The increasing integration of renewable energy sources into modern power systems has elevated the importance of power electronic converters, particularly three-phase grid tied inverters, which serve as the critical interface between distributed generation and the utility grid. The performance of these inverters, especially the quality of the injected current, directly impacts grid stability, power quality, and overall system efficiency. Model Predictive Control (MPC) has emerged as a powerful digital control technique for grid tied inverters due to its fast dynamic response, intuitive handling of constraints, and multi-objective optimization capability. Among various MPC schemes, Finite-Control-Set MPC (FCS-MPC) is widely implemented for its simplicity, as it directly utilizes the discrete switching states of the inverter. However, the inherent single-vector application per control cycle in FCS-MPC leads to significant current ripple and variable switching frequency.
To mitigate this issue, multi-vector MPC strategies, such as Double-Vector MPC (DV-MPC), have been developed. DV-MPC synthesizes an output voltage by applying two active voltage vectors within one sampling period, effectively approximating a modulated output and reducing current ripple. The conventional DV-MPC strategy relies heavily on an accurate discrete-time mathematical model of the grid tied inverter system, primarily the filter inductance (L) and resistance (R). This dependency is a major drawback, as parameter mismatches between the model and the actual physical system are inevitable due to aging, temperature variations, or manufacturing tolerances. Such mismatches degrade control performance, leading to increased current total harmonic distortion (THD), steady-state error, and potentially system instability.
This paper presents an improved Double-Vector Model-Free Predictive Control (DV-MFPC) strategy designed to eliminate the dependency on precise system parameters. The core innovation lies in replacing the analytical model with a dynamically updated data model. By leveraging a recursive estimation technique with an adaptive forgetting factor, the proposed method online identifies the input-output relationship of the inverter, enabling accurate current prediction and optimal vector duration calculation without prior knowledge of L and R. This approach significantly enhances the robustness of the grid tied inverter control system against parameter variations and operational uncertainties.
System Modeling of the Three-Phase Grid-Tied Inverter
The topology of a standard two-level three-phase grid tied inverter is considered. It consists of a DC-link voltage source \(V_{dc}\), a three-phase bridge with six switches, and an L-type output filter (inductance \(L\) and equivalent series resistance \(R\)) connected to the three-phase grid voltage \(\mathbf{e_g}\). The switching states define eight basic voltage vectors (\(\mathbf{v_0}, \mathbf{v_1}, …, \mathbf{v_7}\)) in the \(\alpha\beta\) stationary reference frame.
The continuous-time dynamic model in the \(\alpha\beta\) frame is given by:
$$ L \frac{d\mathbf{i_g}(t)}{dt} = \mathbf{v}(t) – R \mathbf{i_g}(t) – \mathbf{e_g}(t) $$
where \(\mathbf{i_g} = [i_{g\alpha}, i_{g\beta}]^T\) is the grid current vector, and \(\mathbf{v} = [v_{\alpha}, v_{\beta}]^T\) is the inverter output voltage vector.
Using the forward Euler discretization method with a sampling period \(T_s\), the discrete-time model used for prediction in conventional MPC is derived:
$$ \mathbf{i_g}(k+1) = \left(1 – \frac{R T_s}{L}\right)\mathbf{i_g}(k) + \frac{T_s}{L}\left( \mathbf{v}(k) – \mathbf{e_g}(k) \right) $$
Equation (2) is the foundation of predictive control but explicitly contains the parameters \(L\) and \(R\), making it sensitive to their accuracy.
Conventional Double-Vector MPC and Its Robustness Limitations
Conventional DV-MPC operates by applying two of the eight basic voltage vectors, \(\mathbf{v}_i\) and \(\mathbf{v}_j\), within one control cycle \(T_s\). The synthesized output voltage \(\mathbf{v}_{s}\) is calculated based on the principle of volt-second balance:
$$ \mathbf{v}_{s} = \frac{t_i}{T_s}\mathbf{v}_i + \frac{t_j}{T_s}\mathbf{v}_j $$
where \(t_i\) and \(t_j\) are the application times for the two vectors, with \(t_i + t_j = T_s\). A common method to determine these times is based on the inverse proportionality principle, where the duration is inversely related to the cost function value of each individual vector.
$$ t_i = \frac{G_j}{G_i + G_j}T_s, \quad t_j = \frac{G_i}{G_i + G_j}T_s $$
The cost functions \(G_i\) and \(G_j\) evaluate the tracking error if only vector \(\mathbf{v}_i\) or \(\mathbf{v}_j\) were applied for the entire period, computed using the model from Eq. (2):
$$ G_i = |\mathbf{i^*_g}(k+1) – \mathbf{i_{g,i}}(k+1)|^2 $$
where \(\mathbf{i^*_g}\) is the reference current and \(\mathbf{i_{g,i}}(k+1)\) is the predicted current using Eq. (2) with \(\mathbf{v}(k)=\mathbf{v}_i\).
