In this work, I present a comprehensive investigation into the grid-connected control strategy for quasi-Z-source cascaded multilevel photovoltaic solar inverters. The growing demand for high-efficiency and low-cost renewable energy conversion has motivated the adoption of advanced solar inverters that can perform both dc-dc boosting and dc-ac inversion in a single stage. The quasi-Z-source cascaded multilevel inverter (QZS-CMI) topology combines the advantages of the quasi-Z-source impedance network with the modular structure of cascaded H-bridge cells, resulting in improved reliability, independent maximum power point tracking (MPPT), balanced dc-link voltages, and high-quality grid current. My research focuses on modeling, modulation, control, and power-balance compensation under partial shading conditions.
In this paper, I first analyze the operational principles of the quasi-Z-source inverter and derive the corresponding boost mechanism. Then, I construct the state-space average model of a single quasi-Z-source H-bridge inverter (QZS-HBI) unit, establish the equivalent model of the whole cascaded system, and investigate the over-modulation mechanism caused by power imbalance. Based on the switch-state analysis, I propose an improved carrier phase-shifted sinusoidal pulse-width modulation (SPWM) strategy that fuses the shoot-through duty cycle into the conventional single-leg chopping modulation. For the overall control, I adopt a disturbance-observer-based MPPT method for each PV panel, followed by a voltage-current double-loop controller for the dc-dc stage. In the grid-side stage, a power outer loop and a current inner loop are designed to regulate the dc-link voltages and inject a sinusoidal grid current. To tackle severe power imbalance, I introduce an optimal third-harmonic compensation strategy that extends the modulation range while eliminating the third-harmonic component in the grid current. Finally, real-time simulation results from RT-LAB and a two-cell experimental platform verify the correctness and effectiveness of the proposed schemes.
1. Operating Principle and Modeling of the QZS-CMI System
The quasi-Z-source inverter topology is shown in the schematic diagram below. It consists of two inductors \(L_1,L_2\), two capacitors \(C_1,C_2\), one diode, and an H-bridge inverter. Compared with the traditional two-stage solar inverter, the quasi-Z-source solar inverter allows shoot-through operation of the same-phase bridge legs, which eliminates the dead-time effect and provides a single-stage boosting capability. The key relations are derived from the steady-state analysis of the shoot-through and non-shoot-through modes.
Let \(D\) be the shoot-through duty cycle, and let \(f_s=1/T\) be the switching frequency. During the shoot-through interval \(T_0=DT\), the H-bridge is shorted and the inductors store energy. During the non-shoot-through interval \(T_1=(1-D)T\), the impedance network delivers energy to the load. Applying KVL and KCL in both modes and using the volt-second and ampere-second balance principle, the capacitor voltages and the dc-link peak voltage are obtained as:
$$
\begin{aligned}
v_{C1} &= \frac{1-D}{1-2D}V_{\mathrm{PV}}, \\
v_{C2} &= \frac{D}{1-2D}V_{\mathrm{PV}}, \\
v_{\mathrm{PN}} &= v_{C1}+v_{C2} = \frac{1}{1-2D}V_{\mathrm{PV}} = B V_{\mathrm{PV}},
\end{aligned}
$$
where \(B\) is the boost factor. The peak ac-side output voltage of the H-bridge is then:
$$
\hat{v}_{\mathrm{out}} = M v_{\mathrm{PN}} = M B V_{\mathrm{PV}},
$$
where \(M\) is the modulation ratio. In order to ensure that the shoot-through states do not affect the active states, the following constraint must be satisfied:
$$
M + D \leq 1.
$$
For the design of the impedance network, I select the capacitance and inductance values based on the allowable voltage ripple and current ripple. The capacitor voltage ripple \(\Delta v_{C1}\) is related to the average inductor current \(I_{L2}\) by:
$$
\Delta v_{C1} = \frac{D}{2 f_s C_1} I_{L2}.
$$
Thus, with a ripple limit of \(\alpha\), the capacitor parameter is chosen as:
$$
C_1 \geq \frac{D I_{L2}}{2 f_s \alpha v_{C1}}.
$$
Similarly, the inductor current ripple \(\Delta i_{L2}\) is given by:
$$
\Delta i_{L2} = \frac{D v_{C1}}{2 f_s L_2},
$$
leading to the inductance selection rule:
$$
L_2 \geq \frac{D v_{C1}}{2 f_s \beta i_{L2}}.
$$
In my design, with \(f_s=20\,\mathrm{kHz}\), \(D=0.316\), \(V_{\mathrm{PV}}=36.8V\), and a desired ripple below \(5\%\), I choose \(C_1=C_2=5\,\mathrm{mF}\) and \(L_1=L_2=1\,\mathrm{mH}\). The diode is selected as a fast-recovery type with a peak reverse voltage higher than \(v_{\mathrm{PN}}\) and a current rating that covers the maximum inductor current.
