We propose an improved virtual flux direct power control (VF-DPC) strategy for a cascaded three‑phase grid‑connected solar inverter. By establishing the equivalent model of the modular cascaded system, we analyze the overall equivalent switching states. In the grid side, a modified virtual flux observer is employed to suppress the DC offset, enabling accurate power estimation. On the DC side, the maximum power point tracking (MPPT) algorithm provides the power reference. Based on the steady-state vector diagram, we design a decoupled power controller. To achieve multilevel output and constant switching frequency, we combine carrier phase shifting with space vector modulation to realize phase-shifted space vector modulation (PSSVM). Taking a three-module cascaded system as an example, we validate the effectiveness of the proposed strategy through Simulink simulations. The results show that the solar inverter can omit grid voltage sensors, and the photovoltaic (PV) arrays can directly connect to the three-phase line voltage cascaded inverter without any boost converter. The cascaded three-phase solar inverter outputs five-level voltage, operates with unity power factor, and delivers the rated maximum power of 4667 W. The total harmonic distortion (THD) of the grid current is only 2.01%. Moreover, the modular cascaded structure offers high engineering feasibility.
1. Introduction
Photovoltaic (PV) power generation is not restricted by energy resources, raw materials, or application environments, and thus has vast development prospects. Grid-connected PV systems account for nearly 99% of existing PV installations, with an annual growth rate of 25%–30% worldwide. Typical grid-connected PV systems employ either a two-stage inverter with a DC/DC converter or a cascaded boost inverter. However, the added DC/DC stage reduces the overall system efficiency, and the high switching frequency of the inverter requires large filter inductors and capacitors. The cascaded multilevel inverter, on the other hand, generates a staircase output voltage waveform by summing the outputs of multiple power units, thereby reducing voltage distortion and allowing low switching frequency while maintaining high output quality. The most common cascaded multilevel topology is based on H-bridge cells, which lack inherent boosting capability and require many switching devices and DC sources. For three-phase applications, star or delta connections are typically needed. A three-phase line voltage cascaded multilevel converter with boosting capability has been proposed, using three-phase VSCs as basic modules. This paper applies such a topology to grid-connected PV systems, eliminating the need for step-up transformers and enabling direct injection of DC power into the three-phase grid. Using carrier phase-shifted PWM or space vector modulation in cascaded inverters can achieve multilevel outputs. However, many approaches require many measurements. We adopt a phase-shifted space vector modulation (PSSVM) that combines the advantages of carrier phase shifting and space vector modulation to fix the switching frequency while generating a multilevel output. In high-performance control strategies for grid-connected inverters, virtual flux oriented direct power control (VF-DPC) offers a simple structure, reduces sensor requirements, and provides excellent dynamic and static instantaneous power regulation. This paper extends VF-DPC to the cascaded three-phase solar inverter, using a modified virtual flux observer to eliminate the DC offset, and designing a decoupled power controller. The overall system is modular, flexible, and suitable for PV integration.
