The pursuit of efficient and reliable energy conversion from photovoltaic (PV) sources has driven significant advancements in power electronics. Central to these systems is the solar inverter, the critical component responsible for converting the DC power generated by PV panels into grid-compliant AC power. Traditional two-stage inverter topologies, while effective, often involve complex cascaded structures. The Quasi-Z-Source Inverter (qZSI) has emerged as a promising single-stage alternative for solar inverter applications, offering unique advantages like inherent buck-boost capability and improved reliability due to its impedance network. However, the intermittent nature of solar power remains a significant challenge. Integrating an energy storage unit directly into the qZSI structure, forming an Energy-Storage qZSI (ES-qZSI), presents an elegant solution for power smoothing and grid support. This integration allows the system to manage power flow between the PV array, the battery, and the grid seamlessly within a single power conversion stage. The control of such a three-port system is crucial for optimal performance, demanding strategies that ensure fast dynamic response, precise current tracking, and low harmonic distortion.
The operational principle of the ES-qZSI hinges on its two distinctive states: the shoot-through (ST) state and the non-shoot-through (NST) state. During the NST state, the inverter operates similarly to a conventional voltage source inverter (VSI), applying one of the six active voltage vectors or two zero vectors to the load. The ST state, unique to impedance source networks, is created by simultaneously gating on both switches in at least one phase leg, short-circuiting the inverter bridge. This state energizes the impedance network, boosting the DC-link voltage beyond the input voltage, which is essential for a solar inverter when the PV voltage is lower than the grid voltage requirement. The equivalent circuits for these states form the basis of its mathematical model.
The dynamics of the quasi-Z network inductors and capacitors are described by the following equations. For the shoot-through state:
$$L_1 \frac{di_{L1}}{dt} = u_{C2} + u_{PV}$$
$$L_2 \frac{di_{L2}}{dt} = u_{C1}$$
For the non-shoot-through state:
$$L_1 \frac{di_{L1}}{dt} = u_{PV} – u_{C1}$$
$$L_2 \frac{di_{L2}}{dt} = – u_{C2}$$
Where \(L_1, L_2\) are the inductances, \(C_1, C_2\) are the capacitances, \(i_{L1}, i_{L2}\) are the inductor currents, \(u_{C1}, u_{C2}\) are the capacitor voltages, and \(u_{PV}\) is the PV input voltage. For a balanced network with \(L_1 = L_2 = L\) and \(C_1 = C_2 = C\), the average capacitor voltages and the peak DC-link voltage (\(u_{PN}\)) across the inverter bridge relate to the input voltage and the shoot-through duty ratio \(D\) as:
$$U_{C1} = \frac{1-D}{1-2D} U_{PV}, \quad U_{C2} = \frac{D}{1-2D} U_{PV}, \quad U_{PN} = \frac{1}{1-2D} U_{PV} = B \cdot U_{PV}$$
Here, \(B\) is the boost factor provided by the qZSI network, a vital feature for a solar inverter to handle varying input conditions. The output AC voltage is then synthesized from this boosted DC-link voltage.
The three-port power balance is the cornerstone of the ES-qZSI system’s operation. The power from the PV panels (\(P_{PV}\)), the power exchanged with the battery (\(P_B\)), and the output power delivered to the grid (\(P_{out}\)) must satisfy:
$$P_{PV} – P_{out} – P_B – P_{loss} \approx 0$$
where \(P_{loss}\) represents system losses. By controlling any two of these power flows, the third is automatically determined. Typically, the PV is controlled to operate at its Maximum Power Point (MPPT), and the inverter output is controlled to deliver the required grid current. The battery then naturally compensates for the power difference, charging when \(P_{PV} > P_{out}\) and discharging when \(P_{PV} < P_{out}\). This elegant power balancing act is managed within the control framework of the solar inverter.
