In the context of global energy transition and the increasing integration of renewable sources, the grid connected inverter has emerged as a critical interface for converting direct current from distributed generation systems into alternating current compatible with the utility grid. The performance and stability of these inverters directly influence power quality, system reliability, and overall grid health. Traditional control strategies, such as linear proportional-integral methods, often face challenges in handling nonlinearities and dynamic disturbances. Among advanced techniques, Finite Control Set Model Predictive Control (FCS-MPC) has gained prominence due to its intuitive design, fast dynamic response, and ability to handle multiple constraints. However, conventional FCS-MPC for grid connected inverters suffers from inherent issues like computational delays, sensitivity to parameter variations, and difficulties in weighting coefficient assignment for multi-objective optimization. This article, from my research perspective, proposes a novel multi-objective co-optimization model predictive control strategy specifically designed for LC-filtered grid connected inverters. The approach integrates Lyapunov-based stability analysis, delay compensation mechanisms, and a systematic method for balancing control objectives, thereby enhancing steady-state and dynamic performance while reducing switching losses. Through comprehensive mathematical formulation, simulation validation, and comparative analysis, I demonstrate the efficacy of this strategy in improving the operational robustness of grid connected inverter systems.
The topology of a three-phase two-level LC-filtered grid connected inverter is fundamental to understanding its control. It typically consists of a DC voltage source, six switching devices (e.g., IGBTs), an LC filter (inductor L and capacitor C), and a connection to the three-phase grid. The inverter’s output voltages are synthesized by switching states, which generate discrete voltage vectors. In the stationary αβ-reference frame, the dynamic equations governing the inverter can be derived using Kirchhoff’s laws. For the grid connected inverter, the state variables include the inverter-side inductor current \( \mathbf{i}_1 = [i_{1\alpha}, i_{1\beta}]^T \) and the capacitor voltage \( \mathbf{u}_c = [u_{c\alpha}, u_{c\beta}]^T \). The grid voltage is denoted as \( \mathbf{e} = [e_{\alpha}, e_{\beta}]^T \), and the inverter output voltage as \( \mathbf{u} = [u_{\alpha}, u_{\beta}]^T \). The continuous-time model is given by:
$$ L \frac{d\mathbf{i}_1}{dt} = \mathbf{u} – R_{re}\mathbf{i}_1 – \mathbf{e} $$
$$ C \frac{d\mathbf{u}_c}{dt} = \mathbf{i}_1 – \mathbf{i}_g $$
where \( R_{re} \) represents the equivalent resistance of the inductor, \( \mathbf{i}_g = [i_{g\alpha}, i_{g\beta}]^T \) is the grid current, and \( L \) and \( C \) are the filter inductance and capacitance, respectively. For digital implementation, discretization using the forward Euler method with a sampling period \( T_s \) yields the discrete predictive model:
$$ \mathbf{i}_1^{pre}(k+1) = \left(1 – \frac{R_{re} T_s}{L}\right) \mathbf{i}_1(k) + \frac{T_s}{L} \left( \mathbf{u}(k) – \mathbf{e}(k) \right) $$
$$ \mathbf{u}_c^{pre}(k+1) = \mathbf{u}_c(k) + \frac{T_s}{C} \left( \mathbf{i}_1(k) – \mathbf{i}_g(k) \right) $$
Here, \( k \) denotes the current sampling instant, and \( pre \) indicates predicted values. This model forms the basis for predicting future states in FCS-MPC. The grid connected inverter can produce eight switching states, resulting in seven distinct voltage vectors (including two zero vectors), as summarized in Table 1. These vectors are essential for the prediction and optimization process.
