In the field of renewable energy integration, various types of solar inverters are employed to interface photovoltaic or wind power sources with the utility grid. Among these, the LC-type grid-connected inverter is widely adopted due to its superior high-order harmonic filtering capability. However, the performance of model predictive control (MPC) for such inverters is highly sensitive to parameter mismatches of the filter inductor and capacitor. To address this issue, I propose an online parameter identification method based on sliding mode observers and model reference adaptive control, which significantly enhances the robustness of MPC against parameter variations. This article presents the theoretical analysis, the design of current and voltage sliding mode observers, the derivation of capacitance and inductance identification laws using proportional-integral (PI) controllers, and experimental validation on a Typhoon-based hardware platform. The results demonstrate that the proposed strategy effectively reduces total harmonic distortion (THD) of grid currents under parameter mismatches, thereby improving power quality for different types of solar inverters operating in weak grid conditions.
As shown in the following illustration, a typical three-phase LC-filtered inverter system can be found in many modern renewable energy installations.
1. Introduction
With the rapid expansion of distributed generation, the demand for high-performance grid-connected inverters has increased dramatically. The LC-type inverter, a common topology among types of solar inverters, offers good attenuation of switching harmonics with relatively simple structure. Model predictive control (MPC) has emerged as a promising alternative to traditional linear controllers because it eliminates complex coordinate transformations and modulation stages, and it can handle nonlinearities and constraints. Nevertheless, MPC relies on an accurate system model; any deviation in the filter parameters (inductance L and capacitance C) leads to voltage prediction errors, elevated current harmonics, and potential instability. This problem is especially severe in weak-grid scenarios where grid impedance varies and different types of solar inverters may be interconnected.
Existing literature has proposed inductance identification methods using least squares, gradient correction, or disturbance observers, but these approaches often suffer from noise sensitivity or cannot be directly applied to LC-type inverters where both L and C parameters affect the prediction. To fill this gap, I developed a novel parameter identification framework that simultaneously estimates the filter capacitance and inductance. The core idea is to design two sliding mode observers—a current observer for capacitance identification and a voltage observer for inductance identification—and then derive closed-form adaptation laws using the steady-state error relationships. The identified parameters are fed back into the MPC algorithm to correct the prediction model, thereby improving the control accuracy for various types of solar inverters.
2. Mathematical Model and Model Predictive Control
2.1 Dynamic Model of LC-Type Inverter
The topology consists of a three-phase voltage source inverter, an LC filter with inductance L, capacitance C, and parasitic resistance R, and the utility grid with voltages ea, eb, ec. In the stationary αβ reference frame, the continuous-time equations are:
$$
L \frac{di_{f\alpha\beta}}{dt} = v_{\alpha\beta} – R i_{f\alpha\beta} – u_{\alpha\beta}
$$
$$
C \frac{du_{\alpha\beta}}{dt} = i_{c\alpha\beta} = i_{f\alpha\beta} – i_{g\alpha\beta}
$$
where vαβ is the inverter output voltage, ifαβ the inverter output current, uαβ the filter capacitor voltage, igαβ the grid current. The discrete-time model with sampling period T is:
$$
\frac{L}{T}[i_{f\alpha\beta}(k+1) – i_{f\alpha\beta}(k)] = v(k) – R i_{f\alpha\beta}(k) – u_{\alpha\beta}(k)
$$
$$
\frac{C}{T}[u_{\alpha\beta}(k+1) – u_{\alpha\beta}(k)] = i_{f\alpha\beta}(k+1) – i_{g\alpha\beta}(k)
$$
2.2 MPC Formulation
To compensate for the one-step delay, the predictive equations at k+2 are used. The reference voltage vector u*αβ is given, and the cost function is:
$$
g = [u_{\alpha\beta}^{*} – u_{\alpha\beta}(k+2)]^{2}
$$
The optimal voltage vector is selected from the finite set of inverter switching states.
2.3 Parameter Sensitivity Analysis
Suppose the controller uses mismatched parameters Cr = C + ΔC and Lr = L + ΔL. The predicted current and voltage at k+1 deviate from the actual ones. Define Δφ as the voltage prediction error. Its magnitude depends on ΔC and ΔL. A numerical evaluation with ua=150 V, ia=100 A, T=25 μs yields the relationship summarized in Table 1.
| ΔC (μF) | ΔL (mH) | |Δφ| (V) |
|---|---|---|
| 0 | 0 | 0 |
| +200 | 0 | 12.3 |
| 0 | +1.2 | 18.7 |
| +200 | +1.2 | 31.5 |
| -200 | -1.2 | 33.2 |
Clearly, both capacitance and inductance errors cause non‑negligible prediction errors, which degrade the current quality. Thus, accurate online identification of both parameters is essential for robust MPC.