Finite Control Set Model Predictive Control (FCS-MPC) has gained prominence in power electronics for its intuitive concept, fast dynamic response, and ability to handle nonlinearities and constraints. For a solar inverter like the ES-qZSI, a typical FCS-MPC strategy involves predicting the future behavior of control variables (like grid currents) for all possible switching states of the inverter. A cost function is then evaluated for each prediction, and the switching state that minimizes this cost function is applied. The standard predictive model for the grid-connected inverter in the synchronous rotating (d-q) reference frame is:
$$\frac{d}{dt}\begin{bmatrix} i_d \\ i_q \end{bmatrix} = \begin{bmatrix} -\frac{R}{L} & \omega \\ -\omega & -\frac{R}{L} \end{bmatrix} \begin{bmatrix} i_d \\ i_q \end{bmatrix} + \frac{1}{L}\begin{bmatrix} u_d – e_d \\ u_q – e_q \end{bmatrix}$$
where \(i_d, i_q\) are the grid currents, \(u_d, u_q\) are the inverter output voltages, \(e_d, e_q\) are the grid voltages, \(R\) and \(L\) are the filter resistance and inductance, and \(\omega\) is the grid angular frequency. Discretizing this model using a forward Euler approximation with sampling time \(T_s\) yields the prediction equation:
$$i_{dq}(k+1) = \left(1 – \frac{R T_s}{L}\right) i_{dq}(k) + \frac{T_s}{L} \left( u_{dq}(k) – e_{dq}(k) \right) + \omega T_s \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} i_{dq}(k)$$
The cost function for the classical single-vector FCS-MPC is often:
$$g = |i_d^*(k+1) – i_d^{p}(k+1)| + |i_q^*(k+1) – i_q^{p}(k+1)|$$
where the superscript \(*\) denotes the reference and \(p\) denotes the predicted value. While effective, this method inherently applies only one voltage vector per control period, leading to significant current ripple and variable switching frequency, which can degrade the performance and efficiency of the solar inverter.
To overcome the limitations of single-vector FCS-MPC, the Three-Vector Model Predictive Current Control (TV-MPCC) strategy is proposed for the ES-qZSI. This method aims to approximate the operation of continuous modulation schemes by applying three voltage vectors within one control period, thereby reducing current ripple and improving the steady-state performance of the solar inverter. The core idea is to synthesize a virtual voltage vector whose amplitude and direction can be more flexibly controlled, offering a finer resolution than the eight discrete vectors available from a two-level inverter.
The control process for the TV-MPCC can be broken down into several key steps, executed every sampling period \(T_s\).
Step 1: Sector Identification and Vector Selection. First, the required reference voltage vector \(\mathbf{u}^*\) in the stationary \(\alpha\)-\(\beta\) frame is estimated. This can be derived from the dead-beat principle or, more commonly, by using the system model and the current error. A simplified dead-beat calculation ignoring resistance gives:
$$\mathbf{u}^*(k) \approx \mathbf{e}(k) + \frac{L}{T_s} \left( \mathbf{i}^*(k+1) – \mathbf{i}(k) \right) – \omega L \begin{bmatrix} -i_q(k) \\ i_d(k) \end{bmatrix}$$
Based on the phase angle of \(\mathbf{u}^*(k)\), the sector of the complex plane in which it lies is identified. The “nearest three vectors” principle is then applied. For any sector, the three nearest vectors are the two active vectors bounding the sector and one of the zero vectors (\(u_0\) or \(u_7\)). For instance, if \(\mathbf{u}^*\) lies in Sector I, the selected vectors are \(u_1\), \(u_2\), and \(u_0\) (or \(u_7\)). This selection is consistent across all six sectors, leading to six predefined virtual vector combinations (\(u_{v1}\) to \(u_{v6}\)).
