With the advancement of photovoltaic (PV) technology and the increasing demand for renewable energy integration, the role of solar inverters in modern microgrid systems has become more critical than ever. The transition from single-unit power generation to multi-inverter parallel systems has been driven by the need for higher capacity, enhanced reliability, and improved system flexibility. In my research, I focus on the finite control set model predictive control (FCS-MPC) strategy based on mixed logical dynamical (MLD) modeling for single-phase solar inverters. The primary objective is to improve the output power quality of solar inverters, accelerate the grid-connection process, and suppress circulating currents in parallel inverter systems. This paper presents a comprehensive investigation into the mathematical modeling, controller design, and simulation validation of the proposed strategies.
The research begins with an analysis of existing inverter topologies and parallel connection methods. I selected a non-isolated two-stage single-phase full-bridge inverter topology as the main研究对象 due to its high efficiency, compact design, and widespread applicability in distributed generation systems. The increasing penetration of solar inverters into the power grid requires precise control algorithms to ensure stability and power quality. Traditional control methods, such as proportional-integral (PI) control, hysteresis control, and sliding mode control, have been widely used. However, these methods often require pulse width modulation (PWM) or suffer from slow dynamic responses. To address these limitations, I adopted the FCS-MPC approach, which directly manipulates the switching states of solar inverters without the need for a separate modulation stage.
The structure of this paper is organized as follows. First, I introduce the fundamental concepts of MLD modeling and FCS-MPC. Second, I present the application of these methods to a single-phase grid-connected solar inverter system. Third, I extend the methodology to a parallel configuration of multiple solar inverters, focusing on circulating current suppression. Finally, I provide simulation results to verify the effectiveness of the proposed control strategies.
1. Introduction and Background
The global energy landscape is undergoing a significant transformation. According to the BP Statistical Review of World Energy (2020), the consumption of renewable energy in 2019 increased by 3.2 exajoules compared to the previous year, accounting for 5.0% of global primary energy consumption. This trend highlights the growing importance of renewable energy sources, particularly solar energy, in mitigating the depletion of fossil fuels and reducing environmental pollution. Solar inverters serve as the core interface between PV panels and the power grid, directly influencing the efficiency, reliability, and safety of the entire photovoltaic system.
In recent years, the concept of microgrids has gained considerable attention as a solution for integrating distributed energy resources (DERs) into the existing power infrastructure. Microgrids offer several advantages, including reduced transmission losses, improved power quality, and enhanced resilience against grid disturbances. However, the intermittent nature of solar energy and the complexity of multi-inverter parallel systems pose significant technical challenges. The parallel operation of solar inverters is particularly attractive because it allows for modular system expansion, higher total power capacity, and redundant operation. Nevertheless, parallel-connected solar inverters inevitably suffer from circulating currents due to mismatched circuit parameters, which can lead to uneven current distribution, increased losses, and even system failure.
2. MLD Modeling and FCS-MPC Fundamentals
Before delving into the specific implementation details, it is essential to establish a solid theoretical foundation for the concepts used throughout this research. The MLD modeling framework provides a systematic approach for describing hybrid systems that contain both continuous dynamics and discrete events. In the context of power electronics, solar inverters represent a classic example of hybrid systems because they combine continuous state variables, such as currents and voltages, with discrete switching states determined by the ON/OFF status of power semiconductor devices.
The MLD model can be expressed in the following general form:
$$ \begin{cases} x(t+1) = A x(t) + B_1 u(t) + B_2 \delta(t) + B_3 z(t) \\ y(t) = C x(t) + D_1 u(t) + D_2 \delta(t) + D_3 z(t) \\ E_2 \delta(t) + E_3 z(t) \leq E_1 u(t) + E_4 x(t) + E_5 \end{cases} $$
In this formulation, x(t) represents the state variables, u(t) is the input vector, δ(t) denotes auxiliary logical variables, z(t) represents auxiliary continuous variables, and y(t) is the output vector. The matrices A, B, C, D, and E are constant coefficient matrices that encapsulate the system dynamics and constraints. The key advantage of MLD modeling lies in its ability to transform logical relationships and switching rules into linear inequalities, which can then be incorporated into the predictive control framework.
