In modern renewable energy systems, the integration of photovoltaic (PV) generation with efficient power conversion is crucial. Among various inverter topologies, the quasi-Z-source inverter (qZSI) has gained significant attention due to its single-stage buck-boost capability and enhanced reliability. However, PV power output is inherently intermittent, influenced by environmental factors such as solar irradiance and temperature. To address this, energy storage systems are often incorporated to stabilize power delivery. In this context, the energy-stored quasi-Z-source inverter (ES-qZSI) emerges as a promising solution, combining a qZSI with a battery storage unit directly paralleled to one of its capacitors. This configuration simplifies the system by avoiding additional DC-DC converters, reducing losses and complexity. As a key component in PV applications, the solar inverter must ensure high-quality grid connection with minimal current distortion and fast dynamic response. Traditional control methods, such as PI controllers or sliding mode control, have been applied but may suffer from limitations in precision and adaptability. Model predictive control (MPC), particularly finite control set MPC (FCS-MPC), offers a flexible alternative with excellent dynamic performance. However, conventional FCS-MPC for solar inverters often results in substantial current ripples due to the limited output voltage vectors of two-level inverters, leading to reduced current control accuracy. To overcome this, we propose a three-vector-based model predictive current control (TV-MPCC) strategy for the ES-qZSI. This approach leverages virtual voltage vectors constructed from three nearest vectors, enabling the inverter to output voltage vectors covering any direction and amplitude, thereby improving current tracking and reducing distortion. In this article, we detail the mathematical modeling of the ES-qZSI, derive the TV-MPCC algorithm, and validate its superiority through simulations, highlighting its potential for advanced solar inverter applications.
The ES-qZSI topology, as shown in the reference, integrates a PV source, a quasi-Z network with inductors L1, L2 and capacitors C1, C2, a battery storage unit connected across C1, and a three-phase inverter feeding an RL load or grid. The battery is modeled as an ideal voltage source with an internal resistance. The inverter operates in two states: the shoot-through state and the non-shoot-through state. In the shoot-through state, both switches in one or more phases are turned on simultaneously, causing the DC-link voltage to boost. In the non-shoot-through state, the inverter behaves like a conventional voltage source inverter (VSI). The mathematical model is derived based on these states. For the quasi-Z network, the inductor voltages during shoot-through and non-shoot-through states are given by:
$$L_1 \frac{di_{L1}}{dt} = u_{C2} + u_{PV} \quad \text{(shoot-through)}$$
$$L_1 \frac{di_{L1}}{dt} = u_{PV} – u_{C1} \quad \text{(non-shoot-through)}$$
Discretizing these using forward Euler method with sampling time \(T_s\) yields the inductor current predictions:
$$i_{L1}(k+1) = i_{L1}(k) + \frac{T_s}{L_1} [u_{C2}(k) + u_{PV}(k)] \quad \text{(shoot-through)}$$
$$i_{L1}(k+1) = i_{L1}(k) + \frac{T_s}{L_1} [u_{PV}(k) – u_{C1}(k)] \quad \text{(non-shoot-through)}$$
Similarly, for the three-phase inverter in the synchronous rotating dq-frame, the dynamics are:
$$\frac{di_d}{dt} = \frac{1}{L} (u_d – e_d – R i_d + \omega L i_q)$$
$$\frac{di_q}{dt} = \frac{1}{L} (u_q – e_q – R i_q – \omega L i_d)$$
Discretization leads to:
$$i_d(k+1) = \left(1 – \frac{R T_s}{L}\right) i_d(k) + \frac{T_s}{L} [u_d(k) – e_d(k) + \omega L i_q(k)]$$
$$i_q(k+1) = \left(1 – \frac{R T_s}{L}\right) i_q(k) + \frac{T_s}{L} [u_q(k) – e_q(k) – \omega L i_d(k)]$$
These equations form the basis for the predictive current control. The ES-qZSI system involves three power ports: PV input, battery storage, and grid output. Power balance is maintained as:
$$P_{PV} – P_{out} – P_B = 0$$
where \(P_{PV}\) is the PV power, \(P_{out}\) is the output power to the grid, and \(P_B\) is the battery power. The battery power compensates for mismatches between PV generation and load demand, ensuring stable operation of the solar inverter. To achieve this, control references are generated: the PV maximum power point tracking (MPPT) provides the inductor current reference \(i_{L1ref}\), and the battery current reference \(i_{Bref}\) is derived from power balance:
$$i_{Bref} = \frac{u_{PV} i_{L1} – P_{out}}{u_B}$$
The d-axis current reference \(i_{dref}\) is obtained from the battery current error via a PI controller, while the q-axis reference \(i_{qref}\) is set to zero for unity power factor. This multi-port control framework enables the solar inverter to manage energy flow efficiently.