Step 2: Calculation of Vector Duty Cycles (Dwell Times). The next step is to calculate the time duration (\(t_1\), \(t_2\), \(t_0\)) for which each of the three selected vectors should be applied, such that their time-averaged effect equals the desired \(\mathbf{u}^*\). This is governed by the volt-second balance equation:
$$\mathbf{u}^* \cdot T_s = \mathbf{u}_1 \cdot t_1 + \mathbf{u}_2 \cdot t_2 + \mathbf{u}_0 \cdot t_0$$
$$T_s = t_1 + t_2 + t_0$$
Solving these equations in the \(\alpha\)-\(\beta\) frame yields the duty cycles. A more systematic approach uses the current dynamics. The current change produced by a voltage vector \(\mathbf{u}_x\) is its slope \(\boldsymbol{\delta}_x = d\mathbf{i}/dt|_{\mathbf{u}=\mathbf{u}_x}\). The prediction with three vectors is:
$$\mathbf{i}^{p}(k+1) = \mathbf{i}(k) + \boldsymbol{\delta}_1 t_1 + \boldsymbol{\delta}_2 t_2 + \boldsymbol{\delta}_0 t_0$$
Enforcing dead-beat control (\(\mathbf{i}^{p}(k+1) = \mathbf{i}^*(k+1)\)) and the total time constraint provides three equations to solve for the three unknown times. The general solution can be expressed in matrix form. If the calculated \(t_1 + t_2 > T_s\), the zero vector time is set to zero, and the active vector times are rescaled:
$$t_1^* = \frac{t_1}{t_1+t_2}T_s, \quad t_2^* = \frac{t_2}{t_1+t_2}T_s, \quad t_0^* = 0$$
Step 3: Synthesis of Virtual Voltage Vector and Optimal Selection. Using the calculated times, a virtual voltage vector \(\mathbf{u}_v\) is defined for the sector. For Sector I:
$$\mathbf{u}_{v1} = \frac{t_1}{T_s}\mathbf{u}_1 + \frac{t_2}{T_s}\mathbf{u}_2 + \frac{t_0}{T_s}\mathbf{u}_0$$
This \(\mathbf{u}_{v1}\) represents the optimal continuous voltage that would be applied in an ideal modulator. In the TV-MPCC framework, this vector is not directly modulated. Instead, the algorithm evaluates the consequence of applying the sequence \(\mathbf{u}_1\), \(\mathbf{u}_2\), \(\mathbf{u}_0\) for durations \(t_1\), \(t_2\), \(t_0\). A cost function (e.g., the current tracking error at \(k+1\)) is computed for this specific three-vector combination. This process is repeated for all six possible virtual vector combinations (\(u_{v1}\) to \(u_{v6}\)). The combination that yields the minimum cost function value is chosen as the optimal output for the next control period. This reduces the exhaustive search from 7 NST vectors to a fixed 6 evaluations, streamlining the computation for the solar inverter controller.
Step 4: Integration with Shoot-Through Control. For the ES-qZSI, the control must also decide when to insert a shoot-through state to achieve the necessary boost. This is managed via a separate, secondary cost function or a dedicated control loop for the impedance network. A common approach is to use the inductor current \(i_{L1}\) as a control variable. A reference for \(i_{L1}\) is generated from the PV MPPT algorithm or a DC-link voltage controller. The predicted \(i_{L1}\) for the upcoming period under both a candidate NST state and a corresponding ST state (often created by modifying a zero state into an ST state) is calculated. A simple cost function decides:
$$g_{ST} = |i_{L1}^* – i_{L1, NST}^p| – |i_{L1}^* – i_{L1, ST}^p|$$
If \(g_{ST} < 0\), it indicates that applying the ST state would bring the inductor current closer to its reference, so the ST state is selected. Otherwise, the optimal NST sequence from the TV-MPCC algorithm is applied. This seamlessly integrates the boost control with the advanced output current control.
