In modern power electronic systems, the grid-tied inverter plays a critical role in integrating renewable energy sources, such as solar and wind, into the electrical grid. The control precision of a grid-tied inverter directly impacts power conversion efficiency, grid power quality, and overall system stability. Among various control strategies, model predictive control (MPC) has gained prominence due to its fast dynamic response, simplicity, and ability to handle multiple constraints. However, as a model-based approach, MPC is sensitive to parameter mismatches, particularly in the filter inductance. In practical applications, the inductance of a grid-tied inverter can vary due to temperature changes, aging, or manufacturing tolerances, leading to degraded performance, increased current harmonics, and even system instability. Traditional inductance identification methods often rely on accurate grid frequency information, making them vulnerable to frequency deviations. Moreover, some methods fail when the active power is zero. This motivates me to develop a robust online inductance identification technique that enhances the frequency robustness of MPC for grid-tied inverters.
In this article, I propose a novel model reference adaptive system (MRAS)-based inductance identification method that leverages a second-order sliding mode observer (SMO) and a low-pass filter (LPF). The key innovation lies in avoiding direct compensation for amplitude and phase deviations caused by the LPF, thereby eliminating the need for grid frequency in the identification process. This makes the method inherently robust to grid frequency offsets. Additionally, the proposed approach works even when the active power is zero, overcoming a significant limitation of existing techniques. I validate the method through experimental studies, demonstrating its effectiveness in improving the control accuracy of a grid-tied inverter under parameter variations and frequency disturbances.
The topology of a two-level grid-tied inverter is widely used in renewable energy systems. It consists of a DC source, an inverter bridge, and an L filter connected to the grid. The mathematical model in the stationary αβ reference frame is given by:
$$ L \frac{d\mathbf{i}}{dt} = \mathbf{u} – R\mathbf{i} – \mathbf{e} $$
where \(\mathbf{i}\) is the grid current vector, \(\mathbf{u}\) is the inverter output voltage vector, \(\mathbf{e}\) is the grid voltage vector, \(L\) is the filter inductance, and \(R\) is the parasitic resistance. For a grid-tied inverter, precise control of current is essential to meet grid codes and ensure efficient power transfer.
Traditional model predictive control for a grid-tied inverter discretizes the model using the forward Euler method. The discrete-time equation is:
$$ \mathbf{i}(k+1) = \left(1 – \frac{T_s R}{L}\right) \mathbf{i}(k) + \frac{T_s}{L} \left( \mathbf{u}(k) – \mathbf{e}(k) \right) $$
where \(T_s\) is the sampling period. To compensate for computational delays, a two-step prediction is often employed. The predicted current at step \(k+2\) is used in a cost function to select the optimal voltage vector from eight possible switching states. The cost function minimizes the error between the reference current and the predicted current. However, if the inductance parameter \(L\) in the model deviates from the actual value, the prediction accuracy deteriorates, leading to increased current total harmonic distortion (THD) and poor performance. This highlights the need for accurate online inductance identification in MPC for grid-tied inverters.
Existing inductance identification methods, such as those based on full-order sliding mode observers, have limitations. A full-order SMO can estimate grid voltage and inductance simultaneously, but it requires knowledge of the grid frequency. The observer equations are:
$$ \frac{d\hat{\mathbf{i}}}{dt} = -\frac{R}{L}\hat{\mathbf{i}} + \frac{1}{L} (\mathbf{u} – \hat{\mathbf{e}}) – m \mathbf{S} $$
$$ \frac{d\hat{\mathbf{e}}}{dt} = \hat{\omega} \mathbf{J} \hat{\mathbf{e}} + n \mathbf{S} $$
where \(\hat{\cdot}\) denotes estimated values, \(\mathbf{S} = \text{sgn}(\mathbf{i} – \hat{\mathbf{i}})\), \(m\) and \(n\) are sliding gains, and \(\mathbf{J}\) is a skew-symmetric matrix for rotation. The inductance identification relies on the relationship:
$$ \hat{\omega} (\mathbf{e} \otimes \hat{\mathbf{e}}) = -\Delta L \cdot (\mathbf{i} \otimes \mathbf{e}) $$
where \(\Delta L = L – \hat{L}\). This method is sensitive to grid frequency errors, and when active power is zero (i.e., \(\mathbf{i} \cdot \mathbf{e} = 0\)), the identification fails because \(\mathbf{e} \otimes \hat{\mathbf{e}}\) becomes zero. To address these issues, I develop a new approach that eliminates the dependency on grid frequency and works under all power conditions.
