The integration of photovoltaic (PV) systems into the grid relies heavily on the performance and reliability of power electronic converters. Among various topologies, the Cascaded H-Bridge (CHB) inverter has emerged as a prominent candidate for medium-voltage grid-connected applications due to its modular structure, scalability, and the inherent advantage of independent maximum power point tracking (MPPT) for each PV string. However, a significant operational challenge arises from the natural variability and potential mismatch in power generation across the different H-bridge modules. Factors such as partial shading, uneven soiling, module aging, or manufacturing tolerances can lead to severe power imbalances. Under such conditions, a conventional control strategy focused solely on unity power factor operation can force modules with higher irradiance into over-modulation, leading to distorted grid current and potential instability. This work addresses this critical issue by proposing and validating an optimized control strategy based on reactive power compensation, effectively expanding the stable operating region of cascaded solar inverters under highly unbalanced power conditions.
The core of the problem lies in the inherent constraint of the system. In a single-phase CHB solar inverter with ‘n’ modules, the control system typically aims to regulate ‘n+1’ variables: the ‘n’ individual DC-link voltages and the single grid current. Under perfect power balance, this is manageable. However, when power is imbalanced, maintaining all DC voltages at their respective maximum power point (MPP) while injecting purely active current at unity power factor may require a modulation index for the highest-power module that exceeds the linear modulation limit. This over-modulation distorts the synthesized multilevel voltage waveform, leading to harmonic injection into the grid current.
My investigation into this problem begins with establishing a clear mathematical model. The system consists of ‘n’ H-bridge cells, each fed by an independent PV array, cascaded through their output terminals and connected to the grid via a filter inductor. The key equations governing the instantaneous dynamics can be derived from Kirchhoff’s laws. For a single-phase system, it is advantageous to analyze steady-state power flow using phasor representation in a synchronous reference frame (dq-frame) aligned with the grid current. This approach provides profound insight into the stability boundaries.
Let us define the grid voltage phasor as $$\mathbf{U}_s$$ and the grid current phasor as $$\mathbf{I}_s$$. The total inverter output voltage phasor is $$\mathbf{U}_{AB} = \sum_{x=1}^{n} \mathbf{U}_{Hx}$$, where $$\mathbf{U}_{Hx}$$ is the fundamental output voltage of the x-th H-bridge. The relationship is given by:
$$\mathbf{U}_s + j\omega L_s \mathbf{I}_s = \mathbf{U}_{AB}$$
where $$L_s$$ is the filter inductance and $$\omega$$ is the grid angular frequency. Aligning the d-axis with the grid current phasor ($$\mathbf{I}_s = I_{sd} + j0$$) simplifies the power analysis. The active power delivered from the x-th module to the grid is:
$$P_x = U_{Hx\_d} I_{sd}$$
where $$U_{Hx\_d}$$ is the d-axis component of $$\mathbf{U}_{Hx}$$. Since $$U_{Hx\_d} = m_{x\_d} u_{dcx}$$, where $$m_{x\_d}$$ is the d-axis (active) modulation index and $$u_{dcx}$$ is the DC-link voltage, we get:
$$m_{x\_d} = \frac{P_x}{I_{sd} u_{dcx}} = \frac{P_x U_s}{P_T u_{dcx}}$$
Here, $$P_T = \sum P_x$$ is the total active power. Under conventional unity power factor control with no reactive compensation, the q-axis (reactive) voltage component for each module is determined by the need to counteract the inductor voltage drop, distributed proportionally to the active power. This leads to a q-axis modulation index:
$$m_{x\_q} = \frac{\omega L_s I_{sd}}{u_{dcx}} \propto \frac{P_x}{u_{dcx}}$$
The total modulation index for module x is:
$$M_x = \sqrt{m_{x\_d}^2 + m_{x\_q}^2} = \frac{P_x}{P_T} \frac{\sqrt{U_s^2 + (\omega L_s I_{sd})^2}}{u_{dcx}} \approx \frac{P_x}{P_T} \frac{U_s}{u_{dcx}}$$
The condition to avoid over-modulation in a linear modulation scheme (like Sinusoidal PWM) is $$M_x \leq \frac{\sqrt{2}}{2}$$. This yields the crucial stability limit for unity power factor operation:
$$\frac{P_x u_{dcx}}{P_T U_s} \leq \frac{\sqrt{2}}{2}$$
When a module’s power $$P_x$$ is too high relative to the total power $$P_T$$, this inequality is violated, leading to over-modulation. This clearly defines the operational limit of traditional control strategies for cascaded solar inverters.