For the FCS-MPC algorithm, the control action is selected from a finite set of possible switching states that solar inverters can assume. Unlike continuous control set model predictive control (CCS-MPC), which requires a PWM modulator, FCS-MPC directly applies the optimal switching state to the inverter. This characteristic significantly simplifies the control architecture and improves the dynamic response speed. The basic structure of a single-step FCS-MPC system is illustrated in the following diagram:
The fundamental principle of FCS-MPC is to evaluate all possible switching states through a predefined cost function and select the state that minimizes this function. The cost function typically includes the tracking error of the controlled variable and additional system constraints such as switching frequency limits or circulating current limits. This optimization process is performed at each sampling instant, enabling real-time adaptation to system changes.
3. Single-Phase Photovoltaic Inverter MLD Modeling and FCS-MPC
3.1 System Topology and Simplification
In this section, I focus on a single-phase grid-connected solar inverter system. The topology I selected is a non-isolated two-stage structure, consisting of a front-end Boost converter and a back-end full-bridge inverter. The input from the PV array is first boosted to a stable DC bus voltage, which is then converted into an AC output through the inverter stage. A second-order LC filter is employed at the output to attenuate high-frequency switching harmonics and ensure a smooth AC waveform that meets the grid-connection requirements.
For the purpose of control design, I simplified the two-stage converter into a single full-bridge inverter model with an equivalent DC voltage source. This simplification is justified because the front-end Boost converter, regulated by maximum power point tracking (MPPT) control, maintains a relatively constant DC bus voltage under steady-state operating conditions. The simplified equivalent circuit includes four IGBT switches (S1-S4), a filter inductor L, a filter capacitor C, the line equivalent resistance r, and the grid voltage source.
3.2 MLD Modeling of the Single-Phase Inverter
The MLD modeling process for the single-phase solar inverter begins with the definition of logical variables to represent the switching states. I introduce a logical variable λ to distinguish between two valid switching combinations: when S1 and S4 are ON and S2 and S3 are OFF, λ = 1; conversely, when S2 and S3 are ON and S1 and S4 are OFF, λ = 0. Additionally, I introduce another logical variable δ to denote the direction of the inductor current iL. When the current flows in the positive direction (from the inductor left side to the right side), δ = 1, otherwise δ = 0. The relationship between the bridge output voltage uab, the DC bus voltage Udc, and the logical variables can be expressed as follows:
$$ \begin{cases} u_{ab} = U_{dc} \Longleftrightarrow (\lambda = 1, \delta = 1) \vee (\lambda = 1, \delta = 0) \\ u_{ab} = -U_{dc} \Longleftrightarrow (\lambda = 0, \delta = 1) \vee (\lambda = 0, \delta = 0) \end{cases} $$
Substituting these logical relationships into the circuit differential equations yields the continuous-time MLD model for the solar inverter:
$$ \begin{cases} \dot{x} = A x + B_1 i_g + B_2 u_{ab} \\ y = C x \end{cases} $$
where x = [iL, uC]T represents the state vector, and the coefficient matrices are defined as follows:
$$ A = \begin{bmatrix} 0 & -\frac{1}{C} \\ \frac{1}{L} & -\frac{r}{L} \end{bmatrix}, \quad B_1 = \begin{bmatrix} -\frac{1}{C} \\ 0 \end{bmatrix}, \quad B_2 = \begin{bmatrix} 0 \\ \frac{1}{L} \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 1 \end{bmatrix} $$
To implement the FCS-MPC controller on a digital processor, I discretized the continuous-time MLD model using the forward Euler method. With a sampling period Ts, the discrete-time MLD model becomes:
$$ \begin{cases} x(k+1) = A^* x(k) + B_1^* i_g(k) + B_2^* u_{ab}(k) \\ y(k) = C x(k) \end{cases} $$
where A* = eATs, B1* = ∫₀TseAτB1dτ, and B2* = ∫₀TseAτB2dτ. This discrete model accurately captures the hybrid behavior of solar inverters and serves as the basis for the prediction module in the FCS-MPC framework.
3.3 Full-Dimensional State Observer Design
In practical implementations, the grid-connected current ig may not be directly measurable due to sensor limitations or noise contamination. To address this issue, I designed a full-dimensional state observer to estimate the output current of solar inverters. The observer dynamics are described by the following equations:
$$ \begin{cases} \dot{\hat{x}} = A \hat{x} + B u + J(y – \hat{y}) \\ \hat{y} = C \hat{x} \end{cases}$$
where Ω represents the estimated state vector, and J is the observer gain matrix. The observer gain is designed to ensure that the estimation error dynamics are stable and converge sufficiently fast. Substituting the system matrices, the observer for the grid current ig can be written as:
$$ \begin{bmatrix} \dot{\hat{i}}_L \\ \dot{\hat{u}}_C \\ \dot{\hat{i}}_g \end{bmatrix} = A_{obs} \begin{bmatrix} \hat{i}_L \\ \hat{u}_C \\ \hat{i}_g \end{bmatrix} + B_{obs} u_{ab} $$
From this observer, the estimated grid current is expressed as:
$$ \hat{i}_g = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \hat{i}_L \\ \hat{u}_C \\ \hat{i}_g \end{bmatrix} $$
The performance of the observer was verified through simulation, and the estimated current closely tracks the actual value with minimal error. This accurate estimation is crucial for the proper functioning of the FCS-MPC controller.
