In recent years, the rapid development of solar photovoltaic (PV) power generation has significantly alleviated energy shortages and environmental pollution, owing to its advantages of being不受地域限制 and having simple energy conversion. Among PV systems, grid-connected systems are the primary form for effectively utilizing solar energy, as they can supply power to loads and feed excess electricity directly into the grid. As a bridge connecting PV generation devices and the grid, the grid tied inverter is closely related to the stable, reliable, and efficient operation of the power generation system, and its control strategy directly affects the quality of the grid-connected current.
Traditional control methods, such as PI control, PR control, and hysteresis control, often struggle to meet the diverse demands of application scenarios simultaneously. With the advancement of digital processors, model predictive control (MPC) has been widely adopted in grid tied inverters due to its simple structure, no need for complex parameter tuning, fast dynamic response, and multi-objective optimization. However, conventional model predictive current control (MPCC) relies heavily on an accurate mathematical model of the grid tied inverter. In practical applications, any uncertainties can affect the parameters of inverter components, leading to model parameter mismatches and degraded system performance. To enhance system robustness, extensive research has been conducted, resulting in various model-free predictive current control (MFPCC) methods.
Currently, MFPCC can be categorized into three types: based on an ideal model, based on current gradient, and based on an ultra-local model. The ideal model-based approach still depends on the mathematical model and requires additional observers to monitor parameter changes, offering no significant improvement in robustness and involving extensive parameter tuning. The current gradient-based method eliminates the need for system parameters by storing sampled current gradients in a lookup table for prediction. However, this method suffers from lookup table update stagnation, which can lead to non-optimal vector selection and deteriorate steady-state performance, potentially failing to meet national grid standards. In contrast, the ultra-local model-based approach is more reliable, reducing dependence on the mathematical model and avoiding lookup table update issues. Yet, it typically requires a linear extended state observer to estimate system parameters and predefined input gains during offline stages, which involves complex observer design and parameter optimization.
Moreover, most MFPCC methods employ a single vector, selecting one optimal voltage vector from the eight basic vectors and applying it throughout a switching cycle. This results in unchanged switch states during the cycle, preventing the actual current from strictly tracking the reference current and causing significant current ripple. To address this, a dual-vector MFPCC method has been proposed, applying two voltage vectors to reduce current ripple. However, this method has limitations in extending the active voltage vector region, offering limited suppression of current pulsations.
Based on these challenges, this paper comprehensively considers the robustness and steady-state characteristics of grid-connected systems and proposes a three-vector-based MFPCC method for grid tied inverters. The method utilizes an ultra-local model to calculate unknown system parameters online using real-time measured current and voltage information, enabling accurate prediction of future current values. Compared to traditional methods, the proposed approach eliminates the need for complex observer design and avoids tedious parameter tuning. Additionally, three voltage vectors—two adjacent active vectors and one zero vector—are applied within a control period. The duty cycles of these vectors are calculated using the Lagrange multiplier method to improve grid-connected current quality. Finally, the effectiveness of the proposed method is validated through experimental results.
The mathematical model of a two-level voltage source inverter (VSI) for grid tied applications is fundamental to understanding the control strategy. The equivalent circuit topology of a two-level VSI is shown below, where it consists of three legs with switches that generate eight switching states, corresponding to eight voltage vectors in the space vector diagram.
According to Kirchhoff’s voltage law, the mathematical model of a two-level VSI in the natural coordinate system can be expressed as:
$$ L \frac{d}{dt} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} = \begin{bmatrix} u_{aN} \\ u_{bN} \\ u_{cN} \end{bmatrix} – R \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} – \begin{bmatrix} e_a \\ e_b \\ e_c \end{bmatrix} $$
where \( L \) is the output filter inductance, \( R \) is the parasitic resistance, \( i_x \) and \( e_x \) (with \( x = a, b, c \)) are the grid-side current and voltage, respectively, and \( u_{xN} \) is the inverter output voltage determined by the switch state \( S_x \). The two-level VSI can produce eight switch states, as illustrated in the space voltage vector distribution. When \( S_x = 1 \), the upper switch of phase \( x \) is on, and when \( S_x = 0 \), the lower switch is on, allowing output current control by changing \( S_x \).
