The integration of renewable energy sources into the power grid has become a critical pathway towards achieving energy sustainability and security. As the primary interface between distributed generation resources, such as photovoltaic arrays and wind turbines, and the utility grid, the performance of the grid tied inverter is paramount. Its control system directly dictates power quality, stability, and overall system efficiency. Consequently, developing advanced control strategies for the current output of a grid tied inverter is a subject of significant research importance.
Traditional linear controllers, like the ubiquitous PI regulator in the synchronous rotating (dq) frame, are commonly employed for their simplicity in decoupling active and reactive power control. However, their performance is inherently limited. The gain of a PI controller diminishes as the frequency of the reference signal increases, leading to steady-state tracking errors for sinusoidal signals unless complex resonant terms are added. Furthermore, their dynamic response is often sluggish, and meticulous parameter tuning is required, which relies heavily on an accurate system model. Various nonlinear control strategies have been explored to overcome these limitations, including Lyapunov-based control, adaptive control combined with fuzzy logic, and sliding mode control. While offering improved robustness or dynamics, these methods often introduce new challenges such as complex stability proof constructions, heuristic design of membership functions, or the persistent issue of chattering in sliding mode implementations.
Among modern control techniques, Finite Control Set Model Predictive Control (FCS-MPC) has garnered considerable attention for power electronics applications. Its appeal lies in its intuitive concept, fast dynamic response, and inherent ability to handle multiple control objectives and constraints simultaneously. For a grid tied inverter, FCS-MPC typically involves using a discrete-time model of the system to predict the future behavior of the output currents for all possible voltage vectors generated by the inverter. A cost function, which encodes the control goals (e.g., current tracking error), is then evaluated for each prediction. The voltage vector that minimizes this cost function is selected and applied during the next sampling interval. This direct handling of the converter’s discrete nature leads to excellent performance.
Despite its advantages, a fundamental vulnerability of FCS-MPC is its high dependence on the accuracy of the predictive model. The core algorithm assumes that the model parameters (e.g., filter inductance and resistance, DC-link voltage) used for prediction match the actual physical parameters of the system. In practice, this assumption is frequently violated. The inductance of the grid-side filter in a grid tied inverter is susceptible to variations due to magnetic core saturation, manufacturing tolerances, and temperature changes. Similarly, the DC-link voltage may fluctuate due to source variations or load changes in upstream stages. Such parameter mismatches between the model and the plant lead to inaccurate current predictions, which subsequently cause steady-state tracking errors, increased current distortion, and can even compromise system stability. Therefore, enhancing the robustness of MPC against parameter uncertainties is a crucial research direction for reliable operation of a grid tied inverter.
Previous efforts to address this issue include online parameter identification methods, such as recursive least squares algorithms. While effective, these methods often impose a substantial computational burden on the digital controller, making them less suitable for high-switching-frequency applications. Other approaches involve recalculating model gradients or reference values, but they also face similar computational complexity challenges. Observer-based methods present a more computationally efficient alternative. By treating the aggregate effect of all parameter mismatches and unmodeled dynamics as a lumped disturbance, an observer can estimate this disturbance in real-time. The estimate is then fed forward to compensate the predictive model, thereby restoring its accuracy.
In this context, we present a robust model predictive control scheme for a three-phase two-level grid tied inverter. The proposed strategy centers on the design of a Sliding Mode Disturbance Observer (SMDO). The sliding mode approach is selected for its inherent robustness to modeling uncertainties and its strong disturbance rejection capabilities compared to linear observers like Luenberger or Kalman filters. We first derive the discrete-time state-space model of the inverter system in the dq reference frame. The discrepancies between the nominal parameters used in the controller and the actual system parameters are lumped into a generalized disturbance term. A discrete-time sliding mode observer is then meticulously designed to estimate both components (d and q) of this lumped disturbance. These estimates are seamlessly incorporated into the predictive model used by the MPC algorithm. The cost function remains focused on minimizing the current tracking error. By compensating the model with the observed disturbance, the predictive accuracy is maintained despite parameter variations, leading to robust performance. The effectiveness and robustness of the proposed control method are validated through comprehensive analysis and experimental results on a 10 kW prototype.