3.4 Reference Voltage Prediction
In order to achieve precise output voltage tracking, the controller must also predict the reference voltage at the next sampling instant. For this purpose, I employed a second-order Lagrange extrapolation formula to estimate the reference voltage uC*(k+1) based on the current and previous reference values:
$$ u_C^*(k+1) = 3 u_C^*(k) – 3 u_C^*(k-1) + u_C^*(k-2) $$
This extrapolation method provides a good trade-off between prediction accuracy and computational complexity. By incorporating the predicted reference voltage into the cost function, the controller can minimize the tracking error more effectively in the next sampling period.
3.5 Cost Function and Control Strategy
For the single-phase grid-connected solar inverter system, the cost function is defined as the absolute error between the predicted output voltage and the reference voltage:
$$ g = \left| u_C^*(k+1) – u_C(k+1) \right| $$
The FCS-MPC algorithm systematically evaluates all four possible switching states of the full-bridge inverter. For each switching state, the predicted voltage at time k+1 is computed using the discrete-time MLD model. The switching state that produces the minimum cost function value is selected and applied to the solar inverter at the next sampling instant. The complete control algorithm is summarized in the flowchart below.
To validate the effectiveness of the proposed MLD-FCS-MPC strategy, I conducted comprehensive simulations using MATLAB/Simulink. The system parameters used in the simulations are listed in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| DC bus voltage | Udc | 300 V |
| Filter inductance | L | 6.8 mH |
| Filter capacitance | C | 24 μF |
| Grid voltage | Ug | 220 V |
| Grid frequency | f | 50 Hz |
| Line resistance | r | 0.6 Ω |
| Sampling period | Ts | 5 × 10−6 s |
The simulation results demonstrate significant improvements over the traditional switching-function-based MPC method. The output current and voltage waveforms under the MLD-based FCS-MPC strategy reach steady state within the first fundamental cycle, whereas the traditional MPC approach requires approximately four cycles to stabilize. This faster convergence enables solar inverters to achieve grid synchronization more rapidly, reducing the transition time and improving the overall system availability.
The total harmonic distortion (THD) analysis further confirms the superiority of the proposed method. As illustrated in Table 2, the THD of the output current was reduced from 2.82% under the traditional MPC method to 0.52% under the MLD-based FCS-MPC strategy. This reduction in harmonic distortion significantly improves the power quality of solar inverters and ensures compliance with the IEEE 1547 standard for grid interconnections, which mandates a THD below 5%.
| Method | Fundamental Amplitude (A) | THD (%) |
|---|---|---|
| Traditional MPC | 10.12 | 2.82 |
| MLD-FCS-MPC | 10.05 | 0.52 |
4. Parallel Single-Phase Solar Inverter Systems: MLD Modeling and FCS-MPC
4.1 System Topology and MLD Modeling
In the previous section, I successfully demonstrated the effectiveness of the MLD-based FCS-MPC strategy for a single solar inverter. However, as mentioned in the introduction, modern power systems increasingly require the parallel operation of multiple solar inverters to achieve higher capacity and reliability. In this section, I extend the proposed modeling and control methodology to a parallel configuration of solar inverters.
Figure 4.1 depicts the schematic diagram of a single-phase photovoltaic multi-inverter parallel system in islanded operation mode. Each inverter subunit includes a full-bridge DC/AC converter followed by an LC filter. The outputs of all inverter subunits are connected to a common AC bus, which supplies power to a resistive load. For simplicity, I assumed that the inverter units are identical in terms of their semiconductor devices, and the front-end Boost converters are represented by equivalent DC voltage sources.
