We present an improved model predictive control (MPC) strategy based on a dual-objective cost function, specifically designed for T-type three-level grid-connected inverters in photovoltaic systems. The rapid adoption of renewable energy sources such as solar and wind power has intensified the demand for high-efficiency, high-quality power converters. Among the various types of solar inverters, the T-type three-level topology stands out for its advantages in terms of reduced harmonic distortion, lower voltage stress on devices, and higher power quality. However, practical implementations face significant challenges: control delays in digital systems cause tracking errors, and the neutral-point potential of the DC-link capacitor divider inevitably drifts, leading to voltage unbalance. To address these issues, we propose a finite control set model predictive control (FCS-MPC) framework that integrates a deadbeat current prediction model with Lagrangian interpolation and a dual-objective cost function. The first objective ensures fast and accurate grid current tracking, while the second minimizes neutral-point voltage imbalance. Simulations in Matlab/Simulink demonstrate that compared to single-objective MPC, the total harmonic distortion (THD) of the grid current is reduced from 4.23% to 1.37%, and the DC-link capacitor voltages become balanced within 0.04 s with a fluctuation rate of only 2.545%.
Our work focuses on a class of types of solar inverters commonly employed in medium-voltage grid-connected systems. The T-type three-level inverter is particularly attractive because it provides three output voltage levels, which significantly reduces the harmonic content of the AC side current. Nevertheless, the control system must overcome two practical hurdles: the inherent one-step delay in digital signal processors (DSPs) that degrades prediction accuracy, and the natural tendency of the DC-link midpoint voltage to drift due to asymmetrical loading or non-ideal switching patterns. We combine a deadbeat current predictor with a multi-step Lagrangian extrapolation to compensate for the delay, and we embed the neutral-point balance term directly into the cost function to eliminate the need for external balancing circuits or complex modulation schemes.
System Modeling and Deadbeat Current Prediction
We begin by analyzing the operating principle of the T-type three-level grid-connected inverter. The main circuit topology consists of a DC-link split into two equal capacitors \(C_1\) and \(C_2\), and three-phase legs each containing four IGBT switches (plus antiparallel diodes). Each leg can generate three voltage levels: \(+U_{dc}/2\) (P state), 0 (O state), and \(-U_{dc}/2\) (N state). The switching states are defined by a variable \(S_x \in \{1,0,-1\}\) for phase \(x = a,b,c\). The AC side is connected to the grid via a filter inductor \(L\) and an equivalent resistor \(R\).
The equivalent circuit in the three-phase stationary abc frame is:
\[
\begin{cases}
u_{an} = R i_a + L \frac{di_a}{dt} + e_a \\
u_{bn} = R i_b + L \frac{di_b}{dt} + e_b \\
u_{cn} = R i_c + L \frac{di_c}{dt} + e_c
\end{cases}
\]
where \(u_{xn}\) is the inverter output voltage with respect to the grid neutral point, and \(e_x\) is the grid phase voltage. Applying Clarke transformation, we obtain the model in the \(\alpha\beta\) stationary frame:
\[
\begin{cases}
u_{\alpha n} = L \frac{di_{\alpha}}{dt} + R i_{\alpha} + e_{\alpha} \\
u_{\beta n} = L \frac{di_{\beta}}{dt} + R i_{\beta} + e_{\beta}
\end{cases}
\]
To implement digital control, we discretize the model using forward Euler approximation with sampling period \(T_s\):
\[
\begin{cases}
i_{\alpha}(k+1) = \frac{T_s}{L}[u_{\alpha n}(k)-e_{\alpha}(k)] + \left(1-\frac{R}{L}T_s\right)i_{\alpha}(k) \\
i_{\beta}(k+1) = \frac{T_s}{L}[u_{\beta n}(k)-e_{\beta}(k)] + \left(1-\frac{R}{L}T_s\right)i_{\beta}(k)
\end{cases}
\]
Equation (4) is the deadbeat current prediction model. It allows us to compute the required output voltage vector that makes the current at the next sampling instant equal to a given reference. However, because the controller executes with one-step delay, we must predict the reference current at time \(k+1\) using a multi-step Lagrangian interpolation (n=2):
\[
\begin{cases}
i_{\alpha}^*(k+1) = 3i_{\alpha}^*(k) – 3i_{\alpha}^*(k-1) + i_{\alpha}^*(k-2) \\
i_{\beta}^*(k+1) = 3i_{\beta}^*(k) – 3i_{\beta}^*(k-1) + i_{\beta}^*(k-2)
\end{cases}
\]
The predicted optimal voltage is then derived from the inverse of the discrete model:
\[
\begin{cases}
u_{\alpha n}^*(k) = \frac{L}{T_s}[i_{\alpha}^*(k+1)-i_{\alpha}(k)] + R i_{\alpha}(k) + e_{\alpha}(k) \\
u_{\beta n}^*(k) = \frac{L}{T_s}[i_{\beta}^*(k+1)-i_{\beta}(k)] + R i_{\beta}(k) + e_{\beta}(k)
\end{cases}
\]
This voltage reference serves as the input to the finite control set optimization. In traditional FCS-MPC, the output voltage is directly selected from the 27 possible switching combinations of the T-type inverter. The challenge lies in efficiently choosing the best vector that simultaneously satisfies current tracking and neutral-point balance.
