The escalating integration of photovoltaic (PV) generation into the power grid has accentuated its impact on grid stability and power quality. The on grid inverter, serving as the critical interface for energy conversion between PV panels and the utility grid, plays a pivotal role in this dynamic. Among various topologies, the Cascaded H-Bridge (CHB) on grid inverter offers distinct advantages for PV applications, including multilevel voltage output, reduced switching device stress, lower grid current harmonic content, smaller filter size, and inherent modularity for scalability. Furthermore, by appropriately configuring the number of cascaded cells, its output voltage can match the grid level, enabling transformerless connection and thereby reducing overall system cost. Consequently, the CHB topology is exceptionally well-suited for PV on grid inverter systems.
In a typical CHB PV on grid inverter system, each H-bridge module is independently connected to a PV string. Ideally, all strings operate at their individual Maximum Power Point (MPP). However, in practical scenarios, PV modules experience disparate irradiance levels, temperatures, and aging effects, leading to significant power imbalances among the modules. Under such conditions, H-bridge modules with higher power output are prone to overmodulation—where their modulation index exceeds 1—causing severe distortion in the grid-injected current and degrading power quality. While strategies like the Third Harmonic Compensation Strategy (THCS) can extend the linear modulation range to 1.155 and mitigate issues for moderate imbalances, they fail under severe power imbalance where the modulation index of an over-modulated cell surpasses this limit. This paper addresses this limitation by proposing a novel overmodulation control strategy. The core idea is to actively adjust the modulation wave amplitude and inject specific harmonic components for over-modulated cells while implementing harmonic suppression control in non-overmodulated cells, thereby preventing overmodulation even when cell modulation indices exceed 1.155 and ensuring high-quality grid current.
System Configuration and Overmodulation Problem Formulation
The topology of a single-phase CHB PV on grid inverter is illustrated above. It comprises ‘n’ series-connected H-bridge modules. The DC side of each module is connected to a PV string, and the AC sides are cascaded, feeding into the grid through a filtering inductor \(L_s\). Here, \(I_{PVn}\) and \(V_{dcn}\) represent the output current and voltage of the n-th PV string, \(C_n\) is the DC-link capacitor, \(V_{Hn}\) is the output voltage of the n-th H-bridge, and \(v_g\) and \(i_g\) denote the grid voltage and current, respectively. For stable operation, the modulation wave \(m_i\) and the modulation index \(M_i\) for the i-th H-bridge cell are defined as:
$$m_i = \frac{v_i}{V_{dci}}$$
$$M_i = \frac{V_i}{V_{dci}}$$
where \(v_i\) and \(V_i\) are the fundamental component and its amplitude of the output voltage \(V_{Hi}\).
Analysis of the Overmodulation Problem
Consider a scenario where PV strings 1 to \(x\) experience reduced irradiance, decreasing their output powers \((P_1 \ldots P_x)\), while strings \(x\) to \(n\) maintain normal irradiance and output powers \((P_x \ldots P_n)\). The total system output power \(P_T\) decreases. Since the grid voltage \(v_g\) is constant, the grid current \(i_g\) must decrease proportionally. However, the current through all series-connected H-bridge cells remains identical (equal to \(i_g\)). Consequently, the H-bridge cells with unchanged input power must increase their output voltage to maintain their power level, leading to an increase in their modulation wave amplitude. This can drive the modulation index \(M_i\) beyond 1, causing overmodulation. A necessary condition for stable operation without overmodulation in any cell is given by:
$$I_g \ge \frac{\sqrt{2} P_i}{V_{dci}}$$
for all cells \(i = 1, 2, \ldots, n\), where \(I_g\) is the RMS value of \(i_g\). When this condition is violated for any cell, that cell enters overmodulation, injecting significant low-order harmonics into the grid current. Traditional methods like THCS extend the stable range by compensating the third harmonic, effectively increasing the maximum linear modulation index to 1.155. However, for more severe power imbalances, this range is insufficient. This work proposes a strategy to handle modulation indices greater than 1.155, pushing the effective operational boundary further.
Proposed Overmodulation Control Strategy
The proposed strategy intelligently manages the modulation waves across different H-bridge cells based on their instantaneous modulation index \(M_i\), calculated from the cell’s output power and the total system power. The system categorizes cells into three operational states and applies distinct control actions to each group.
Mathematical Foundation and Modulation Wave Reformulation
The fundamental modulation wave for each cell, derived from current control loops, is \(M_i \cos(\omega t + \theta_r)\). The proposed method modifies this wave depending on the cell’s state.
