In recent years, the rapid integration of renewable energy sources, such as wind and photovoltaic systems, has intensified the focus on power quality in electrical grids. As a critical interface, the on-grid inverter must ensure efficient and stable operation to mitigate adverse impacts on grid performance. Among various control strategies, Finite Control Set Model Predictive Control (FCS-MPC) has gained prominence for its simplicity, fast dynamic response, and ability to handle multiple control objectives. However, the precision of FCS-MPC is often compromised by prediction errors arising from discrete model inaccuracies, parameter mismatches, and external disturbances. Specifically, the mismatch in filter inductance parameters can exacerbate current prediction errors, leading to increased harmonic distortion in grid-side currents. This article addresses these challenges by proposing an adaptive error correction method for model predictive control in on-grid inverters. By incorporating an error correction link and real-time inductance parameter identification using an Extended Kalman Filter (EKF), the method enhances prediction accuracy and suppresses harmonics, even under parameter variations. Through comprehensive analysis and simulation, the effectiveness of this approach is validated, contributing to improved power quality in renewable energy systems.
The on-grid inverter serves as a pivotal component in converting DC power from renewable sources into AC power synchronized with the grid. Its control strategy directly influences grid stability and efficiency. Traditional control methods, such as proportional-integral (PI) controllers, often struggle with nonlinearities and dynamic changes. In contrast, FCS-MPC offers a predictive framework by evaluating a finite set of voltage vectors each switching period, optimizing performance based on a cost function. Nonetheless, the discrete nature of FCS-MPC introduces prediction errors, which are further amplified when system parameters, like filter inductance, deviate from nominal values. This not only degrades current tracking but also increases total harmonic distortion (THD), posing risks to grid compliance. Therefore, developing adaptive mechanisms to correct these errors is essential for robust inverter operation. This work delves into the root causes of prediction errors, proposes a correction scheme, and integrates online parameter identification to maintain control accuracy. The subsequent sections detail the modeling, error analysis, adaptive correction methodology, and simulation results, emphasizing the role of on-grid inverters in modern power systems.
To understand the prediction errors in FCS-MPC for on-grid inverters, we first establish a mathematical model. Consider a three-phase on-grid inverter with an L filter, where the inverter-side current dynamics in the stationary αβ-reference frame are described by:
$$ \begin{bmatrix} u_{\alpha} \\ u_{\beta} \end{bmatrix} = L \frac{d}{dt} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} + R \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} + \begin{bmatrix} u_{g\alpha} \\ u_{g\beta} \end{bmatrix} $$
Here, \( u_{\alpha} \) and \( u_{\beta} \) represent the inverter output voltages, \( i_{\alpha} \) and \( i_{\beta} \) are the inverter-side currents, \( L \) is the filter inductance, \( R \) is the parasitic resistance, and \( u_{g\alpha} \) and \( u_{g\beta} \) denote the grid voltages. Discretizing this equation using a forward Euler method with a sampling period \( T_s \) yields the prediction model for the next sampling instant \( k+1 \):