The proposed strategy introduces reactive power compensation as an additional control degree of freedom to overcome this limit. By allowing the inverter to inject a controlled amount of reactive current $$I_{sq}$$, the phase angle $$\phi$$ between the grid current and voltage is no longer zero. This alters the voltage phasor diagram. The key is to calculate the minimum reactive current needed to bring all modules back within the linear modulation region. The active current component remains tied to total power: $$I_{sd} = P_T / U_s$$. The new d-axis component of the total inverter voltage becomes:
$$U_{AB\_d} = U_s – \omega L_s I_{sq}$$
However, from the perspective of power balance per module, the active voltage component for module x must satisfy $$U_{Hx\_d} I_{sd} = P_x$$. Therefore, the required active modulation indices become:
$$m_{x\_d} = \frac{P_x}{I_{sd} u_{dcx}} = \frac{P_x U_s}{P_T u_{dcx}}$$
Crucially, these are identical to the indices in the unity power factor case. The reactive voltage component $$U_{AB\_q}$$ that must be synthesized is:
$$U_{AB\_q} = \omega L_s I_{sd} – U_s \sin \phi$$
where $$\sin \phi = I_{sq} / |\mathbf{I}_s|$$. The phase angle $$\phi$$ (and hence $$I_{sq}$$) is determined by the geometric constraint that the vector sum of the per-module voltages, each with its fixed $$m_{x\_d}$$ and a share of the total reactive voltage, must have a d-component equal to $$U_{AB\_d}$$. A practical calculation method involves first finding the required $$U_{AB\_d}$$ from the module with the highest active power (the most stressed module) if it were to operate at the maximum allowable modulation index $$M_{max}$$. The required phase angle is then:
$$\phi = \arccos\left(\frac{U_{AB\_d}}{U_s}\right)$$
And the reactive current reference is:
$$I_{sq}^* = I_{sd} \tan \phi$$
The distribution of the total reactive power generation among the modules is a critical design choice. A fair and effective strategy is to allocate the reactive voltage (or q-axis modulation index) according to each module’s “capacity” to handle it without over-modulating. The module with the highest active power (closest to over-modulation) is assigned zero reactive compensation. For other modules, the maximum allowable q-axis modulation index is:
$$m_{x\_q}^{max} = \sqrt{M_{max}^2 – m_{x\_d}^2}$$
The total required q-axis voltage is distributed proportionally to this available capacity:
$$m_{x\_q} = \frac{m_{x\_q}^{max}}{\sum_{k=1, k \neq x_{max}}^{n} m_{k\_q}^{max}} \cdot \frac{U_{AB\_q}}{u_{dcx}}$$
This ensures that after allocation, no module exceeds the linear modulation limit. The following table summarizes the key parameters and their role in the control strategy:
| Symbol | Description | Role in Control Strategy |
|---|---|---|
| $$P_x$$ | Active power from module x | Measured, used to calculate $$m_{x\_d}$$ and stability limit. |
| $$u_{dcx}$$ | DC-link voltage of module x | Regulated to MPP voltage via a dedicated PI controller. |
| $$M_{max}$$ | Max linear modulation index (e.g., $$\sqrt{2}/2$$) | Threshold for over-modulation detection. |
| $$m_{x\_d}$$ | d-axis (active) modulation index for module x | Calculated as $$P_x U_s / (P_T u_{dcx})$$. |
| $$m_{x\_q}^{max}$$ | Max allowed q-axis modulation index for module x | Determines reactive power capacity: $$\sqrt{M_{max}^2 – m_{x\_d}^2}$$. |
| $$I_{sq}^*$$ | Reactive current reference | Calculated from stability condition, input to current controller. |
| $$\phi$$ | Grid current phase shift | Result of reactive compensation, $$\phi = \arctan(I_{sq}^*/I_{sd})$$. |
A significant improvement in the implementation of this strategy, compared to prior arts, lies in the control block diagram architecture. In some previous methods, the reactive compensation angle was applied by directly rotating the current reference vector. This inadvertently changes the effective active current reference seen by the DC-link voltage controllers, affecting their dynamics and creating cross-coupling. In the optimized strategy proposed here, the active and reactive current references are decoupled. The active current reference $$I_{sd}^*$$ is generated solely by the aggregated output of the DC-link voltage PI controllers (after MPPT processing), ensuring it directly represents the total PV power. The reactive current reference $$I_{sq}^*$$ is calculated independently based on the over-modulation condition check and the power imbalance. These two orthogonal references are then used to synthesize the final current reference for a high-bandwidth current controller, such as a Proportional-Resonant (PR) or a synchronous frame PI controller. This architecture keeps the voltage control loop unaffected by the reactive power compensation, simplifying tuning and improving robustness.