For a cell whose original modulation index \(M_a\) is between 1 and 1.155, the goal is to inject a third harmonic to bring the peak of the composite wave down to 1. The modified modulation wave \(m_x\) is:
$$m_x = M_b \cos(\omega t) + k_x \cos(3\omega t)$$
where \(k_x\) is the compensation coefficient and \(M_b = M_a\). To minimize harmonic injection, \(k_x\) is chosen such that \(\max(m_x) = 1\). The relationship between \(k_x\) and the achievable fundamental modulation index \(M_i\) (where \(M_i\) is the amplitude of the fundamental component in \(m_x\)) is derived from:
$$M_i = [\max(\cos(\omega t + \theta_r) + k_i \cos(3\omega t + 3\theta_r))]^{-1}$$
A fifth-order polynomial provides a precise fit for this relationship, enabling real-time calculation of \(k_i\):
$$k_i = B_5 M_i^5 + B_4 M_i^4 + B_3 M_i^3 + B_2 M_i^2 + B_1 M_i + B_0$$
The polynomial coefficients obtained via fitting are listed in Table 1.
| Coefficient | Value | Coefficient | Value |
|---|---|---|---|
| \(B_5\) | -0.1676 | \(B_2\) | 2.0432 |
| \(B_4\) | 0.8958 | \(B_1\) | -1.0902 |
| \(B_3\) | -1.9137 | \(B_0\) | 0.2326 |
For a cell where \(M_a > 1.155\), exceeding the capability of simple third-harmonic injection, a two-step approach is used. First, a fixed third harmonic component corresponding to the maximum compensation (\(k_i = -1/6\), \(M_i=1.155\)) is injected. Second, the fundamental component’s amplitude is reduced. The modifications are:
Fixed third harmonic injection: \(m_{hi} = -0.1925 \cos(3\omega t + 3\theta_r)\)
Fundamental amplitude adjustment: \(m_{fi} = -(M_a – 1.155) \cos(\omega t + \theta_r)\)
The combined modified modulation wave ensures its peak is clamped at 1.
For cells with \(M_i < 1\) (non-overmodulated), a harmonic suppression control is applied to cancel the unwanted harmonic currents (primarily the 3rd order) generated by the actions taken in the overmodulated cells. The compensation needed is distributed among these cells based on their available modulation margin \(M_{coi} = 1 – M_i\).
System Control Architecture and Modulation Wave Allocation
The overall control system for the CHB on grid inverter consists of a central controller and individual H-bridge controllers, as shown in the conceptual block diagram.
Central Controller and Cell State Classification
The central controller performs grid synchronization, total current control, and calculates the global modulation reference. Using phase-locked loop (PLL) and Park transformations, it generates the total reference modulation voltage amplitude \(V_r\) and its phase angle \(\theta_r\) relative to the grid. The modulation index for each cell is computed based on its share of the total power:
$$M_i = \frac{P_i}{P_T} \frac{V_r}{V_{dci}}$$
Based on the calculated \(M_i\), the central controller classifies the ‘n’ cells into three groups:
- Group 1 (Cells 1 to p): \(1 \le M_i \le 1.155\)
- Group 2 (Cells p+1 to p+q): \(M_i > 1.155\)
- Group 3 (Cells p+q+1 to n): \(M_i < 1\)
Modulation Wave Synthesis per Group
For Group 1: The compensation coefficient \(k_i\) is calculated using the polynomial in Eq. (9) and Table 1. The total third harmonic to be injected by this group is \(m_T = \sum_{i=1}^{p} k_i M_i \cos(3\omega t + 3\theta_r)\).
For Group 2: Each cell applies the fixed harmonic injection \(m_{hi}\) and the fundamental adjustment \(m_{fi}\). The total adjustments from this group are summed.
For Group 3 (Harmonic Suppression): A harmonic extractor (e.g., using a Second-Order Generalized Integrator or a 150 Hz notch filter) isolates the harmonic component \(i_{gh}\) from the grid current \(i_g\). A quasi-Proportional-Resonant (PR) controller, tuned at 150 Hz, regulates this harmonic current to zero. Its output is a corrective harmonic voltage \(v_{PR}\). This total corrective voltage is distributed to each non-overmodulated cell in proportion to its available margin:
$$m_{hoi} = \frac{M_{coi}}{\sum_{i=p+q+1}^{n} M_{coi}} \cdot \frac{v_{PR}}{V_{dci}}$$
The final modulation wave for a Group 3 cell is: \(m_{xi} = m_{hoi} + M_i \cos(\omega t + \theta_r)\). This active harmonic suppression in the non-overmodulated cells ensures that the distortion caused by the harmonic injection in the overmodulated cells is canceled out at the point of grid connection, maintaining high-quality sinusoidal current.