Dual-Objective Cost Function Design
We consider two control objectives: (1) accurate tracking of the grid current reference, and (2) balancing the voltages \(U_{C1}\) and \(U_{C2}\) across the two DC-link capacitors. The single-objective cost function only includes the current error:
\[
g_1(k) = |i_{\alpha}^*(k+1)-i_{\alpha}(k+1)| + |i_{\beta}^*(k+1)-i_{\beta}(k+1)|
\]
However, using only \(g_1(k)\) ignores neutral-point drift, which leads to increased harmonic distortion and potential overvoltage stress on the switching devices. To incorporate balance, we model the effect of each switching state on the capacitor voltages. The midpoint current \(I_o\) is related to the capacitor currents by:
\[
I_o = I_{C1} – I_{C2}
\]
Using the switch model, the predicted capacitor voltages are:
\[
\begin{cases}
U_{C1}^*(k+1) = U_{C1}(k) + \frac{T_s}{C_1}[I_{dc}(k) – H_{1a}i_a – H_{1b}i_b – H_{1c}i_c] \\
U_{C2}^*(k+1) = U_{C2}(k) + \frac{T_s}{C_2}[I_{dc}(k) + H_{2a}i_a + H_{2b}i_b + H_{2c}i_c]
\end{cases}
\]
where \(H_{1x}=1\) if \(S_x=1\) else 0, and \(H_{2x}=1\) if \(S_x=-1\) else 0. The dual-objective cost function then becomes:
\[
g_2(k) = |i_{\alpha}^*(k+1)-i_{\alpha}(k+1)| + |i_{\beta}^*(k+1)-i_{\beta}(k+1)| + \lambda_{v1}\left|U_{C1}^*(k+1)-U_{C2}^*(k+1)\right|
\]
The weighting factor \(\lambda_{v1}\) determines the relative priority between current tracking and neutral-point balancing. Tuning this parameter is straightforward and can be done offline by observing the trade-off in THD and voltage ripple. In our design, we set \(\lambda_{v1}=0.5\) after a sensitivity study.
The overall control system works as follows: At each sampling instant \(k\), we measure the grid currents, grid voltages, DC-link voltage, and both capacitor voltages. We compute the reference currents in the \(dq\) frame using an outer voltage loop, then transform to \(\alpha\beta\). Using (5), we obtain the reference currents at \(k+1\). For each of the 27 valid switching states of the T-type inverter, we compute the predicted currents from (4) and the predicted capacitor voltages from (9). We evaluate the cost function \(g_2(k)\) for each state and select the one that minimizes it. The corresponding switching signals are applied to the inverter in the next control period. The cycle repeats, forming a receding horizon control.
Simulation Results and Comparative Analysis
We validated our proposed dual-objective MPC against the single-objective alternative using a 600 W T-type three-level grid-connected inverter model built in Matlab/Simulink. The system parameters are summarized in Table I.
| Parameter | Symbol | Value |
|---|---|---|
| Grid phase voltage amplitude | \(E\) | 60 V |
| DC-link voltage | \(U_{dc}\) | 150 V |
| DC-link series resistance | \(R_{dc}\) | 0.001 Ω |
| DC-link capacitors | \(C_1, C_2\) | 1000 μF |
| Grid frequency | \(f\) | 50 Hz |
| Sampling frequency | \(f_s\) | 10 kHz |
| Filter inductance | \(L\) | 20 mH |
| Equivalent resistance | \(R\) | 0.01 Ω |
The DC source was emulated by a three-phase diode rectifier with a smoothing capacitor. We compared two cases: (a) using only single-objective cost function \(g_1(k)\), and (b) using the proposed dual-objective cost function \(g_2(k)\). The grid current waveforms and their harmonic spectra were analyzed.