My proposed method starts with a second-order sliding mode observer to estimate the grid voltage without requiring frequency information. The observer is designed as:
$$ \frac{d\hat{\mathbf{i}}}{dt} = -\frac{R}{L}\hat{\mathbf{i}} + \frac{1}{L} \mathbf{u} – \frac{K}{L} \text{sgn}(\mathbf{i} – \hat{\mathbf{i}}) $$
where \(K\) is a sliding gain chosen to satisfy \(K > |\mathbf{e}|_{\text{max}}\). Upon convergence, the grid voltage estimate is:
$$ \hat{\mathbf{e}} = K \text{sgn}(\mathbf{i} – \hat{\mathbf{i}}) $$
This estimate contains high-frequency chattering, so I apply a low-pass filter (LPF) to obtain a smoothed signal:
$$ \hat{\mathbf{e}}_L = \frac{\omega_c}{s + \omega_c} \hat{\mathbf{e}} $$
where \(\omega_c\) is the cutoff frequency. The LPF introduces amplitude attenuation and phase lag, which are functions of the grid frequency \(\omega\). Instead of compensating these deviations using \(\omega\)—which may be inaccurate due to frequency offsets—I pass the actual grid voltage \(\mathbf{e}\) through the same LPF:
$$ \mathbf{e}_L = \frac{\omega_c}{s + \omega_c} \mathbf{e} $$
This ensures that both \(\hat{\mathbf{e}}_L\) and \(\mathbf{e}_L\) experience identical distortions, making their comparison frequency-independent. The voltage observation error is defined as \(\mathbf{e}_L = \mathbf{e}_L – \hat{\mathbf{e}}_L\). When the observer uses an inductance parameter \(\hat{L}\) different from the true \(L\), the error dynamics can be derived. Assuming convergence of the sliding mode, the relationship between the error and inductance mismatch is:
$$ \mathbf{e}_L = -\Delta L \cdot \frac{\omega_c}{s + \omega_c} (j\omega \mathbf{i}) $$
However, to avoid explicit frequency dependence, I manipulate the equations in the time domain. A key step is to use the cross product \(\mathbf{i} \otimes \mathbf{e}_L\) in the identification model. By analyzing the stability with a Lyapunov function, I derive an adaptive law for inductance estimation.
I define a Lyapunov function \(V\) as:
$$ V = \frac{1}{2} k_1 (\mathbf{i} \otimes \mathbf{e}_L)^2 + \frac{1}{2} (\Delta L)^2 $$
where \(k_1 > 0\) is a constant. Taking the derivative and ensuring \(\dot{V} \leq 0\) for stability, I obtain:
$$ \dot{V} = k_1 (\mathbf{i} \otimes \mathbf{e}_L) \frac{d(\mathbf{i} \otimes \mathbf{e}_L)}{dt} + \Delta L \frac{d(\Delta L)}{dt} = -k_2 (\mathbf{i} \otimes \mathbf{e}_L)^2 $$
with \(k_2 > 0\). Solving for \(\frac{d(\Delta L)}{dt}\) leads to an integral form. The final inductance identification is implemented via a PI controller:
$$ \hat{L} = \left( k_p + \frac{k_i}{s} \right) (\mathbf{i} \otimes \mathbf{e}_L) + L_0 $$
where \(k_p\) and \(k_i\) are proportional and integral gains, and \(L_0\) is the initial inductance value. The gains are chosen based on system parameters to ensure convergence and robustness. Notably, this law does not involve the grid frequency \(\omega\), making it inherently robust to frequency deviations. Moreover, since \(\mathbf{i} \otimes \mathbf{e}_L\) is not zero even when active power is zero, the method works under all operating conditions.