For the parallel system, the state vector is defined as x = [UC, iL1, iL2, …, iLn]T, where UC represents the common bus voltage, and iL1, iL2, …, iLn represent the inductor currents of each inverter unit. Following a similar MLD modeling procedure as described in Section 3, the continuous-time model of the multi-inverter parallel system can be expressed as:
$$ \begin{cases} \dot{x} = A x + B u_{ab} \\ y = C x \end{cases} $$
where the system matrices for n parallel solar inverters are given by the following block matrix forms:
$$ A = \begin{bmatrix} -\frac{1}{C_n} \sum_{j=1}^{n} \frac{1}{r_{0j}} & \frac{1}{C_n} & \frac{1}{C_n} & \cdots & \frac{1}{C_n} \\ -\frac{1}{L_1} & -\frac{r_1}{L_1} & 0 & \cdots & 0 \\ -\frac{1}{L_2} & 0 & -\frac{r_2}{L_2} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -\frac{1}{L_n} & 0 & 0 & \cdots & -\frac{r_n}{L_n} \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ \frac{1}{L_1} \\ \frac{1}{L_2} \\ \vdots \\ \frac{1}{L_n} \end{bmatrix} $$
The output matrix C is defined as C = [0 0 0 … 1], and the output vector is y = [iL1, iL2, …, iLn]T. This MLD model encapsulates the interactions between parallel-connected solar inverters and the common AC bus, providing a solid foundation for predicting the system behavior under different switching states.
4.2 Circulating Current Analysis
To understand the impact of parameter mismatches on the performance of parallel solar inverters, I conducted a detailed circulating current analysis using a two-inverter parallel system as an example. The circulating current iH is defined as half the difference between the output currents of the two inverters:
$$ i_H = \frac{1}{2}(i_1 – i_2) $$
By applying Kirchhoff’s voltage law to the output loops of both solar inverters and neglecting the small differences in the inverter output impedances, I derived the relationship between the circulating current, the output voltage difference, and the equivalent impedance mismatch:
$$ i_H(k+1) = \frac{1}{2}(i_1(k+1) – i_2(k+1)) $$
The discrete-time equations for the output currents of the two solar inverters can be expressed as:
$$ \begin{cases} i_1(k+1) = \frac{T_s}{L_1} \left[ u_{ab1}(k) – u_C(k) – i_{L1}(k) r_1 \right] + i_1(k) \\ i_2(k+1) = \frac{T_s}{L_2} \left[ u_{ab2}(k) – u_C(k) – i_{L2}(k) r_2 \right] + i_2(k) \end{cases} $$
From this analysis, it becomes evident that the circulating current in parallel-connected solar inverters primarily depends on the difference in output voltages and the mismatch in equivalent output impedances. Since the load impedance does not directly influence the circulating current, the primary control objective is to equalize the inverter output voltages to minimize the circulating current.
4.3 Cost Function with Circulating Current Constraint
In the parallel solar inverter system, the FCS-MPC controller must achieve two control objectives simultaneously: accurate output voltage tracking and circulating current suppression. These two objectives correspond to different physical quantities with different units, so I introduced weighting factors into the cost function to balance their relative importance. The cost function I designed for the parallel system is:
$$ J = Q_1 \sum_{j=1}^{n} \left| U_{Cj}^*(k+1) – U_{Cj}(k+1) \right|^2 + Q_2 \sum_{j=1}^{n} \left| i_{Hj}(k+1) \right|^2 $$
where Q1 and Q2 are the weighting factors, UCj*(k+1) and UCj(k+1) are the reference and predicted voltages, respectively, and iHj(k+1) is the predicted circulating current. In my simulations, I determined these weighting factors through a systematic trial-and-error approach, selecting Q1 = 5 and Q2 = 1. The larger weighting on the voltage tracking error emphasizes the importance of maintaining high output power quality, while the smaller weighting on the circulating current still ensures sufficient suppression without compromising the tracking performance.
The integration of the circulating current constraint into the cost function also provides a useful mechanism for real-time optimization. As the system operating conditions change, the controller can dynamically adjust the switching states to maintain the circulating current within acceptable limits. This online optimization capability is a key advantage of the FCS-MPC approach compared to passive circulating current suppression techniques such as output impedance matching or isolation transformers.
