In the field of high-voltage and high-power photovoltaic grid-connected generation, the solar inverter plays a central role in energy conversion and system stability. Among various multilevel topologies, the open-end winding dual inverter structure offers significant advantages in terms of reduced harmonic content, higher DC-link voltage utilization, and the ability to independently control two separate photovoltaic arrays. This article focuses on a novel control and reactive power compensation strategy for an open-end winding dual solar inverter system. The proposed method ensures that two photovoltaic arrays can operate at their respective maximum power points even under severe power imbalance conditions, while avoiding over-modulation of the inverters. Mathematical modeling, voltage space vector 180° decoupling modulation, double closed-loop control, and a simple reactive power compensation scheme are presented in detail. Simulation and experimental results validate the effectiveness of the proposed approach.
The topology studied here consists of two two-level inverters connected to an open-end winding three-phase transformer. Each inverter is supplied by an independent photovoltaic array. Compared with traditional diode-clamped or flying-capacitor multilevel converters, this solar inverter topology reduces hardware cost and control complexity. The two photovoltaic arrays can be controlled separately, which is beneficial for maximizing energy harvest under partial shading or mismatch conditions. However, when the output powers of the two arrays are significantly unbalanced, the inverter connected to the higher-power array tends to enter over-modulation. This degrades the grid current quality and may even cause system instability. Therefore, a reactive power compensation strategy is required to extend the stable operation range of the solar inverter system.
1. Mathematical Model of Open-End Winding Dual Solar Inverter
The equivalent circuit of the open-end winding dual solar inverter system is shown in Figure 1 (topology omitted for brevity). The two photovoltaic arrays are represented by DC voltage sources \(V_{dc1}\) and \(V_{dc2}\), respectively. The AC side is connected through a filter inductor \(L\) and resistance \(R\) to the grid voltage \(E\). The phase voltages of the open-end winding transformer are \(v_a, v_b, v_c\). The switching functions \(s_{a1}, s_{b1}, s_{c1}\) and \(s_{a2}, s_{b2}, s_{c2}\) define the states of the two inverters.
In the three-phase stationary coordinate system, the voltage equations can be expressed as:
\[
\begin{cases}
v_a = E_a + L \frac{di_a}{dt} + R i_a = s_{a1} V_{dc1} – s_{a2} V_{dc2} \\
v_b = E_b + L \frac{di_b}{dt} + R i_b = s_{b1} V_{dc1} – s_{b2} V_{dc2} \\
v_c = E_c + L \frac{di_c}{dt} + R i_c = s_{c1} V_{dc1} – s_{c2} V_{dc2}
\end{cases}
\]
Applying the Clarke and Park transformations, the equations in the synchronous rotating \(dq\) reference frame become:
\[
\begin{cases}
V_d = E_d – \omega L i_q + L \frac{di_d}{dt} + R i_d = V_{d1} – V_{d2} \\
V_q = E_q + \omega L i_d + L \frac{di_q}{dt} + R i_q = V_{q1} – V_{q2}
\end{cases}
\]
where \(\omega\) is the grid angular frequency, \(E_d, E_q\) are the grid voltage dq components, \(i_d, i_q\) are the grid current dq components, and \(V_{d1}, V_{q1}, V_{d2}, V_{q2}\) are the dq components of the output voltages of inverter 1 and inverter 2, respectively. The total equivalent voltage vector of the open-end winding dual solar inverter system is the vector difference between the two inverter voltage vectors.
2. Voltage Space Vector 180° Decoupled Modulation
Each two-level inverter produces eight basic voltage space vectors. By combining the vectors of inverter 1 and inverter 2, the resultant solar inverter produces 19 distinct voltage space vectors, as shown in Figure 2 (not reproduced). With the 180° decoupled modulation strategy, the reference voltage vector \(V^*\) is split into two components \(V_1^*\) and \(V_2^*\), which are phase-shifted by 180° with respect to each other. This approach fully utilizes the largest hexagon of the combined vector space, thereby increasing the DC voltage utilization compared with conventional 120° decoupling.
Let the reference voltage vector be \(V^* = V_1^* – V_2^*\). If \(V_{dc1} = V_{dc2} = V_{dc}\), then the maximum achievable output phase voltage amplitude is \(V_{dc}\) for the dual inverter system, whereas a single two-level inverter can only achieve \(V_{dc}/2\). This is a major advantage of the open-end winding topology. The switching states are redundant, allowing optimized switching sequences to reduce common-mode voltage and switching losses.
For the implementation of SVPWM, the reference vector magnitude and angle are computed from the dq components. The dwell times for adjacent active vectors and zero vectors are calculated in the same manner as conventional SVPWM, but the actual switching signals are generated by distributing the PWM pulses to the two inverters in a complementary way. A detailed analysis of the modulation algorithm is presented in the literature and is not repeated here.