It is important to quantify the side-effect of reactive power compensation: increased voltage stress on the DC-link capacitors. The instantaneous power flow from the DC to the AC side in a single-phase system has a double-line-frequency ripple. This ripple causes a corresponding double-line-frequency voltage ripple on the DC-link. When the inverter provides reactive power, the apparent power $$S_x$$ of a module increases for the same active power $$P_x$$, as $$S_x = \sqrt{P_x^2 + Q_x^2}$$. The peak-to-peak DC voltage ripple can be approximated by:
$$U_{pp\_x} \approx \frac{S_x}{2 \pi f_{grid} C_{dcx} u_{dcx}}$$
where $$f_{grid}$$ is the grid frequency and $$C_{dcx}$$ is the DC capacitance. The maximum DC-link voltage stress is thus:
$$u_{dcx}^{max} \approx U_{mpp\_x} + \frac{U_{pp\_x}}{2} = U_{mpp\_x} + \frac{\sqrt{P_x^2 + Q_x^2}}{4 \pi f_{grid} C_{dcx} U_{mpp\_x}}$$
where $$U_{mpp\_x}$$ is the MPP voltage. While reactive compensation increases this stress, the analysis shows the increase is often manageable for typical capacitor selections. The benefit of maintaining MPPT and low grid current distortion usually outweighs this minor increase in voltage ripple.
To validate the proposed control strategy for cascaded solar inverters, comprehensive simulation and experimental studies were conducted. A detailed simulation model of a 9-level single-phase CHB inverter was built. The results clearly demonstrated the limitation: under significant power imbalance, modules with higher power violated the stability condition $$M_x > \frac{\sqrt{2}}{2}$$, causing severe grid current distortion (THD >16%) when using unity power factor control. Upon activating the proposed reactive compensation strategy, the DC-link voltages of all modules remained stable at their MPP references, and the grid current THD was restored to below 2%, while the inverter operated with a non-unity power factor as required. The decoupled control architecture was also verified, showing no transient disturbance in the DC-link voltage loops when reactive compensation was initiated.
Experimental validation was performed on a 3-cell CHB prototype connected to the grid. The setup used programmable DC sources to emulate mismatched PV strings. The results mirrored the simulations. Under high imbalance, the traditional method led to visible current distortion and audible noise from over-modulation. With the reactive compensation algorithm enabled, the system maintained excellent current waveform quality with THD below 4%, even when one module delivered more than twice the power of another. The DC-link voltages were well-regulated, confirming the stability of the approach. The effectiveness of the decoupled current reference generation was also confirmed experimentally, as the transition into reactive compensation mode did not cause any significant deviation in the regulated DC voltages.
In conclusion, the operational stability of cascaded H-bridge solar inverters under realistic, uneven irradiance conditions is a pivotal concern for system reliability and power quality. The stability boundary for unity power factor operation is clearly defined by the derived inequality. The proposed reactive power compensation strategy successfully expands this stable operating region by utilizing the inverter’s inherent capability to generate reactive power as an additional control freedom. The core of the method involves calculating the minimum required reactive current to prevent any module from entering over-modulation, followed by an equitable distribution of the reactive load among modules based on their individual capacity. Furthermore, an optimized control structure that decouples the active and reactive current references eliminates undesirable interaction with the vital DC-link voltage control loops. While introducing a measurable increase in DC-link voltage ripple, the benefits of maintaining maximum power extraction and high-quality grid current are substantial. This strategy, validated through simulation and experiment, provides a robust, efficient, and cost-effective solution for enhancing the fault tolerance and performance of modular multilevel solar inverters in the face of inevitable power imbalances, contributing significantly to the advancement of control methodologies for modern solar inverters.