For the complete QZS-CMI system, each cell contains one PV panel, one quasi-Z-source impedance network, and one H-bridge. With \(N\) cells connected in series, the total output voltage is the sum of the individual cell voltages:
$$
v_o = \sum_{n=1}^{N} v_{Hn} = \sum_{n=1}^{N} S_n v_{\mathrm{PN}n},
$$
where \(S_n \in \{-1,0,1\}\) is the switching function of the \(n\)-th H-bridge. I establish the state-space model of a single QZS-HBI cell using the averaging technique. The state vector is \(\mathbf{x} = [i_{L1}, i_{L2}, v_{C1}, v_{C2}, v_{\mathrm{PV}}]^T\), and the input vector is \(\mathbf{u} = [i_{\mathrm{PV}}, i_{\mathrm{PN}}]^T\). After applying the small-signal perturbation, the transfer function from the shoot-through duty cycle to the inductor current is derived as:
$$
G_{i_L,d}(s) = \frac{V_{C1}+V_{C2}+(I_{C1}+I_{C2})(1-2D)}{LC s^2 (1-2D) + 2(1-2D)},
$$
and the transfer function from the inductor current to the PV voltage is:
$$
G_{v_{\mathrm{PV}},i_L}(s) = \frac{LC s^2 + (1-2D)^2}{LC s^2 [(2D-1)] + (1-2D) C s + \cdots}.
$$
These transfer functions form the basis for designing the dc-dc double-loop controller.
2. Power Imbalance and Over-Modulation Analysis
In a QZS-CMI photovoltaic system, all series-connected cells carry the same grid current. Therefore, the output power of each cell is proportional to its output voltage. If some PV panels are shaded or degraded, their output powers decrease, and the total system power \(P_T\) also decreases. Since the grid voltage \(V_g\) is fixed, the grid current \(I_g\) must decrease. The modulation index of the \(n\)-th cell is given by:
$$
M_n = \frac{P_n}{P_T}\frac{V_H^*}{V_{\mathrm{PN}n}},
$$
where \(V_H^*\) is the amplitude of the total inverter output voltage. If \(M_n\) exceeds \(1-D_0\), the cell enters over-modulation. The condition for safe operation is:
$$
\frac{P_n}{P_T} \le \frac{(1-D_0)V_g}{2 V_{\mathrm{PN}n}}.
$$
When this inequality is violated, the output voltage waveform becomes clipped, and the grid current cannot track its reference, causing distortion and instability. Figure below shows the over-modulation mechanism in a two-cell example. During intervals when the reference current is larger than the actual current, the closed-loop controller increases the modulation voltage, but the modulation wave is already saturated, leading to a persistent current error.
3. Proposed Modulation Strategy
The conventional carrier phase-shifted SPWM can be classified into vertical and horizontal shifting methods. For cascaded multilevel inverters, the horizontal phase-shifted SPWM is preferred because it gives equal power sharing and reduces the output harmonic content. In my system, I use a unipolar single-leg chopping modulation with complementary switching on the same leg. By combining this technique with the shoot-through insertion, the H-bridge switches can be driven as shown in the timing diagram below.
The modulation principle for one cell is described as follows. Let \(m(t)=M\sin(\omega t)\) be the sinusoidal reference. A shoot-through comparison level is set to \(1-D\). In the positive half-cycle, when the triangular carrier is larger than \(1-D\), a shoot-through signal is generated for the upper switch of the left leg. In the negative half-cycle, the comparison with \(-(1-D)\) generates the shoot-through signal for the lower switch. The high-frequency legs are switched with the conventional sinusoidal PWM, while the low-frequency legs receive the superimposed shoot-through pulses. For \(N\) cells, the carriers are shifted by \(\pi/N\), yielding an equivalent switching frequency of \(2N f_s\). This method is called the improved carrier phase-shifted SPWM with fused shoot-through duty cycle.
The proposed modulation strategy has the following advantages:
- It fully utilizes the zero states to insert shoot-through without disturbing the active states.
- It reduces switching losses because only one leg operates at high frequency in a given half-cycle.
- It achieves a seven-level output voltage for three cells, which lowers the total harmonic distortion.