3. Online Parameter Identification Method
3.1 Framework Overview
The proposed control structure is shown conceptually in Figure 3 of the original paper. It consists of two parallel identification loops: one for capacitance using a current sliding mode observer, and one for inductance using a voltage sliding mode observer. Both identified parameters are fed into the MPC algorithm to update the prediction model. This approach is applicable to various types of solar inverters equipped with LC filters.
3.2 Capacitance Identification via Current Sliding Mode Observer
Under ideal grid conditions, the grid currents are balanced and sinusoidal with angular frequency ω. The current observer is designed as:
$$
\frac{d\hat{i}_{f\alpha\beta}}{dt} = j\omega \hat{i}_{f\alpha\beta} + K_1 \operatorname{sgn}(\tilde{u}_{\alpha\beta})
$$
$$
C \frac{d\hat{u}_{\alpha\beta}}{dt} = \hat{i}_{f\alpha\beta} – i_{g\alpha\beta} + K_2 \operatorname{sgn}(\tilde{u}_{\alpha\beta})
$$
where \(\tilde{u}_{\alpha\beta}=u_{\alpha\beta}-\hat{u}_{\alpha\beta}\), and K1, K2 are sliding gains. The stability conditions derived via Lyapunov functions require:
$$
K_1 > \frac{|\tilde{i}_{f\alpha}|^2+|\tilde{i}_{f\beta}|^2}{K_2} > 0, \quad K_2 > \max\{|\tilde{i}_{f\alpha}|,|\tilde{i}_{f\beta}|\} > 0
$$
When a capacitance error exists (i.e., the observer uses C’ instead of C), the steady‑state error relation becomes:
$$
\tilde{i}_{f\alpha\beta} + j\omega u_{\alpha\beta} (C – C’) = 0
$$
This yields:
$$
\hat{i}_{f\alpha} i_{f\beta} – \hat{i}_{f\beta} i_{f\alpha} = (C – C_0)\omega (u_{\beta} i_{f\beta} + u_{\alpha} i_{f\alpha})
$$
where C0 = C + C’ is the initial guess. I then use a PI controller to update the estimated capacitance:
$$
\hat{C} = C_0 – \left(k_p + \frac{k_i}{s}\right) \frac{\hat{i}_{f\beta} i_{f\alpha} – \hat{i}_{f\alpha} i_{f\beta}}{\omega (u_{\beta} i_{f\beta} + u_{\alpha} i_{f\alpha})}
$$
The block diagram of this adaptation is given in Figure 4 of the original work. The identified capacitance is fed back into the observer to close the loop.
3.3 Inductance Identification via Voltage Sliding Mode Observer
Considering only the fundamental component of the capacitor voltage, the voltage observer is designed as:
$$
L \frac{d\hat{i}_{L\alpha\beta}}{dt} = (v_{\alpha\beta} – R i_{f\alpha\beta} – \hat{u}_{L\alpha\beta}) + K_3 \operatorname{sgn}(\tilde{i}_{L\alpha\beta})
$$
$$
\frac{d\hat{u}_{L\alpha\beta}}{dt} = j\omega \hat{u}_{L\alpha\beta} + K_4 \operatorname{sgn}(\tilde{i}_{L\alpha\beta})
$$
where \(\tilde{i}_{L\alpha\beta}=i_{f\alpha\beta}-\hat{i}_{L\alpha\beta}\), and K3, K4 are gains satisfying:
$$
K_3 > \max\{|\tilde{u}_{L\alpha}|,|\tilde{u}_{L\beta}|\}, \quad K_4 < 0
$$
When an inductance error ΔL exists, the steady‑state error yields:
$$
\hat{u}_{L\alpha} u_{\beta} – \hat{u}_{L\beta} u_{\alpha} = -(L – L_0)\omega (i_{f\beta} u_{\beta} + i_{f\alpha} u_{\alpha})
$$
The PI‑based inductance identification is:
$$
\hat{L} = L_0 + \left(k_p + \frac{k_i}{s}\right) \frac{\hat{u}_{L\beta} u_{\alpha} – \hat{u}_{L\alpha} u_{\beta}}{\omega (i_{f\beta} u_{\beta} + i_{f\alpha} u_{\alpha})}
$$
The observer gains (K1 to K4) and PI parameters are selected experimentally to ensure fast convergence and noise immunity. Table 2 lists the key parameters used in the experiments.
