Assuming the grid voltage \( u_s = U_m \cos(\omega t + \phi) \) and inverter output current \( i_o = I_m \cos(\omega t) \), the active power \( P \) and reactive power \( Q \) are:
$$ P = U I \cos \phi $$
$$ Q = U I \sin \phi $$
Transforming these into the dq coordinate system with the grid voltage aligned to the d-axis yields:
$$ i_d = \frac{2P}{U_m} $$
$$ i_q = -\frac{2Q}{U_m} $$
Thus, by controlling \( i_d \) and \( i_q \), precise regulation of active and reactive power is achieved. The control structure includes current loops with PI controllers and decoupling terms to ensure independence between d and q axes. Simulation and experimental results under varying power setpoints confirm the accuracy of this method, with the single phase inverter maintaining unit power factor operation and responding dynamically to changes in power references.
The overall control strategy integrates all operational modes, with mode switching logic to transition between islanded, pre-synchronization, and grid-connected states. Initially, the single phase inverter operates in V/f mode, providing a stable AC output. When grid connection is required, pre-synchronization ensures phase alignment, and upon grid connection, the system switches to PQ control. This comprehensive approach simplifies the control of single phase inverters across different scenarios, enhancing their applicability in renewable energy systems.
In conclusion, the adoption of the dq coordinate system for single phase inverter control offers significant advantages over traditional methods. By leveraging coordinate transformation, time-varying signals are converted to DC quantities, enabling the use of PI controllers for precise tracking. The strategies for islanded operation, pre-synchronization, and grid-connected operation are streamlined, with experimental validation confirming their feasibility. Future work will focus on optimizing transition dynamics, such as reducing voltage overshoot and frequency deviations during mode switching, to further improve the performance of single phase inverters in practical applications.
| Operational Mode | Control Method | Key Equations | Performance Metrics |
|---|---|---|---|
| Islanded Operation | V/f Control | $$ u_d^{ref} = U_m, \quad u_q^{ref} = 0 $$ | Stable 50 Hz output, low THD |
| Pre-synchronization | Phase Tracking | $$ \theta_{ref} = \omega_g t $$ | Rapid phase alignment with grid |
| Grid-connected Operation | PQ Control | $$ i_d = \frac{2P}{U_m}, \quad i_q = -\frac{2Q}{U_m} $$ | Accurate power control, unit power factor |
The mathematical derivations and control designs presented herein underscore the versatility of the dq coordinate system in enhancing the performance of single phase inverters. Through continuous refinement, these strategies can contribute to the efficient integration of renewable energy sources into the power grid, addressing the growing demand for sustainable electricity generation.