Discrete-Time Modeling of the Three-Phase Grid-Tied Inverter
The topology of a standard three-phase two-level voltage source inverter (VSI) connected to the grid through an L-filter is considered. The DC-link is supplied by a voltage source \(V_{dc}\). The inverter outputs are connected to the grid voltages \(v_a, v_b, v_c\) through filter inductors \(L_o\) and their equivalent series resistors \(R_o\). The output currents are denoted as \(i_a, i_b, i_c\). Each inverter leg has two complementary switching states, defining the phase voltage relative to the DC-link midpoint.
Applying Kirchhoff’s voltage law (KVL) to each phase yields the continuous-time dynamics:
$$
\begin{aligned}
u_a &= L_o \frac{di_a}{dt} + R_o i_a + v_a \\
u_b &= L_o \frac{di_b}{dt} + R_o i_b + v_b \\
u_c &= L_o \frac{di_c}{dt} + R_o i_c + v_c
\end{aligned}
$$
where \(u_a, u_b, u_c\) are the inverter output phase voltages. These voltages are determined by the switching states and the DC-link voltage. By applying the Park transformation with grid voltage orientation (aligning the d-axis with the grid voltage vector), the model is transformed into the synchronous rotating dq frame. This transformation simplifies the control objectives to regulating DC quantities: the d-axis current \(i_d\) controls reactive power, and the q-axis current \(i_q\) controls active power.
The state-space representation in the dq frame is given by:
$$
\frac{d}{dt}\begin{bmatrix} i_d \\ i_q \end{bmatrix} = \mathbf{A} \begin{bmatrix} i_d \\ i_q \end{bmatrix} + \mathbf{B_1} \begin{bmatrix} M_d \\ M_q \end{bmatrix} + \mathbf{B_2} \begin{bmatrix} v_d \\ v_q \end{bmatrix}
$$
with the matrices defined as:
$$
\mathbf{A} = \begin{bmatrix} -\frac{R_o}{L_o} & \omega \\ -\omega & -\frac{R_o}{L_o} \end{bmatrix}, \quad
\mathbf{B_1} = \begin{bmatrix} \frac{V_{dc}}{L_o} & 0 \\ 0 & \frac{V_{dc}}{L_o} \end{bmatrix}, \quad
\mathbf{B_2} = \begin{bmatrix} -\frac{1}{L_o} & 0 \\ 0 & -\frac{1}{L_o} \end{bmatrix}
$$
Here, \(\omega\) is the grid angular frequency, and \(M_d, M_q\) are the dq components of the modulation vector, which are directly related to the inverter’s switching states. For digital implementation, this continuous model must be discretized. Using a zero-order hold (ZOH) method with a sampling period \(T_s\), the exact discrete-time model is obtained:
$$
\mathbf{x}(k+1) = \mathbf{A_d} \mathbf{x}(k) + \mathbf{B_{1d}} \mathbf{M_{dq}}(k) + \mathbf{B_{2d}} \mathbf{v_{dq}}(k)
$$
where \(\mathbf{x}(k) = [i_d(k), i_q(k)]^T\), and the discrete matrices are:
$$
\mathbf{A_d} = e^{\mathbf{A}T_s}, \quad \mathbf{B_{1d}} = \int_0^{T_s} e^{\mathbf{A}(T_s-\tau)} \mathbf{B_1} d\tau, \quad \mathbf{B_{2d}} = \int_0^{T_s} e^{\mathbf{A}(T_s-\tau)} \mathbf{B_2} d\tau
$$
This discrete equation forms the foundation for the predictive control algorithm. However, this model uses the true parameters \(L_o, R_o, V_{dc}\). In the controller, only nominal values \(L_{on}, R_{on}, V_{dcn}\) are available. The mismatch between the actual and nominal parameters leads to an erroneous prediction. If we denote the discrete matrices calculated with nominal parameters as \(\mathbf{A_{dn}}, \mathbf{B_{1dn}}, \mathbf{B_{2dn}}\), the actual discrete system can be expressed in terms of these nominal matrices plus error terms:
$$
\mathbf{x}(k+1) = \mathbf{A_{dn}} \mathbf{x}(k) + \mathbf{B_{1dn}} \mathbf{M_{dq}}(k) + \mathbf{B_{2dn}} \mathbf{v_{dq}}(k) + \mathbf{N}(k)
$$
The term \(\mathbf{N}(k) = [N_d(k), N_q(k)]^T\) encapsulates the total lumped disturbance caused by all parameter mismatches:
$$
\mathbf{N}(k) = (\mathbf{A_d} – \mathbf{A_{dn}})\mathbf{x}(k) + (\mathbf{B_{1d}} – \mathbf{B_{1dn}})\mathbf{M_{dq}}(k) + (\mathbf{B_{2d}} – \mathbf{B_{2dn}})\mathbf{v_{dq}}(k)
$$
This disturbance \(\mathbf{N}(k)\) is the key source of degradation in conventional MPC performance for the grid tied inverter. Our objective is to estimate \(\mathbf{N}(k)\) in real-time and cancel its effect.