| Vector Index | Switching State [S_a, S_b, S_c] | Voltage Vector \( \mathbf{u} \) in αβ-Frame |
|---|---|---|
| 0 | [0, 0, 0] | 0 |
| 1 | [1, 0, 0] | \(\frac{2}{3}U_{dc}\) |
| 2 | [1, 1, 0] | \(\frac{1}{3}U_{dc} + j\frac{\sqrt{3}}{3}U_{dc}\) |
| 3 | [0, 1, 0] | \(-\frac{1}{3}U_{dc} + j\frac{\sqrt{3}}{3}U_{dc}\) |
| 4 | [0, 1, 1] | \(-\frac{2}{3}U_{dc}\) |
| 5 | [0, 0, 1] | \(-\frac{1}{3}U_{dc} – j\frac{\sqrt{3}}{3}U_{dc}\) |
| 6 | [1, 0, 1] | \(\frac{1}{3}U_{dc} – j\frac{\sqrt{3}}{3}U_{dc}\) |
| 7 | [1, 1, 1] | 0 |
Conventional FCS-MPC for a grid connected inverter often employs a single-objective cost function, such as minimizing the tracking error of the output current. The cost function is typically defined as:
$$ g_1 = \| \mathbf{i}^{ref}(k+1) – \mathbf{i}^{pre}(k+1) \|^2 $$
where \( \mathbf{i}^{ref} \) is the reference current. While simple, this approach neglects other critical aspects like capacitor voltage regulation (due to LC filter coupling) and switching frequency, leading to suboptimal performance. Moreover, inherent computational delays in digital control platforms cause a one-step delay between calculation and actuation, degrading stability and increasing current harmonics. To address these limitations, I propose a multi-objective co-optimization strategy that synergistically manages inductor current, capacitor voltage, and switching transitions, backed by Lyapunov stability guarantees and advanced delay compensation.
The core innovation lies in the formulation of a multi-objective cost function that explicitly accounts for the coupled dynamics of the LC filter in the grid connected inverter. Instead of ad-hoc weighting, I derive stability conditions using Lyapunov’s direct method to ensure robust control of the primary objective—inductor current tracking. Let the current tracking error be \( \mathbf{e}_i = \mathbf{i}_1^{ref} – \mathbf{i}_1^{pre} \). Consider a Lyapunov function candidate:
$$ V = \frac{1}{2} \mathbf{e}_i^T \mathbf{e}_i $$
Its derivative along the system trajectories is:
$$ \dot{V} = \mathbf{e}_i^T \dot{\mathbf{e}}_i = \mathbf{e}_i^T \left( \dot{\mathbf{i}}_1^{ref} – \dot{\mathbf{i}}_1^{pre} \right) $$
By substituting the discrete dynamics and ensuring \( \dot{V} < 0 \) through proper voltage vector selection, asymptotic stability of the current tracking error is guaranteed. This Lyapunov constraint guides the primary objective, allowing the secondary objectives—capacitor voltage tracking and switching reduction—to be incorporated with weighted penalties. The overall cost function becomes:
$$ g_2 = \| \mathbf{i}_1^{ref}(k+3) – \mathbf{i}_1^{pre}(k+3) \|^2 + \lambda_1 \| \mathbf{u}_c^{ref}(k+3) – \mathbf{u}_c^{pre}(k+3) \|^2 + \lambda_2 n_s $$
where \( \lambda_1 \) and \( \lambda_2 \) are weighting coefficients for capacitor voltage error and switching transitions, respectively, and \( n_s \) is the number of switching changes per sampling period, calculated as:
$$ n_s = \sum_{x=a,b,c} |S_x(k+1) – S_x(k)| $$
The reference values \( \mathbf{i}_1^{ref}(k+3) \), \( \mathbf{u}_c^{ref}(k+3) \), and \( \mathbf{e}(k+2) \) are estimated using vector angle compensation to account for future variations:
$$ \mathbf{e}(k+2) = \mathbf{e}(k) e^{2j\omega T_s} $$
$$ \mathbf{i}_1^{ref}(k+3) = \mathbf{i}_1^{ref}(k) e^{3j\omega T_s} $$
$$ \mathbf{u}_c^{ref}(k+3) = \mathbf{u}_c^{ref}(k) e^{3j\omega T_s} $$
Here, \( \omega \) is the grid angular frequency. This compensation mitigates errors caused by reference prediction lag. Furthermore, to address computational delay, I employ a three-step prediction horizon. The state variables are predicted for instants \( k+1 \), \( k+2 \), and \( k+3 \) using the discretized model iteratively. For example, the inductor current prediction extends as:
$$ \mathbf{i}_1^{pre}(k+2) = \left(1 – \frac{R_{re} T_s}{L}\right) \mathbf{i}_1^{pre}(k+1) + \frac{T_s}{L} \left( \mathbf{u}(k+1) – \mathbf{e}(k+1) \right) $$
$$ \mathbf{i}_1^{pre}(k+3) = \left(1 – \frac{R_{re} T_s}{L}\right) \mathbf{i}_1^{pre}(k+2) + \frac{T_s}{L} \left( \mathbf{u}(k+2) – \mathbf{e}(k+2) \right) $$
Similar equations apply for capacitor voltage. By evaluating the cost function \( g_2 \) at \( k+3 \), the optimal voltage vector that minimizes the multi-objective criteria while satisfying Lyapunov stability is selected for actuation at \( k+1 \), effectively compensating the delay. This integrated approach ensures that the grid connected inverter operates with enhanced precision and efficiency.
