The pursuit of efficient and high-power solar energy harvesting continues to drive innovations in power electronic converter topologies. Multi-level solar inverters are pivotal in high-voltage grid-connected photovoltaic (PV) systems due to their superior output waveform quality and reduced stress on switching devices. Among various multi-level structures, the open-winding dual solar inverter configuration presents a compelling alternative. This topology essentially synthesizes a three-level output by connecting two standard two-level voltage source inverters (VSIs) to either end of a three-phase open-winding transformer. A key advantage of this system is the ability to feed each solar inverter from an independent PV array, allowing for individual Maximum Power Point Tracking (MPPT) and potentially increasing the overall energy yield of the installation. However, a significant challenge arises during partial shading or module failures, which cause a substantial imbalance in the output power of the two PV arrays. This imbalance can lead to over-modulation in the solar inverter connected to the higher-power array, distorting grid current and threatening system stability. This paper delves into the mathematical modeling and control of such a system and proposes a reactive power compensation strategy to maintain stable, high-quality operation under these unbalanced conditions, thereby enhancing the robustness and efficiency of the solar inverter system.
System Topology and Mathematical Foundation
The fundamental structure of the open-winding dual solar inverter system is shown in the conceptual diagram below. Two separate PV arrays provide DC power to Inverter 1 and Inverter 2, respectively. The AC outputs of these solar inverters are connected to an open-winding transformer, whose secondary side is coupled to the grid.
The system’s behavior in the three-phase stationary frame (abc) is described by:
$$V_a = E_a + L\frac{di_a}{dt} + i_a R = s_{a1}V_{dc1} – s_{a2}V_{dc2}$$
$$V_b = E_b + L\frac{di_b}{dt} + i_b R = s_{b1}V_{dc1} – s_{b2}V_{dc2}$$
$$V_c = E_c + L\frac{di_c}{dt} + i_c R = s_{c1}V_{dc1} – s_{c2}V_{dc2}$$
where \(V_{a,b,c}\) are the synthesized phase voltages, \(E_{a,b,c}\) are the grid voltages, \(i_{a,b,c}\) are the grid currents, \(L\) and \(R\) are the filter inductance and resistance, \(V_{dc1}\) and \(V_{dc2}\) are the DC-link voltages, and \(s_{x1}, s_{x2}\) are the switching states (0 or 1) for the respective solar inverter legs.
Transforming these equations into the synchronous rotating reference frame (dq) aligned with the grid voltage vector simplifies the control design:
$$V_d = E_d – \omega L i_q + L\frac{di_d}{dt} + i_d R = V_{d1} – V_{d2}$$
$$V_q = E_q + \omega L i_d + L\frac{di_q}{dt} + i_q R = V_{q1} – V_{q2}$$
Here, \(V_d\) and \(V_q\) are the synthesized dq-axis voltages, \(V_{d1,q1}\) and \(V_{d2,q2}\) are the dq-axis output voltages of solar inverter 1 and solar inverter 2, respectively. Crucially, the total voltage vector of the dual solar inverter system is the vector difference between the outputs of the two individual inverters.
Modulation Strategy: 180° Decoupled Space Vector PWM
To effectively control the dual solar inverter, a modulation strategy that leverages its inherent structure is required. The 180° Decoupled Space Vector Pulse Width Modulation (SVPWM) is particularly suitable. Each two-level solar inverter can produce eight voltage space vectors. The composite system, by taking the vector difference between all possible combinations from both inverters, generates 19 distinct voltage vectors, offering high redundancy and superior harmonic performance compared to classic three-level topologies.
In this strategy, the total reference voltage vector \(V^*\) for the system is decomposed into two reference vectors for the individual solar inverters, \(V_1^*\) and \(V_2^*\), which are phase-shifted by 180°:
$$V^* = V_1^* – V_2^*$$
This approach maximizes DC voltage utilization by effectively using the outermost hexagon of the composite vector space.
Dual-Loop Control for Independent MPPT
A primary advantage of this solar inverter topology is independent MPPT for each PV array. A dual-loop control scheme is employed to achieve this. The outer voltage loop regulates the DC-link voltages \(V_{dc1}\) and \(V_{dc2}\) to their MPPT reference values \(V_{dc1}^*\) and \(V_{dc2}^*\). The errors are processed to generate the system’s d-axis current reference \(I_d^*\) and a power distribution factor \(k\) (0 ≤ k ≤ 1).
The inner current loop then controls the grid currents \(i_d\) and \(i_q\) to follow their references. The outputs of the current controllers, after decoupling and grid voltage feedforward, form the total dq-axis reference voltages \(V_d^*\) and \(V_q^*\). These are then transformed into a reference frame aligned with the grid current vector, yielding the active voltage component \(V_{d’}^*\) (in-phase with current) and the reactive voltage component \(V_{q’}^*\) (quadrature to current):
$$
\begin{bmatrix}
V_{d’}^* \\
V_{q’}^*
\end{bmatrix} =
\begin{bmatrix}
\cos\phi & \sin\phi \\
-\sin\phi & \cos\phi
\end{bmatrix}
\begin{bmatrix}
V_{d}^* \\
V_{q}^*
\end{bmatrix}
$$
where \(\phi\) is the angle between grid voltage and current. By appropriately distributing \(V_{d’}^*\) using factor \(k\), the two solar inverters can be controlled to draw different levels of active power from their respective PV arrays, enabling independent MPPT.