$$ \begin{bmatrix} i_{\alpha}(k+1) \\ i_{\beta}(k+1) \end{bmatrix} = \begin{bmatrix} i_{\alpha}(k) \\ i_{\beta}(k) \end{bmatrix} + \frac{T_s}{L} \begin{bmatrix} u_{\alpha}(k) – u_{g\alpha}(k) – R i_{\alpha}(k) \\ u_{\beta}(k) – u_{g\beta}(k) – R i_{\beta}(k) \end{bmatrix} $$
This model is fundamental to FCS-MPC, where the cost function minimizes the error between predicted and reference currents, along with other objectives like DC-link voltage balance. However, errors emerge due to discretization approximations and parameter inaccuracies. To analyze the discretization error, we divide the prediction period into M intervals and derive the true current \( i_{z\alpha\beta}(k+1) \) as an average over these intervals. The prediction error \( \Delta i_{dis\alpha\beta}(k+1) \) caused by discretization is approximated as:
$$ \Delta i_{dis\alpha\beta}(k+1) = \frac{T_s}{L} \left[ \bar{u}_{g\alpha\beta}(k) – u_{g\alpha\beta}(k) \right] $$
where \( \bar{u}_{g\alpha\beta}(k) \) is the average grid voltage over the period. This error remains constant between consecutive cycles under steady-state grid conditions, affecting current tracking. More critically, parameter mismatch, especially in inductance \( L \), introduces additional errors. If the actual inductance \( L_z \) differs from the nominal \( L \) by \( \Delta L \), the prediction error due to parameter mismatch \( \Delta i_{para\alpha\beta} \) is given by:
$$ \frac{\Delta i_{para\alpha\beta}}{i_{\alpha\beta}} = \frac{T_s}{L} \left(1 – \frac{1}{L / L_z}\right) u_{\alpha\beta}(k) – \frac{T_s}{L} \left(1 – \frac{1}{L / L_z}\right) u_{g\alpha\beta}(k) – \frac{T_s}{L} \left(1 – \frac{1}{L / L_z}\right) R i_{\alpha\beta}(k) $$
This expression shows that the error scales with \( \Delta L \) and depends on instantaneous currents and voltages, emphasizing the need for accurate parameter knowledge in on-grid inverters. To quantify the impact, assume typical values: \( T_s = 50 \, \mu s \), \( L = 2 \, mH \), and voltage errors of 10 V. The relationship between inductance mismatch and current prediction error can be summarized in the following table, highlighting the sensitivity of FCS-MPC performance to parameter variations.
| Inductance Mismatch (\( \Delta L / L \)) | Current Prediction Error Increase | Effect on Grid Current THD |
|---|---|---|
| 10% | Approx. 5-10% | Significant rise in harmonics |
| 20% | Approx. 10-20% | Potential instability |
| 50% | Approx. 25-50% | Severe distortion |
Given these challenges, an adaptive error correction method is proposed to enhance the robustness of model predictive control for on-grid inverters. The core idea involves integrating an error correction term into the prediction model and dynamically identifying the filter inductance using an EKF. Starting with error correction, we define the correction terms \( \Delta i_{\alpha}(k+1) \) and \( \Delta i_{\beta}(k+1) \) based on past prediction errors:
$$ \begin{bmatrix} \Delta i_{\alpha}(k+1) \\ \Delta i_{\beta}(k+1) \end{bmatrix} = \eta \begin{bmatrix} 2(i_{p\alpha}(k) – i_{\alpha}(k)) + (i_{p\alpha}(k-1) – i_{\alpha}(k-1)) \\ 2(i_{p\beta}(k) – i_{\beta}(k)) + (i_{p\beta}(k-1) – i_{\beta}(k-1)) \end{bmatrix} $$
Here, \( i_{p\alpha}(k) \) and \( i_{p\beta}(k) \) are predicted currents, \( i_{\alpha}(k) \) and \( i_{\beta}(k) \) are sampled currents, and \( \eta \) is a correction coefficient typically set between 0.1 and 0.5 for stability. Incorporating this into the discrete model yields the corrected prediction:
$$ \begin{bmatrix} i_{c\alpha}(k+1) \\ i_{c\beta}(k+1) \end{bmatrix} = \left(1 – \frac{R T_s}{L} – \frac{\eta}{2}\right) \begin{bmatrix} i_{\alpha}(k) \\ i_{\beta}(k) \end{bmatrix} + \left(\frac{R T_s}{2L} + \frac{\eta}{2}\right) \begin{bmatrix} i_{p\alpha}(k) \\ i_{p\beta}(k) \end{bmatrix} + \frac{T_s}{L} \begin{bmatrix} u_{\alpha}(k) – u_{g\alpha}(k) \\ u_{\beta}(k) – u_{g\beta}(k) \end{bmatrix} + \frac{\eta}{2} \begin{bmatrix} i_{p\alpha}(k-1) – i_{\alpha}(k-1) \\ i_{p\beta}(k-1) – i_{\beta}(k-1) \end{bmatrix} $$