To validate the proposed strategy, I conducted extensive simulations using MATLAB/Simulink, focusing on a three-phase grid connected inverter with LC filter. The system parameters are listed in Table 2. Comparative analyses were performed against conventional single-objective FCS-MPC and a dual-objective FCS-MPC (considering only inductor current and capacitor voltage). The performance metrics included steady-state current total harmonic distortion (THD), dynamic response to reference changes, and average switching frequency.
| Parameter | Symbol | Value |
|---|---|---|
| DC Bus Voltage | \( U_{dc} \) | 600 V |
| Grid Voltage (RMS) | \( e \) | 220 V |
| Grid Frequency | \( f_n \) | 50 Hz |
| Filter Inductance | \( L \) | 10 mH |
| Filter Capacitance | \( C \) | 20 μF |
| Equivalent Resistance | \( R_{re} \) | 2 Ω |
| Sampling Period | \( T_s \) | 10 μs |
| Switching Frequency | \( f_s \) | 20 kHz |
In steady-state operation with a reference current of 15 A, the proposed multi-objective co-optimization strategy demonstrated superior performance. The grid current waveforms were nearly sinusoidal with minimal distortion. Quantitative analysis revealed a THD of 1.71% for the proposed method, compared to 5.76% for conventional FCS-MPC and 7.23% for dual-objective FCS-MPC. This improvement stems from the effective decoupling of inductor current and capacitor voltage dynamics through Lyapunov-based stabilization and appropriate weighting. The fundamental current amplitude aligned closely with the reference (15.51 A vs. 15 A), indicating accurate tracking. In contrast, dual-objective control without stability constraints exhibited significant coupling effects, causing current畸变 and instability. The results underscore the importance of holistic multi-objective design in grid connected inverter control.
Dynamic performance was evaluated by applying a step change in reference current from 15 A to 25 A at t = 0.1 s. Both conventional and proposed strategies showed fast response, but the proposed method achieved smoother and quicker settling with negligible overshoot. The local enlarged view confirmed that the grid current tracked the reference almost instantaneously, thanks to the three-step prediction and delay compensation. This robustness is crucial for grid connected inverter applications where rapid power adjustments are required, such as in renewable energy systems with fluctuating generation.
Switching frequency reduction is another key advantage. By incorporating the switching transition term \( n_s \) in the cost function with weight \( \lambda_2 \), the proposed strategy actively minimizes unnecessary switchings. Over one grid cycle, the average switching count decreased from 6441 (conventional FCS-MPC) to 5157 (proposed method), representing a reduction of approximately 20%. This directly translates to lower switching losses, improved thermal management, and enhanced longevity of the grid connected inverter hardware. The weighting coefficients \( \lambda_1 \) and \( \lambda_2 \) were tuned via iterative simulation to balance performance metrics; typical values ranged from 0.1 to 1 for \( \lambda_1 \) and 0.01 to 0.1 for \( \lambda_2 \), depending on system priorities. This empirical tuning, guided by Lyapunov stability, avoids the complexity of theoretical weight derivation while ensuring reliable operation.
Further insights can be drawn from frequency-domain analysis. The output impedance of the grid connected inverter under the proposed control exhibits improved damping characteristics, reducing resonance risks associated with LC filters. The Nyquist plots indicate enhanced phase margins, confirming stability across operational ranges. Moreover, the strategy’s computational burden remains manageable for modern digital signal processors (DSPs), as the three-step prediction only involves linear extrapolations and the evaluation of seven voltage vectors per sampling cycle. For real-time implementation, lookup tables for voltage vectors and precomputed trigonometric terms for angle compensation can optimize execution speed.
The proposed multi-objective co-optimization model predictive control strategy offers a comprehensive solution for LC-filtered grid connected inverters. By integrating Lyapunov stability theory, advanced delay compensation, and balanced weighting of control objectives, it addresses key limitations of traditional FCS-MPC. The strategy ensures precise current and voltage tracking, low harmonic distortion, robust dynamic response, and reduced switching losses—all critical for modern power systems with high renewable penetration. Future work could explore adaptive weighting mechanisms, extension to three-level inverters, and hardware-in-the-loop validation. As grid connected inverters become ubiquitous in smart grids and microgrids, such advanced control methodologies will be instrumental in achieving efficient, stable, and sustainable energy integration.