Simulation Verification and Comparative Analysis
To validate the effectiveness of the proposed overmodulation control strategy for the CHB on grid inverter under significant power imbalance, a detailed simulation model of a single-phase seven-level inverter (n=3) was developed in MATLAB/Simulink. The system parameters for the PV modules and the inverter are summarized in Table 2 and Table 3, respectively.
| Parameter | Value |
|---|---|
| Maximum Power (\(P_{max}\)) | 189 W |
| Voltage at MPP (\(V_{mpp}\)) | 36.7 V |
| Current at MPP (\(I_{mpp}\)) | 5.14 A |
| Short-Circuit Current (\(I_{sc}\)) | 5.5 A |
| Open-Circuit Voltage (\(V_{oc}\)) | 44.8 V |
| Parameter | Value |
|---|---|
| Number of H-bridge Cells (N) | 3 |
| Grid Frequency (\(f_g\)) | 50 Hz |
| Grid Voltage Amplitude (\(V_M\)) | 70 V |
| Switching/Carrier Frequency (\(f_{car}\)) | 3 kHz |
| Filter Inductance (\(L_s\)) | 3 mH |
| DC-link Capacitance (\(C_i\)) | 8 mF |
Two distinct irradiation conditions were simulated to create different levels of power imbalance, as defined in Table 4. The system performance was evaluated at t=0.5s after the irradiance change.
| Condition | Before 0.5s (String1, String2, String3) | After 0.5s (String1, String2, String3) |
|---|---|---|
| Condition 1 (Moderate Imbalance) | 800, 800, 1000 | 360, 350, 1000 |
| Condition 2 (Severe Imbalance) | 800, 800, 1000 | 250, 350, 1000 |
Performance under Condition 1 (Moderate Imbalance)
Under Condition 1, the modulation index of the over-modulated cell (String 3) remains below 1.155. A baseline simulation with no power balancing control shows that after t=0.5s, Cell 3 enters overmodulation (\(M_3>1\)), causing severe grid current distortion with a Total Harmonic Distortion (THD) of 20.25%. When the Third Harmonic Compensation Strategy (THCS) is applied, it successfully brings the modulation wave peak below 1, reducing the grid current THD to 3.57%. This confirms THCS’s effectiveness within its designed linear range. The proposed strategy also performs excellently in this condition.
Performance under Condition 2 (Severe Imbalance) and Comparative Analysis
Condition 2 introduces a more severe irradiance drop, pushing the modulation index of Cell 3 well beyond 1.155 (simulated value >1.4). This scenario critically tests the limitation of THCS and the capability of the proposed method.
With THCS: Although THCS injects the third harmonic, the modulation wave for Cell 3 remains above 1 after compensation because the required reduction exceeds what the third harmonic alone can provide. Consequently, the on grid inverter system experiences overmodulation, leading to a distorted grid current with a THD of 10.6%, which fails to meet power quality standards.
With the Proposed Overmodulation Control Strategy: The controller correctly identifies Cell 3 as belonging to Group 2 (\(M_i > 1.155\)). It applies the combined fixed third-harmonic injection and fundamental amplitude reduction, resulting in a modified modulation wave for Cell 3 with a peak value precisely at 1. Simultaneously, the harmonic suppression control in Cells 1 and 2 (Group 3) actively cancels the induced harmonic currents. The result is a clean, sinusoidal grid current with a significantly lower THD of 2.03%, demonstrating successful avoidance of overmodulation.
The key outcomes from the simulation study are consolidated in Table 5. The results unequivocally show that while THCS is effective within its limited modulation range (up to 1.155), it fails under more severe power imbalances. The proposed overmodulation control strategy successfully handles these severe cases, maintaining excellent grid current quality (THD ~2%) even when individual cell modulation indices exceed 1.155, thereby significantly extending the stable operating range of the cascaded H-bridge on grid inverter.
| Irradiance Condition | Control Strategy | THD before 0.5s | THD after 0.5s | Effective Linear Range |
|---|---|---|---|---|
| Condition 1 (Moderate) | No Balancing Control | 2.68% | 20.25% | M_i < 1 |
| THCS | 2.68% | 3.57% | M_i ≤ 1.155 | |
| Condition 2 (Severe) | Proposed Strategy | 0.56% | 2.03% | M_i > 1.155 (Handled) |
| THCS | 2.68% | 10.6% | Fails for M_i > 1.155 |
Conclusion
This paper has addressed a critical limitation in the operation of single-phase cascaded H-bridge photovoltaic on grid inverters under severe module-level power imbalance. While existing methods like the Third Harmonic Compensation Strategy offer improved performance, their linear modulation range is capped at 1.155. The proposed novel overmodulation control strategy overcomes this barrier. By strategically classifying H-bridge cells based on their instantaneous modulation index and applying tailored actions—amplitude adjustment with third-harmonic injection for severely over-modulated cells and active harmonic suppression for non-overmodulated cells—the strategy ensures that no cell enters overmodulation. This is achieved even when the calculated modulation index of a cell exceeds 1.155. Comprehensive simulation studies confirm the superiority of the proposed method. Under severe imbalance conditions where THCS fails and causes significant current distortion (THD=10.6%), the proposed strategy maintains high-quality grid current with a THD of 2.03%. Therefore, this strategy significantly enhances the operational robustness and power quality of CHB-based on grid inverter systems in real-world environments with uneven irradiance, contributing to more reliable and efficient solar energy integration.