In modern power systems, particularly within microgrid architectures, the seamless and high-performance integration of distributed energy resources is paramount. The on-grid inverter serves as the critical power electronic interface, tasked with converting DC power from sources like photovoltaics or batteries into high-quality AC power synchronized with the utility grid. The fidelity of the injected current directly governs grid stability, power quality, and compliance with stringent grid codes. Consequently, developing advanced current control strategies that are simultaneously precise, robust, and computationally efficient remains a central focus of research and development in power electronics.
Among various control paradigms, Finite Control Set Model Predictive Current Control (FCS-MPCC) has gained significant traction for on-grid inverter applications. Its appeal lies in a straightforward conceptual framework: utilizing a discrete-time model of the system to predict future behavior for all possible switching states, and subsequently selecting the state that minimizes a user-defined cost function. This approach offers intrinsic handling of system constraints, fast dynamic response, and ease of multi-objective optimization. However, conventional FCS-MPCC suffers from two well-documented drawbacks. First, its implementation with a single active vector per control cycle leads to variable switching frequency and considerable current ripple, necessitating oversized output filters. Second, and more critically, its performance is acutely sensitive to the accuracy of the system model parameters, namely the filter inductance (L) and resistance (R). Mismatches between the parameters used in the control law and the actual hardware values—arising from manufacturing tolerances, temperature variations, or aging—degrade prediction accuracy. This results in steady-state current error, increased total harmonic distortion (THD), and can potentially compromise system stability.
To address the issue of current ripple, modulated or multi-vector MPCC strategies have been developed. Instead of applying a single voltage vector, these methods synthesize a reference voltage over the control period using a combination of two or more basic inverter vectors and null vectors. Techniques like dual-vector and triple-vector MPCC effectively reduce ripple and fix the effective switching frequency. However, the calculation of the optimal duration for each vector within the sequence remains fundamentally dependent on an accurate system model. Therefore, while they solve the ripple problem, they do not mitigate the core vulnerability to parameter mismatches.
An alternative pathway to robustness is the pursuit of model-free control strategies. The core idea is to discard the explicit mathematical model and instead derive control actions directly from measured input-output data. One promising avenue within predictive control is Model-Free Predictive Current Control (MFPCC). Early MFPCC methods relied on building and continuously updating a lookup table that mapped applied voltage vectors to the resulting measured current gradients. While eliminating explicit parameter dependence, these methods face challenges in accurately and efficiently updating gradients under dynamic conditions and are susceptible to measurement noise.
A powerful framework for data-driven control is the ultra-local model. This approach models a system’s input-output dynamics using a very simple structure: a scaling gain on the input plus a lumped “disturbance” term that encapsulates all unknown dynamics, nonlinearities, and external perturbations. The disturbance is estimated in real-time from input-output measurements. This elegant formulation provides a robust basis for prediction without requiring detailed physical parameters.
This article presents a novel, comprehensive control strategy that synergizes the benefits of multi-vector modulation for superior steady-state performance with the robustness of a model-free approach. We propose an Adaptive Triple-Vector Model-Free Predictive Current Control (ATV-MFPCC) strategy for on-grid inverters. The method is built upon an ultra-local model of the inverter system. A key innovation is the use of an Extended State Observer (ESO) enhanced with an adaptive gain mechanism to accurately estimate the time-varying lumped disturbance. This adaptive ESO ensures precise disturbance rejection across varying operating conditions and strengthens the system’s immunity to sensor noise. The estimated disturbance, along with the simple ultra-local model, is then used to perform current predictions for candidate triple-vector sequences. Finally, a cost function evaluates these predictions to select the optimal voltage vector sequence for the next switching period.