Clearly, the controlled object is a high-order time-varying nonlinear system. To simplify analysis, its mathematical model is decoupled using Clark and Park transformations. The transformation matrices \( T_{\text{Clark}} \) and \( T_{\text{Park}} \) are:
$$ T_{\text{Clark}} = \begin{bmatrix} 2/3 & -1/2 & -1/2 \\ 0 & \sqrt{3}/3 & -\sqrt{3}/3 \end{bmatrix} $$
$$ T_{\text{Park}} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} $$
where \( \theta \) is the angle between the d-axis and the a-axis. The expression in the synchronous rotating coordinate system is:
$$ L \frac{d}{dt} \begin{bmatrix} i_d \\ i_q \end{bmatrix} = \begin{bmatrix} u_d \\ u_q \end{bmatrix} + \begin{bmatrix} -R & \omega \\ -\omega & -R \end{bmatrix} \begin{bmatrix} i_d \\ i_q \end{bmatrix} – \begin{bmatrix} e_d \\ e_q \end{bmatrix} $$
where the subscripts \( d \) and \( q \) denote d-q axis components, and \( \omega \) is the grid angular frequency. This model forms the basis for traditional MPCC, but its reliance on accurate parameters poses challenges in real-world applications, especially for grid tied inverters subject to parameter variations.
Traditional MPCC aims to obtain a discrete solution for predicting future grid currents. Using the forward Euler method, the mathematical model is discretized to derive the current at time \( k+1 \):
$$ \begin{bmatrix} i_d(k+1) \\ i_q(k+1) \end{bmatrix} = \begin{bmatrix} 1 – RT_s/L & \omega T_s \\ 1 – RT_s/L & -\omega T_s \end{bmatrix} \begin{bmatrix} i_d(k) \\ i_q(k) \end{bmatrix} + \begin{bmatrix} T_s/L & 0 \\ 0 & T_s/L \end{bmatrix} \begin{bmatrix} u_d(k) – e_d(k) \\ u_q(k) – e_q(k) \end{bmatrix} $$
where \( T_s \) is the sampling period, and \( k \) and \( k+1 \) represent variables at time \( k \) and \( k+1 \), respectively. Then, an optimal voltage vector is selected by evaluating a cost function, which can be expressed as:
$$ g = |i_{d}^{\text{ref}}(k+1) – i_d(k+1)| + |i_{q}^{\text{ref}}(k+1) – i_q(k+1)| $$
where the superscript \( \text{ref} \) denotes reference values. However, this method is highly sensitive to parameter inaccuracies in the grid tied inverter, such as variations in \( L \) and \( R \), which can degrade performance.
To address this, the proposed MFPCC method is based on an ultra-local model, which reduces dependence on precise system parameters. In modern control theory, the ultra-local model for a first-order single-variable system can be expressed as:
$$ \dot{y} = F + \alpha u $$
where \( y \) and \( u \) are the system output and input, respectively, \( \alpha \) is the input gain, and \( F \) represents the sum of known and unknown parts of the system. For a grid tied inverter, based on Equation (1) and Equation (7), the ultra-local model can be expressed as:
$$ \frac{di_x}{dt} = F + \alpha u_x $$
where \( i_x \), \( F \), and \( \alpha \) are given by:
$$ i_x = i_d + j \cdot i_q $$
$$ F = -\frac{R \cdot i_x + e_x}{L} $$
$$ \alpha = \frac{1}{L} $$
Due to high sampling frequencies and short control periods, it is assumed that \( F \) and \( \alpha \) remain constant over the past two control cycles. Thus, the stator current relationship at past times can be expressed as:
$$ i_x(k-1) = i_x(k-2) + T_s (F + \alpha u_x(k-2)) $$
$$ i_x(k) = i_x(k-1) + T_s (F + \alpha u_x(k-1)) $$
Defining the current differentials:
$$ \Delta i_x(k-2) = \frac{i_x(k-1) – i_x(k-2)}{T_s} = F + \alpha u_x(k-2) $$
$$ \Delta i_x(k-1) = \frac{i_x(k) – i_x(k-1)}{T_s} = F + \alpha u_x(k-1) $$
Combining these equations yields:
$$ \alpha = \frac{\Delta i_x(k-1) – \Delta i_x(k-2)}{u_x(k-1) – u_x(k-2)} $$
$$ F = \Delta i_x(k-1) – \alpha u_x(k-1) $$
