In modern renewable energy systems, on-grid inverters play a pivotal role in interfacing distributed generation sources, such as solar and wind power, with the main electrical grid. As the penetration of these sources increases, the demand for high-performance control strategies that ensure stability, efficiency, and reliability becomes paramount. Among various advanced control techniques, model predictive control (MPC) has gained significant attention due to its fast dynamic response, simplicity in handling constraints, and ability to optimize multiple objectives. However, traditional MPC implementations often rely on voltage and current sensors to measure grid parameters, which can increase hardware costs and reduce system robustness due to potential sensor failures. This has spurred research into sensorless control methods, particularly for grid voltage estimation, to enhance the reliability of on-grid inverters.
In this context, we address a critical challenge in sensorless MPC for on-grid inverters: the susceptibility to grid frequency deviations and DC-offsets in voltage measurements. These disturbances are common in real-world grid conditions due to imbalances, harmonics, or measurement errors, and they can severely degrade the accuracy of grid voltage estimation, leading to increased current control errors and compromised power quality. Traditional approaches, such as those based on sliding mode observers with low-pass filters, have shown limitations in mitigating these issues, especially when DC-offsets are present, as they can amplify such offsets and cause significant phase and amplitude errors. Therefore, we propose an improved grid voltage observation method that combines a sliding mode observer with dual quasi-resonant filters, enabling robust sensorless MPC for on-grid inverters even under non-ideal grid conditions.
Our work begins by analyzing the conventional sensorless MPC strategy for on-grid inverters. The discrete-time model of an on-grid inverter in the αβ reference frame is given by:
$$ \begin{bmatrix} i_\alpha(k+1) \\ i_\beta(k+1) \end{bmatrix} = \left(1 – \frac{RT}{L}\right) \begin{bmatrix} i_\alpha(k) \\ i_\beta(k) \end{bmatrix} + \frac{T}{L} \begin{bmatrix} u_\alpha(k) – e_\alpha(k) \\ u_\beta(k) – e_\beta(k) \end{bmatrix} $$
where \(i_\alpha\) and \(i_\beta\) are the grid currents, \(u_\alpha\) and \(u_\beta\) are the inverter output voltages, \(e_\alpha\) and \(e_\beta\) are the grid voltages, \(R\) and \(L\) are the filter resistance and inductance, and \(T\) is the sampling period. For MPC, the future current at time \(k+2\) is predicted to compensate for control delays, and a cost function minimizes the error between reference and predicted currents. However, this requires accurate knowledge of grid voltages, which are typically measured using sensors. In sensorless approaches, these voltages must be estimated, and any inaccuracies can propagate through the control loop, affecting the performance of the on-grid inverter.
To estimate grid voltages without sensors, sliding mode observers have been widely employed due to their robustness to parameter variations and disturbances. A conventional sliding mode observer for an on-grid inverter can be designed as:
$$ \dot{\hat{i}}_\alpha = -\frac{R}{L} \hat{i}_\alpha + \frac{1}{L} (u_\alpha – \hat{e}_\alpha) + K \text{sgn}(\tilde{i}_\alpha) $$
$$ \dot{\hat{i}}_\beta = -\frac{R}{L} \hat{i}_\beta + \frac{1}{L} (u_\beta – \hat{e}_\beta) + K \text{sgn}(\tilde{i}_\beta) $$
where \(\hat{i}_\alpha\) and \(\hat{i}_\beta\) are the estimated currents, \(\tilde{i}_\alpha = \hat{i}_\alpha – i_\alpha\) and \(\tilde{i}_\beta = \hat{i}_\beta – i_\beta\) are the current errors, \(K\) is the sliding gain, and \(\text{sgn}(\cdot)\) is the sign function. The estimated grid voltages \(\hat{e}_\alpha\) and \(\hat{e}_\beta\) are derived from the sliding mode terms, but they contain high-frequency chattering noise and are sensitive to DC-offsets. To mitigate noise, low-pass filters (LPFs) are often applied, but they introduce phase delays and do not suppress DC-offsets. Moreover, in scenarios with grid frequency deviations, LPFs can cause additional errors, as their frequency response is fixed. We analyze this by considering a DC-offset \(u_{d\alpha}\) injected into the voltage measurements, which leads to estimated grid voltages with distorted components. For instance, if the actual grid voltage is \(e_\alpha = E \cos(\theta_g)\), the observed voltage after an LPF becomes:
$$ e_{\alpha1} = E \cdot \Delta E \cos(\theta_g – \Delta\theta) + e_{\alpha d} $$
where \(\Delta E\) and \(\Delta\theta\) are the attenuation and phase shift due to the LPF, and \(e_{\alpha d}\) is the DC-offset. This demonstrates that DC-offsets persist and can affect subsequent compensation steps, ultimately degrading the accuracy of sensorless control for on-grid inverters.
