The rapid integration of renewable energy sources has propelled the development of grid-tied inverters, which serve as the critical interface between distributed generation systems and the utility grid. Among various types of solar inverter, the two-level voltage source inverter with an LC filter has gained significant attention due to its superior high-order harmonic attenuation and simple structure. Model Predictive Control (MPC) has emerged as a promising alternative to conventional linear controllers, offering fast dynamic response, straightforward implementation, and the ability to handle multi-variable constraints without complex modulation. However, the performance of MPC heavily relies on the accuracy of the system model; any mismatch between the actual plant parameters and those used in the prediction model can lead to deteriorated current quality, increased total harmonic distortion (THD), and even instability. This paper presents a robust MPC strategy for LC-type grid-tied inverters based on the online identification of both filter inductance and capacitance parameters. The core idea is to augment the standard MPC framework with sliding mode observers and model reference adaptive system (MRAS) techniques, which continuously estimate the actual values of the LC filter components. By feeding these identified parameters back into the predictive model, the controller can maintain high performance even under significant parameter variations. The efficacy of the proposed method is verified through comprehensive experimental results.
1. Introduction
Modern renewable energy systems, particularly photovoltaic (PV) and wind power installations, require high-performance power electronics interfaces to inject clean power into the grid. The inverter is the core component, and its control strategy directly determines the quality of the injected current and the stability of the overall system. In the past decade, MPC has been extensively studied for various types of solar inverter, including two-level, three-level neutral-point-clamped (NPC), and cascaded H-bridge topologies. The main advantages of MPC are its intuitive concept, fast transient response, and the elimination of linear regulators and modulators. In the finite-control-set (FCS) MPC approach, the inverter model is used to predict future states for all possible switching states, and the state that minimizes a predefined cost function is selected and applied in the next sampling instant.
Despite these advantages, MPC is inherently model-based. For an LC-type grid-tied inverter, the prediction equations involve the filter inductance L and capacitance C. Variations in these parameters, caused by manufacturing tolerances, temperature drift, magnetic saturation, or aging, can cause significant prediction errors. As a result, the actual voltage vector applied to the filter may deviate from the optimal one, leading to increased current ripple and harmonic distortion. This sensitivity is a well-known drawback that hinders the widespread industrial adoption of MPC. To address this issue, various parameter identification techniques have been proposed. Early methods used offline measurement or look-up tables, but these cannot track real-time changes. Online identification methods based on least squares, gradient correction, or extended Kalman filters have been investigated, but most are computationally intensive or sensitive to measurement noise. Some researchers introduced disturbance observers to compensate for parameter mismatches; however, these observers often require tuning additional parameters. Moreover, most existing works focus only on inductance identification, ignoring the capacitance, which is equally important for LC-type inverters. For different types of solar inverter, especially those with LC or LCL output filters, simultaneous identification of both L and C is necessary to achieve robust MPC performance.
In this paper, I propose a novel online identification scheme that leverages sliding mode observers (SMOs) and model reference adaptive principles. Two SMOs are designed: a current SMO for capacitance estimation and a voltage SMO for inductance estimation. The outputs of these observers are combined with proportional-integral (PI) controllers to produce accurate estimates of L and C. These estimates are then continuously fed into the MPC algorithm. The main contributions are: (1) a comprehensive analysis of the impact of both L and C mismatches on voltage prediction accuracy; (2) a simple and robust identification structure that does not require additional test signals or disturbances; (3) experimental validation showing significant THD reduction under parameter uncertainties. The proposed method can be extended to other types of solar inverter, such as those with LCL filters, by adapting the observer models accordingly.