Therefore, by measuring current and voltage information from the past two control cycles, \( F \) and \( \alpha \) can be directly estimated online, eliminating the need for complex observer design and parameter tuning. Consequently, the current at time \( k+1 \) can be predicted as:
$$ \begin{bmatrix} i_d(k+1) \\ i_q(k+1) \end{bmatrix} = T_s F \begin{bmatrix} 1 \\ 1 \end{bmatrix} + T_s \alpha \begin{bmatrix} u_d(k) \\ u_q(k) \end{bmatrix} + \begin{bmatrix} i_d(k) \\ i_q(k) \end{bmatrix} $$
This online parameter calculation enhances the robustness of the grid tied inverter against parameter variations, making it suitable for dynamic PV system environments.
To further improve steady-state performance, the concept of space vector modulation (SVM) is integrated into the predictive control framework. In the proposed three-vector MFPCC, three voltage vectors are applied within one control period: two adjacent active vectors and one zero vector. This approach better synthesizes the desired voltage vector and reduces current ripple in the grid tied inverter. The eight voltage vectors of a two-level VSI divide the control space into six sectors. After determining the sector of the desired voltage vector, appropriate vectors are selected. First, based on traditional MPC, one optimal vector \( u_{\text{sub}} \) is selected from the six active vectors. Second, a sub-optimal vector \( u_{\text{opt}} \) is determined from the vectors adjacent to the optimal one. Finally, a zero vector \( u_0 \) is chosen; since both zero vectors \( V_0 \) and \( V_7 \) have the same effect on current, \( V_0 \) is used uniformly. The selection process is summarized in Table 1.
| Optimal Vector | Sub-optimal Candidate Vectors |
|---|---|
| V1 | V2, V6 |
| V2 | V1, V3 |
| V3 | V2, V4 |
| V4 | V3, V5 |
| V5 | V4, V6 |
| V6 | V5, V1 |
Assuming the applied voltage vectors \( u_0 \), \( u_{\text{opt}} \), and \( u_{\text{sub}} \) have duty cycles \( t_0 \), \( t_{\text{opt}} \), and \( t_{\text{sub}} \) respectively, the total current error function can be expressed as:
$$ g_1 = g(u_0)^2 t_0^2 + g(u_{\text{opt}})^2 t_{\text{opt}}^2 + g(u_{\text{sub}})^2 t_{\text{sub}}^2 $$
These duty cycles must satisfy the following constraints:
$$ t_0 + t_{\text{opt}} + t_{\text{sub}} = T_s $$
$$ 0 \leq t_0 \leq T_s $$
$$ 0 \leq t_{\text{opt}} \leq T_s $$
$$ 0 \leq t_{\text{sub}} \leq T_s $$
Combining the error function and constraints, the Lagrange multiplier method is used to solve for the optimal duty cycles \( t_0^* \), \( t_{\text{opt}}^* \), and \( t_{\text{sub}}^* \):
$$ t_0^* = \frac{g(u_{\text{opt}})^2 g(u_{\text{sub}})^2 T_s}{g(u_0)^2 g(u_{\text{opt}})^2 + g(u_{\text{opt}})^2 g(u_{\text{sub}})^2 + g(u_{\text{sub}})^2 g(u_0)^2} $$
$$ t_{\text{opt}}^* = \frac{g(u_0)^2 g(u_{\text{sub}})^2 T_s}{g(u_0)^2 g(u_{\text{opt}})^2 + g(u_{\text{opt}})^2 g(u_{\text{sub}})^2 + g(u_{\text{sub}})^2 g(u_0)^2} $$
$$ t_{\text{sub}}^* = \frac{g(u_0)^2 g(u_{\text{opt}})^2 T_s}{g(u_0)^2 g(u_{\text{opt}})^2 + g(u_{\text{opt}})^2 g(u_{\text{sub}})^2 + g(u_{\text{sub}})^2 g(u_0)^2} $$
Finally, PWM modulation generates the drive signals for the grid tied inverter. Different vector combinations produce specific switching sequences. For example, if the selected active vectors are V1 and V2, the switching sequence is V1—V2—V7—V2—V1—V0, with only one phase switch state changed at any time to reduce switching losses. The overall control block diagram of the proposed three-vector MFPCC for grid tied inverters integrates these elements, showcasing the online parameter calculation and multi-vector application for improved performance.