4. Control Strategy of the QZS-CMI System
4.1 Independent MPPT Control
Each PV panel is connected to its own quasi-Z-source network, so independent MPPT is possible. I adopt the perturb-and-observe (P&O) method. The flowchart of the algorithm is shown above. The PV voltage \(v_{\mathrm{PV}}\) and current \(i_{\mathrm{PV}}\) are sampled at a low rate (20 Hz), and the reference voltage \(v_{\mathrm{PV}}^*\) is adjusted in steps of \(\Delta v = 0.1\,\mathrm{V}\) to move the operating point toward the maximum power point. The output of the MPPT block provides the reference voltage for the dc-dc voltage loop.
4.2 DC-DC Double-Loop Control
For each QZS-HBI cell, an inner current loop regulates the inductor current \(i_{L1}\), while an outer voltage loop regulates the PV voltage. The control structure is shown in the block diagram above. The open-loop transfer functions are given by:
$$
T_{i\_ol}(s) = G_{\mathrm{PI2}}(s) G_{i_L,d}(s) M(s) H_{i}(s),
$$
$$
T_{v\_ol}(s) = G_{\mathrm{PI1}}(s) T_{i\_cl}(s) G_{v_{\mathrm{PV}},i_L}(s) H_{v}(s),
$$
where \(H_i(s)\) and \(H_v(s)\) are sampling gains (set to unity). I design the PI controllers to achieve a crossover frequency of 2400 Hz for the current loop with a phase margin of 78°, and a crossover frequency of 0.4 Hz for the voltage loop with a phase margin of 45°. The selected parameters are:
| Controller | Proportional gain | Integral gain |
|---|---|---|
| Current inner loop \(G_{\mathrm{PI2}}(s)\) | 0.15 | 500 |
| Voltage outer loop \(G_{\mathrm{PI1}}(s)\) | 0.045 | 0.1 |
| Grid current loop \(G_{\mathrm{PIg}}(s)\) | 40 | 400 |
| DC-link voltage balancing loop | 0.1 | 0.001 |
4.3 Grid-Current Control and DC-Link Voltage Balancing
The grid-side control consists of a power outer loop and a current inner loop. The total active power \(P_T\) is divided by the grid voltage amplitude \(V_M\) to obtain the amplitude of the grid current reference. A phase-locked loop (PLL) provides the phase angle \(\theta\), so the instantaneous current reference is:
$$
i_g^* = \frac{P_T}{V_M}\sqrt{2}\,\sin\theta.
$$
The error between \(i_g^*\) and the measured grid current \(i_g\) is processed by a PI controller. The controller output is added to the grid voltage \(v_g\) to generate the total voltage command \(v_H^*\). The current-loop open-loop transfer function is:
$$
G_{gc\_ol}(s) = \frac{K_p s + K_i}{L_g s^2},
$$
where \(L_g\) is the filter inductance. With \(K_p=40\) and \(K_i=400\), the closed-loop Bode plot shows zero steady-state error and negligible phase lag at 50 Hz.
For dc-link voltage balancing, each cell’s capacitor voltages are summed and compared with the common reference \(v_{\mathrm{PN}}^*\). The error is passed through a PI controller, and the output is multiplied by the measured dc-link peak voltage to obtain the individual power command \(P_n\). This ensures that all dc-link voltages are equal even under unequal PV power output.
4.4 Optimal Third-Harmonic Compensation for Power Imbalance
When severe power imbalance occurs, the modulation index of a high-power cell may exceed \(1-D_0\). To solve this problem, I propose to inject an optimal third harmonic into the modulation wave of the overmodulated cell. The original modulation wave is:
$$
m(t) = M \sin \omega t + kM \sin 3\omega t,
$$
where \(k\) is the third-harmonic compensation coefficient. For a given \(M\), the maximum value of \(m(t)\) depends on \(k\). The relationship between \(M\) and \(k\) is calculated by solving \(\max\{ \sin\varphi + k \sin 3\varphi \}=0.684\). To obtain the optimal \(k\) that maximizes the modulation range, I use a fifth-order polynomial fitting:
$$
k = A_0 + A_1 M + A_2 M^2 + A_3 M^3 + A_4 M^4 + A_5 M^5,
$$
where the coefficients are listed in the table below.
| Coefficient | Value |
|---|---|
| \(A_0\) | -5686.1898 |
| \(A_1\) | 40173.8322 |
| \(A_2\) | -113456.7850 |
| \(A_3\) | 160087.3579 |
| \(A_4\) | -112849.4621 |
| \(A_5\) | 31793.5754 |
With \(k=1/6\), the modulation range is extended from \(M=0.684\) to \(M=0.790\). For any overmodulated cell, the computed \(k\) makes the new modulation wave peak exactly equal to \(0.684\). To avoid injecting third harmonic into the grid current, the same amount of negative third harmonic is allocated to the non-overmodulated cells. The total third-harmonic voltage injected by the overmodulated cells is:
$$
V_{3T} = \sum_{n=1}^{x} k_n M_n v_{\mathrm{PN}n}.