Mathematical Foundation and Problem Formulation
System Model of a Three-Phase On-Grid Inverter
The standard topology of a two-level, three-phase on-grid inverter is considered. The system dynamics in the stationary αβ-reference frame are given by:
$$ L \frac{d\mathbf{i}_{\alpha\beta}}{dt} = \mathbf{v}_{\alpha\beta} – R \mathbf{i}_{\alpha\beta} – \mathbf{e}_{\alpha\beta} $$
where \( \mathbf{i}_{\alpha\beta} = [i_\alpha, i_\beta]^T \) is the grid current vector, \( \mathbf{v}_{\alpha\beta} = [v_\alpha, v_\beta]^T \) is the inverter output voltage vector (one of eight possible space vectors, \( V_0…V_7 \)), \( \mathbf{e}_{\alpha\beta} = [e_\alpha, e_\beta]^T \) is the grid voltage vector, \( L \) is the filter inductance, and \( R \) is the filter resistance.
Using the forward Euler discretization method with a sampling period \( T_s \), the discrete-time model for prediction is:
$$ \mathbf{i}_{\alpha\beta}(k+1) = \mathbf{i}_{\alpha\beta}(k) + \frac{T_s}{L} [ \mathbf{v}_{\alpha\beta}(k) – R \mathbf{i}_{\alpha\beta}(k) – \mathbf{e}_{\alpha\beta}(k) ] $$
This equation forms the basis for traditional MPCC. Its explicit dependence on \( L \) and \( R \) is evident.
Limitations of Traditional Triple-Vector MPCC
In a triple-vector MPCC scheme, the applied voltage over one control period \( T_s \) is synthesized as:
$$ \mathbf{v}_{t_j}(k) = \frac{t_0}{T_s}\mathbf{v}_0 + \frac{t_1}{T_s}\mathbf{v}_1 + \frac{t_2}{T_s}\mathbf{v}_2 $$
where \( t_0, t_1, t_2 \) are the dwell times for a null vector \( \mathbf{v}_0 \), and two active vectors \( \mathbf{v}_1, \mathbf{v}_2 \), with \( t_0+t_1+t_2=T_s \). The optimal times are typically calculated by solving an optimization problem that minimizes current error, which involves the model parameters. The predicted current under this synthesized vector is:
$$ \mathbf{i}_{\alpha\beta}^{j}(k+1) = \mathbf{i}_{\alpha\beta}(k) + \frac{T_s}{L} [ \mathbf{v}_{t_j}(k) – R \mathbf{i}_{\alpha\beta}(k) – \mathbf{e}_{\alpha\beta}(k) ] $$
The cost function for delay compensation and vector selection is:
$$ g_j = |i^*_\alpha – i^j_\alpha(k+2)|^2 + |i^*_\beta – i^j_\beta(k+2)|^2 $$
where \( i^*_{\alpha\beta} \) is the reference current.
The critical vulnerability is two-fold:
1. Vector Duration Sensitivity: The calculation of \( t_0, t_1, t_2 \) depends on \( L \) and \( R \). Model mismatch leads to erroneous dwell times, causing the actual applied voltage to deviate from the intended optimal value.
2. Current Prediction Sensitivity: The prediction of \( \mathbf{i}_{\alpha\beta}(k+2) \) itself relies on \( L \) and \( R \). An inaccurate model leads to poor prediction, causing the cost function \( g_j \) to select a non-optimal vector sequence.
The following table summarizes the impact of parameter deviation on the performance of a traditional triple-vector MPCC for an on-grid inverter.