Design of the Discrete-Time Sliding Mode Disturbance Observer
To estimate the lumped disturbance \(\mathbf{N}(k)\), we design a Sliding Mode Disturbance Observer (SMDO). The sliding mode technique is chosen for its robustness and finite-time convergence properties. We first augment the system state to include the disturbance as a state variable. Assuming the disturbance varies slowly relative to the sampling period \(( \mathbf{N}(k+1) \approx \mathbf{N}(k) )\), the augmented discrete state-space model is:
$$
\begin{aligned}
\begin{bmatrix} \mathbf{x}(k+1) \\ \mathbf{N}(k+1) \end{bmatrix} &=
\begin{bmatrix} \mathbf{A_{dn}} & \mathbf{I} \\ \mathbf{0} & \mathbf{I} \end{bmatrix}
\begin{bmatrix} \mathbf{x}(k) \\ \mathbf{N}(k) \end{bmatrix} +
\begin{bmatrix} \mathbf{B_{1dn}} \\ \mathbf{0} \end{bmatrix} \mathbf{M_{dq}}(k) +
\begin{bmatrix} \mathbf{B_{2dn}} \\ \mathbf{0} \end{bmatrix} \mathbf{v_{dq}}(k) \\
\mathbf{y}(k) &= \begin{bmatrix} \mathbf{I} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{x}(k) \\ \mathbf{N}(k) \end{bmatrix}
\end{aligned}
$$
Or, more compactly:
$$
\mathbf{X}(k+1) = \mathbf{\Phi} \mathbf{X}(k) + \mathbf{G_1} \mathbf{U_1}(k) + \mathbf{G_2} \mathbf{U_2}(k)
$$
$$
\mathbf{y}(k) = \mathbf{C} \mathbf{X}(k)
$$
where \(\mathbf{X}(k) = [i_d(k), i_q(k), N_d(k), N_q(k)]^T\), \(\mathbf{U_1}(k)=[M_d(k), M_q(k), 0, 0]^T\), \(\mathbf{U_2}(k)=[v_d(k), v_q(k), 0, 0]^T\), and the matrices \(\mathbf{\Phi}, \mathbf{G_1}, \mathbf{G_2}, \mathbf{C}\) are defined accordingly.
Based on this augmented model, the discrete-time sliding mode observer is proposed as follows:
$$
\hat{\mathbf{X}}(k) = \mathbf{\Phi} \hat{\mathbf{X}}(k-1) + \mathbf{G_1} \mathbf{U_1}(k-1) + \mathbf{G_2} \mathbf{U_2}(k-1) + \mathbf{L} \cdot \text{sat}\left( \frac{\mathbf{e_y}(k-1)}{\mathbf{h}} \right)
$$
where:
- \(\hat{\mathbf{X}}(k)\) is the estimated state vector.