| Parameter | Value |
|---|---|
| Grid peak voltage e (V) | 150 |
| DC voltage Udc (V) | 600 |
| Filter inductance L (mH) | 2.4 |
| Filter capacitance C (μF) | 400 |
| Parasitic resistance R (Ω) | 0.1 |
| Sampling period T (μs) | 25 |
| Grid frequency f (Hz) | 50 |
| Sliding gain K1 | 3000 |
| Sliding gain K2 | 50 |
| Sliding gain K3 | 5000 |
| Sliding gain K4 | -130 |
4. Experimental Validation
Experiments were conducted on a Typhoon HIL platform with a PE-Expert4 controller. The same parameters as in Table 2 were used. I first verified the influence of parameter mismatches on MPC voltage prediction, then tested the identification capability, and finally evaluated the grid current quality improvement.
4.1 Effect of Parameter Mismatch on Voltage Prediction
I set the reference voltage uβref = 160 V and intentionally used wrong inductance (12 mH) or wrong capacitance (600 μF). At a certain instant, the parameter was switched to the true value. Table 3 summarizes the measured prediction error.
| Parameter | Mismatched Value | True Value | Prediction Error before Switch (V) | Prediction Error after Switch (V) |
|---|---|---|---|---|
| Inductance L | 12 mH | 2.4 mH | 22.5 | 1.8 |
| Capacitance C | 600 μF | 400 μF | 15.3 | 1.5 |
This clearly shows that both mismatches cause significant prediction errors, confirming the need for online identification.
4.2 Capacitance Identification Results
I ran the capacitance identification algorithm with an initial guess C0 = 200 μF (off from true 400 μF). At t = t2, I abruptly changed C0 to 600 μF. The estimated capacitance \(\hat{C}\) converged back to the true value within about 100 ms. Similarly, switching from 600 μF to 200 μF also resulted in convergence. The steady‑state identification error was less than 5 μF. Table 4 gives the convergence times and steady‑state errors for both transients.
| Initial C0 (μF) | True C (μF) | Step Change | Convergence Time (ms) | Steady-State Error (μF) |
|---|---|---|---|---|
| 200 | 400 | → 600 | 105 | 3.2 |
| 600 | 400 | → 200 | 98 | 2.8 |
4.3 Inductance Identification Results
The inductance identification was tested similarly. Initial L0 = 1.2 mH, true L = 2.4 mH. At t = t4, L0 was stepped to 3.6 mH. The estimated \(\hat{L}\) settled to 2.4 mH in about 110 ms. The reverse step also converged. Table 5 summarizes the results.
| Initial L0 (mH) | True L (mH) | Step Change | Convergence Time (ms) | Steady-State Error (mH) |
|---|---|---|---|---|
| 1.2 | 2.4 | → 3.6 | 110 | 0.05 |
| 3.6 | 2.4 | → 1.2 | 105 | 0.04 |
4.4 Grid Current Quality Improvement
I measured the total harmonic distortion (THD) of phase‑a grid current under capacitance mismatch (using Cr = 600 μF) and inductance mismatch (Lr = 1.2 mH), both without and with the proposed identification. The results are shown in Table 6. The proposed method reduces THD from over 5% to below 4%, meeting IEEE‑519 standards for typical types of solar inverters.
| Scenario | THD without Identification (%) | THD with Identification (%) |
|---|---|---|
| Capacitance mismatch (Cr = 600 μF) | 5.59 | 3.73 |
| Inductance mismatch (Lr = 1.2 mH) | 5.93 | 3.82 |
These experimental results confirm that the proposed parameter identification method effectively restores the MPC performance for LC-type grid-connected inverters, which represent one of the common types of solar inverters used in distributed generation systems.
5. Conclusion
In this work, I have presented a robust model predictive control strategy for LC-filtered grid-connected inverters that integrates online identification of both filter inductance and capacitance. The method relies on two dedicated sliding mode observers and model‑reference adaptive PI controllers. The theoretical analysis and experimental validations confirm that parameter mismatches cause significant voltage prediction errors and current distortion, but the proposed identification scheme can accurately track the true parameters within 100 ms after a step change. Consequently, the THD of grid currents is reduced from above 5.5% to around 3.8%, ensuring high power quality. The approach is independent of the specific operating point and can be applied to various types of solar inverters with LC output filters, thereby enhancing the reliability and efficiency of renewable energy systems.