1. Introduction
In the context of the global energy transition, solar energy has become a critical component of renewable energy systems. Photovoltaic (PV) power generation, as a clean and sustainable alternative to conventional energy sources, is expanding rapidly worldwide. According to data released by the National Energy Administration, the cumulative installed capacity of solar power has surpassed other renewable energy technologies. From a residential perspective, distributed PV systems provide an effective solution for reducing electricity costs and improving energy self-sufficiency.
The inverter is the core component of a PV system. It converts DC electricity generated by solar panels into grid-compatible AC power. Due to its modular structure, lower switching losses, and high-quality output waveforms, the cascaded H-bridge (CHB) multilevel inverter is particularly well suited for grid-connected PV applications. One of the important advantages of this topology is its ability to achieve module-level maximum power point tracking (MPPT). Nevertheless, when the irradiance across PV modules is not uniform, the system suffers from power imbalance among H-bridge cells, which can cause over-modulation and degraded performance. Addressing this issue is essential for improving the overall efficiency and reliability of PV systems.
This paper focuses on the single-phase CHB PV grid-connected inverter and its power balance control strategies. The research includes a comparative study of different modulation techniques, examination of power imbalance mechanisms, and development of harmonic compensation-based and hybrid modulation-based control strategies to widen the linear modulation range and ensure stable operation under non-uniform illumination conditions.
| Parameter | Value |
|---|---|
| PV module model | STP270-24-Vd |
| Open-circuit voltage | 44.5 V |
| Short-circuit current | 8.2 A |
| MPP voltage | 35 V |
| MPP current | 7.71 A |
| Parameter | Value |
|---|---|
| Number of CHB cells | 4 |
| Grid voltage amplitude | 115 V |
| Grid frequency | 50 Hz |
| Carrier frequency | 3 kHz |
| DC-link capacitance | 18.8 mF |
| Filter inductance | 5.6 mH |
2. Topology and Modulation Strategies of CHB Inverters
The CHB inverter consists of several H-bridge cells connected in series at the AC side, each powered by an isolated DC source. The output voltage is the sum of each cell’s output. For \(n\) cells, the AC output can reach \(2n-1\) levels. This design provides high modularity, easier fault tolerance, and offers excellent scalability for medium- and high-voltage applications. The H-bridge has four primary switching states, which are listed in the following table.
| Switching State (\(S_{n1}\), \(S_{n2}\), \(S_{n3}\), \(S_{n4}\)) | Output Voltage \(u_{on}\) |
|---|---|
| (1, 0, 0, 1) | +E |
| (1, 1, 0, 0) | 0 |
| (0, 0, 1, 1) | 0 |
| (0, 1, 1, 0) | −E |
The modulation strategy plays a decisive role in harmonic performance and power distribution. Two high-frequency carrier-based strategies, namely carrier phase-shifted PWM (CPS-PWM) and in-phase disposition PWM (IPD-PWM), have been extensively compared in our study. With CPS-PWM, each cell uses a phase-shifted carrier, which naturally distributes both active and reactive power evenly across cells. This is particularly important for PV applications where each module has an independent MPPT controller. In contrast, IPD-PWM presents lower output voltage THD due to its harmonic cancellation properties but causes unequal power sharing among cells under partial shading conditions, which is undesirable in PV applications.
To further simplify the DC-side configuration, a single DC-source seven-level CHB topology was studied. This topology replaces two of the three DC sources with floating capacitors, resulting in a DC-link voltage ratio of 1:1:1. The total output voltage is expressed as:
\[
u_o(t) = u_{o1}(t) + u_{o2}(t) + u_{o3}(t)
\]
The load current is given by:
\[
i_o(t) = I_m \sin(\omega t – \varphi)
\]
where \(I_m\) is the current amplitude, and \(\varphi\) is the impedance angle. The amplitude can be calculated as:
\[
I_m = \frac{ME}{Z} = \frac{3ME}{R + j\omega L}
\]
where \(M\) is the modulation index, \(E\) is the rated DC voltage of each cell, and \(Z\) is the load impedance.