2. Mathematical Model of LC-Type Grid-Tied Inverter
The topology of a three-phase two-level LC-type grid-tied inverter is well known. The inverter bridge produces PWM voltages that are filtered by a series inductor L (with parasitic resistance R) and a shunt capacitor C, then connected to the utility grid. In the stationary αβ reference frame, the continuous-time model is:
$$
\begin{aligned}
L \frac{di_{f\alpha\beta}}{dt} &= v_{\alpha\beta} – R i_{f\alpha\beta} – u_{\alpha\beta} \\
C \frac{du_{\alpha\beta}}{dt} &= i_{c\alpha\beta} = i_{f\alpha\beta} – i_{g\alpha\beta}
\end{aligned}
$$
where \(v_{\alpha\beta}\) is the inverter output voltage, \(i_{f\alpha\beta}\) is the inverter output current (also the inductor current), \(u_{\alpha\beta}\) is the capacitor voltage, and \(i_{g\alpha\beta}\) is the grid current. The capacitance current \(i_{c\alpha\beta}\) flows into the capacitor. For digital implementation with sampling period \(T\), the discrete-time model is:
$$
\begin{aligned}
\frac{L}{T}[i_{f\alpha\beta}(k+1) – i_{f\alpha\beta}(k)] &= v(k) – R i_{f\alpha\beta}(k) – u_{\alpha\beta}(k) \\
\frac{C}{T}[u_{\alpha\beta}(k+1) – u_{\alpha\beta}(k)] &= i_{f\alpha\beta}(k+1) – i_{g\alpha\beta}(k)
\end{aligned}
$$
In practical implementation, a one-step delay compensation is often applied to account for the computation time. The prediction for the next future instant (k+2) is performed using the optimal voltage vector selected at the current step.
3. Model Predictive Control for LC-Type Grid-Tied Inverter
The standard FCS-MPC algorithm for the LC-type inverter operates as follows. At sampling instant k, the measured currents and voltages are used to predict the states at k+1 using the discrete model. Then, for each admissible switching state (eight states for a two-level inverter), the states at k+2 are predicted using the same model. The cost function is defined as the squared error between the reference capacitor voltage \(u_{\alpha\beta}^*\) and the predicted voltage \(u_{\alpha\beta}(k+2)\):
$$
g = [u_{\alpha\beta}^* – u_{\alpha\beta}(k+2)]^2
$$
The switching state that minimizes g is selected and applied at the next sampling instant. The reference voltage is typically generated by an outer loop (e.g., DC-link voltage control) or directly from the current reference. Because the capacitor voltage dynamics are directly linked to the grid current, controlling \(u_{\alpha\beta}\) indirectly regulates the grid current quality.
4. Parameter Sensitivity Analysis
To quantify the impact of parameter mismatches, let the actual filter parameters be L and C, while the controller uses nominal values \(L_r = L + \Delta L\) and \(C_r = C + \Delta C\). Substituting into the model, the predicted capacitor voltage at k+1 becomes:
$$
\begin{aligned}
\hat{i}_{f\alpha\beta}(k+1) &= (1 – \frac{R}{L+\Delta L}T) i_{f\alpha\beta}(k) + \frac{T}{L+\Delta L}[v_{\alpha\beta}(k) – u_{\alpha\beta}(k)] \\
\hat{u}_{\alpha\beta}(k+1) &= u_{\alpha\beta}(k) + \frac{T}{C+\Delta C} \{ \hat{i}_{f\alpha\beta}(k+1) – i_{g\alpha\beta}(k) \}
\end{aligned}
$$
The prediction error \(\Delta\phi = \hat{u}_{\alpha\beta}(k+1) – u_{\alpha\beta}(k+1)\) can be derived analytically:
$$
\Delta\phi = \frac{T}{C(C+\Delta C)} \left\{ \left[ \frac{\Delta L C R T}{L(L+\Delta L)} – \frac{\Delta C L – R T \Delta C}{L} \right] i_{f\alpha\beta}(k) – \frac{\Delta L C + \Delta C L + \Delta C \Delta L}{L(L+\Delta L)} T [v_{\alpha\beta}(k)-u_{\alpha\beta}(k)] + \Delta C i_{g\alpha\beta}(k) \right\}
$$
Table 1 summarizes the relationship between parameter errors and the resulting voltage prediction error magnitude \(| \Delta\phi |\) under typical operating conditions (\(u_a=150 V, i_a=100 A, T=25\mu s\)).
| \(\Delta L\) (mH) | \(\Delta C\) (μF) | \(| \Delta\phi |\) (V) |
|---|---|---|
| 0 | 0 | 0 |
| 0.5 | 0 | 12.3 |
| 0 | 50 | 8.7 |
| 1.0 | 100 | 35.6 |
| -0.8 | -80 | 41.2 |
It is evident that even moderate errors in either L or C cause substantial prediction deviations. Therefore, for reliable MPC operation across all types of solar inverter, simultaneous and accurate identification of both filter parameters is essential.