Impact of Parameter Mismatch: The vulnerability of this method stems from two model-dependent calculations:
- Current Prediction: The prediction \(\mathbf{i_{g,i}}(k+1)\) in Eq. (5) directly uses the mismatched parameters \(L_0\) and \(R_0\).
- Vector Duration: The durations \(t_i\) and \(t_j\) in Eq. (4) are derived from \(G_i\) and \(G_j\), which are erroneous due to flawed current predictions.
This dual dependency amplifies the control error. The table below summarizes the effects of inductance mismatch on key performance metrics in a conventional DV-MPC for a grid tied inverter.
| Inductance Mismatch (\(L_0/L\)) | Current THD (%) | RMSE (A) @ 10A ref. | Observation |
|---|---|---|---|
| 0.2 (80% under-estimated) | >5.0 | ~1.22 | Severe distortion, poor tracking. |
| 0.5 | ~4.5 | ~0.59 | Significant performance degradation. |
| 1.0 (Accurate) | 3.94 | 0.36 | Nominal performance. |
| 1.5 (50% over-estimated) | ~4.2 | ~0.40 | Increased ripple and error. |
| 1.8 | ~4.5 | ~0.43 | Degraded performance similar to under-estimation. |
Proposed Double-Vector Model-Free Predictive Control (DV-MFPC)
The proposed DV-MFPC strategy circumvents the need for an explicit mathematical model by constructing and continuously updating a data model of the grid tied inverter. The core idea is to learn the relationship between the applied voltage and the resulting current change directly from measured operational data.
Data Model Establishment
Rearranging the discrete model from Eq. (2), the current change from step \(k-1\) to \(k\) can be expressed as:
$$ \Delta \mathbf{i_g}(k) = \mathbf{i_g}(k) – \mathbf{i_g}(k-1) = \frac{T_s}{L} (\mathbf{v}(k-1) – \mathbf{e_g}(k-1)) – \frac{R T_s}{L} \mathbf{i_g}(k-1) $$
This equation can be generalized into a linear data model form, where the complex terms \(T_s/L\) and \(-R T_s/L\) are treated as unknown coefficients to be identified:
$$ \Delta \mathbf{i_g}(k) = \mathbf{X}(k) \cdot \mathbf{V_{diff}}(k-1) + \mathbf{Y}(k) \cdot \mathbf{i_g}(k-1) $$
where \(\mathbf{V_{diff}}(k-1) = \mathbf{v}(k-1) – \mathbf{e_g}(k-1)\) is the effective voltage applied to the filter, and \(\mathbf{X}(k)\) and \(\mathbf{Y}(k)\) are the time-varying model parameters to be identified. For a three-phase system in \(\alpha\beta\) coordinates, these are 2×2 matrices, but can often be simplified to scalar gains if cross-coupling is neglected or separately compensated.
Adaptive Online Model Update with Forgetting Factor
To accurately track the system’s behavior, including parameter drifts, a Recursive Least Squares (RLS) algorithm with an adaptive forgetting factor \(\lambda(k)\) is employed. The data model is written in standard linear regression form:
$$ \mathbf{y}(k) = \boldsymbol{\phi}^T(k) \boldsymbol{\theta}(k) $$
where:
\(\mathbf{y}(k) = \Delta \mathbf{i_g}(k)\) (output),
\(\boldsymbol{\phi}(k) = [\mathbf{V_{diff}}^T(k-1), \mathbf{i_g}^T(k-1)]^T\) (regressor),
\(\boldsymbol{\theta}(k) = [\mathbf{X}(k), \mathbf{Y}(k)]^T\) (parameter vector to be identified).