Four modulation modes were defined to handle the charging and discharging of capacitor cells. Mode 4 is described in the following table.
| Mode | Output voltage | \(u_{o1}\) | \(u_{o2}\) | \(u_{o3}\) |
|---|---|---|---|---|
| Mode 4 | +3E | 1 | 1 | 1 |
| +2E | 1 | 0 | 1 | |
| +E | 1 | 1 | −1 | |
| 0 | 1 | −1 | 0 | |
| −E | 1 | −1 | −1 | |
| −2E | 0 | −1 | −1 | |
| −3E | −1 | −1 | −1 |
The mode selection is based on capacitor voltage errors and current direction. The capacitor voltage deviation is defined as:
\[
\Delta u_{ci} = u_{ci}^* – u_{ci}
\]
The charging and discharging conditions are evaluated using \(\Delta u_{ci} \cdot i_o\). When \(\Delta u_{ci} i_o < 0\), the capacitor is charged; when \(\Delta u_{ci} i_o > 0\), it is discharged. The dynamic control ensures the capacitor voltages remain balanced at the reference value.
To ensure capacitor voltage ripple remains within 10% of the rated value, the minimum capacitance value is derived from the maximum continuous discharge interval:
\[
C_{\text{min}} = \frac{\Delta Q}{0.1E}
\]
\[
\Delta Q = \int_{\theta_1}^{\theta_2} I_m |\sin(\omega t – \varphi)| D_2(\theta)\, d\theta + \int_{\theta_2}^{\pi-\theta_1} I_m |\sin(\omega t – \varphi)| D_3(\theta)\, d\theta
\]
where \(D_2\) and \(D_3\) are duty cycles for the respective voltage levels. The capacitance value of 300 μF was calculated for a modulation index of 0.75, and the derived value was validated via simulations, which showed that the ripple coefficient was below the required limit.
3. Single-Phase CHB PV System and Control Strategies
In this section, the single-phase CHB PV inverter topology is first established. Then, its mathematical model is derived. Each H-bridge module is connected to a PV module and a DC-link capacitor. The Kirchhoff current law applied at the DC side is:
\[
i_{pvi} = i_{ci} + i_{oi}
\]
The power balance equation that links the DC input power and AC output power can be expressed as:
\[
\sum_{i=1}^{n} P_i = P_{\Sigma} = V_g I_g
\]
where \(V_g\) and \(I_g\) are the amplitudes of the grid voltage and current. According to the power conservation principle, when some PV cells receive reduced irradiance, their output power drops. Consequently, the total power \(P_{\Sigma}\) decreases, which reduces the grid current \(I_g\). However, the unaffected cells still need to deliver their maximum power. Since the grid voltage remains constant, the output voltage of those healthy cells increases, leading to a higher modulation index:
\[
M_i = \frac{V_{oi}}{V_{dci}}
\]
This behavior explains why over-modulation occurs in healthy H-bridge units when other units experience peak power mismatch.
3.1 MPPT Control
The MPPT control ensures that each PV module operates at its maximum power point under varying irradiance and temperature conditions. Two methods were examined: perturb and observe (P&O) and incremental conductance (INC). The P&O method continuously perturbs the PV voltage and compares the instantaneous power before and after perturbation. The step size of the reference voltage is adjusted based on the sign of \(\Delta P\) and \(\Delta V\). Although the method is straightforward to implement, it suffers from steady-state oscillation around the MPP.
The INC method is based on the derivative of power with respect to voltage:
\[
\frac{dP}{dV} = I + V \frac{dI}{dV} = 0
\]
The working point can be identified by comparing the instantaneous conductance \(I/V\) and the incremental conductance \(dI/dV\). This approach achieves higher MPPT accuracy, particularly during rapid changes in irradiance, at the cost of a more complex control algorithm. For the CHB-based PV system, each module has its own MPPT controller, which enables module-level maximum power point tracking and minimizes mismatch losses due to partial shading.