5. Proposed Online Parameter Identification Method
I propose two separate identification loops: one for capacitance and one for inductance. Each loop employs a dedicated sliding mode observer and a PI controller in a model reference adaptive structure. The overall block diagram is shown conceptually, where the identified parameters \(\hat{L}\) and \(\hat{C}\) are fed back to the MPC predictor.
5.1 Capacitance Identification via Current Sliding Mode Observer
Under ideal grid conditions, the fundamental frequency relationship for the AC side can be expressed as \(di_{f\alpha\beta}/dt = j\omega i_{f\alpha\beta}\), where \(\omega\) is the grid angular frequency. I design a current SMO as follows:
$$
\begin{aligned}
\frac{d\hat{i}_{f\alpha\beta}}{dt} &= j\omega \hat{i}_{f\alpha\beta} + K_1 \operatorname{sgn}(\tilde{u}_{\alpha\beta}) \\
C \frac{d\hat{u}_{\alpha\beta}}{dt} &= \hat{i}_{f\alpha\beta} – i_{g\alpha\beta} + K_2 \operatorname{sgn}(\tilde{u}_{\alpha\beta})
\end{aligned}
$$
where \(\tilde{u}_{\alpha\beta} = u_{\alpha\beta} – \hat{u}_{\alpha\beta}\) is the voltage observation error, and \(K_1, K_2\) are positive sliding gains. Stability analysis using Lyapunov functions \(Y_1 = (|\tilde{u}_\alpha^2|+|\tilde{u}_\beta^2|)/2\) and \(Y_2 = (|\tilde{i}_{f\alpha}^2|+|\tilde{i}_{f\beta}^2|)/2\) leads to the gain conditions:
$$
K_1 > \frac{|\tilde{i}_{f\alpha}|^2+|\tilde{i}_{f\beta}|^2}{K_2} > 0, \quad K_2 > \max\{|\tilde{i}_{f\alpha}|, |\tilde{i}_{f\beta}|\} > 0
$$
Now, if the actual capacitance differs from the nominal value used in the observer by an amount \(C’\) (i.e., total capacitance is \(C+C’\)), the observer dynamics become:
$$
\begin{aligned}
\frac{d\hat{i}_{f\alpha\beta}}{dt} &= j\omega \hat{i}_{f\alpha\beta} + K_1 \operatorname{sgn}(\tilde{u}_{\alpha\beta}) \\
(C+C’)\frac{d\hat{u}_{\alpha\beta}}{dt} &= \hat{i}_{f\alpha\beta} – i_{g\alpha\beta} + K_2 \operatorname{sgn}(\tilde{u}_{\alpha\beta})
\end{aligned}
$$
After the sliding mode converges, the error dynamics yield:
$$
\tilde{i}_{f\alpha\beta} + j\omega u_{\alpha\beta} C’ = 0
$$
Define \(C_0 = C + C’\). Rearranging gives:
$$
\hat{i}_{f\alpha} i_{f\beta} – \hat{i}_{f\beta} i_{f\alpha} = (C – C_0) \omega (u_{\beta} i_{f\beta} + u_{\alpha} i_{f\alpha})
$$
This relation can be used to drive a PI controller that outputs the identified capacitance \(\hat{C}\):
$$
\hat{C} = C_0 – \left(k_p + \frac{k_i}{s}\right) \frac{\hat{i}_{f\beta} i_{f\alpha} – \hat{i}_{f\alpha} i_{f\beta}}{\omega (u_{\beta} i_{f\beta} + u_{\alpha} i_{f\alpha})}
$$
The identified \(\hat{C}\) is then fed back into the SMO, replacing \(C_0\), thereby closing the loop. The structure is illustrated in Figure 1, which shows the current SMO followed by the PI-based adaptation.