3. Dual Closed-Loop Control Scheme for Independent MPPT
To achieve independent maximum power point tracking (MPPT) for the two photovoltaic arrays, the actual DC-link voltages \(V_{dc1}\) and \(V_{dc2}\) must follow their respective reference values \(V_{dc1}^*\) and \(V_{dc2}^*\). The proposed double closed-loop control strategy for the open-end winding dual solar inverter is described below.
The error signals of the two DC-link voltages are defined as:
\[
e_1 = V_{dc1}^* – V_{dc1}, \quad e_2 = V_{dc2}^* – V_{dc2}
\]
The sum of the errors is fed into a sum voltage controller (denoted as \(\Sigma\) controller) to generate the d-axis current reference \(I_d^*\). The difference of the errors is fed into a difference voltage controller (denoted as \(\Delta\) controller) to generate a power distribution coefficient \(k\), where \(0 \le k \le 1\). The current references are then processed by the inner current loop.
The complete control block diagram is shown in Figure 3 (omitted). The d-axis current reference \(I_d^*\) is compared with the actual d-axis current \(i_d\), and the error is passed through a PI controller. The feedforward terms \(-\omega L i_q\) and grid voltage \(E_d\) are added to compute the d-axis voltage reference \(V_d^*\). Similarly, the q-axis current reference \(I_q^*\) is compared with \(i_q\), and the error is regulated by a PI controller, with feedforward terms \(\omega L i_d\) and \(E_q\), to produce \(V_q^*\).
In the synchronous frame oriented to the grid voltage vector, \(V_d^*\) and \(V_q^*\) are not necessarily aligned with the active and reactive power axes when the power factor is not unity. Therefore, a coordinate rotation is introduced to transform \(V_d^*, V_q^*\) into the active voltage reference \(V_{d’}^*\) and reactive voltage reference \(V_{q’}^*\), using the angle \(\varphi\) between the grid voltage vector and the grid current vector:
\[
\begin{bmatrix}
V_{d’}^* \\
V_{q’}^*
\end{bmatrix}
=
\begin{bmatrix}
\cos\varphi & \sin\varphi \\
-\sin\varphi & \cos\varphi
\end{bmatrix}
\begin{bmatrix}
V_d^* \\
V_q^*
\end{bmatrix}
\]
The voltage reference vector \(V^*\) is then decomposed into the two inverter reference vectors according to the power distribution coefficient \(k\):
\[
V_1^* = k V^*, \quad V_2^* = (1-k) V^*
\]
For the open-end winding dual solar inverter, the grid current \(i_g\) is the common variable linking the two inverters. Under balanced operation (\(P_1 = P_2\)), the coefficient \(k\) is approximately 0.5, and the two inverters share the total output voltage equally. When \(P_1 \neq P_2\), the voltage share must be adjusted so that each inverter operates within its linear modulation range. The coefficient \(k\) is determined by the difference voltage controller based on the MPPT voltage errors.
4. Over-Modulation Mechanism and Reactive Power Compensation
4.1 Over-Modulation Mechanism
Consider the case where photovoltaic array 1 maintains its maximum power \(P_1\), while photovoltaic array 2 experiences a reduction in power due to partial shading or damage. Since the two inverters are series-connected on the AC side, they carry the same AC current \(I_g\). The power balance for each inverter is:
\[
P_1 = \frac{3}{2} V_1 I_g \cos\delta_1, \quad P_2 = \frac{3}{2} V_2 I_g \cos\delta_2
\]
where \(\delta_1, \delta_2\) are the angles between the inverter voltage vectors and the current vector. If \(P_2\) decreases while \(P_1\) remains constant, then \(V_1\) must increase to maintain the power balance, assuming the current magnitude remains the same. When \(V_1\) exceeds the maximum voltage that inverter 1 can produce with the given DC-link voltage \(V_{dc1}\), over-modulation occurs.
For SPWM modulation, the maximum fundamental AC voltage amplitude that a two-level inverter can produce is \(V_{dc}/2\). Therefore, the condition to avoid over-modulation is:
\[
V_1^* \le \frac{V_{dc1}}{2}, \quad V_2^* \le \frac{V_{dc2}}{2}
\]
In terms of AC current amplitude \(I_g\), the power balance yields:
\[
I_g \ge \frac{4}{3} I_{dc1}, \quad I_g \ge \frac{4}{3} I_{dc2}
\]
where \(I_{dc1} = P_1/V_{dc1}\) and \(I_{dc2} = P_2/V_{dc2}\) are the DC-side currents of the two photovoltaic arrays. The minimum AC current required to avoid over-modulation is thus:
\[
I_{\min} = \max\left\{ \frac{4}{3} I_{dc1}, \frac{4}{3} I_{dc2} \right\}
\]
Under unity power factor operation, the grid current amplitude is:
\[
I_{gd} = \frac{P_1 + P_2}{\frac{3}{2} V_g}
\]
where \(V_g\) is the grid line-to-neutral voltage amplitude. If \(I_{gd} \le I_{\min}\), the system will experience over-modulation, resulting in distorted grid current and potential instability.