3.2 DC-link Voltage Control
The control strategy used in this study is the centralized DC-link voltage control. The outer voltage loop generates the DC current reference by comparing the MPPT voltage with the actual PV voltage. The error is passed through a PI controller. The output power reference of each unit is then summed to obtain the total power reference \(P_{dc}^*\). According to the power balance principle:
\[
P_{dc}^* = P_1^* + P_2^* + \dots + P_n^*
\]
Upon dividing the total power reference by the grid voltage amplitude \(V_m\), the amplitude of the grid current reference \(I_{ref}\) is obtained. Since the single-phase system has an inherent double-frequency ripple in the DC voltage, a 100 Hz notch filter is inserted in the voltage control loop to suppress this unwanted component. The transfer function of the notch filter is:
\[
G_{\text{notch}}(s) = \frac{s^2 + \omega_0^2}{s^2 + 2 \xi \omega_0 s + \omega_0^2}
\]
where \(\omega_0 = 2\pi \cdot 100\) rad/s, and \(\xi\) is the bandwidth parameter. Our simulation results confirmed that the notch filter effectively removes the second harmonic component from the DC-side voltage. The introduction of this filter prevents the propagation of third-order harmonics into the grid current, thereby significantly improving the output power quality.
3.3 Grid Synchronization based on SOGI-PLL
Grid synchronization is essential for grid-connected inverters. The second-order generalized integrator-based phase-locked loop (SOGI-PLL) generates two orthogonal signals \(u_\alpha\) and \(u_\beta\). These signals are transformed to the \(dq\) frame, and the \(q\)-axis component is driven to zero by a PI regulator. The closed-loop transfer functions of the SOGI are given by:
\[
H_\alpha(s) = \frac{k\omega s}{s^2 + k\omega s + \omega^2}
\]
\[
H_\beta(s) = \frac{k\omega^2}{s^2 + k\omega s + \omega^2}
\]
The parameter \(k\) determines the bandwidth of the SOGI and influences the filtering performance and dynamic response. The Bode plot analysis indicates that the SOGI-PLL provides a fast and accurate phase lock capability while maintaining harmonic immunity. The PLL output frequency quickly stabilizes at the grid frequency, and the phase error converges to zero in steady state.
3.4 Current Control using Proportional Quasi-Resonant (PQR) Controller
For the inner current loop, the grid current must track a sinusoidal reference without steady-state error. The conventional PI controller is not appropriate here since the reference is sinusoidal. Hence, a proportional quasi-resonant (PQR) controller is adopted. Its transfer function is:
\[
G_{PR}(s) = k_p + \frac{2k_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2}
\]
where \(k_p\) and \(k_r\) are the proportional and resonant gains, respectively, \(\omega_c\) is the cutoff frequency, and \(\omega_0\) is the resonant frequency. By tuning these parameters, the current controller provides high gain at the fundamental frequency, ensuring accurate tracking of the sinusoidal current reference and rejecting disturbances. The current inner loop output is added to the grid voltage feedforward term, resulting in the modulation voltage reference \(V_r\). This combined approach enhances the dynamic response and robustness of the system against grid voltage distortions.
4. Power Balance Control Based on Harmonic Compensation
To mitigate over-modulation in high-power cells under unbalanced power conditions, harmonic compensation strategies were studied. Our investigation began with the third harmonic compensation strategy (THCS), which is based on the injection of a third harmonic component into the modulation waveform. The compensated modulation waveform is expressed as:
\[
S_i(\omega t) = M_i \sin(\omega t) + k M_i \sin(3\omega t)
\]
where \(M_i\) is the fundamental amplitude of the modulation wave and \(k\) is the third harmonic compensation coefficient. By examining the three-dimensional surface of this function, we found that the minimum peak value occurs when \(k = 1/6\). At this optimal value, the resulting waveform reaches a minimum amplitude of approximately 0.866. Therefore, the linear modulation range of THCS is limited to:
\[
M_{\max,\text{THCS}} = \frac{1}{0.866} = 1.155
\]
However, the THCS exhibits an inherent drawback. The unmodulated cells must inject the opposite phase of the injected third harmonic, which increases their peak amplitude and may cause these cells to become over-modulated. This disadvantage restricts the applicability of THCS in cases of severe power imbalance.