For the single-objective case, the grid current was symmetrical and in phase with the grid voltage, achieving unity power factor. However, the total harmonic distortion (THD) of the phase-a current was 4.23%, which exceeds typical grid standards (e.g., IEEE 519 recommends below 5%, but often lower is desired). The fundamental current amplitude was 5.921 A.
For the dual-objective case, the grid current waveform quality improved dramatically. The THD dropped to 1.37%, and the fundamental amplitude became 5.974 A. The current tracking was accurate with minimal overshoot. Table II summarizes the comparative performance.
| Metric | Single-Objective MPC | Dual-Objective MPC |
|---|---|---|
| Phase-a current THD | 4.23% | 1.37% |
| Fundamental current amplitude | 5.921 A | 5.974 A |
| DC capacitor voltage balance time | Not achieved | 0.04 s |
| Capacitor voltage ripple (p-p) | >5 V | ~3 V (±1.5 V) |
| Voltage fluctuation rate | Not applicable | 2.545% |
We further examined the neutral-point balancing behavior. In the single-objective case, the capacitor voltages \(U_{C1}\) and \(U_{C2}\) drifted continuously, reaching a deviation of more than 10 V after 0.5 s. At 0.5 s, we activated the dual-objective term by adding the balance component to the cost function. Figure 1 shows the transient response: the voltages converged to within ±1.5 V of each other within 0.04 s and remained stable thereafter. This is a marked improvement, as many types of solar inverters rely on additional hardware circuits or complex modulation to achieve balance.
The image above illustrates a modern energy storage inverter, which shares similar topology and control challenges with the T-type three-level inverter studied in this work.
Discussion on Weight Factor Selection and Computational Complexity
The weighting factor \(\lambda_{v1}\) plays a critical role in the dual-objective cost function. A small value gives priority to current tracking but may lead to insufficient voltage balance; a large value forces quick balancing at the expense of current quality. We performed a sweep of \(\lambda_{v1}\) from 0.1 to 2.0 and observed that values between 0.3 and 0.8 provided the best trade-off. Below 0.3, the voltage ripple exceeded 4 V; above 0.8, the current THD increased above 2%. Our chosen value of 0.5 yielded a balanced performance. This simple manual tuning is acceptable for offline design, but in future work we may explore adaptive weighting based on real-time operating conditions.
Another important aspect is the computational burden. The traditional FCS-MPC for a T-type three-level inverter needs to evaluate 27 candidate vectors per sampling period. Our dual-objective cost function adds only one additional term (two multiplications and one absolute value) to each evaluation, which does not significantly increase the cycle time. With modern DSPs or FPGAs operating at 10 kHz, the algorithm is feasible for real-time implementation. Nevertheless, further reductions in computation can be achieved by eliminating redundant vectors or adopting a sequential optimization approach, as reported in the literature for other types of solar inverters.
We also compared the performance of our scheme with a conventional PI-based current control plus carrier-based PWM with neutral-point balancing. PI control requires careful tuning and often exhibits slower response and higher THD under transient conditions. Our MPC naturally handles system nonlinearities and constraints, providing superior dynamic performance. For instance, when a step change in reference current occurred, the MPC settled within one or two sampling periods (0.2 ms), whereas PI control needed several milliseconds.
Conclusion and Future Work
We have developed an improved model predictive control strategy that effectively solves the two major issues of T-type three-level grid-connected inverters: control delay and neutral-point voltage imbalance. By integrating a deadbeat current predictor enhanced with Lagrangian interpolation and a dual-objective cost function that explicitly penalizes capacitor voltage deviation, we achieved a grid current THD of only 1.37% and balanced the DC-link capacitors within 0.04 s with a ripple of ±1.5 V. The method requires no additional sensors or hardware, making it cost-effective for many types of solar inverters used in distributed generation systems.
Our future work will focus on extending the approach to handle grid faults (e.g., low-voltage ride-through) and parameter uncertainty (e.g., filter inductance variation). We also plan to test the algorithm on a hardware prototype to validate the simulation findings in a real-world environment. Furthermore, we aim to reduce the computational cost by developing a vector preselection method that only considers the most promising candidates, which is especially beneficial when controlling larger multilevel topologies such as the five-level T-type inverter.
In summary, the proposed dual-objective MPC offers a promising solution for improving the power quality and reliability of grid-connected inverters, contributing to the wider adoption of clean energy and the transition toward carbon neutrality goals such as China’s “2030 carbon peak, 2060 carbon neutrality” targets.