The proposed TV-MPCC strategy enhances current control precision by constructing virtual voltage vectors. In traditional FCS-MPCC, only the eight discrete voltage vectors (including six active vectors and two zero vectors) are used, leading to limited resolution and higher current ripples. Our approach synthesizes six virtual voltage vectors \(u_{v1}\) to \(u_{v6}\) based on the nearest three vectors principle. Each virtual vector is composed of two active vectors and one zero vector, with their durations calculated to minimize current errors. Consider a virtual vector \(u_{v1}\) in sector I, formed by vectors \(u_1\), \(u_2\), and \(u_{0,7}\). According to volt-second balance:
$$u_{v1} = \frac{t_i}{T_s} u_1 + \frac{t_j}{T_s} u_2 + \frac{t_z}{T_s} u_{0,7}$$
where \(t_i\), \(t_j\), and \(t_z\) are the durations of the vectors. To determine these durations, we use the deadbeat control principle, aiming for zero error between predicted and reference currents at the end of the sampling period. The current slopes for the dq-axes are defined as:
$$\delta_d^s = \left. \frac{di_d}{dt} \right|_{u=u_s}, \quad \delta_q^s = \left. \frac{di_q}{dt} \right|_{u=u_s}$$
for \(s \in \{i, j, z\}\). The predicted currents are:
$$i_d(k+1) = i_d(k) + \delta_d^i t_i + \delta_d^j t_j + \delta_d^z t_z$$
$$i_q(k+1) = i_q(k) + \delta_q^i t_i + \delta_q^j t_j + \delta_q^z t_z$$
Setting errors \(\Delta i_d = i_{dref} – i_d(k+1)\) and \(\Delta i_q = i_{qref} – i_q(k+1)\) to zero yields a system of equations. Solving for \(t_i\), \(t_j\), and \(t_z\) gives:
$$t_i = \frac{\Delta i_d (\delta_q^j – \delta_q^z) + \Delta i_q (\delta_d^z – \delta_d^j) + T_s (\delta_q^z \delta_d^j – \delta_q^j \delta_d^z)}{\delta_q^z \delta_d^j + \delta_q^i \delta_d^z + \delta_q^j \delta_d^i – \delta_q^i \delta_d^j – \delta_q^j \delta_d^z – \delta_q^z \delta_d^i}$$
$$t_j = \frac{\Delta i_d (\delta_q^z – \delta_q^i) + \Delta i_q (\delta_d^i – \delta_d^z) + T_s (\delta_q^i \delta_d^z – \delta_q^z \delta_d^i)}{\delta_q^z \delta_d^j + \delta_q^i \delta_d^z + \delta_q^j \delta_d^i – \delta_q^i \delta_d^j – \delta_q^j \delta_d^z – \delta_q^z \delta_d^i}$$
$$t_z = T_s – t_i – t_j$$
If \(t_i + t_j > T_s\), the zero vector is omitted, and the active vector durations are scaled:
$$t_i^* = \frac{t_i}{t_i + t_j} T_s, \quad t_j^* = \frac{t_j}{t_i + t_j} T_s, \quad t_z^* = 0$$
The dq-components of the virtual voltage vector are then:
$$u_d = \frac{t_i}{T_s} u_{di} + \frac{t_j}{T_s} u_{dj}, \quad u_q = \frac{t_i}{T_s} u_{qi} + \frac{t_j}{T_s} u_{qj}$$
These components are used in the prediction model to compute future currents for each virtual vector. The control algorithm involves two cost functions: one for shoot-through state selection and another for non-shoot-through optimization. The shoot-through cost function is based on inductor current error:
$$g_{i_{L1}}(x) = |i_{L1ref}(k+1) – i_{L1}^{ST}(k+1)| – |i_{L1ref}(k+1) – i_{L1}^{NST}(k+1)|$$
If \(g_{i_{L1}}(x) < 0\), the shoot-through state is applied; otherwise, the non-shoot-through state is evaluated. For non-shoot-through, the cost function minimizes dq-current errors:
$$g(x) = |i_{dref} – i_d(k+1)| + |i_{qref} – i_q(k+1)|$$
The TV-MPCC algorithm iterates over the six virtual vectors, predicting currents and selecting the one with the minimum cost. This reduces computational burden compared to evaluating all eight discrete vectors, while improving accuracy. The overall control structure for the ES-qZSI solar inverter integrates PV MPPT, battery power management, and TV-MPCC, ensuring robust performance under varying conditions.