To extend this range further, the multiple harmonic compensation control strategy (MHCCS) was developed. The MHCCS replaces the over-modulated cell’s sinusoidal modulation with a square-wave-like waveform \(g(x)\) expressed as:
\[
g(x) =
\begin{cases}
0 & 2k\pi – \pi/2 \leq x < 2k\pi – \pi/2 + \alpha \\
1 & 2k\pi – \pi/2 + \alpha \leq x < 2k\pi + \pi/2 – \alpha \\
0 & 2k\pi + \pi/2 – \alpha \leq x < 2k\pi + \pi/2 + \alpha \\
-1 & 2k\pi + \pi/2 + \alpha \leq x < 2k\pi + 3\pi/2 – \alpha \\
0 & 2k\pi + 3\pi/2 – \alpha \leq x < 2k\pi + 3\pi/2 + \alpha
\end{cases}
\]
This waveform is expressed in a Fourier series as:
\[
g(x) = \sum_{n=1}^{\infty} b_n \sin(nx)
\]
where the fundamental amplitude is:
\[
M_1 = \frac{4}{\pi} \cos(\alpha)
\]
For a fundamental amplitude \(M\) of the sinusoidal modulation wave, the conduction angle \(\alpha\) is chosen as:
\[
\alpha = \arccos\left( \frac{M}{4/\pi} \right)
\]
Thus, MHCCS is capable of extending the linear modulation range to \(4/\pi \approx 1.273\). Moreover, the anti-phase harmonic voltage should be injected into the unaffected cells to cancel the injected harmonics from the total output, which can be expressed as:
\[
v_{ohm} = \sum_{i=1}^{x} v_{hm_i} = \sum_{i=1}^{x} \left( g_i – S_i \right) V_{dci}
\]
The maximum compensation voltage for the non-over-modulated cells is:
\[
v_{hi_{\max}} = (1 – M_i) V_{dci}
\]
A self-adaptive power balance control strategy was further developed to handle cases where the modulation index exceeds \(4/\pi\). When the modulation index exceeds \(4/\pi\), the reference power of the corresponding cell is dynamically switched to a fixed power value \(P_h\). During this condition, the modulation waveform becomes a pure square wave with \(\alpha = 0\). The maximum power that can be delivered by the cell is given by:
\[
P_h = \frac{4}{\pi} \frac{V_{dc} V_r}{V_{dci}}
\]
By replacing the MPPT reference power with \(P_h\), the cell avoids over-modulation, and the power imbalance is effectively redistributed among the remaining cells. The adaptive strategy thus ensures stable operation under extreme shading conditions.
The overall control block diagram of the adaptive power balance control strategy was established. It consists of the MPPT control loop, the voltage control loop, the current control loop, and the adaptive control module. The module continuously monitors the modulation index of each cell and determines which compensation mode should be applied. When \(M_i < 1\), no harmonic compensation is needed; when \(1 < M_i < 4/\pi\), MHCCS is applied; and when \(M_i > 4/\pi\), the fixed power reference mode is activated.
Extensive simulations were carried out in MATLAB/Simulink environment on a four-cell CHB PV inverter. The operating scenario was designed to simulate gradual changes in irradiance. At \(t=0.6\) s, the irradiance of cells 3 and 4 were reduced to 450 W/m² and 300 W/m², respectively, and at \(t=1.0\) s, it was further reduced to 140 W/m² for both cells. The simulation results demonstrated that MHCCS can keep the grid current THD below 1.7% in the moderate shading case, and the adaptive strategy maintains THD at only 1.79% even in severe shading conditions. In contrast, without additional control, THD rose to 11.11% under the same shading scenario, confirming the effectiveness of the proposed control methods.
5. Power Balance Control Based on Hybrid Modulation Strategy
In addition to harmonic compensation techniques, hybrid modulation strategy (HMS) was also investigated in our research. HMS combines high-frequency SPWM and low-frequency square-wave modulation. Compared to the conventional CPS-SPWM method, HMS can significantly extend the linear modulation range by taking advantage of the square-wave operation available in some cells.
The control block diagram of the HMS-based power balance control is shown. The DC voltage external loop generates the grid current reference amplitude, while the inner current loop produces the total modulation voltage reference \(V_r\). The HMS module then receives \(V_r\), the grid current \(i_g\), the DC-link voltage \(v_{dci}\), and the MPPT voltage \(v_{mppti}\). The DC voltage deviation of each cell is calculated as:
\[
\Delta V_{dvi} = v_{dci} – v_{mppti}
\]
The DC voltages are sorted in ascending order by \(\Delta V_{dvi}\). Then, the total modulation voltage level \(k\) is determined by:
\[
\sum_{i=1}^{k-1} V_{dci} \leq |V_r| \leq \sum_{i=1}^{k} V_{dci}
\]
The switching states are assigned based on \(V_r\), \(i_g\), and the parity of \(n-k\). The cell with the lowest \(\Delta V_{dvi}\) is set to charge, and the cell with the highest \(\Delta V_{dvi}\) is set to discharge. The use of square-wave operation allows cells to output either \(+V_{dci}\), \(-V_{dci}\), or PWM reference, thereby raising the PWM band without exceeding the linear limit. In HMS, the maximum modulation index is expanded to \(4/\pi\).