2. System Topology and Equivalent Model
The modular cascaded three-phase grid-connected PV system is shown conceptually in the block diagram (the image above illustrates a typical solar inverter). The system consists of \(m=3\) three-phase inverter modules, each supplied by a PV array. The grid-side voltages are \(e_{abc}\), filter inductors \(L_{abc}\), and line resistances \(r_{abc}\). The grid current RMS is \(I\). From Kirchhoff’s current law and the three-phase relationships, the currents in each module are given by equations (1)–(3). Although the three bridge arm currents of each sub-module are unbalanced, the maximum amplitude appears in the phase currents feeding into the grid. Therefore, when selecting the current rating of the switching devices, only the maximum phase current amplitude needs to be considered. The line voltages on the grid side are obtained as:
$$u_{ab}=u_{ab1}+u_{ab2}+u_{ab3}$$
$$u_{bc}=u_{bc1}+u_{bc2}+u_{bc3}$$
$$u_{ca}=u_{ca1}+u_{ca2}+u_{ca3}$$
The output line voltage of the cascaded inverter is twice the line voltage of a single inverter module. Therefore, the entire cascaded system can be equivalently modeled as a conventional three-phase inverter with a DC bus voltage equal to twice that of a single module. The equivalent switching signals \(S_A, S_B, S_C\) are derived from the switching signals of each module \(S_{ma}, S_{mb}, S_{mc}\) and the DC voltage per module \(U_{mdc}\). Assuming all modules have the same DC voltage \(U_{dc}\), the equivalent inverter output voltage components in the stationary frame can be expressed as:
$$u_{\alpha} = \frac{2}{3}U_{dc}\left(S_A – \frac{1}{2}S_B – \frac{1}{2}S_C\right) – L\frac{di_{\alpha}}{dt}$$
$$u_{\beta} = \frac{1}{\sqrt{3}}U_{dc}(S_B – S_C) – L\frac{di_{\beta}}{dt}$$
The effective switching signals \(S_A, S_B, S_C\) take values in \(\{-2,-1,0,1,2\}\), realizing five-level line voltage output.
| Parameter | Symbol | Value |
|---|---|---|
| Grid voltage (RMS) | \(E\) | 220 V |
| Grid frequency | \(f\) | 50 Hz |
| PV module open-circuit voltage | \(V_{oc}\) | 44 V |
| PV module short-circuit current | \(I_{sc}\) | 8.4 A |
| DC bus voltage of each module | \(U_{dc}\) | 40 V (MPPT) |
| Number of cascaded modules | \(m\) | 3 |
| Filter inductance (grid side) | \(L\) | 5 mH |
| Line resistance | \(R\) | 0.03 Ω |
| Switching frequency | \(f_s\) | 10 kHz |
| Power controller parameters (PI) | \(K_p, K_i\) | 2, 10 |
3. Proposed Control Strategy
3.1 Virtual Flux Observer with DC Offset Elimination
The virtual flux of the grid is defined as \(\Psi = \int e\, dt\). In the synchronous \(dq\) frame aligned with the virtual flux vector, the grid voltage vector \(E\) is aligned with the \(q\)-axis. The spatial angle \(\gamma\) is obtained from the stationary frame flux components:
$$\gamma = \arctan\left(\frac{\psi_{\beta}}{\psi_{\alpha}}\right)$$
The flux components are estimated from the measured currents and the reconstructed inverter voltages:
$$\psi_{\alpha} = \int (u_{\alpha} – L\frac{di_{\alpha}}{dt}) dt$$
$$\psi_{\beta} = \int (u_{\beta} – L\frac{di_{\beta}}{dt}) dt$$
To avoid the DC offset problem inherent in pure integrators, we use a modified observer that combines a low-pass filter (LPF) and a high-pass filter (HPF). The transfer function is:
$$\Psi^{*} = \frac{1}{j\omega + k_1 \omega_0} \cdot \frac{j\omega}{j\omega + k_2 \omega_0} E$$
The equivalent integration in the time domain yields the flux components without saturation or phase error caused by DC offsets.
3.2 Power Control Loop Design
The instantaneous active and reactive powers in the virtual flux oriented frame are:
$$p_s = \omega (\psi_{\alpha} i_{\beta} – \psi_{\beta} i_{\alpha})$$
$$q_s = \omega (\psi_{\alpha} i_{\alpha} + \psi_{\beta} i_{\beta})$$
The power reference is derived from the MPPT algorithm (perturb and observe), and the reactive power reference is set to zero for unity power factor operation. The decoupled control law in the \(dq\) frame is obtained from the system dynamic model:
$$\frac{d}{dt}p_s = -\frac{R}{L}p_s + \omega q_s + \frac{1}{L}(-u_d e_q)$$
$$\frac{d}{dt}q_s = -\frac{R}{L}q_s – \omega p_s + \frac{1}{L}(-u_q e_q – e_q^2)$$
Using PI controllers with feedforward compensation, we obtain:
$$u_d = \left(K_p + \frac{K_i}{s}\right)(p_s^{*} – p_s) + \omega L q_s – \frac{R}{e_q} p_s$$
$$u_q = -\left(K_p + \frac{K_i}{s}\right)(q_s^{*} – q_s) – \omega L p_s + \frac{R}{e_q} q_s + e_q$$
where \(e_q = |E|\). The closed-loop transfer function for the active power loop is:
$$\frac{p_s}{p_s^{*}} = \frac{K_p s + K_i}{L s^2 + (R + K_p) s + K_i}$$
With proper tuning, the system achieves zero steady-state error and fast dynamic response.