- \(\mathbf{e_y}(k-1) = \mathbf{C}\hat{\mathbf{X}}(k-1) – \mathbf{y}(k-1) = [\hat{i}_d(k-1)-i_d(k-1), \hat{i}_q(k-1)-i_q(k-1)]^T\) is the output estimation error.
- \(\mathbf{L}\) is the observer gain matrix to be designed.
- \(\text{sat}(\cdot)\) is the vector saturation function, used instead of the sign function to reduce chattering. The saturation function for a scalar \(x\) is defined with boundary layer thickness \(h\):
$$
\text{sat}(x) = \begin{cases}
1, & x \ge h \\
x/h, & -h < x < h \\
-1, & x \le -h
\end{cases}
$$
The gain matrix \(\mathbf{L}\) and the boundary layer vector \(\mathbf{h} = [h_d, h_q]^T\) are critical design parameters. The gains can be designed using pole placement techniques for the linear part of the error dynamics to ensure stability and desired convergence speed. The boundary layer represents a trade-off: a larger \(h\) reduces chattering but increases the observation error, while a smaller \(h\) improves accuracy but may induce more high-frequency components. The estimates of the lumped disturbances are simply extracted from the augmented state estimate:
$$
\hat{\mathbf{N}}(k) = \begin{bmatrix} \hat{N}_d(k) \\ \hat{N}_q(k) \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \hat{\mathbf{X}}(k)
$$
The structure of this observer allows it to dynamically track the combined effect of parameter mismatches in the grid tied inverter model. Its robust nature ensures that the estimation remains accurate even in the presence of the very disturbances it is observing.
Robust Model Predictive Control Algorithm with Disturbance Compensation
The core of the Finite Control Set Model Predictive Control (FCS-MPC) algorithm for a grid tied inverter involves evaluating a cost function for each possible voltage vector (switch state) and selecting the one that minimizes it. The conventional cost function \(J\) for current tracking is:
$$
J = |i_d^{ref}(k+1) – i_d^p(k+1)| + |i_q^{ref}(k+1) – i_q^p(k+1)|
$$
where \(i_d^{ref}, i_q^{ref}\) are the reference currents, and \(i_d^p(k+1), i_q^p(k+1)\) are the predicted currents for the next sampling instant using the nominal model. Under parameter mismatch, this prediction is inaccurate: \(i^p(k+1) \neq i(k+1)\).
The proposed robust MPC algorithm integrates the disturbance observer to correct this prediction. The prediction step is now performed using the nominal model compensated by the estimated disturbance:
$$
\begin{aligned}
i_d^p(k+1) &= A_{dn11} i_d(k) + A_{dn12} i_q(k) + B_{1dn11} M_d(k) + B_{1dn12} M_q(k) + B_{2dn11} v_d(k) + \hat{N}_d(k) \\
i_q^p(k+1) &= A_{dn21} i_d(k) + A_{dn22} i_q(k) + B_{1dn21} M_d(k) + B_{1dn22} M_q(k) + B_{2dn21} v_d(k) + \hat{N}_q(k)
\end{aligned}
$$
Note that for a balanced grid, \(v_q = 0\). The cost function \(J\) is calculated using these compensated predictions. The algorithm iterates through all seven distinct voltage vectors (excluding the zero vectors which are identical) available from a two-level inverter. The mapping between switching states \(S_a, S_b, S_c\) and the corresponding \(M_d, M_q\) values is pre-calculated and stored in a lookup table for efficiency.
| Switching State [Sa, Sb, Sc] | Voltage Vector [Md, Mq]T |
|---|---|
| [0, 0, 0] | [0, 0]T |
| [1, 0, 0] | \([\sin(\theta), \cos(\theta)]^T\) |
| [1, 1, 0] | \([\sin(\theta-2\pi/3), \cos(\theta-2\pi/3)]^T\) |
| [0, 1, 0] | \([\sin(\theta+2\pi/3), \cos(\theta+2\pi/3)]^T\) |
| [0, 1, 1] | \([-\sin(\theta), -\cos(\theta)]^T\) |
| [0, 0, 1] | \([-\sin(\theta-2\pi/3), -\cos(\theta-2\pi/3)]^T\) |
| [1, 0, 1] | \([-\sin(\theta+2\pi/3), -\cos(\theta+2\pi/3)]^T\) |
| [1, 1, 1] | [0, 0]T |
Where \(\theta = \omega t\) is the phase angle of the grid voltage vector.