To overcome these limitations, we propose replacing LPFs with quasi-resonant filters in the grid voltage observation scheme. A quasi-resonant filter has a transfer function:
$$ G_R(s) = \frac{2\omega_{c1} s}{s^2 + 2\omega_{c1} s + \omega_e^2} $$
where \(\omega_{c1}\) is the cutoff frequency and \(\omega_e\) is the resonant frequency. Unlike an LPF, which has a gain of 1 at DC, the quasi-resonant filter has zero gain at DC, effectively eliminating DC-offsets. Additionally, at the resonant frequency \(\omega_e\), the filter has unity gain and zero phase shift, making it ideal for extracting fundamental grid voltage components without distortion. However, if the grid frequency deviates from \(\omega_e\), the filter introduces amplitude and phase errors. To address this, we employ a dual quasi-resonant filter structure in series, coupled with an adaptive compensation mechanism. The first quasi-resonant filter removes high-frequency chattering and DC-offsets from the sliding mode observer output, while the second filter, along with compensation, adjusts for frequency deviations. The compensation parameters are derived from the amplitudes and phases of the filtered signals, allowing real-time correction without prior knowledge of the exact grid frequency.
The proposed grid voltage observation method is integrated into an MPC framework for on-grid inverters. The block diagram of our approach is summarized as follows: the sliding mode observer provides initial voltage estimates, which are processed by the dual quasi-resonant filters to produce clean and accurate grid voltage signals \(\hat{e}_{\alpha6}\) and \(\hat{e}_{\beta6}\), along with a phase estimate \(\theta_5\). These are then fed into the MPC predictor, which computes future currents and selects the optimal voltage vector to minimize the cost function. This enables robust sensorless operation of on-grid inverters, even in the presence of DC-offsets and frequency variations. To illustrate the system topology, consider a typical on-grid inverter setup with a DC source, filter inductors, and grid connection, as shown below:
This configuration highlights the importance of reliable control in practical applications, such as solar energy systems with battery storage, where on-grid inverters must maintain synchronization and power quality under varying grid conditions.
We validate our method through extensive experimental comparisons with conventional techniques. The performance is evaluated under different scenarios, including DC-offset injection, grid frequency deviations, and harmonic distortions. For instance, when a DC-offset of -1 V is introduced, traditional methods exhibit significant errors in grid voltage estimation, leading to distorted currents with high total harmonic distortion (THD). In contrast, our quasi-resonant filter-based approach suppresses the DC-offset effectively, preserving current waveform quality. Similarly, under a grid frequency shift of 5 Hz (e.g., 45 Hz instead of 50 Hz), single quasi-resonant filters fail due to frequency misalignment, while our dual-filter method maintains accurate voltage estimation and current control. The results are summarized in the following tables, which compare key metrics such as voltage error, current THD, and power factor.