$$
This voltage is distributed among the remaining cells proportionally to their modulation margins. The non-overmodulated cell \(j\) receives a third harmonic of amplitude:
$$
V_{3j} = \frac{(1-D_0 – M_j) v_{\mathrm{PN}j}}{\sum_{m=x+1}^{N}(1-D_0 – M_m) v_{\mathrm{PN}m}} V_{3T}.
$$
Thus, the modulation wave of the \(j\)-th non-overmodulated cell becomes:
$$
m_j(t) = M_j \sin \omega t – \frac{V_{3j}}{v_{\mathrm{PN}j}}\sin 3\omega t.
$$
This method guarantees that no cell exceeds the modulation limit while the grid current remains free of low-order harmonic distortion.
5. Real-Time Simulation and Experimental Validation
I built a three-cell QZS-CMI photovoltaic system in MATLAB/Simulink and executed it on the RT-LAB real-time simulation platform. The system parameters are listed below:
| Parameter | Value |
|---|---|
| Quasi-Z-source inductor \(L_1,L_2\) | 1 mH |
| Quasi-Z-source capacitor \(C_1,C_2\) | 5 mF |
| Filter inductance \(L_f\) | 2 mH |
| Grid voltage peak \(V_M\) | 120 V |
| Grid frequency | 50 Hz |
| PV panel open-circuit voltage | 44.3 V |
| PV panel short-circuit current | 8.69 A |
| Maximum power of one panel | 300 W |
Initially, all three PV panels receive identical irradiance of 1000 W/m² at 25°C, and the system reaches the maximum power point in about 0.35 s. At \(t=2\,\mathrm{s}\), the irradiance of cell 2 is reduced to 700 W/m² and cell 3 to 500 W/m², while cell 1 remains at 1000 W/m². The MPPT controller is able to track the new maximum power points within 0.14 s, and the dc-link voltages of all cells return to the reference value of 100 V. The capacitor voltages \(v_{C1}\) and \(v_{C2}\) show a ripple of about 18 mV, which meets the design specification.
When the irradiance is further reduced (cell 2 to 500 W/m², cell 3 to 300 W/m²), the original control strategy leads to a modulation index of cell 1 exceeding 0.684, causing severe over-modulation. The grid current THD increases from 1.87% to 10.98%, and the current waveform is visibly distorted. After applying the proposed optimal third-harmonic compensation, the modulation wave of cell 1 no longer saturates, and all cells operate within the linear modulation region. The grid current THD is reduced to 2.39%, and the system maintains unity power factor. The comparative results are summarized below:
| Condition | Grid current THD | Max modulation index | System status |
|---|---|---|---|
| Balanced power | 1.87% | 0.62 | Normal |
| Severe imbalance without compensation | 10.98% | >0.684 | Overmodulation |
| Severe imbalance with compensation | 2.39% | 0.684 | Normal |
To further validate the theoretical analysis, I constructed a two-cell QZS-CMI experimental prototype. The dc-link voltages of the two cells are balanced at about 90 V, as shown in the measured waveforms. The output voltage exhibits two-level and three-level patterns in the positive/negative half-cycles, and the load current has a sinusoidal waveform with low distortion. The experimental results are in good agreement with the simulations.
6. Conclusion
In this thesis, I have presented a complete study on the grid-connected control strategy of quasi-Z-source cascaded multilevel photovoltaic solar inverters. The main contributions include:
- A detailed modeling framework for the QZS-CMI system, including the state-space averaging model and the power-imbalance over-modulation analysis.
- An improved carrier phase-shifted SPWM strategy that fuses the shoot-through duty cycle with unipolar single-leg chopping, achieving high efficiency and low harmonics.
- A hierarchical control scheme consisting of independent MPPT, dc-dc double-loop control, dc-link voltage balancing, and grid current control.
- An optimal third-harmonic compensation method that expands the linear modulation range and eliminates the third harmonic from the grid current under severe power imbalance.
- Real-time simulation and experimental verification using RT-LAB and a two-cell prototype, demonstrating the effectiveness and robustness of the proposed methods.
Future work could focus on extending the proposed strategy to three-phase systems, investigating model predictive control for improved dynamic response, and addressing fault-tolerant operation of the QZS-CMI solar inverters.