| Parameter Error | Effect on Vector Duration | Effect on Current Prediction | Overall System Impact |
|---|---|---|---|
| Inductance \( L \) too high (e.g., 150% of actual) | Active vector times underestimated. | Predicted current change underestimated. | Increased current ripple, potential steady-state error. |
| Inductance \( L \) too low (e.g., 50% of actual) | Active vector times overestimated. | Predicted current change overestimated. | Overshoot in transients, possible instability, higher THD. |
| Resistance \( R \) mismatch | Minor effect on optimal timing. | Steady-state offset in prediction. | Steady-state current tracking error. |
Proposed Adaptive Triple-Vector Model-Free Control (ATV-MFPCC)
Ultra-Local Model of the On-Grid Inverter
To liberate the controller from parameter dependence, we adopt an ultra-local model. The complex dynamics of the on-grid inverter are represented by a first-order model with a lumped disturbance:
$$ \frac{d\mathbf{i}_{\alpha\beta}}{dt} = \alpha \mathbf{v}_{\alpha\beta} + \mathbf{f}_{\alpha\beta} $$
Here, \( \alpha \) is a non-physical scaling gain (a simple tuning parameter), and \( \mathbf{f}_{\alpha\beta} \) is the lumped disturbance term. This term absorbs everything not captured by the simple \( \alpha \mathbf{v}_{\alpha\beta} \) term: the back-EMF \( \mathbf{e}_{\alpha\beta} \), the resistive drop \( R\mathbf{i}_{\alpha\beta} \), parameter variations, cross-coupling, and unmodeled nonlinearities. The discrete form is:
$$ \mathbf{i}_{\alpha\beta}(k+1) = \mathbf{i}_{\alpha\beta}(k) + T_s [ \alpha \mathbf{v}_{\alpha\beta}(k) + \mathbf{f}_{\alpha\beta}(k) ] $$
If \( \alpha \) and \( \mathbf{f}_{\alpha\beta}(k) \) can be accurately determined/estimated, this model can predict future current without knowledge of \( L \), \( R \), or \( \mathbf{e}_{\alpha\beta} \).
Estimation of the Ultra-Local Model Components
From two consecutive measurements, we can derive estimates. Defining the current gradient as \( \Delta \mathbf{i}_{\alpha\beta}(k) = \mathbf{i}_{\alpha\beta}(k) – \mathbf{i}_{\alpha\beta}(k-1) \), and assuming the disturbance changes slowly \( (\mathbf{f}_{\alpha\beta}(k-1) \approx \mathbf{f}_{\alpha\beta}(k-2)) \), we obtain:
$$ \alpha = \frac{ \Delta \mathbf{i}_{\alpha\beta}(k-1) – \Delta \mathbf{i}_{\alpha\beta}(k-2) }{ T_s ( \mathbf{v}_{\alpha\beta}(k-1) – \mathbf{v}_{\alpha\beta}(k-2) ) } $$
$$ \mathbf{f}_{\alpha\beta}(k-1) = \frac{ \Delta \mathbf{i}_{\alpha\beta}(k-1) }{ T_s } – \alpha \mathbf{v}_{\alpha\beta}(k-1) $$
While this direct calculation works in principle, it is highly sensitive to measurement noise in both current and voltage. To achieve robust and accurate estimation of the critical disturbance term \( \mathbf{f}_{\alpha\beta} \), we employ an Extended State Observer (ESO).
Enhanced Adaptive Extended State Observer (ESO)
We treat \( \mathbf{f}_{\alpha\beta} \) as an extended state. The system for the ESO is defined as:
$$ \begin{cases}
\frac{d \hat{\mathbf{i}}_{\alpha\beta}}{dt} = \alpha \mathbf{v}_{\alpha\beta} + \hat{\mathbf{f}}_{\alpha\beta} + \beta_1 (\mathbf{i}_{\alpha\beta} – \hat{\mathbf{i}}_{\alpha\beta}) \\
\frac{d \hat{\mathbf{f}}_{\alpha\beta}}{dt} = \beta_2 (\mathbf{i}_{\alpha\beta} – \hat{\mathbf{i}}_{\alpha\beta})
\end{cases} $$
where \( \hat{\mathbf{i}}_{\alpha\beta} \) and \( \hat{\mathbf{f}}_{\alpha\beta} \) are the estimated current and disturbance, and \( \beta_1, \beta_2 \) are observer gains. Discretizing with Euler method:
$$ \begin{cases}
\hat{\mathbf{i}}_{\alpha\beta}(k+1) = \hat{\mathbf{i}}_{\alpha\beta}(k) + T_s[ \alpha \mathbf{v}_{\alpha\beta}(k) + \hat{\mathbf{f}}_{\alpha\beta}(k) + \beta_1 \mathbf{e}_o(k) ] \\
\hat{\mathbf{f}}_{\alpha\beta}(k+1) = \hat{\mathbf{f}}_{\alpha\beta}(k) + T_s \beta_2 \mathbf{e}_o(k)
\end{cases} $$
with \( \mathbf{e}_o(k) = \mathbf{i}_{\alpha\beta}(k) – \hat{\mathbf{i}}_{\alpha\beta}(k) \). For stability and placing the observer poles at \( p \) in the discrete domain, the gains are chosen as: \( \beta_1 = 1-p \), \( \beta_2 = (1-p)^2 / T_s \).