3.3 Phase-Shifted Space Vector Modulation (PSSVM)
To generate the multilevel output while maintaining a fixed switching frequency, we combine conventional two-level space vector modulation with carrier phase shifting. For \(m\) cascaded modules, the carrier of module \(k\) is phase-shifted by \(\theta_k = 2\pi (k-1)/m\). The central carrier signal for module 2 and module 3 are shifted by \(120^\circ\) and \(240^\circ\) respectively (for \(m=3\)). The resulting line voltage is the sum of the individual modules’ line voltages, producing a five-level waveform. The overall switching signals \(S_A, S_B, S_C\) are the sum of the corresponding module switching states, as given by:
$$S_A = S_{1a} – S_{3a}$$
$$S_B = S_{2b} – S_{1b}$$
$$S_C = S_{3c} – S_{2c}$$
The PSSVM method ensures that each module switches at a fixed frequency, while the equivalent output frequency is effectively multiplied.
4. Simulation Results and Analysis
We built a simulation model of the three-module cascaded solar inverter in MATLAB/Simulink. The parameters are summarized in the table. The PV arrays are modeled as a controlled current source with I-V characteristics. The MPPT converges quickly to the maximum power point (400 V total DC voltage across each module, i.e., 40 V per module).
The equivalent switching signals \(S_A, S_B, S_C\) over one switching period are shown in the simulation results. They take values of \(-2,-1,0,1,2\), confirming the five-level operation. The inverter output line voltage reaches 800 V peak, which is twice the module DC voltage (400 V), as expected.
The virtual flux observer performance is compared. The modified observer (LPF+HPF) produces flux waveforms that perfectly match the reference obtained from pure integration, while the simple LPF introduces phase and amplitude errors.
The active and reactive power responses are depicted. The active power quickly reaches the reference value of 4667 W with small ripple, and the reactive power is maintained near zero, confirming decoupled control. The grid currents are balanced and sinusoidal. The phase relationship between the grid voltage and current shows unity power factor. The total harmonic distortion (THD) of the grid current is only 2.01%, well within the IEEE 1547 standard.
| Performance Metric | Value |
|---|---|
| Maximum power delivered | 4667 W |
| Power factor | 1.0 |
| Grid current THD | 2.01% |
| Number of output voltage levels | 5 |
| Switching frequency per module | 10 kHz |
| DC voltage of each module | 40 V |
| Equivalent line voltage (peak) | 800 V |
5. Conclusion
In this paper, we have presented a virtual flux direct power control strategy for a cascaded three-phase solar inverter. The proposed system uses a modular line-voltage cascaded topology, which reduces the number of switching devices and DC sources compared to traditional H-bridge cascaded structures. By applying a modified virtual flux observer, we eliminate the need for grid voltage sensors and achieve robust power estimation. The decoupled power controller together with phase-shifted space vector modulation ensures unity power factor operation, fixed switching frequency, and a five-level output voltage waveform. Simulation results confirm that the cascaded solar inverter can deliver the rated PV power of 4667 W with low current distortion (THD 2.01%). The modular design makes the system highly practical for engineering implementation. Future work may extend this strategy to higher-level cascaded systems and experimental validation.
We have demonstrated that the combination of cascaded topology and VF-DPC is a promising solution for high‑power grid‑connected solar inverters, offering both performance and cost advantages.