The rapid integration of photovoltaic and wind power into the electrical grid has significantly increased the demand for high-performance power conversion interfaces. The stability and power quality of the grid are profoundly influenced by the control strategies of the interfacing on grid inverters. To achieve efficient and high-quality grid connection, extensive research has been conducted globally on the topologies and control methodologies for these inverters. Compared to the conventional three-leg topology, the three-level four-leg on grid inverter, despite its additional leg, offers distinct advantages. It inherently handles unbalanced output currents, improves DC-link voltage utilization, and does not substantially increase the complexity of the DC-side control, making it a suitable candidate for modern distributed generation systems.
Control strategies for three-phase four-leg on grid inverters can be broadly categorized into linear and nonlinear methods. Traditional linear controllers like PI and PR are susceptible to modeling inaccuracies. Variations in system parameters and external disturbances in practical applications can detrimentally affect the control precision of such linear schemes. To overcome the limitations associated with linear control, Finite Control Set Model Predictive Control (FCS-MPC) has been applied to on grid inverters. This method evaluates all possible switching states within a predefined set within each control cycle, selecting the optimal one that minimizes a defined cost function. This approach inherently offers superior dynamic response and enhanced robustness against disturbances. However, a significant challenge arises when applying conventional FCS-MPC to a three-level four-leg on grid inverter. The total number of available voltage vectors is 81 (3^4), leading to a substantial computational burden due to the exhaustive enumeration and prediction required for each vector every sampling period.
To address this critical issue of computational load, this work proposes an improved FCS-MPC strategy. The proposed method incorporates a two-step prediction horizon to effectively compensate for the inevitable delays caused by digital sampling and computation. Furthermore, it introduces a novel voltage vector preselection scheme. The current tracking constraints in the cost function are transformed into equivalent voltage vector constraints. The vast set of 81 vectors is then intelligently organized into layers and sectors, drastically reducing the number of candidate vectors that need to be evaluated in each control cycle, thereby alleviating the computational burden without compromising control performance.
Mathematical Model of the Four-Leg On-Grid Inverter
Figure 1 shows the main circuit topology of the three-level four-leg on grid inverter, neglecting the front-end boost stage. The DC-link capacitors C1 and C2 are equal, each with capacitance C. The switches and clamping diodes form the three-level neutral-point-clamped (NPC) legs. The grid-side filter for each phase consists of an inductor L (with parasitic resistance R). The grid voltages are denoted by ek, and the injected currents are ij.
The continuous-time model of the inverter’s AC side in the abc-frame is given by:
$$
\begin{aligned}
u_a &= L\frac{d(i_a – i_n)}{dt} + R(i_a – i_n) + e_a \\
u_b &= L\frac{d(i_b – i_n)}{dt} + R(i_b – i_n) + e_b \\
u_c &= L\frac{d(i_c – i_n)}{dt} + R(i_c – i_n) + e_c \\
i_a + i_b + i_c + i_n &= 0
\end{aligned}
$$
where ua, ub, uc are the output phase voltages of the inverter. Applying the Clarke transformation to the αβ0-frame simplifies the model and decouples the components. The transformed model is:
$$
\begin{bmatrix} u_{\alpha} \\ u_{\beta} \\ u_{0} \end{bmatrix} = L \frac{d}{dt} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ 4i_{0} \end{bmatrix} + R \begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ 4i_{0} \end{bmatrix} + \begin{bmatrix} e_{\alpha} \\ e_{\beta} \\ e_{0} \end{bmatrix}
$$
For a three-level NPC inverter, maintaining the balance of the DC-link midpoint voltage is crucial. The dynamic of the capacitor voltage difference Δvc = vc1 – vc2 is derived as:
$$
C \frac{d \Delta v_c}{dt} = \left(-|S_o|-|S_a|\right)i_a + \left(-|S_o|-|S_b|\right)i_b + \left(-|S_o|-|S_c|\right)i_c
$$
where Sz (z = a, b, c, o) represents the switching state of each leg, mapping to its output voltage level.