The standard RLS update equations are:
$$
\begin{aligned}
\mathbf{K}(k) &= \frac{\mathbf{P}(k-1)\boldsymbol{\phi}(k)}{\lambda(k) + \boldsymbol{\phi}^T(k)\mathbf{P}(k-1)\boldsymbol{\phi}(k)} \\
\boldsymbol{\hat{\theta}}(k) &= \boldsymbol{\hat{\theta}}(k-1) + \mathbf{K}(k)\left[\mathbf{y}(k) – \boldsymbol{\phi}^T(k)\boldsymbol{\hat{\theta}}(k-1)\right] \\
\mathbf{P}(k) &= \lambda(k)^{-1} \left[ \mathbf{I} – \mathbf{K}(k)\boldsymbol{\phi}^T(k) \right] \mathbf{P}(k-1)
\end{aligned}
$$
The key innovation is the adaptive tuning of the forgetting factor \(\lambda(k)\) to balance convergence speed and noise immunity:
$$ \lambda(k) = \begin{cases}
\lambda_{min} & \text{if } |e(k)| > e_0 \\
1 – (1-\lambda_{min}) e^{-\gamma |e(k)|} & \text{if } |e(k)| \le e_0
\end{cases} $$
where \(e(k) = \mathbf{y}(k) – \boldsymbol{\phi}^T(k)\boldsymbol{\hat{\theta}}(k-1)\) is the prediction error, \(e_0\) is an error boundary, \(\gamma\) is a positive constant, and \(\lambda_{min}\) is the lower bound (e.g., 0.8). This design ensures fast adaptation when the model error is large (\(|e|>e_0\)) and high immunity to measurement noise when the model is accurate (\(|e|\approx 0\), \(\lambda \rightarrow 1\)).
Model-Free Current Prediction and Vector Selection
With the updated data model parameters \(\mathbf{\hat{X}}(k)\) and \(\mathbf{\hat{Y}}(k)\), the current gradient (increment) for any candidate basic voltage vector \(\mathbf{v}_i\) for the next step is predicted as:
$$ \Delta \mathbf{i_{g,i}^{p}}(k+1) = \mathbf{\hat{X}}(k) \cdot (\mathbf{v}_i – \mathbf{e_g}(k)) + \mathbf{\hat{Y}}(k) \cdot \mathbf{i_g}(k) $$
The predicted current at \(k+1\) if only \(\mathbf{v}_i\) is applied becomes:
$$ \mathbf{i_{g,i}^{p}}(k+1) = \mathbf{i_g}(k) + \Delta \mathbf{i_{g,i}^{p}}(k+1) $$
For DV-MFPC, the optimal durations \(t_i\) and \(t_j\) for a candidate pair (\(\mathbf{v}_i, \mathbf{v}_j\)) are calculated using the inverse proportionality principle from Eq. (4), but with cost functions \(G_i\) and \(G_j\) computed using the model-free predictions from Eq. (11).
The synthesized current gradient for the double-vector combination is then:
$$ \Delta \mathbf{i_{g,ij}^{p}}(k+1) = \frac{t_i}{T_s}\Delta \mathbf{i_{g,i}^{p}}(k+1) + \frac{t_j}{T_s}\Delta \mathbf{i_{g,j}^{p}}(k+1) $$
Finally, the predicted current at \(k+2\) (to compensate computational delay) is:
$$ \mathbf{i_{g,ij}^{p}}(k+2) = \mathbf{i_g}(k) + \Delta \mathbf{i_{g,ij}^{p}}(k+1) + \Delta \mathbf{i_{g,ij}^{p}}(k+2) $$
where \(\Delta \mathbf{i_{g,ij}^{p}}(k+2)\) is estimated similarly using the same data model, assuming \(\mathbf{e_g}(k+1)\approx \mathbf{e_g}(k)\). The optimal vector pair and its durations are selected by minimizing a cost function, typically the squared error between the reference and predicted current at \(k+2\):
$$ G_{ij}^* = \min_{i,j} \left\| \mathbf{i^*_g}(k+2) – \mathbf{i_{g,ij}^{p}}(k+2) \right\|^2 $$
The selected vectors \(\mathbf{v}_i^*\) and \(\mathbf{v}_j^*\) with durations \(t_i^*\) and \(t_j^*\) are then applied in the next control cycle via appropriate PWM generation.