The complete algorithmic flow for each control cycle \(k\) is as follows:
- Measurement: Sample the grid currents \(i_d(k), i_q(k)\) and grid voltage \(v_d(k)\).
- Disturbance Observation: Update the Sliding Mode Disturbance Observer using Eq. (6) to obtain the latest estimates \(\hat{N}_d(k), \hat{N}_q(k)\).
- Prediction & Optimization Loop: For each candidate voltage vector \(j\) (from 1 to 7):
- Obtain \(M_d^j, M_q^j\) from the lookup table.
- Calculate the compensated current predictions \(i_d^{p,j}(k+1), i_q^{p,j}(k+1)\) using Eq. (7).
- Evaluate the cost function \(J^j = |i_d^{ref} – i_d^{p,j}(k+1)| + |i_q^{ref} – i_q^{p,j}(k+1)|\).
- Selection: Identify the voltage vector \(j_{opt}\) that yields the minimum cost \(J^{min}\).
- Actuation: Apply the corresponding switching state \(S^{j_{opt}}\) to the grid tied inverter for the next interval.
This cycle repeats at every sampling instant. By integrating the real-time disturbance estimate, the predictive model is effectively adapted, making the control law robust to parameter variations. This is the essence of the proposed robust control scheme for the grid tied inverter.
Analysis of Robustness and Performance
The proposed control scheme enhances the system’s robustness by actively rejecting the lumped disturbance \(\mathbf{N}(k)\). The role of the SMDO is critical. Its stability and convergence conditions must be ensured through proper selection of the observer gains \(\mathbf{L}\). The error dynamics of the observer, defined as \(\mathbf{e}(k) = \hat{\mathbf{X}}(k) – \mathbf{X}(k)\), can be derived. Under the sliding mode condition, the output error \(\mathbf{e_y}(k)\) is driven into the boundary layer \(|\mathbf{e_y}| \leq \mathbf{h}\). Once inside this layer, the disturbance estimation error \(\hat{\mathbf{N}}(k) – \mathbf{N}(k)\) converges with dynamics dictated by the eigenvalues of \((\mathbf{\Phi} – \mathbf{L}\mathbf{C})\). Placing these poles inside the unit circle ensures asymptotic estimation of the states and disturbances.
The compensation mechanism can be analyzed by substituting the ideal case where \(\hat{\mathbf{N}}(k) = \mathbf{N}(k)\) into the prediction equation (7). The predicted current then becomes:
$$
\mathbf{x}^p(k+1) = \mathbf{A_{dn}} \mathbf{x}(k) + \mathbf{B_{1dn}} \mathbf{M_{dq}}(k) + \mathbf{B_{2dn}} \mathbf{v_{dq}}(k) + \mathbf{N}(k)
$$
This is identical to the actual system dynamics under parameter mismatch as expressed in Eq. (3). Therefore, with perfect disturbance observation, the prediction becomes exact, and the current tracking error in the cost function minimization is based on the true future behavior of the grid tied inverter. This theoretically eliminates the steady-state error caused by parameter mismatch. In practice, the accuracy depends on the observer’s performance, but even a non-ideal estimate significantly improves robustness compared to no compensation.
The computational burden of the proposed method is moderate. The added complexity consists of the observer update (a few matrix multiplications and additions) and the addition of \(\hat{N}_d, \hat{N}_q\) to the two prediction calculations within the main MPC loop. This is generally acceptable for modern digital signal processors (DSPs) at typical switching frequencies for a grid tied inverter (e.g., 10-50 kHz). The SMDO offers a favorable trade-off between robust performance and computational efficiency compared to online parameter identification methods.