Discrete Predictive Model and FCS-MPC Formulation
To implement FCS-MPC, a discrete-time model is required for prediction. Using forward Euler discretization with a sampling period Ts, the current prediction for the next sampling instant (k+1) is:
$$
\begin{aligned}
i_{\alpha}(k+1) &= \left(1-\frac{R T_s}{L}\right)i_{\alpha}(k) + \frac{T_s}{L}\left(u_{\alpha}(k) – e_{\alpha}(k)\right) \\
i_{\beta}(k+1) &= \left(1-\frac{R T_s}{L}\right)i_{\beta}(k) + \frac{T_s}{L}\left(u_{\beta}(k) – e_{\beta}(k)\right) \\
i_{0}(k+1) &= \left(1-\frac{R T_s}{4L}\right)i_{0}(k) + \frac{T_s}{4L}\left(u_{0}(k) – e_{0}(k)\right)
\end{aligned}
$$
To compensate for the computational delay, a two-step prediction is employed. The current at (k+2) is predicted using the voltage applied at (k+1):
$$
\begin{aligned}
i_{\alpha}(k+2) &= \left(1-\frac{R T_s}{L}\right)i_{\alpha}(k+1) + \frac{T_s}{L}\left(u_{\alpha}(k+1) – e_{\alpha}(k+1)\right) \\
i_{\beta}(k+2) &= \left(1-\frac{R T_s}{L}\right)i_{\beta}(k+1) + \frac{T_s}{L}\left(u_{\beta}(k+1) – e_{\beta}(k+1)\right) \\
i_{0}(k+2) &= \left(1-\frac{R T_s}{4L}\right)i_{0}(k+1) + \frac{T_s}{4L}\left(u_{0}(k+1) – e_{0}(k+1)\right)
\end{aligned}
$$
Here, eα(k+1), eβ(k+1), e0(k+1) are estimated by rotating the measured grid voltages at (k) by ωTs. Similarly, the midpoint voltage is predicted one step ahead:
$$
\Delta v_c(k+1) = \Delta v_c(k) – \frac{T_s}{C} \left[ \left(|S_a(k)|+|S_o(k)|\right)i_a(k) + \left(|S_b(k)|+|S_o(k)|\right)i_b(k) + \left(|S_c(k)|+|S_o(k)|\right)i_c(k) \right]
$$
The control objectives—accurate current tracking and midpoint voltage balance—are combined into a single cost function J with a weighting factor λ:
$$
J = |i^*_{\alpha}(k+2) – i_{\alpha}(k+2)| + |i^*_{\beta}(k+2) – i_{\beta}(k+2)| + |i^*_{0}(k+2) – i_{0}(k+2)| + \lambda |\Delta v_c(k+2)|
$$
In traditional FCS-MPC, this cost function is evaluated for all 81 voltage vectors to find the minimizing switch state, which is computationally intensive.
Proposed Voltage Vector Preselection-Based FCS-MPC
The core innovation of the proposed method lies in transforming the problem and reducing the search space. First, by substituting the desired current references into the two-step prediction model, we can solve for the ideal or “reference” voltage vector u* needed at instant (k+1) to achieve the references at (k+2):
$$
\begin{aligned}
u^*_{\alpha}(k+1) &= \left(R – \frac{L}{T_s}\right) i_{\alpha}(k+1) + e_{\alpha}(k) + \frac{L}{T_s} i^*_{\alpha}(k+2) \\
u^*_{\beta}(k+1) &= \left(R – \frac{L}{T_s}\right) i_{\beta}(k+1) + e_{\beta}(k) + \frac{L}{T_s} i^*_{\beta}(k+2) \\
u^*_{0}(k+1) &= \left(R – \frac{4L}{T_s}\right) i_{0}(k+1) + e_{0}(k) + \frac{4L}{T_s} i^*_{0}(k+2)
\end{aligned}
$$
From the discrete model, the relationship between current error and voltage error can be derived as:
$$
| \mathbf{i^*}(k+2) – \mathbf{i}(k+2) | \propto \frac{T_s}{L} | \mathbf{u^*}(k+1) – \mathbf{u}(k+1) |
$$
Therefore, minimizing the current tracking error is equivalent to minimizing the distance between the reference voltage vector u*(k+1) and the actual applied inverter voltage vector u(k+1). This allows the cost function to be reformulated as:
$$
J_{new} = |u^*_{\alpha}(k+1) – u_{\alpha}(k+1)| + |u^*_{\beta}(k+1) – u_{\beta}(k+1)| + |u^*_{0}(k+1) – u_{0}(k+1)| + \lambda |\Delta v_c(k+2)|
$$
This reformulation bypasses the need for 81 individual current predictions. However, searching through 81 voltage vectors to minimize Jnew remains costly. To solve this, we propose a layered sector partition of the voltage vector space.
