The global energy landscape is undergoing a significant transformation, driven by the depletion of fossil fuels and the urgent need to address climate change. In this context, distributed generation systems based on renewable energy sources, such as photovoltaic and wind power, have become a central focus of research and development worldwide. The grid connected inverter serves as the critical interface between these renewable energy sources and the utility grid. Its control performance directly dictates the power quality, stability, and efficiency of the entire energy injection system. Therefore, developing advanced control strategies for grid connected inverter topologies is of paramount importance.
Conventional control techniques for grid connected inverter systems include methods like direct power control (DPC) and field-oriented control (FOC). While effective, these methods often involve cascaded control loops and modulator stages, which can limit dynamic response and design flexibility. In recent years, Model Predictive Control (MPC) has emerged as a powerful alternative for power electronics applications. Its strengths lie in its intuitive concept, fast dynamic response, inherent ability to handle nonlinearities and constraints, and ease of implementing multi-objective optimization. The basic Finite Control Set MPC (FCS-MPC) evaluates a finite number of possible switching states of the grid connected inverter and selects the one that minimizes a defined cost function, typically the future current tracking error.
However, the classical single-vector FCS-MPC approach, where only one voltage vector is applied per control period, exhibits several inherent drawbacks when applied to a grid connected inverter. The limited control set leads to variable switching frequency, which results in a dispersed harmonic spectrum, making filter design challenging. Furthermore, the current ripple is often significant because the output voltage of the grid connected inverter is restricted to one of the seven discrete vectors for the entire sampling interval. To mitigate these issues, multi-vector MPC schemes have been proposed. These methods synthesize a reference voltage by applying a combination of active and zero vectors within one control cycle, effectively increasing the number of available voltage levels and providing more precise control. This significantly improves the steady-state performance of the grid connected inverter.
Nevertheless, challenges remain in the design of multi-vector MPC for grid connected inverter applications. The selection of optimal vector combinations and the precise calculation of their respective application times are critical. Moreover, as the switching sequence becomes more complex—involving multiple vector transitions—the traditional cost function, which evaluates tracking error only at the end of the prediction horizon, becomes inadequate. A single-point evaluation fails to capture the current trajectory’s deviation from its reference during the intermediate switching instants within the control period. This can lead to the selection of a sequence that has a low error at the end-point but exhibits large ripples in between, ultimately degrading the current quality of the grid connected inverter.
This article presents a refined Optimal Switching Sequence Model Predictive Control (OSS-MPC) strategy specifically designed for two-level voltage source grid connected inverter systems. The core contributions address the aforementioned limitations: First, a set of predefined, optimized switching sequences is introduced to ensure a fixed and reduced switching frequency per device while maintaining a symmetrical output pattern. Second, the duty cycles for the vectors within a sequence are calculated using a deadbeat principle to achieve precise reference tracking. Third, and most importantly, a novel dynamic cost function is constructed. This function accumulates the current tracking errors at multiple sampling instants within the prediction horizon, corresponding to each switch transition in the sequence. This multi-point evaluation provides a holistic measure of the sequence’s performance over the entire control cycle, leading to superior current waveform quality for the grid connected inverter.