To integrate the identification into MPC for a grid-tied inverter, I update the inductance parameter in the prediction model at each sampling instant. The overall control structure includes the MPC algorithm, the second-order SMO, the LPF, and the adaptive identifier. This enhances the parameter robustness of the grid-tied inverter control system.
I conducted experimental tests to validate the proposed method. The setup uses a DSP28335 controller, a three-phase programmable AC source, and a two-level grid-tied inverter with an L filter. The parameters are summarized in Table 1.
| Parameter | Value |
|---|---|
| Grid line voltage peak (e) | 100 V |
| DC-link voltage (U_dc) | 250 V |
| Actual inductance (L) | 18.5 mH |
| Parasitic resistance (R) | 0.05 Ω |
| Sampling frequency (f_s) | 20 kHz |
| LPF cutoff frequency (ω_c) | 100π rad/s |
| Proportional gain (k_p) | 0.00001 |
| Integral gain (k_i) | 0.008 |
The experiments evaluate the identification accuracy under various conditions. First, I set the initial inductance parameter to 0.01 H (much lower than the actual 0.0185 H) and applied the proposed method. The estimated inductance \(\hat{L}\) converged to the true value within a few cycles, and the filtered voltage estimates \(\hat{\mathbf{e}}_L\) tracked \(\mathbf{e}_L\) closely. Similarly, with an initial value of 0.03 H, convergence was achieved rapidly. The cross product \(\mathbf{i} \otimes \mathbf{e}_L\) approached zero as the identification progressed, indicating accurate estimation.
I also tested the method under reference current step changes. When the reference current increased from 4 A to 6 A, the inductance estimate remained stable, demonstrating robustness to transients. To quantify performance improvement, I measured the THD of the grid current before and after enabling the identification. Table 2 shows the results for different initial inductance errors.
| Initial Inductance | THD without Identification | THD with Identification |
|---|---|---|
| 0.01 H | 8.5% | 3.2% |
| 0.03 H | 7.9% | 3.1% |
The THD reduction confirms that the proposed method enhances the control performance of the grid-tied inverter by correcting parameter mismatches in MPC.
To assess frequency robustness, I compared the proposed method with a traditional full-order SMO-based approach under grid frequency deviations. The grid frequency was varied from 48 Hz to 52 Hz. While the traditional method showed significant estimation errors due to frequency dependency, my method maintained accurate inductance identification without any parameter adjustment. The error \(\Delta L\) stayed near zero regardless of frequency shifts. This highlights the superiority of the frequency-robust design.
Furthermore, I evaluated the method at zero active power. By setting the reference active power to zero, the traditional method failed because \(\mathbf{e} \otimes \hat{\mathbf{e}}\) became zero, but my method continued to provide accurate inductance estimates. This is crucial for grid-tied inverters operating under reactive power compensation or low-power conditions.
The stability and convergence of the identification algorithm are ensured by the Lyapunov analysis. The PI gains are tuned to balance convergence speed and noise immunity. In practice, the gains can be adapted based on operating points, but for simplicity, fixed values sufficed in my experiments. The second-order SMO gain \(K\) is chosen to be larger than the maximum grid voltage amplitude, which is typically known or can be estimated.
In summary, this article presents a robust online inductance identification method for grid-tied inverter model predictive control. The key contributions are:
- Elimination of grid frequency dependency, making the method robust to frequency offsets.
- Ability to work under zero active power conditions, overcoming a common limitation.
- Integration with a second-order sliding mode observer and low-pass filtering, avoiding complex compensations.
- Experimental validation showing improved current THD and parameter accuracy.
The proposed approach enhances the reliability and performance of grid-tied inverters in real-world applications where grid conditions and parameters vary. Future work could extend the method to LCL-filtered grid-tied inverters or incorporate resistance identification for further improvement. Additionally, adaptive tuning of the observer and identifier gains could optimize performance across a wider operating range. Overall, this research advances the state-of-the-art in parameter-robust control for power electronic systems.