Performance Evaluation and Comparative Analysis
The effectiveness of the proposed DV-MFPC strategy for a grid tied inverter is validated through simulation and experimental tests, comparing it against conventional DV-MPC and a basic DV-MFPC method with a fixed forgetting factor. Key system parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| DC-Link Voltage | \(V_{dc}\) | 300 V |
| Grid Phase Voltage (peak) | \(E_m\) | 100 V |
| Grid Frequency | \(f_g\) | 50 Hz |
| Filter Inductance (Actual) | \(L\) | 8 mH |
| Filter Resistance (Actual) | \(R\) | 0.5 Ω |
| Sampling / Control Frequency | \(f_s\) | 10 kHz |
Robustness Against Parameter Mismatch
The primary advantage of the proposed method is its insensitivity to parameter errors. The following table compares the steady-state current Root Mean Square Error (RMSE) for different control methods under significant inductance mismatch, with a 10 A reference current.
| Control Method | \(L_0 = 0.2L\) (1.6 mH) | \(L_0 = 0.5L\) (4 mH) | \(L_0 = L\) (8 mH) | \(L_0 = 1.8L\) (14.4 mH) |
|---|---|---|---|---|
| Conventional DV-MPC | 1.22 A | 0.59 A | 0.36 A | 0.43 A |
| DV-MFPC (Fixed \(\lambda\)) | 0.38 A | 0.38 A | 0.38 A | 0.38 A |
| Proposed DV-MFPC (Adaptive \(\lambda\)) | 0.34 A | 0.34 A | 0.34 A | 0.34 A |
The results clearly show that while conventional DV-MPC performance degrades severely with parameter error, both MFPC methods maintain consistent performance. The proposed DV-MFPC with adaptive forgetting factor achieves the lowest RMSE, demonstrating its superior noise suppression capability leading to better steady-state accuracy.
Steady-State and Dynamic Performance
Under accurate nominal parameters, the proposed method achieves comparable and even slightly better steady-state performance than conventional DV-MPC, as it continuously fine-tunes its internal model. The Total Harmonic Distortion (THD) of the grid current is a critical metric for any grid tied inverter.
| Control Method | THD @ 10 A (%) | THD @ 15 A (%) | Dynamic Response (to step change) |
|---|---|---|---|
| Conventional DV-MPC | 3.94 | 2.70 | Fast (~2 ms) |
| DV-MFPC (Fixed \(\lambda\)) | 4.08 | 2.86 | Fast (~2 ms) |
| Proposed DV-MFPC (Adaptive \(\lambda\)) | 3.87 | 2.68 | Fast (~2 ms) |
The dynamic response during a reference current step change from 10 A to 15 A is equally fast for all methods, as the predictive control inherently offers excellent transient performance.
Computational Burden Analysis
A practical concern for digital implementation is the computational load. The proposed method replaces complex model-based calculations with simpler arithmetic operations and a fixed-complexity RLS update. A comparison of the execution time within a DSP control cycle shows the feasibility of the approach.
| Algorithm Segment | Conventional DV-MPC | Proposed DV-MFPC |
|---|---|---|
| Model-Based Prediction (12 vectors) | ~12 μs | — |
| Data Model Update (RLS) | — | ~8 μs |
| Model-Free Gradient Calc. (8 vectors) | — | ~6 μs |
| Vector Duration & Synthesis (12 comb.) | ~10 μs | ~10 μs |
| Total Estimated Computation | ~29.7 μs | ~31.0 μs |
The total computation times are similar and well within a 100 μs control cycle, confirming the real-time implementability of the proposed DV-MFPC strategy for a grid tied inverter.
Conclusion
This paper has presented an improved Double-Vector Model-Free Predictive Control strategy for three-phase grid tied inverters. The method successfully addresses the critical weakness of parameter sensitivity in conventional model-based predictive control. By establishing a dynamic data model and updating it online using a Recursive Least Squares algorithm with an adaptive forgetting factor, the controller learns the system’s behavior directly from operational data, eliminating dependency on the precise values of filter inductance and resistance. The adaptive forgetting factor optimally balances the need for fast convergence during transients or parameter changes with high noise immunity during steady-state operation.
Comparative analysis demonstrates that the proposed DV-MFPC strategy maintains excellent current quality (low THD and RMSE) even under severe parameter mismatches, where conventional DV-MPC fails. It also shows superior steady-state performance compared to basic MFPC with a fixed forgetting factor, thanks to its enhanced noise rejection. The computational overhead is manageable for modern digital signal processors, ensuring practical feasibility. Therefore, the proposed control strategy represents a significant step towards more robust, reliable, and maintenance-friendly control of grid tied inverters in renewable energy systems, where operating conditions and component parameters are subject to change.