4.4 Simulation Results and Discussion
To validate the performance of the proposed MLD-based FCS-MPC strategy for parallel solar inverters, I conducted simulations using the parameters listed in Table 3. The two inverter units in the parallel system are intentionally designed with slightly different parameters to create realistic mismatched conditions.
| Parameter | Symbol | Inverter 1 | Inverter 2 |
|---|---|---|---|
| DC bus voltage | Udc | 300 V | 300 V |
| Filter inductance | L | 6.8 mH | 5.9 mH |
| Filter capacitance | C | 24 μF | 26 μF |
| Line resistance | r | 0.56 Ω | 0.6 Ω |
| Load resistance | r0 | 50 Ω | 50 Ω |
| Sampling period | Ts | 5 × 10−6 s | 5 × 10−6 s |
As a benchmark for comparison, I also simulated a conventional droop control method. The simulation results clearly demonstrate the advantages of the proposed MLD-FCS-MPC approach. Under the traditional droop control, the output voltage of the parallel solar inverter system contains significant harmonic distortions, as reflected by the THD value of 8.08%. In contrast, the proposed strategy achieves a significantly smoother output voltage waveform with a THD value of 3.90%. This improvement surpasses the recommended industry threshold and demonstrates the superior harmonic suppression capability of the proposed method.
| Performance Metric | Droop Control | MLD-FCS-MPC |
|---|---|---|
| Output voltage THD (%) | 8.08 | 3.90 |
| Fundamental amplitude (V) | 154.1 | 157.8 |
| Circulating current magnitude | Large fluctuations | Effectively suppressed |
| Dynamic response | Slower | Fast |
Furthermore, the dynamic response of the parallel solar inverter system under the proposed control strategy was evaluated through a load step test. At t = 0.02 s, the load resistance was increased from 50 Ω to 100 Ω. The simulation results show that the output voltage remains stable across the load transient, while the output current quickly converges to a new steady state within a short time. This excellent dynamic response is attributable to the predictive nature of the FCS-MPC algorithm, which anticipates future system states and selects the optimal switching actions accordingly.
5. Conclusion
In this research, I have comprehensively investigated the application of MLD-based FCS-MPC strategies for single-phase solar inverters, ranging from a single grid-connected unit to a parallel multi-inverter system. The main contributions and findings of this work can be summarized as follows:
First, I successfully developed an MLD model for solar inverters that accurately captures the hybrid dynamics of the system, encompassing both continuous state variables and discrete switching states. The MLD model transforms the logical relationships between switch states and circuit variables into a unified set of algebraic constraints, which can be efficiently handled within the predictive control optimization framework.
Second, for the single-phase grid-connected solar inverter system, I designed a full-dimensional state observer to estimate the grid current, eliminating the need for an additional current sensor and improving the system robustness against measurement noise. The short prediction horizon of the FCS-MPC strategy combined with the accurate MLD model enables fast convergence to steady state, reducing the grid-connection time compared to conventional methods. The THD of the output current was measured at 0.52%, which not only meets the grid code requirements but also significantly improves the power quality of solar inverters.
Third, I extended the MLD-FCS-MPC methodology to parallel-connected solar inverter systems. The circulating current between parallel solar inverters was analyzed theoretically and incorporated into the cost function as an additional constraint term. Through careful selection of weighting factors, the controller achieved a balanced trade-off between voltage tracking accuracy and circulating current suppression. The simulation results demonstrated a significant reduction in output voltage THD from 8.08% (droop control) to 3.90% (MLD-FCS-MPC), confirming the superior performance of the proposed strategy.
Fourth, the dynamic response of the parallel solar inverter system under load transients was shown to be exceptionally fast, with the output voltage remaining stable and the output current adapting to new conditions within a very short transient period. This rapid dynamic response is essential for maintaining power supply continuity and quality in microgrid applications.
Despite the promising results, there are several avenues for future research that could further enhance the performance and applicability of the proposed methods. One area of investigation is the extension to three-phase solar inverter systems, where the MLD model dimensionality increases and the circulating current suppression becomes more complex due to the interaction between phase sequence and zero-sequence current. Another valuable research direction is the incorporation of model parameter estimation techniques to make the MLD-FCS-MPC strategy more robust against parameter variations caused by temperature aging or component tolerances. Additionally, the experimental validation of the proposed strategy on a hardware test bench would provide valuable insights into the practical implementation challenges, such as computational delay, switching losses, and sensor noise, which are not fully captured in simulation studies.
In conclusion, the MLD-based FCS-MPC strategy presents a powerful and versatile control approach for solar inverters. Its ability to handle multiple control objectives, incorporate system constraints, and achieve fast dynamic responses makes it particularly well-suited for modern microgrid applications where solar inverters play a pivotal role. The findings of this research contribute to the ongoing efforts to enhance the efficiency, reliability, and power quality of renewable energy systems, thereby accelerating the transition towards a sustainable energy future.