Over-Modulation Problem and Reactive Power Compensation Strategy
Under significant power imbalance (e.g., \(P_1 \gg P_2\)), a fundamental constraint is challenged. The grid current \(I_g\), common to both solar inverters, must carry the total active power \(P_T = P_1 + P_2\). From power balance per solar inverter:
$$P_1 = \frac{3}{2} V_1 I_g \cos\delta, \quad P_2 = \frac{3}{2} V_2 I_g \cos\delta$$
where \(V_1, V_2\) are the fundamental output voltage magnitudes of the inverters and \(\delta\) is a small angle. For a solar inverter using SPWM, the maximum fundamental AC voltage it can generate is \(V_{dc}/2\). Therefore, to avoid over-modulation, we require \(V_1 \leq V_{dc1}/2\) and \(V_2 \leq V_{dc2}/2\). This imposes a minimum grid current, \(I_{min}\), necessary to transfer the higher power without over-modulating the corresponding solar inverter:
$$I_{min} = \max\left(\frac{4P_1}{3V_{dc1}}, \frac{4P_2}{3V_{dc2}}\right)$$
If the current required for unity power factor operation, \(I_{gd} = 2P_T / (3V_g)\), is less than \(I_{min}\), the system will be forced into over-modulation, degrading current quality.
The proposed solution injects reactive current. By making the grid current lag or lead the voltage (\(\phi > 0\)), the magnitude of the grid current \(I_g\) increases for the same active power transfer (\(P_T = \frac{3}{2} V_g I_g \cos\phi\)). This higher \(I_g\) allows the solar inverter with higher power (e.g., Inverter 1) to operate at a lower modulation index while still transferring its power, thus avoiding over-modulation. The required compensation is determined by solving for the minimum \(I_g\) and phase angle \(\phi\) that satisfy three constraints simultaneously: 1) Inverter 1 power balance at its voltage limit (\(V_1 = V_{dc1}/2\)), 2) Total active power balance, and 3) Grid reactive power balance.
$$
\begin{cases}
\frac{3}{2} \cdot \frac{V_{dc1}}{2} \cdot I_g \cos\phi_1 = P_1 \\
\frac{3}{2} V_g I_g \cos\phi = P_1 + P_2 \\
V_g I_g \sin\phi = \omega L I_g^2 + \frac{V_{dc1}}{2} I_g \left(\frac{P_1+P_2}{P_1}\right) \sin\phi_1
\end{cases}
$$
Here, \(\phi_1\) is the angle between Inverter 1’s voltage and the grid current. Solving this system yields the necessary grid current magnitude \(I_g\) and phase \(\phi\), from which the q-axis current reference for compensation is set: \(I_q^* = I_g \sin\phi\).
| Operating Condition | PV Power P1 (kW) | PV Power P2 (kW) | Unity PF Current \(I_{gd}\) (A) | Min. Current \(I_{min}\) (A) | Over-modulation Risk | Compensation Action | Resulting Current THD |
|---|---|---|---|---|---|---|---|
| Balanced | 15.76 | 15.76 | 53.1 | 34.1 | No | None | < 3% |
| Light Imbalance | 15.76 | 12.11 | 46.9 | 34.1 | No | None | ~4.1% |
| Heavy Imbalance (No Comp.) | 15.76 | 10.61 | 44.4 | 34.1 | Yes | None | ~6.5% |
| Heavy Imbalance (With Comp.) | 15.76 | 10.61 | → \(I_g\)=48.2 | 34.1 | No | Reactive Injection | ~3.9% |
Simulation and Experimental Verification
To validate the proposed control and compensation strategy, detailed simulations and a laboratory-scale experiment were conducted. The system parameters are summarized below:
| Parameter | Value |
|---|---|
| DC Source Voltage (Vdc1, Vdc2) | ~615 V (simulated PV) |
| Grid Voltage (Line-to-Line RMS) | 396 V |
| Filter Inductance (L) | 3.5 mH |
| Switching Frequency | 5 kHz |
| Rated Power | 40 kW |
Simulation Results: Under balanced power (P1=P2=15.76 kW), the system operated at unity power factor with a three-level output voltage waveform, confirming the equivalence of the dual solar inverter to a three-level converter. For a heavy imbalance (P1=15.76 kW, P2=10.61 kW), the unity-power-factor control caused significant current distortion (THD 5.53%) and visible over-modulation in the modulation wave of the first solar inverter. Activating the reactive compensation algorithm at t=0.4s immediately improved the current waveform, reducing THD to 3.72% and eliminating over-modulation, while maintaining both PV arrays at their MPPs.
Experimental Results: The hardware platform utilized two programmable DC sources and a DSP controller. The experiments mirrored the simulation scenarios. Under heavy imbalance without compensation, the grid current was distorted with a THD of 6.47%. With the reactive compensation strategy enabled, the current quality was restored (THD 3.93%), demonstrating the practical efficacy of the method in a real solar inverter system.
Conclusion
This paper has presented a thorough analysis of the open-winding dual solar inverter topology for grid-connected PV systems. The dual solar inverter structure offers the significant benefit of independent MPPT control for two PV sources, enhancing energy harvest. The mathematical model and the 180° decoupled SVPWM strategy form the basis for its control. The core challenge of power imbalance-induced over-modulation was analytically defined through a stability criterion involving the minimum grid current \(I_{min}\). A systematic reactive power compensation strategy was then derived and implemented to inject the minimal necessary reactive current, thereby increasing the grid current magnitude just enough to prevent the higher-power solar inverter from over-modulating. This ensures stable operation and high-quality grid currents even under severe partial shading conditions, maximizing the operational range of the MPPT controllers. Both simulation and experimental results conclusively validate the proposed control framework, confirming its effectiveness and practical value for improving the reliability and efficiency of advanced solar inverter systems in real-world applications with uneven irradiance.