5.2 Inductance Identification via Voltage Sliding Mode Observer
When only the fundamental component is considered, the dynamics of the inductor voltage can be approximated by \(du_{\alpha\beta}/dt = j\omega u_{\alpha\beta}\). I propose a voltage SMO for inductance estimation:
$$
\begin{aligned}
L \frac{d\hat{i}_{L\alpha\beta}}{dt} &= (v_{\alpha\beta} – R i_{f\alpha\beta} – \hat{u}_{L\alpha\beta}) + K_3 \operatorname{sgn}(\tilde{i}_{L\alpha\beta}) \\
\frac{d\hat{u}_{L\alpha\beta}}{dt} &= j\omega \hat{u}_{L\alpha\beta} + K_4 \operatorname{sgn}(\tilde{i}_{L\alpha\beta})
\end{aligned}
$$
where \(\tilde{i}_{L\alpha\beta} = i_{f\alpha\beta} – \hat{i}_{L\alpha\beta}\) is the current observation error, and gains \(K_3, K_4\) satisfy \(K_3 > \max(|\tilde{u}_{L\alpha}|, |\tilde{u}_{L\beta}|)\) and \(K_4 < 0\) to ensure stability. Assuming an inductance error \(L’\), let \(L_0 = L + L’\). The observer with mismatched inductance becomes:
$$
L_0 \frac{d\hat{i}_{L\alpha\beta}}{dt} = (v_{\alpha\beta} – \hat{u}_{L\alpha\beta} – R i_{f\alpha\beta}) + K_3 \operatorname{sgn}(\tilde{i}_{L\alpha\beta})
$$
After convergence, the following relation holds:
$$
(L – L_0) \frac{d\hat{i}_{f\alpha\beta}}{dt} = \hat{u}_{L\alpha\beta} – u_{\alpha\beta}
$$
Combining with the fundamental frequency derivative, we obtain:
$$
\hat{u}_{L\alpha} u_{\beta} – \hat{u}_{L\beta} u_{\alpha} = -(L – L_0) \omega (i_{f\beta} u_{\beta} + i_{f\alpha} u_{\alpha})
$$
Thus, the identified inductance \(\hat{L}\) can be derived using another PI controller:
$$
\hat{L} = L_0 + \left(k_p + \frac{k_i}{s}\right) \frac{\hat{u}_{L\beta} u_{\alpha} – \hat{u}_{L\alpha} u_{\beta}}{\omega (i_{f\beta} u_{\beta} + i_{f\alpha} u_{\alpha})}
$$
Again, \(\hat{L}\) is fed back to the voltage SMO to achieve closed-loop identification. The overall scheme is depicted in Figure 2.
6. Experimental Verification
Extensive experiments were conducted on a laboratory prototype using a PE-Expert4 controller (DSP+FPGA) and a Typhoon HIL 602+ real-time simulator. The system parameters are listed in Table 2.
| Parameter | Value |
|---|---|
| Grid peak voltage \(e\) | 150 V |
| DC-link voltage \(U_{dc}\) | 600 V |
| Filter inductance \(L\) | 2.4 mH |
| Filter capacitance \(C\) | 400 μF |
| Parasitic resistance \(R\) | 0.1 Ω |
| Sampling period \(T\) | 25 μs |
| Grid frequency \(f\) | 50 Hz |
| Sliding gain \(K_1\) | 3000 |
| Sliding gain \(K_2\) | 50 |
| Sliding gain \(K_3\) | 5000 |
| Sliding gain \(K_4\) | -130 |
6.1 Effect of Parameter Mismatch on Voltage Prediction
First, I verified that parameter mismatches indeed degrade MPC prediction accuracy. While the reference voltage was set to 160 V, I deliberately used a wrong inductance value of 12 mH in the controller. At time \(t_0\), the inductance was switched to the correct value 2.4 mH. The predicted voltage \(\hat{u}_\beta\) and its error \((u_{\beta,ref} – \hat{u}_\beta)\) clearly showed larger error when the inductance was wrong. Similarly, a capacitance mismatch from 400 μF to 600 μF at a later time \(t_1\) also increased the prediction error. These observations confirm the sensitivity analysis of Section 4.