Mathematical Model and Conventional FCS-MPC
The foundation of any model-based control strategy is an accurate discrete-time model of the plant. For a standard two-level Voltage Source Inverter (2L-VSI) acting as a grid connected inverter, the system dynamics in the stationary αβ-reference frame are given by:
$$ v_{\alpha\beta} = R i_{\alpha\beta} + L \frac{di_{\alpha\beta}}{dt} + e_{\alpha\beta} $$
where \( v_{\alpha\beta} = [v_\alpha, v_\beta]^T \) is the inverter output voltage vector, \( i_{\alpha\beta} = [i_\alpha, i_\beta]^T \) is the filter inductor current vector (also the grid current), \( e_{\alpha\beta} = [e_\alpha, e_\beta]^T \) is the grid voltage vector, \( R \) is the equivalent resistance, and \( L \) is the filter inductance. Using the forward Euler discretization method with a sampling period \( T_s \), this continuous-time model can be approximated for digital implementation:
$$ i_{\alpha\beta}(k+1) = \left(1 – \frac{R}{L}T_s\right) i_{\alpha\beta}(k) + \frac{T_s}{L} \left( v_{\alpha\beta}(k) – e_{\alpha\beta}(k) \right) $$
In conventional FCS-MPC for a grid connected inverter, the controller uses this discrete model to predict the future current \( i_{\alpha\beta}(k+1) \) for each of the 7 possible voltage vectors \( v_{\alpha\beta}(k) \) generated by the 2L-VSI (vectors V0, V1, …, V7). The optimal vector \( v_{opt} \) is selected as the one that minimizes a cost function \( G \), commonly defined as the squared error between the predicted current and its reference:
$$ G = \left[ i^{*}_{\alpha}(k+1) – i_{\alpha}(k+1) \right]^2 + \left[ i^{*}_{\beta}(k+1) – i_{\beta}(k+1) \right]^2 $$
where \( i^{*}_{\alpha\beta}(k+1) \) is the reference current at the next sampling instant. This simple approach, while computationally light, applies only one voltage vector during the entire control period \( T_s \), leading to the performance limitations discussed earlier, namely high current ripple and variable switching frequency for the grid connected inverter.
Proposed OSS-MPC Strategy for Grid Connected Inverter
The proposed strategy enhances the performance of the grid connected inverter by orchestrating the application of multiple voltage vectors in a predefined, optimal sequence within each control cycle. The methodology consists of four integrated components: the design of the switching sequence, the calculation of the current trajectory, the precise computation of vector duty cycles, and the reconstruction of the cost function for accurate sequence evaluation.
1. Design of Optimal Switching Sequences
To achieve a fixed switching frequency and reduce the average device switching count, symmetrical seven-segment switching sequences are designed. For a two-level grid connected inverter, six unique sequences are defined, each corresponding to one of the six sectors formed by the active voltage vectors (V1 to V6). Each sequence applies two adjacent active vectors and a zero vector in a symmetrical pattern. The general form of a sequence for sector \( i \) is: \( \{ V_0, V_n, V_m, V_7, V_m, V_n, V_0 \} \), where \( V_n \) and \( V_m \) are the two adjacent active vectors bounding the sector, and \( V_0 \) and \( V_7 \) are the null vectors. The specific assignments are summarized in Table 1.
| Sequence Index (i) | Sector (Reference Vector Location) | Switching Sequence {V0, Vn, Vm, V7, Vm, Vn, V0} |
|---|---|---|
| 1 | I [0°, 60°] | {V0, V1, V2, V7, V2, V1, V0} |
| 2 | II [60°, 120°] | {V0, V3, V2, V7, V2, V3, V0} |
| 3 | III [120°, 180°] | {V0, V3, V4, V7, V4, V3, V0} |
| 4 | IV [180°, 240°] | {V0, V5, V4, V7, V4, V5, V0} |
| 5 | V [240°, 300°] | {V0, V5, V6, V7, V6, V5, V0} |
| 6 | VI [300°, 360°] | {V0, V1, V6, V7, V6, V1, V0} |
This structure ensures that each switch toggles at most once per half of the control period, leading to a predictable and reduced switching frequency for the grid connected inverter.