| Method | DC-Offset Condition | Grid Voltage Error (V) | Current THD (%) | Power Factor |
|---|---|---|---|---|
| Conventional (LPF-based) | With -1 V offset | ~10 | 11.48 | 0.95 |
| Single Quasi-Resonant Filter | With -1 V offset | < 1 | 3.2 | 0.99 |
| Dual Quasi-Resonant Filter (Proposed) | With -1 V offset | < 0.5 | 2.8 | 1.00 |
| Method | Frequency Deviation | Phase Error (degrees) | Amplitude Error (%) | Current Control Accuracy |
|---|---|---|---|---|
| Conventional (LPF-based) | 45 Hz grid | ~5 | ~8 | High |
| Single Quasi-Resonant Filter | 45 Hz grid | ~15 | ~12 | Low |
| Dual Quasi-Resonant Filter (Proposed) | 45 Hz grid | < 2 | < 3 | High |
These tables demonstrate the superiority of our proposed method in enhancing the robustness of sensorless MPC for on-grid inverters. The dual quasi-resonant filter structure not only mitigates DC-offsets but also adapts to frequency changes, ensuring precise grid voltage estimation. This is crucial for maintaining stable operation of on-grid inverters in real-world environments where grid conditions are often non-ideal.
Further analysis involves the mathematical formulation of the compensation mechanism. Let \(e_{\alpha1}\) and \(e_{\beta1}\) be the outputs after the first quasi-resonant filter, and \(e_{\alpha2}\) and \(e_{\beta2}\) after the second filter. Their amplitudes \(E_1, E_2\) and phases \(\theta_1, \theta_2\) are computed as:
$$ E_1 = \sqrt{e_{\alpha1}^2 + e_{\beta1}^2}, \quad \theta_1 = \arctan\left(\frac{e_{\beta1}}{e_{\alpha1}}\right) $$
$$ E_2 = \sqrt{e_{\alpha2}^2 + e_{\beta2}^2}, \quad \theta_2 = \arctan\left(\frac{e_{\beta2}}{e_{\alpha2}}\right) $$
The compensation factors for amplitude and phase are then derived as:
$$ \Delta E = \frac{E_2}{E_1}, \quad \Delta\theta = \theta_1 – \theta_2 $$
These factors are used to adjust the final estimated grid voltages:
$$ \hat{e}_{\alpha6} = \frac{E_2}{(\Delta E)^2} \cos(\theta_2 + 2\Delta\theta), \quad \hat{e}_{\beta6} = \frac{E_2}{(\Delta E)^2} \sin(\theta_2 + 2\Delta\theta) $$
This adaptive compensation ensures that even with grid frequency deviations, the estimated voltages align closely with the actual grid voltages, enabling accurate current control in the on-grid inverter. The effectiveness of this approach is further verified under harmonic grid conditions, where additional voltage harmonics (e.g., 7th harmonic) are present. Our method maintains low estimation errors, whereas traditional methods suffer from increased distortions due to their limited filtering capabilities.
In terms of implementation, the proposed sensorless MPC strategy for on-grid inverters is computationally efficient, as it relies on simple filter structures and straightforward arithmetic operations. The sliding mode observer provides robustness against model uncertainties, while the quasi-resonant filters offer selective frequency response without complex tuning. This makes it suitable for real-time applications in digital signal processors (DSPs) commonly used in on-grid inverter controls. We have tested our method on a experimental setup with a three-phase on-grid inverter connected to a programmable grid simulator, using a DSP28335 controller. The results confirm that the current waveforms remain sinusoidal and in phase with the grid voltage, even under adverse conditions, highlighting the practical viability of our approach.
Looking ahead, the proposed method can be extended to other sensorless control schemes for on-grid inverters, such as deadbeat predictive control or virtual flux-based strategies. The principles of using quasi-resonant filters for DC-offset suppression and frequency adaptation are generic and can be applied to various estimation problems in power electronics. Additionally, future work could explore integration with grid support functions, like reactive power compensation or harmonic mitigation, to further enhance the role of on-grid inverters in smart grid applications.
In conclusion, we have developed a novel grid voltage sensorless model predictive control strategy for on-grid inverters that is resistant to DC-offsets and grid frequency deviations. By combining a sliding mode observer with dual quasi-resonant filters, we achieve accurate voltage estimation without sensors, leading to improved current control performance and higher system reliability. Experimental comparisons validate the effectiveness of our method over conventional techniques, demonstrating its potential for widespread adoption in renewable energy systems. As the demand for robust and efficient on-grid inverters grows, such advanced control solutions will play a key role in ensuring grid stability and power quality in the face of evolving grid challenges.