The advantages of the TV-MPCC strategy are summarized in the table below, contrasting it with the conventional FCS-MPC approach for the solar inverter application.
| Control Aspect | Conventional FCS-MPC (Single Vector) | Proposed TV-MPCC |
|---|---|---|
| Voltage Vector Resolution | Limited to 8 discrete vectors. | Effectively synthesizes a continuous range of vectors via combination. |
| Current Ripple & THD | Higher ripple and Total Harmonic Distortion (THD). | Significantly reduced ripple and lower output current THD. |
| Switching Frequency | Variable and dependent on operating point. | More constant, approximating fixed-frequency operation (\(f_{sw} \approx 1/T_s\)). |
| Computational Load | Evaluates 7 or 8 states per cycle. | Evaluates 6 predefined virtual vector sequences per cycle. |
| Steady-State Performance | Good dynamic response, lower steady-state precision. | Excellent steady-state performance while retaining fast dynamics. |
| Implementation Complexity | Conceptually simple, straightforward coding. | Higher algorithmic complexity due to duty cycle calculation. |
To validate the effectiveness of the TV-MPCC strategy for the ES-qZSI, a detailed simulation study was conducted. The system parameters were chosen to reflect a realistic small-scale solar inverter application with integrated storage, as shown in the following parameter table.
| System Parameter | Symbol | Value |
|---|---|---|
| Grid Line Voltage (RMS) | \(U_{grid}\) | 70 V |
| PV Input Voltage Range | \(U_{PV}\) | 150 – 250 V |
| Battery Voltage | \(U_B\) | 39.3 V |
| Output Filter (L, R) | \(L_f, R_f\) | 7.1 mH, 5.0 Ω |
| qZSI Inductors | \(L_1, L_2\) | 2100 μH |
| qZSI Capacitors | \(C_1, C_2\) | 2300 μF |
| Switching/Sampling Frequency | \(f_s = 1/T_s\) | 20 kHz |
In the steady-state test, the PV input was set to 150V, delivering approximately 1500W. The grid output power was commanded to be 1106W. According to the power balance, the battery absorbed the difference, charging at about 382W (excluding losses). The TV-MPCC controller successfully regulated all power flows. A key performance metric, the Total Harmonic Distortion (THD) of the grid current, was analyzed. Under the conventional FCS-MPC, the current THD was measured at 4.01%. In contrast, the TV-MPCC strategy reduced the THD to 1.09%, demonstrating a substantial 2.92% improvement. This dramatic reduction in harmonic content is a direct result of the reduced current ripple afforded by the three-vector synthesis, leading to cleaner power injection from the solar inverter.
The dynamic performance was tested by imposing a step change in the grid power reference. Initially, with \(P_{PV}=900W\) and \(P_{out}=400W\), the battery was charging at 488W. At t=0.4s, \(P_{out}\) was stepped to 1200W. This required the battery to instantly switch from charging to discharging at -312W to supply the extra power to the grid. Both control strategies responded quickly to the change. However, the transient current waveform under TV-MPCC exhibited significantly lower overshoot and smoother settling compared to the pronounced ripple and oscillation observed with FCS-MPC. The d-axis and q-axis current components (\(i_d\), \(i_q\)) also showed superior tracking with minimal error in the TV-MPCC case, confirming its enhanced dynamic regulation capability for the solar inverter under rapidly changing conditions.
In conclusion, the integration of an energy storage unit into a quasi-Z-source network creates a highly versatile single-stage power conversion system ideal for modern solar applications. The proposed Three-Vector Model Predictive Current Control strategy represents a significant advancement in the control methodology for such an ES-qZSI based solar inverter. By synthesizing virtual voltage vectors from the three nearest discrete vectors, the TV-MPCC effectively overcomes the inherent limitation of high current ripple associated with traditional finite-set predictive control. The method provides a much finer control resolution, leading to a remarkable reduction in grid current harmonic distortion and improved steady-state waveform quality. Simultaneously, it retains the fast dynamic response and constraint-handling capabilities that make MPC attractive. The structured approach of sector-based vector selection and duty cycle calculation, combined with a dedicated shoot-through control loop, results in a comprehensive control solution. This solution not only ensures precise three-port power management between the PV, battery, and grid but also delivers superior electrical performance, enhancing the overall efficiency, reliability, and power quality of the energy storage-integrated solar inverter system. Future work may focus on further reducing the computational burden of the duty cycle calculations and experimental validation under real-world varying solar irradiance and grid conditions.