2. Current Trajectory Prediction for a Sequence
When a sequence is applied, the current does not follow a straight path from \( i(k) \) to \( i(k+1) \). Instead, it exhibits a piecewise linear trajectory with slopes determined by the applied voltage vector. The current derivative (or slope) for a generic voltage vector \( V_x \) is derived from the system model:
$$ \mathbf{f}_{x} = \frac{d\mathbf{i}_{\alpha\beta}}{dt} = \frac{1}{L} ( \mathbf{v}_{x} – \mathbf{e}_{\alpha\beta} + R \mathbf{i}_{\alpha\beta} ) $$
where \( \mathbf{f}_{x} = [f_{\alpha x}, f_{\beta x}]^T \) and \( \mathbf{v}_{x} \) is the vector \( V_x \) in αβ coordinates. For a given sequence \( i \) with vectors \( V_0, V_n, V_m \), we denote their corresponding slopes as \( \mathbf{f}_{i0}, \mathbf{f}_{in}, \mathbf{f}_{im} \). If the application times for \( V_n \) and \( V_m \) within half of the control period are \( t_{in} \) and \( t_{im} \) respectively (with \( t_{i0} = T_s/2 – t_{in} – t_{im} \) being the time for \( V_0/V_7 \)), the predicted current at the end of the full symmetrical period \( T_s \) can be calculated by summing the contributions from all segments:
$$ \mathbf{i}^{pre}(k+1) = \mathbf{i}(k) + 2 \left( \mathbf{f}_{in} t_{in} + \mathbf{f}_{im} t_{im} + \mathbf{f}_{i0} t_{i0} \right) $$
This equation describes the net effect of the entire switching sequence on the grid connected inverter output current.
3. Deadbeat-Based Vector Duration Calculation
To force the predicted current \( \mathbf{i}^{pre}(k+1) \) to equal the reference current \( \mathbf{i}^{*}(k+1) \) in a deadbeat fashion, we formulate an optimization problem. The goal is to find \( t_{in} \) and \( t_{im} \) that minimize the cost function \( J(t_{in}, t_{im}) = || \mathbf{i}^{*}(k+1) – \mathbf{i}^{pre}(k+1) ||^2 \). This is a constrained quadratic minimization problem. Using the method of Lagrange multipliers or directly solving the system of equations from setting the partial derivatives to zero, the optimal times can be derived. The solution is given by:
$$ t_{in} = \frac{E_{\alpha}(f_{\beta im} – f_{\beta i0}) + E_{\beta}(f_{\alpha i0} – f_{\alpha im})}{2A} + \frac{T_s(f_{\alpha im}f_{\beta i0} – f_{\alpha i0}f_{\beta im})}{2A} $$
$$ t_{im} = \frac{E_{\alpha}(f_{\beta i0} – f_{\beta in}) + E_{\beta}(f_{\alpha in} – f_{\alpha i0})}{2A} + \frac{T_s(f_{\alpha i0}f_{\beta in} – f_{\alpha in}f_{\beta i0})}{2A} $$
$$ t_{i0} = \frac{T_s}{2} – t_{in} – t_{im} $$
where,
$$ E_{\alpha} = i^{*}_{\alpha}(k+1) – i_{\alpha}(k), \quad E_{\beta} = i^{*}_{\beta}(k+1) – i_{\beta}(k) $$
$$ A = f_{\alpha i0}f_{\beta in} – f_{\alpha i0}f_{\beta im} + f_{\alpha in}f_{\beta im} – f_{\alpha in}f_{\beta i0} + f_{\alpha im}f_{\beta i0} – f_{\alpha im}f_{\beta in} $$
This calculation is performed for all six candidate sequences \( i = 1…6 \). The result is a set of six possible duty cycles \( (t_{in}, t_{im}, t_{i0}) \) that, if applied, would theoretically achieve perfect tracking at the sampling instant for each respective sequence of the grid connected inverter.
4. Dynamic Multi-Point Cost Function
The critical innovation lies in evaluating which of the six sequences, with its calculated deadbeat times, produces the best current waveform throughout the entire control cycle—not just at its end. The traditional cost function only checks the error at \( k+1 \). For a multi-step sequence, intermediate deviations matter. Therefore, we propose a dynamic cost function that samples the tracking error at multiple instants \( j \) corresponding to the transitions within the sequence.