This corrected model reduces discretization errors by learning from past discrepancies, thereby improving current tracking in on-grid inverters. However, to address parameter mismatch, real-time inductance identification is crucial. The EKF provides an effective solution due to its ability to handle nonlinear systems and offer self-correcting estimates. For a three-phase on-grid inverter with balanced inductors, we focus on identifying the inductance for one phase (e.g., phase A) using active power \( P_a \) and reactive power \( Q_a \) measurements. The state-space model for EKF is formulated as:
$$ Z(k) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} x(k-1) $$
$$ \begin{bmatrix} P(k) \\ Q(k) \end{bmatrix} = \begin{bmatrix} A & B \\ -B & A \end{bmatrix} x(k) + v(k) $$
where \( Z(k) = \begin{bmatrix} Z_1(k) \\ Z_2(k) \end{bmatrix} = \begin{bmatrix} \frac{R(k)}{R^2(k) + (\omega L(k))^2} \\ \frac{\omega L(k)}{R^2(k) + (\omega L(k))^2} \end{bmatrix} \), \( A = u_a^2 – u_a u_{ga} \cos \theta \), \( B = u_a u_{ga} \sin \theta \), \( \omega \) is the grid angular frequency, \( \theta \) is the phase difference, and \( v(k) \) represents measurement noise. The inductance and resistance are then computed as:
$$ R(k) = \frac{Z_1(k)}{Z_1^2(k) + Z_2^2(k)}, \quad L(k) = \frac{Z_2(k)}{\omega (Z_1^2(k) + Z_2^2(k))} $$
By integrating this EKF-based identification into the control loop, the on-grid inverter can adaptively update the inductance value used in the prediction model, ensuring accuracy even under parameter drifts. The overall control structure combines error correction and parameter identification, forming an adaptive MPC scheme. The cost function in FCS-MPC is modified to include corrected predictions:
$$ g_i = [i^*_{\alpha}(k+1) – i_{c\alpha}(k+1)]^2 + [i^*_{\beta}(k+1) – i_{c\beta}(k+1)]^2 + \lambda [\Delta u_c(k+1)]^2 $$
where \( i^*_{\alpha}(k+1) \) and \( i^*_{\beta}(k+1) \) are reference currents, and \( \lambda \) weights the DC-link voltage balance. This approach enables the on-grid inverter to maintain low harmonic distortion and precise current control, crucial for grid code compliance.
To validate the proposed adaptive error correction method for on-grid inverters, simulations are conducted using MATLAB/Simulink. The system parameters are based on a three-level on-grid inverter configuration, as summarized in the table below. These parameters reflect typical renewable energy applications, where the on-grid inverter interfaces with a grid voltage of 220 V and operates at a switching frequency of 20 kHz.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC-link voltage \( U_{dc} \) | 800 V | Grid frequency \( f \) | 50 Hz |
| Capacitors \( C_1 = C_2 \) | 150 μF | Grid voltage \( U_g \) | 220 V |
| Switching frequency \( f_s \) | 20 kHz | Weight coefficient \( \lambda \) | 10 |
| Nominal inductance \( L \) | 2 mH | Sampling period \( T_s \) | 50 μs |
First, the EKF-based inductance identification is tested under different inductance values to verify its accuracy. The results show that for \( L = 3 \, mH \), the identification error stabilizes at approximately 0.02 mH (0.67% error), and for \( L = 5 \, mH \), the error is about 0.05 mH (1.00% error). This confirms the EKF’s effectiveness in providing real-time parameter estimates for on-grid inverters. The identification process converges quickly within a few grid cycles, ensuring minimal disruption to control performance.