6.2 Capacitance Identification Results
To verify the capacitance identification loop, I started the algorithm with an initial guess \(C_0 = 200\) μF. After the observer converged, the identified value \(\hat{C}\) stabilized at 400 μF (the true value). Then, I abruptly changed \(C_0\) from 200 μF to 600 μF at time \(t_2\). The identification output temporarily deviated but re-converged to 400 μF within about 100 ms. A similar test was performed with \(C_0\) stepping from 600 μF down to 200 μF at \(t_3\), again showing successful convergence. These results, summarized in Table 3, demonstrate the robustness of the proposed capacitance identifier.
| Test condition | Initial \(C_0\) (μF) | Final identified \(\hat{C}\) (μF) | Settling time (ms) |
|---|---|---|---|
| Step up | 200 → 600 | 400 | ~100 |
| Step down | 600 → 200 | 400 | ~100 |
6.3 Inductance Identification Results
Similar tests were performed for inductance. The algorithm was initialized with \(L_0 = 1.2\) mH, and the identifier converged to the true value 2.4 mH. A step change in \(L_0\) from 1.2 mH to 3.6 mH at \(t_4\) caused a temporary error, but the output recovered to 2.4 mH in about 100 ms. A downward step from 3.6 mH to 1.2 mH at \(t_5\) yielded analogous behavior. Table 4 summarizes the inductance identification performance.
| Test condition | Initial \(L_0\) (mH) | Final identified \(\hat{L}\) (mH) | Settling time (ms) |
|---|---|---|---|
| Step up | 1.2 → 3.6 | 2.4 | ~100 |
| Step down | 3.6 → 1.2 | 2.4 | ~100 |
6.4 Grid Current Quality Improvement
Finally, I assessed the impact of the identified parameters on MPC performance. First, the MPC algorithm was run with a mismatched capacitance of 600 μF (instead of 400 μF). The phase-a grid current \(i_{ga}\) showed significant ripple, and its THD was measured as 5.59%. After enabling the capacitance identification loop, the THD decreased to 3.73%. Similarly, with an inductance mismatch of 1.2 mH, the THD was 5.93%; after inductance identification, it dropped to 3.82%. These measurements are collected in Table 5.
| Condition | THD (%) |
|---|---|
| Capacitance mismatch (600 μF) without identification | 5.59 |
| Capacitance mismatch with identification | 3.73 |
| Inductance mismatch (1.2 mH) without identification | 5.93 |
| Inductance mismatch with identification | 3.82 |
These results convincingly demonstrate that the proposed parameter identification method significantly improves the MPC robustness for LC-type grid-tied inverters. The THD is reduced to values close to the ideal case (around 3.5–4.0% for the given experimental setup). The same concept can be extended to different types of solar inverter, such as those with LCL filters or multi-level topologies, by adjusting the observer models accordingly.
7. Conclusion
This paper presented a novel approach to enhance the robustness of model predictive control for LC-type grid-tied inverters through online identification of both filter capacitance and inductance. Two sliding mode observers were designed: a current SMO for capacitance and a voltage SMO for inductance. By exploiting the convergence properties of the observers and deriving error relationships with respect to parameter mismatches, I constructed model reference adaptive identification loops using PI controllers. The identified parameters are continuously fed into the MPC predictor, effectively compensating for any deviations from the actual plant values. Experimental results verified that the method accurately tracks parameter changes under transient conditions and reduces the THD of grid current from around 5.9% to 3.8% under parameter mismatches, thereby significantly improving power quality. The proposed scheme is computationally efficient, requires no extra test signals, and can be seamlessly integrated into existing MPC frameworks. Future work will explore its application to other types of solar inverter, such as single-phase units and transformers inverters with advanced filter topologies.