Adaptive Gain for Noise Suppression: Fixed gains offer a compromise between convergence speed and noise rejection. To dynamically strengthen disturbance estimation against noise and model variations, we introduce an adaptive scaling factor \( \gamma(k) > 1 \) to the correction terms:
$$ \begin{cases}
\hat{\mathbf{i}}_{\alpha\beta}(k+1) = \hat{\mathbf{i}}_{\alpha\beta}(k) + T_s[ \alpha \mathbf{v}_{\alpha\beta}(k) + \hat{\mathbf{f}}_{\alpha\beta}(k) + \gamma(k)\beta_1 \mathbf{e}_o(k) ] \\
\hat{\mathbf{f}}_{\alpha\beta}(k+1) = \hat{\mathbf{f}}_{\alpha\beta}(k) + T_s \gamma(k)\beta_2 \mathbf{e}_o(k)
\end{cases} $$
The adaptive gain \( \gamma(k) \) is updated using a gradient-descent-like rule to minimize the estimation error:
$$ \gamma(k) = \theta_1(k) e_m(k) + \theta_2(k) \gamma(k-1) $$
where \( e_m(k) = |i_\alpha(k)-\hat{i}_\alpha(k)| + |i_\beta(k)-\hat{i}_\beta(k)| \) is the magnitude of the observation error. The parameters \( \theta_1, \theta_2 \) are updated online via a recursive least-squares method with a forgetting factor \( \lambda \):
$$ \boldsymbol{\theta}(k) = \boldsymbol{\theta}(k-1) + \frac{\mathbf{P}(k-1) \boldsymbol{\phi}(k) e_m(k)}{ \lambda + \boldsymbol{\phi}^T(k) \mathbf{P}(k-1) \boldsymbol{\phi}(k) } $$
$$ \mathbf{P}(k) = \frac{1}{\lambda} [ \mathbf{P}(k-1) – \frac{\mathbf{P}(k-1) \boldsymbol{\phi}(k) \boldsymbol{\phi}^T(k) \mathbf{P}(k-1)}{ \lambda + \boldsymbol{\phi}^T(k) \mathbf{P}(k-1) \boldsymbol{\phi}(k) } ] $$
where \( \boldsymbol{\theta} = [\theta_1, \theta_2]^T \), \( \boldsymbol{\phi} = [e_m(k), \gamma(k-1)]^T \), and \( \mathbf{P} \) is the covariance matrix.
Triple-Vector Synthesis and Model-Free Prediction
With a robust estimate \( \hat{\mathbf{f}}_{\alpha\beta}(k+1) \) from the adaptive ESO and the gain \( \alpha \) calculated from Eq. (5), the model-free prediction for any candidate triple-voltage vector \( \mathbf{v}_{t_j}(k) \) is performed:
$$ \mathbf{i}^{j}_{\alpha\beta}(k+1) = \mathbf{i}_{\alpha\beta}(k) + T_s [ \alpha \mathbf{v}_{t_j}(k) + \hat{\mathbf{f}}_{\alpha\beta}(k+1) ] $$
To compensate for the computational delay, the prediction for the cost function evaluation is:
$$ \mathbf{i}^{j}_{\alpha\beta}(k+2) = \mathbf{i}^{j}_{\alpha\beta}(k+1) + T_s [ \alpha \mathbf{v}_{t_j}(k+1) + \hat{\mathbf{f}}_{\alpha\beta}(k+1) ] $$
Note that \( \hat{\mathbf{f}}_{\alpha\beta}(k+1) \) is used again, assuming its change over one step is negligible, which is valid for a sufficiently high control frequency.