For a seven-segment sequence, eight key instants are considered: the start and the end of each of the four half-segments in the first half-period (symmetry handles the second half). The predicted current at these intermediate points \( \mathbf{i}_{j+1} \) is calculated iteratively:
$$ \mathbf{i}_{j+1} = \mathbf{i}_{j} + \mathbf{f}_{i,\sigma(j)} \cdot \Delta t_{j} $$
where \( \sigma(j) \) identifies which vector (0, n, or m) is active during the \( j \)-th subinterval, and \( \Delta t_{j} \) is its corresponding duration (e.g., \( t_{i0}, t_{in}, t_{im} \)). The improved cost function \( G_i \) for sequence \( i \) is then the sum of squared errors at all these evaluation points, including the final point:
$$ G_i = \sum_{j=0}^{7} \left[ \left( i^{*}_{\alpha}(k+1) – i_{\alpha, j+1} \right)^2 + \left( i^{*}_{\beta}(k+1) – i_{\beta, j+1} \right)^2 \right] $$
The sequence \( i_{opt} \) that yields the minimum value of \( G_i \) is selected as the optimal one. This sequence, along with its pre-calculated times \( t_{opn}, t_{opm}, t_{op0} \), is used to generate the PWM duty cycles for the three phases of the grid connected inverter:
$$ d_a = \frac{2}{T_s}\left( S_{a,opn}t_{opn} + S_{a,opm}t_{opm} + t_{op0} \right) $$
$$ d_b = \frac{2}{T_s}\left( S_{b,opn}t_{opn} + S_{b,opm}t_{opm} + t_{op0} \right) $$
$$ d_c = \frac{2}{T_s}\left( S_{c,opn}t_{opn} + S_{c,opm}t_{opm} + t_{op0} \right) $$
where \( S_{x,opn} \) and \( S_{x,opm} \) are the switch states (0 or 1) for phase \( x \) corresponding to the optimal active vectors \( V_n \) and \( V_m \).
System Parameters and Comparative Framework
To validate the proposed OSS-MPC strategy, a standard two-level grid connected inverter system is analyzed. The key electrical and control parameters used for both simulation and experimental validation are listed in Table 2. The performance of the proposed method is compared against three established MPC variants for the grid connected inverter: 1) Conventional Single-Vector FCS-MPC (SV-MPC), 2) A Simplified Three-Vector MPC (S3V-MPC) which uses a heuristic for time calculation, and 3) A Deadbeat Three-Vector MPC (DB3V-MPC) which uses deadbeat calculation but a traditional single-point cost function.
| Parameter | Symbol | Value |
|---|---|---|
| DC-Link Voltage | \( V_{dc} \) | 400 V |
| Grid Voltage (Phase, RMS) | \( e_{grid} \) | 100 V |
| Filter Inductance | \( L \) | 7 mH |
| Filter Resistance | \( R \) | 0.02 Ω |
| Switching/Sampling Frequency | \( f_s \) | 10 kHz |
| Fundamental Frequency | \( f_1 \) | 50 Hz |
Performance Evaluation: Simulation and Experimental Results
The effectiveness of the proposed control strategy for the grid connected inverter is demonstrated through both simulation analysis and experimental validation on a laboratory-scale prototype.
Simulation Analysis
Simulations were conducted to compare the steady-state current quality across different reference current magnitudes. The key metric is the Total Harmonic Distortion (THD) of the output current from the grid connected inverter. Table 3 summarizes the THD results for the S3V-MPC, DB3V-MPC, and the proposed OSS-MPC strategies under varying load conditions.
| Reference Current (A, peak) | S3V-MPC THD (%) | DB3V-MPC THD (%) | Proposed OSS-MPC THD (%) |
|---|---|---|---|
| 2 | 5.12 | 4.95 | 4.08 |
| 5 | 2.69 | 2.62 | 2.05 |
| 8 | 2.52 | 2.45 | 1.91 |
| 10 | 2.48 | 2.43 | 1.88 |
The results clearly indicate that the proposed OSS-MPC strategy consistently yields the lowest current THD across all operating points. The improvement over DB3V-MPC, which uses the same deadbeat time calculation, highlights the distinct benefit of the multi-point dynamic cost function. By penalizing intermediate trajectory errors, the OSS-MPC guides the grid connected inverter to select the sequence that minimizes overall ripple, not just the final-point error.