4.2 Reactive Power Compensation Strategy
To solve the over-modulation problem without sacrificing the independent MPPT capability of the two photovoltaic arrays, we propose to inject reactive power into the grid, thereby increasing the AC current magnitude beyond \(I_{\min}\). The additional reactive current does not affect the active power transfer but reduces the required voltage magnitude of the inverters, thus keeping them in the linear modulation range.
The key is to determine the minimum reactive current reference \(I_q^*\). The following constraints are considered:
- Active power balance of inverter 1:
\[
\frac{3}{2} V_1^* I_g \cos\varphi_1 = P_1
\]
- Total active power balance of the whole system:
\[
\frac{3}{2} V_g I_g \cos\varphi = P_1 + P_2
\]
- Reactive power balance of the system (ignoring grid-side resistance):
\[
V_g I_g \sin\varphi = \omega L I_g^2 + V_1^* I_g \sin\varphi_1 \frac{P_1 + P_2}{P_1}
\]
Here, \(\varphi\) is the angle between the grid voltage and the grid current, \(\varphi_1\) is the angle between the inverter 1 voltage reference and the current, and \(V_1^*\) is set to the maximum permissible amplitude \(V_{dc1}/2\). The equations contain three unknowns: \(\varphi, \varphi_1, I_g\). Solving this system of nonlinear equations yields the required grid current magnitude \(I_g\) and angle \(\varphi\). The reactive current command is then:
\[
I_q^* = I_g \sin\varphi
\]
This value is used as the q-axis current reference in the control block diagram. After compensation, the solar inverter system delivers both active and reactive power to the grid. The reduction of power factor can be compensated by additional reactive power compensation devices at the point of common coupling, ensuring that the entire photovoltaic plant can still operate at unity power factor from the grid perspective.
5. Simulation Results
Simulation studies were carried out using MATLAB/Simulink. The photovoltaic arrays are modeled as DC voltage sources with series resistances. The parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| DC source voltage | 656 V |
| Series resistance | 1.6 Ω |
| DC-link capacitance | 3.36 mF |
| Filter inductance | 3.5 mH |
| Transformer turns ratio | 364/380 |
| Switching frequency | 5 kHz |
| Rated power | 40 kW |
| Grid line-to-line voltage (RMS) | 396 V |
Table 2 lists the operating conditions for three scenarios: balanced, lightly unbalanced, and heavily unbalanced.
| Scenario | \(V_{dc1}\) | \(V_{dc2}\) | \(P_1\) | \(P_2\) |
|---|---|---|---|---|
| Balanced | 615 V | 615 V | 15.76 kW | 15.76 kW |
| Light unbalance | 615 V | 625 V | 15.76 kW | 12.11 kW |
| Heavy unbalance | 615 V | 629 V | 15.76 kW | 10.61 kW |
5.1 Balanced Operation
Under balanced conditions, the two photovoltaic arrays deliver equal power. The grid current is sinusoidal and in phase with the grid voltage, confirming unity power factor operation. The synthesized line voltage \(v_{a1b1} – v_{a2b2}\) has the same waveform as that of a single three-level inverter, which validates the equivalence of the open-end winding dual solar inverter to a three-level topology.
5.2 Light Unbalanced Operation
When the output power of photovoltaic array 2 decreases slightly to 12.11 kW, the grid current amplitude decreases accordingly. The condition \(I_{gd} > I_{\min}\) is still satisfied, so no over-modulation occurs. The grid current waveform remains clean and the system continues to operate at unity power factor. The simulation results show that the independent MPPT control successfully maintains both arrays at their maximum power points.
5.3 Heavy Unbalanced Operation without Compensation
When the output power of photovoltaic array 2 drops to 10.61 kW, the condition \(I_{gd} \le I_{\min}\) is violated. Over-modulation occurs in inverter 1, which is connected to the higher-power array. The grid current waveform becomes distorted, such that the total harmonic distortion (THD) reaches 5.53%. The modulation wave of inverter 1 exceeds the triangular carrier amplitude of 0.5, confirming the over-modulation condition. This leads to increased low-frequency harmonics and reduced power quality, which is unacceptable for grid-connected photovoltaic systems.