The triple-vector sequences are pre-defined (e.g., sequences involving two adjacent active vectors and a null vector). For each candidate sequence \( j \), the corresponding \( \mathbf{v}_{t_j}(k+1) \) is constructed. The dwell times \( t_0, t_1, t_2 \) for the sequence are calculated not from the physical model, but from a geometric or optimal principle based on the model-free predictions of the individual vectors, similar in structure to Eq. (5) but using predictions from the ultra-local model.
The cost function \( g_j \) (Eq. 4) is evaluated using the model-free prediction \( \mathbf{i}^{j}_{\alpha\beta}(k+2) \). The triple-vector sequence \( j \) that yields the minimum cost is applied during the next control period \( k+1 \).
Summary of the ATV-MFPCC Algorithm
The step-by-step control algorithm for the on-grid inverter is as follows:
At sampling instant \( k \):
1. Measure \( \mathbf{i}_{\alpha\beta}(k) \) and \( \mathbf{e}_{\alpha\beta}(k) \).
2. Calculate \( \alpha \) using Eq. (5) from past data.
3. Update the adaptive ESO (Eqs. 9-12) to obtain \( \hat{\mathbf{f}}_{\alpha\beta}(k+1) \) and \( \hat{\mathbf{i}}_{\alpha\beta}(k+1) \).
4. For each candidate triple-vector sequence \( j \):
a. Calculate its dwell times based on model-free predictions of its constituent vectors.
b. Form the synthesized vector \( \mathbf{v}_{t_j}(k) \) and \( \mathbf{v}_{t_j}(k+1) \).
c. Compute the model-free current predictions \( \mathbf{i}^{j}_{\alpha\beta}(k+1) \) and \( \mathbf{i}^{j}_{\alpha\beta}(k+2) \) using Eqs. (13) & (14).
d. Evaluate the cost function \( g_j \).
5. Select the sequence \( j_{opt} \) with the minimum \( g_j \).
6. Apply the switching states corresponding to \( \mathbf{v}_{t_{j_{opt}}}(k+1) \) with its calculated dwell times.
Simulation and Performance Analysis
To validate the proposed ATV-MFPCC strategy, a comprehensive simulation model of a three-phase on-grid inverter was developed in MATLAB/Simulink. The system parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | \( V_{dc} \) | 650 V |
| Grid Voltage (Phase-to-Neutral RMS) | \( E_{grid} \) | 230 V |
| Grid Frequency | \( f_g \) | 50 Hz |
| Filter Inductance | \( L \) | 8 mH |
| Filter Resistance | \( R \) | 0.5 Ω |
| Switching/Sampling Frequency | \( f_s = 1/T_s \) | 10 kHz |
| ESO Pole Location | \( p \) | 0.5 |
| Forgetting Factor | \( \lambda \) | 0.98 |
Steady-State and Dynamic Performance
The proposed ATV-MFPCC was compared against traditional Triple-Vector MPCC (TV-MPCC) and a basic Triple-Vector MFPCC (TV-MFPCC) without the adaptive ESO (using direct calculation from Eq. 6). Under nominal parameters, all three methods track a 10 A peak sinusoidal reference current effectively. The key metrics are current Total Harmonic Distortion (THD%) and the root-mean-square current tracking error \( I_{error,RMS} \).
| Control Method | THD% @ 10A | \( I_{error,RMS} \) (A) | Remarks |
|---|---|---|---|
| Traditional TV-MPCC | 2.85% | 0.051 | Baseline performance with perfect model. |
| Basic TV-MFPCC | 3.40% | 0.068 | Higher THD due to noise in disturbance estimation. |
| Proposed ATV-MFPCC | 2.78% | 0.049 | Best performance; adaptive ESO suppresses noise. |
During a transient where the current reference steps from 5A to 15A peak, the proposed method exhibits a settling time comparable to TV-MPCC and faster than the basic TV-MFPCC, demonstrating its excellent dynamic response maintained by the accurate disturbance estimation.