Experimental Validation and Discussion
The proposed robust MPC strategy with the sliding mode disturbance observer was implemented and tested on a 10 kW three-phase two-level grid tied inverter prototype. The system parameters are summarized in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | Prated | 10 kW |
| Grid Phase Voltage (RMS) | Vgrid | 220 V |
| Grid Frequency | f | 50 Hz |
| DC-link Nominal Voltage | Vdcn | 700 V |
| Filter Inductance (Nominal) | Lon | 8 mH |
| Filter Resistance (Nominal) | Ron | 0.5 Ω |
| Control / Sampling Period | Ts | 20 μs |
A TMS320F28335 DSP was used as the main controller. The observer gains and boundary layer were tuned to achieve a fast yet smooth disturbance estimation. Three key scenarios were tested to validate the performance: steady-state operation with precise parameters, dynamic response to reference steps, and robustness tests under significant parameter mismatch.
1. Steady-State Performance: With the actual filter parameters matching their nominal values, the proposed controller operated as a standard, well-tuned MPC. The three-phase grid currents were sinusoidal, balanced, and achieved a low total harmonic distortion (THD) of approximately 3%, demonstrating the excellent inherent steady-state performance of the base algorithm for a grid tied inverter.
2. Dynamic Performance: The response to step changes in the current reference amplitude was tested. The results showed that the output currents tracked their new references within 2-3 ms, with no overshoot and minimal distortion during the transient. This confirms the fast dynamic response characteristic of FCS-MPC, which is preserved in the proposed robust variant.
3. Robustness Verification: This was the core validation. The inductance and resistance of the grid filter were deliberately varied by ±50% from their nominal values used in the controller’s model. Two comparative tests were conducted:
- Conventional MPC (without observer): Under a +50% parameter mismatch, the conventional controller exhibited significant steady-state tracking errors (around 1-1.2 A) and increased current ripple, clearly showing its sensitivity to model inaccuracies.
- Proposed Robust MPC (with SMDO): Under the same +50% mismatch condition, the proposed controller maintained precise current tracking. The observer successfully generated estimates for \(\hat{N}_d\) and \(\hat{N}_q\). These estimates were non-zero, actively compensating for the model error. The q-axis current \(i_q\) accurately followed its 15 A reference, and the d-axis current \(i_d\) remained at its 0 A reference, demonstrating perfect disturbance rejection. The same level of robustness was confirmed under a -50% parameter mismatch. Even during a step change in current reference under this severe mismatch, the observer estimates adjusted quickly, and the current tracking remained accurate and fast.
A further test involved operating the grid tied inverter with a DC-link voltage subject to 100 Hz ripple (simulating a rectifier front-end). The SMDO successfully estimated the disturbance caused by this voltage variation, and the output currents maintained high quality, further validating the observer’s ability to handle time-varying disturbances beyond simple parameter offsets.
Conclusion
This work has presented a robust model predictive current control strategy for three-phase grid tied inverter systems. The method effectively addresses the critical vulnerability of conventional FCS-MPC: its dependence on precise system parameters. By integrating a discrete-time Sliding Mode Disturbance Observer, the aggregate effect of parameter mismatches (in filter inductance, resistance, and DC-link voltage) is estimated in real-time. This estimate is then used as a feedforward compensation term within the predictive model, thereby correcting the current predictions.
The proposed algorithm retains all the desirable features of standard FCS-MPC, such as fast dynamic response and multi-objective handling, while significantly enhancing system robustness. Experimental results on a 10 kW prototype conclusively demonstrate the effectiveness of the approach. Under severe parameter variations of up to ±50%, the robust controller maintained precise current tracking and low distortion, whereas the conventional MPC failed with substantial steady-state errors. The observer also proved effective in compensating for DC-link voltage ripple.
The proposed scheme offers a practical and computationally manageable solution to improve the reliability and performance of grid tied inverters in real-world applications where parameter uncertainties are inevitable. Future work may explore the integration of higher-order sliding mode observers or adaptive techniques to further refine the disturbance estimation and reduce any residual chattering, potentially extending the framework to LCL-filtered inverters with more complex dynamics.