Experimental Validation
A hardware platform was built using a TMS320F28335 digital signal controller. The steady-state current waveforms and their corresponding frequency spectra were captured for all four control methods at a 5 A reference. The quantitative results are consolidated in Table 4.
| Control Method | Current THD (%) | Switching Frequency Characteristic | Key Spectral Observation |
|---|---|---|---|
| SV-MPC | 4.54 | Variable, non-synchronized | Dispersed harmonics across spectrum |
| S3V-MPC | 2.69 | Fixed (10 kHz) | Harmonics concentrated at 10 kHz and multiples |
| DB3V-MPC | 2.62 | Fixed (10 kHz) | Lower harmonic magnitude at 10 kHz than S3V-MPC |
| Proposed OSS-MPC | 2.05 | Fixed (10 kHz) | Lowest harmonic magnitude at 10 kHz; cleanest spectrum |
The experimental waveforms visually confirm the simulation findings. The proposed method produces the smoothest sinusoidal current. The spectral analysis proves that all three multi-vector methods establish a fixed switching frequency, but the OSS-MPC achieves the most effective harmonic suppression for the grid connected inverter.
To quantify tracking precision, the standard deviation of the αβ-axis current errors was computed over a steady-state period. A lower standard deviation indicates tighter and more consistent tracking.
| Control Method | Std. Dev. on iα (A) | Std. Dev. on iβ (A) |
|---|---|---|
| S3V-MPC | 0.0916 | 0.1244 |
| DB3V-MPC | 0.0891 | 0.1320 |
| Proposed OSS-MPC | 0.0803 | 0.1116 |
The proposed OSS-MPC shows the lowest error deviation on both axes, confirming its superior ability to minimize tracking error throughout the control cycle of the grid connected inverter.
Finally, the dynamic performance was tested by applying a step change in the current reference from 5 A to 10 A. The settling time, defined as the time for the current to reach and stay within a 2% band of the new reference, was measured. The OSS-MPC strategy demonstrated the fastest dynamic response with a settling time of approximately 376 μs, compared to 445 μs for S3V-MPC and 466 μs for DB3V-MPC. This demonstrates that the improved sequence selection does not compromise and can even enhance the dynamic response of the grid connected inverter.
Conclusion
This article has presented a comprehensive and enhanced Optimal Switching Sequence Model Predictive Control (OSS-MPC) strategy for two-level voltage source grid connected inverter systems. The strategy systematically addresses the core limitations of prior MPC methods by integrating three key elements: a set of predefined symmetrical switching sequences for fixed-frequency operation, a deadbeat principle for precise vector duration calculation, and a novel dynamic multi-point cost function for holistic sequence evaluation. The multi-point cost function is the pivotal innovation, as it evaluates the current tracking error at multiple instants corresponding to the switching sequence’s transitions, rather than just at the end of the control period. This ensures the selected sequence minimizes current ripple throughout the entire cycle.
Both simulation and experimental results conclusively validate the superiority of the proposed approach. Compared to conventional single-vector MPC and other three-vector MPC variants, the OSS-MPC strategy delivers significantly lower grid current THD, superior reference tracking precision as quantified by lower error standard deviation, and excellent dynamic response. The method maintains the desirable fixed switching frequency characteristic, facilitating filter design. Therefore, the proposed OSS-MPC constitutes a high-performance control solution for grid connected inverter applications, effectively improving power quality and system performance. The core principle of using a dynamic, trajectory-aware cost function provides a valuable design paradigm for enhancing multi-vector predictive control in a wide range of power electronic systems beyond the grid connected inverter.