Robustness to Parameter Mismatch
The core advantage of the model-free approach is tested by intentionally introducing severe parameter mismatches in the controller. The actual \( L \) and \( R \) remain 8 mH and 0.5 Ω, while the values used in the traditional TV-MPCC are varied.
| Control Method | \( L_{model} = 4 \) mH (50%) | \( L_{model} = 12 \) mH (150%) | ||
|---|---|---|---|---|
| THD% | \( I_{error,RMS} \) (A) | THD% | \( I_{error,RMS} \) (A) | |
| Traditional TV-MPCC | 5.92% | 0.215 | 4.10% | 0.132 |
| Basic TV-MFPCC | 3.45% | 0.071 | 3.42% | 0.069 |
| Proposed ATV-MFPCC | 2.80% | 0.050 | 2.79% | 0.049 |
The results are striking. The traditional TV-MPCC performance degrades significantly, with THD increasing by over 100% in the case of 50% inductance underestimation. In contrast, both MFPCC methods maintain consistent performance, completely insensitive to the parameter error. The proposed ATV-MFPCC consistently delivers the lowest THD and error due to its superior disturbance observation.
Immunity to Measurement Noise
A key practical challenge is sensor noise. To evaluate this, white Gaussian noise with a standard deviation of 0.5 A (approximately 5% of the 10 A reference) was added to the measured current feedback.
| Control Method | THD% (Noisy) | THD Increase | Observation |
|---|---|---|---|
| Traditional TV-MPCC | 3.55% | +0.70% | Degraded due to noisy prediction. |
| Basic TV-MFPCC | 4.88% | +1.48% | Severely affected; noise directly impacts disturbance calculation. |
| Proposed ATV-MFPCC | 3.05% | +0.27% | Most robust; adaptive ESO acts as a low-pass filter for the disturbance estimate. |
The proposed method demonstrates remarkable noise immunity. The adaptive ESO effectively filters the high-frequency noise while retaining the accurate low-frequency estimate of the true lumped disturbance \( \mathbf{f}_{\alpha\beta} \). This leads to the smallest increase in THD, making the ATV-MFPCC highly suitable for real-world implementations where sensor noise is inevitable.
Discussion and Conclusion
This article has presented a novel Adaptive Triple-Vector Model-Free Predictive Current Control (ATV-MFPCC) strategy for on-grid inverters. The method successfully decouples control performance from the knowledge of physical system parameters by employing an ultra-local model. The main contributions and findings are summarized as follows:
1. Unified Robustness and Performance: The proposed strategy solves the twin problems of conventional MPCC: it provides the low-ripple, fixed-frequency performance characteristic of multi-vector modulation while simultaneously achieving absolute robustness to parameter mismatches in the on-grid inverter’s L and R filter components.
2. Advanced Disturbance Estimation: The incorporation of an Extended State Observer (ESO) with an online adaptive gain mechanism is pivotal. It provides a smooth, accurate, and real-time estimate of the lumped system disturbance. The adaptive gain feature dynamically strengthens the observer’s correction, granting it superior noise rejection capabilities compared to basic model-free methods that rely on direct algebraic calculation.
3. Comprehensive Performance Validation: Simulation studies under various conditions—nominal operation, severe parameter mismatch, and significant measurement noise—demonstrate the superiority of ATV-MFPCC. It matches or exceeds the steady-state performance of ideal-model MPCC under nominal conditions, maintains perfect consistency under parameter errors, and exhibits the strongest resilience to sensor noise.
The proposed ATV-MFPCC offers a compelling solution for modern on-grid inverter applications, particularly in microgrids and renewable energy systems where component parameters may be uncertain or variable, and where high power quality and reliability are non-negotiable requirements. Future work may focus on experimental validation on a hardware prototype and extending the ultra-local model framework to handle more complex inverter topologies, such as three-level or modular multilevel on-grid inverters.