The 81 voltage vectors of the four-leg on grid inverter can be visualized in the αβ0 coordinate space. A key observation is that the zero-sequence voltage component u0 takes on distinct, discrete levels. We categorize all vectors into 13 horizontal layers (H1 to H13) based on their u0 value. The mapping is shown in the table below.
| Layer (H) | u0 Level (per unit of Vdc) | Representative Level |
|---|---|---|
| 1 | +1 | +Vdc/2 |
| 2 | +5/6 | +5Vdc/12 |
| 3 | +2/3 | +Vdc/3 |
| 4 | +1/2 | +Vdc/4 |
| 5 | +1/3 | +Vdc/6 |
| 6 | +1/6 | +Vdc/12 |
| 7 | 0 | 0 |
| 8 | -1/6 | -Vdc/12 |
| 9 | -1/3 | -Vdc/6 |
| 10 | -1/2 | -Vdc/4 |
| 11 | -2/3 | -Vdc/3 |
| 12 | -5/6 | -5Vdc/12 |
| 13 | -1 | -Vdc/2 |
The control algorithm first calculates u*0(k+1). By comparing this value against the thresholds defined in the table, the corresponding layer H is immediately identified. This step alone narrows down the candidate vectors from 81 to only those residing in the selected layer. For instance, if u*0(k+1) falls into layer 7 (u0 ≈ 0), only the vectors with a near-zero zero-sequence component are considered.
For layers containing a large number of vectors (e.g., layers 5, 6, 7, 8, 9), a further refinement is applied. Within the identified αβ-plane of that layer, the plane is divided into three 120-degree sectors. The location of the reference vector’s αβ-component, [u*α(k+1), u*β(k+1)], determines the specific sector. The search is then confined to the small subset of vectors within the chosen layer and sector. This two-stage preselection—first by layer (u0), then by sector (uαβ)—dramatically reduces the number of candidate vectors for which the cost function Jnew needs to be evaluated, often to fewer than 10, thereby significantly reducing the computational burden of the on grid inverter controller.
Simulation Verification
To validate the performance of the proposed voltage-vector-preselection FCS-MPC strategy for the four-leg on grid inverter, a simulation model was built in Simulink. The system parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| DC Input Voltage | Vdc | 700 V |
| DC-link Capacitance | C1, C2 | 150 µF |
| Weighting Factor | λ | 20 |
| Control Frequency | fs | 20 kHz |
| Grid Phase Voltage (RMS) | e | 220 V |
| Filter Inductance | L | 3 mH |
| Inductor Parasitic Resistance | R | 0.1 Ω |
| Grid Frequency | f | 50 Hz |
The simulation results demonstrate the effectiveness of the proposed control method for the on grid inverter. The three-phase grid currents are sinusoidal and well-balanced, with a low total harmonic distortion (THD). The midpoint voltage of the DC-link capacitors is effectively balanced, with the voltage difference Δvc confined to a very small range of approximately ±3 V. The system shows stable operation and excellent grid-following performance, confirming that the proposed preselection algorithm maintains high control quality while reducing computational demands.
Conclusion
This work addresses the control optimization challenge for three-level four-leg on grid inverters. A comprehensive mathematical model incorporating both output current dynamics and DC-link midpoint voltage behavior was established. An improved FCS-MPC strategy was proposed to overcome the high computational burden of the conventional method. The strategy features a two-step prediction horizon to mitigate system delays and a novel voltage vector preselection scheme. By transforming the cost function’s current constraints into voltage constraints and organizing the vector space into layers and sectors, the number of candidate evaluations per control cycle is drastically reduced. Simulation results verify that the proposed method maintains excellent grid current quality and effective midpoint voltage balance for the on grid inverter, making the advanced predictive control approach more feasible for practical digital implementation. This contribution enhances the viability of using sophisticated model predictive control in high-performance, multi-level on grid inverter applications.