However, an important problem was observed in the HMS approach: the DC capacitor voltage ripple is relatively large. The capacitor voltage of each unit exhibits noticeable oscillations, especially when all \(\Delta V_{dvi}\) values have the same sign. To reduce this voltage fluctuation, the improved hybrid modulation strategy (IHMS) was proposed. In this method, a “0” state is inserted into the output states of the cells. By assigning one cell the “0” output state, the DC capacitor is neither charged nor discharged during that period, which mitigates the over-charging or over-discharging issue.
The simulation results show that in the HMS case, the DC voltage peak-to-peak error ranges from 3.07 V to 3.44 V. With the IHMS, the maximum peak-to-peak voltage error is reduced to within 1.8 V to 2.4 V. Thus, the IHMS significantly improves the DC voltage stability while maintaining the same linear modulation range of \(4/\pi\).
Both HMS and IHMS were tested under dynamic irradiance changes. The system responses were recorded and compared. Under balanced operating conditions with irradiance of 1000 W/m², the grid current THD values were 2.13% for HMS and 2.38% for IHMS, both of which are within acceptable grid standards. Under non-uniform irradiance, the THD increased to about 2.78% for HMS and 2.83% for IHMS, confirming the robust performance of both methods. The output power waveforms confirm that each PV module can quickly return to its new maximum power point after irradiance changes.
The key contributions of this chapter include the detailed analysis of the PWM lifting principle, the introduction of the “0” state concept in modulation, and a comprehensive comparison of the two hybrid strategies in terms of waveform quality, modulation range, and DC voltage ripple.
6. Conclusion
In this dissertation, the single-phase CHB photovoltaic grid-connected inverter and its key technologies have been systematically investigated. The main research achievements are summarized as follows:
(1) A comprehensive comparison between the conventional CHB inverter and a novel CHB topology with partial DC source replacement by floating capacitors was carried out. A modulation strategy for the latter topology was proposed and validated through both simulations and experiments. The results demonstrate that the proposed topology greatly reduces the number of DC sources while maintaining high output quality.
(2) A complete control framework for the single-phase CHB PV inverter was established. The framework integrates module-level MPPT, centralized DC voltage control with double-frequency ripple suppression, SOGI-PLL synchronization, and proportional quasi-resonant current control. The analytical and simulation results confirm the effectiveness of this framework and reveal the inherent mechanism of power imbalance in a cascaded PV system.
(3) To address the power imbalance problem, the multiple harmonic compensation control strategy (MHCCS) and the adaptive power balance control strategy were studied. MHCCS effectively reduces the peak of the modulation waveform and expands the linear modulation range to \(4/\pi\). The adaptive strategy further extends the operating range beyond \(4/\pi\) by redistributing reference power among cells. Simulation results verified that the grid current THD remains low under severe irradiance mismatch conditions.
(4) Hybrid modulation strategies, including the original HMS and the improved IHMS, were investigated. Both schemes enable a linear modulation range of \(4/\pi\). Compared with HMS, the IHMS introduces a “0” state and significantly reduces DC capacitor voltage fluctuations, thereby improving the reliability and durability of the inverter system.
For future work, the following directions should be considered. First, the coupling effect between the single DC-source topology and the PV grid-connected system deserves further research. Second, the adaptive control strategy can be further refined using machine learning algorithms to achieve faster and more robust dynamic responses. Third, more comprehensive experimental validations should be conducted on a hardware-in-the-loop platform to bridge the gap between simulation and real-world deployment.
In summary, this research contributes valuable insights into the design and control of CHB-based solar inverters, especially regarding power balance under non-uniform conditions. The proposed strategies provide effective and practical solutions for the widespread application of distributed solar PV systems.