To validate the proposed TV-MPCC, we conducted simulations in MATLAB/Simulink. The system parameters are summarized in Table 1. The battery is composed of 12 series-connected 3.2 V, 200 Ah cells, forming a 38.4 V, 200 Ah bank. The PV source is represented by an ideal voltage source with series resistance. The solar inverter operates at 50 Hz grid frequency.
| Parameter | Value |
|---|---|
| Grid line voltage | 70 V |
| Input voltage \(u_{PV}\) | 150–250 V |
| Battery voltage \(u_B\) | 39.3 V |
| RL load (L, R) | 7.1 mH, 5.0 Ω |
| PV series resistance | 2.58 Ω |
| qZSI inductors \(L_1, L_2\) | 2100 μH |
| qZSI capacitors \(C_1, C_2\) | 2300 μF |
Steady-state performance was evaluated with \(u_{PVref} = 150\) V, corresponding to PV power \(P_{PV} \approx 1500\) W. The output power \(P_{out}\) was set to 1106 W, and battery power \(P_B\) was 382 W (charging), with circuit losses around 12 W. The battery current reference was 9.72 A, and output current reference was 7.71 A. Under traditional FCS-MPCC, the battery current tracked well, and the DC-link voltage stabilized at 283.5 V, achieving a boost ratio of 1.89. However, the three-phase output currents exhibited higher total harmonic distortion (THD). In contrast, TV-MPCC significantly reduced current ripples. Table 2 compares key metrics between FCS-MPCC and TV-MPCC for steady-state operation.
| Metric | FCS-MPCC | TV-MPCC |
|---|---|---|
| Output current THD | 4.01% | 1.09% |
| Current ripple magnitude | High | Low |
| d-axis current error | Moderate | Negligible |
| q-axis current error | Moderate | Negligible |
The THD reduction of 2.92% demonstrates the superior harmonic performance of TV-MPCC, crucial for grid-connected solar inverters. The virtual vectors enable smoother current waveforms, enhancing power quality. Dynamic performance was tested by stepping the output power from 400 W to 1200 W at t=0.4 s, with \(u_{PVref} = 200\) V (P_PV = 900 W constant). The battery power shifted from 488 W (charging) to -312 W (discharging), compensating for the load change. The battery current reference changed from 12.42 A to -7.94 A, and output current reference from 3.5 A to 8.2 A. Both control methods tracked the references within 0.02 s, but TV-MPCC showed smaller current overshoot and lower oscillations during transients. This highlights the improved dynamic response of the proposed solar inverter control.
Further analysis involves the impact of TV-MPCC on switching frequency and computational load. Although TV-MPCC requires additional calculations for vector durations, the reduction to six virtual vectors minimizes iterations. The average switching frequency remains comparable to FCS-MPCC, but current control precision is enhanced. For practical solar inverter implementations, this trade-off is beneficial, as it reduces filtering requirements and improves efficiency. Moreover, the integration of energy storage in the ES-qZSI topology allows the solar inverter to provide grid support services, such as frequency regulation and peak shaving, making it a versatile solution for modern PV systems.
The mathematical formulation of the TV-MPCC can be extended to include constraints for battery state-of-charge (SOC) management. By incorporating SOC limits into the cost function, the solar inverter can optimize battery usage, prolonging its lifespan. Additionally, robustness against parameter variations, such as inductance or capacitance drift, can be addressed by adaptive predictive models. Future work may explore multi-objective optimization for the solar inverter, balancing current tracking, switching losses, and thermal management.
In conclusion, we have presented a three-vector model predictive current control strategy for an energy-stored quasi-Z-source solar inverter. The TV-MPCC algorithm constructs virtual voltage vectors from three nearest vectors, enabling precise current control with reduced distortion. Simulation results confirm that TV-MPCC outperforms traditional FCS-MPCC in both steady-state and dynamic scenarios, with a 2.92% lower THD and minimized current ripples. The ES-qZSI topology effectively integrates PV generation, battery storage, and grid connection, managed through a unified predictive framework. This approach advances the performance of solar inverters, contributing to stable and efficient renewable energy systems. As solar power penetration grows, such advanced control methods will be essential for maintaining grid power quality and reliability.