Next, comparative simulations are performed between traditional FCS-MPC, error-corrected MPC without parameter identification, and the proposed adaptive error-corrected MPC with EKF identification. The key metrics include current tracking error and total harmonic distortion (THD) of grid-side currents. For traditional FCS-MPC, the on-grid inverter exhibits significant prediction errors, leading to a current deviation of up to 2.1 A and a THD of 2.82%. The harmonic spectrum reveals prominent 5th, 7th, 11th, and 13th harmonics, as detailed in the following table.
| Harmonic Order | Magnitude (% of Fundamental) | Contribution to THD |
|---|---|---|
| 5th | 0.95% | High |
| 7th | 0.98% | High |
| 11th | 0.66% | Medium |
| 13th | 0.84% | Medium |
With error correction alone (without parameter identification), performance improves, reducing the current deviation to 1.3 A and THD to 1.69%. However, when inductance mismatch is introduced (e.g., actual \( L = 2.5 \, mH \) vs. nominal \( 2 \, mH \)), the error-corrected MPC struggles, as the model inaccuracy persists. In this case, the on-grid inverter shows a current deviation of up to 2.4 A and THD of 5.31%, indicating susceptibility to parameter variations.
In contrast, the proposed adaptive error-corrected MPC with EKF identification maintains robust performance under inductance mismatch. The current deviation is limited to 1.5 A, and THD is reduced to 2.23%, with harmonic components significantly suppressed. The improvement stems from the real-time adjustment of the prediction model using identified parameters, allowing the on-grid inverter to adapt to changes. The simulation results are summarized in the table below, comparing the three methods under parameter mismatch conditions.
| Control Method | Current Deviation (Max) | THD | 5th Harmonic | 7th Harmonic | 11th Harmonic | 13th Harmonic |
|---|---|---|---|---|---|---|
| Traditional FCS-MPC | 2.1 A | 2.82% | 0.95% | 0.98% | 0.66% | 0.84% |
| Error-Corrected MPC (No ID) | 2.4 A | 5.31% | 1.35% | 0.88% | 1.22% | 0.78% |
| Adaptive Error-Corrected MPC (EKF ID) | 1.5 A | 2.23% | 0.48% | 0.52% | 0.61% | 0.49% |
These results underscore the importance of adaptive mechanisms in on-grid inverters, especially in dynamic environments where parameters may drift due to temperature, aging, or manufacturing tolerances. The proposed method not only enhances current tracking but also ensures compliance with grid standards, such as IEEE 1547, which limit THD to below 5%. Furthermore, the computational overhead of EKF is manageable for modern digital signal processors, making it feasible for real-time implementation in on-grid inverter controls.
From a broader perspective, the adaptive error correction approach contributes to the reliability of renewable energy integration. On-grid inverters equipped with such intelligent control can mitigate power quality issues, reduce maintenance costs, and support grid stability. Future work could extend this method to multi-inverter systems or incorporate other parameter identifications, such as grid impedance, for enhanced adaptability. Additionally, machine learning techniques could be explored to optimize the correction coefficient \( \eta \) or to predict errors more accurately.
In conclusion, this article has presented an in-depth study on model predictive control for on-grid inverters, focusing on adaptive error correction to address prediction inaccuracies. By analyzing the sources of error, including discretization effects and inductance mismatch, we developed a correction scheme integrated with real-time parameter identification using EKF. Simulations demonstrate that the proposed method significantly reduces current deviations and harmonic distortion, even under parameter variations. This advancement strengthens the role of on-grid inverters in renewable energy systems, ensuring high power quality and robust operation. As the demand for clean energy grows, such adaptive control strategies will be pivotal in building resilient and efficient power grids.
The continuous evolution of on-grid inverter technology underscores the need for innovative control solutions. The integration of adaptive error correction not only addresses immediate performance issues but also paves the way for smarter grid interactions. For instance, on-grid inverters can participate in ancillary services like voltage regulation or frequency support, leveraging accurate current control. Moreover, the methodology outlined here can be adapted to other power electronic converters, such as those in motor drives or uninterruptible power supplies, broadening its impact. Ultimately, by enhancing the precision and reliability of on-grid inverters, we contribute to a sustainable energy future, where renewable sources seamlessly integrate into the global power infrastructure.