5.4 Heavy Unbalanced Operation with Reactive Power Compensation
At \(t=0.4\) s, the proposed reactive power compensation is activated. The reactive current reference \(I_q^*\) is calculated according to the equations in the previous section. After compensation, the grid current amplitude increases, pushing the operating point toward the linear modulation region. The THD of the grid current decreases to 3.72%. The modulation wave amplitude of inverter 1 remains exactly at the carrier amplitude of 0.5, indicating that the over-modulation is avoided. The active power output of both photovoltaic arrays remains unchanged, confirming that the reactive power compensation does not affect the MPPT operation. When the output power of array 2 recovers at \(t=0.6\) s, the system returns to balanced operation and the reactive current reference becomes zero.
The simulation results are summarized in Table 3.
| Condition | THD of grid current | Modulation index of inverter 1 |
|---|---|---|
| Before compensation | 5.53% | > 0.5 (over-modulated) |
| After compensation | 3.72% | = 0.5 (boundary) |
6. Experimental Validation
A laboratory prototype was built to validate the proposed control and reactive power compensation strategy. The experimental platform consists of two 20 kW rectifiers with series resistances emulating the photovoltaic arrays, a TMS320F28335 DSP controller, and an open-end winding transformer with a delta-connected secondary. The experimental parameters are identical to those in the simulation.
Figure 4 (not shown) illustrates the experimental setup (omitted for brevity). The measured waveforms were recorded using a Yokogawa DLM2024 oscilloscope, high-voltage probes, and current clamps.
6.1 Balanced Operation
The experimental results for balanced operation are consistent with the simulation. The two photovoltaic arrays output equal power, the grid current is sinusoidal and in phase with the grid voltage, and the synthesized line voltage waveform matches that of a three-level inverter. This confirms the correct implementation of the 180° decoupled SVPWM in the open-end winding dual solar inverter.
6.2 Light Unbalanced Operation
When the output power of array 2 is reduced slightly, the grid current amplitude decreases without noticeable distortion. The THD of the grid current is measured at 4.08%, which is acceptable for grid-connected operation. Both arrays continue to track their maximum power points independently.
6.3 Heavy Unbalanced Operation with and without Compensation
Under heavy unbalance, the grid current exhibits severe distortion with a THD of 6.47% before compensation. After applying the reactive power compensation, the THD improves to 3.93%. The experimental results confirm the effectiveness of the proposed method. The dynamic behavior during the transition is stable, and no significant transients are observed when the reactive current reference changes.
Table 4 compares the simulation and experimental THD values for the heavy unbalanced condition.
| Condition | Simulation THD | Experimental THD |
|---|---|---|
| Before compensation | 5.53% | 6.47% |
| After compensation | 3.72% | 3.93% |
The slight difference between simulation and experimental values is attributed to the non-ideal characteristics of the laboratory devices, such as dead time, DC-link voltage ripple, and transformer leakage inductance. Nevertheless, the proposed reactive power compensation strategy significantly improves the grid current quality and prevents over-modulation, thereby broadening the MPPT operating range of the two photovoltaic arrays.
7. Conclusion
This article has presented a comprehensive study of a reactive power compensation strategy for an open-end winding dual solar inverter system. The main contributions are summarized as follows:
- A mathematical model of the open-end winding dual solar inverter was established, and the 180° decoupled space vector PWM strategy was described. This modulation method fully utilizes the DC voltage and provides high equivalent switching frequency without additional hardware.
- A double closed-loop control scheme was introduced to achieve independent MPPT control for two photovoltaic arrays. The power distribution coefficient \(k\) is regulated based on the voltage errors to balance the voltage sharing between the two inverters.
- The over-modulation mechanism under unbalanced power conditions was analyzed in detail. It was found that the over-modulation occurs when the grid current amplitude is lower than a critical threshold \(I_{\min}\).
- A simple and effective reactive power compensation method was proposed. By injecting a minimal amount of reactive current, the grid current magnitude is increased above \(I_{\min}\), thereby eliminating over-modulation without affecting the active power output of the photovoltaic arrays.
- Simulation and experimental results verified the correctness and effectiveness of the proposed strategy. The THD of the grid current is significantly reduced under heavy unbalanced conditions, and the system can maintain stable operation over a wider MPPT range.
The proposed control strategy is especially suitable for large-scale photovoltaic power plants where partial shading and module mismatch are common. The open-end winding dual solar inverter topology, combined with the reactive power compensation technique, provides a promising solution for high-power, high-reliability solar inverter systems. Future work will focus on extending the strategy to more than two photovoltaic arrays and optimizing the reactive power injection under grid faults.