To validate the effectiveness of the proposed three-vector model-free predictive current control for grid tied inverters, a 270W experimental platform was constructed. The setup used an adjustable DC power supply to simulate the output of a front-end DC/DC boost converter, with a maximum output voltage of 300V and a test voltage set to 100V. The inverter comprised three FF300R12ME4 modules, and the grid-side filter inductance consisted of three 5mH/0.7Ω reactors. The AC side was connected to an autotransformer to adjust the output voltage amplitude to 311V. The core controller was a TI TMS320F28335 DSP chip. Stator currents were measured using HAS50-S/SP50 current sensors, and grid voltages were measured using LV25-P voltage sensors. The switching frequency and sampling frequency were both set to 10kHz. To verify the steady-state performance, comparative experiments were conducted with a reference current of 4A.
The experimental results for traditional MPCC showed that the grid voltage and three-phase grid-connected current were in phase, indicating unity power factor operation. However, the total harmonic distortion (THD) of the grid-connected current was approximately 7.23%, which fails to meet grid standards. In contrast, the proposed MFPCC method also demonstrated unity power factor operation, but with a THD of 2.79%, satisfying grid requirements. This improvement is attributed to the three-vector approach, which reduces current ripple by better synthesizing the voltage vector in the grid tied inverter.
Additionally, the dynamic performance of the system is crucial. Comparative experiments between traditional MPCC and the proposed MFPCC were conducted with the reference current stepped down from 4A to 2.5A. Both methods showed fast tracking of the reference current, with grid voltage and current remaining in phase, indicating excellent dynamic response. This highlights the robustness of the proposed method in handling transient conditions for grid tied inverters.
Furthermore, the proposed control method’s strong robustness to inverter parameter variations was verified. In scenarios where actual inverter parameters deviate from ideal values due to特殊运行环境, traditional MPCC resulted in significant current harmonics and deteriorated steady-state performance, posing risks to PV system safety. However, with the proposed MFPCC, the current prediction depends solely on sampled voltage and current, making it immune to model parameter mismatches. The grid-connected current consistently met grid standards, ensuring stable and safe operation of the grid tied inverter system.
In summary, this paper proposes an improved model-free predictive current control method based on an ultra-local model for grid tied inverters in photovoltaic systems. By applying three basic voltage vectors within one control period and calculating their duty cycles using the Lagrange multiplier method, the approach effectively reduces grid-connected current ripple and enhances system stability. Experimental results confirm that the proposed MFPCC achieves a THD of 2.79%, meeting grid requirements, and exhibits strong robustness against parameter changes. This strategy contributes to alleviating energy shortages in regions like western areas and promotes the rapid development of new energy industries, such as solar-powered vehicles and rooftop PV power stations, by ensuring reliable performance of grid tied inverters.
The mathematical formulations and control algorithms presented here emphasize the importance of online parameter estimation and multi-vector modulation in advanced grid tied inverter control. Future work could explore extensions to multi-level inverters or integration with energy storage systems to further enhance the flexibility and efficiency of PV power generation. Overall, the three-vector MFPCC offers a practical solution for improving the quality and reliability of grid-connected currents in diverse applications, making it a valuable contribution to the field of renewable energy power electronics.