In modern power systems, the proliferation of distributed generation resources and non-linear loads has led to the formation of weak grids, characterized by significant grid impedance and voltage distortions at the point of common coupling (PCC). For on-grid inverters, these conditions pose severe challenges to power quality and system stability, primarily manifesting as amplified current harmonics and reduced robustness. Traditional voltage feedforward control methods, while effective in ideal grids, often exacerbate harmonic issues in weak grids due to the interactive effects of grid impedance and control delays. This article, from my perspective as a researcher in power electronics, delves into an optimized harmonic suppression strategy for on-grid inverters that adaptively manages PCC voltage feedforward while compensating for control delays. The core innovation lies in an adaptive control algorithm that dynamically adjusts feedforward signals based on harmonic detection without requiring explicit grid impedance estimation, coupled with a repetitive prediction scheme to mitigate delay-induced instability. Through comprehensive analysis and simulation, I demonstrate that this integrated approach significantly enhances the harmonic performance and stability of on-grid inverters under varying weak grid conditions.
The fundamental structure of a grid-connected LCL-filtered inverter is pivotal to understanding the harmonic dynamics. As shown below, the system comprises an inverter-side inductor \(L_1\), a filter capacitor \(C_f\), and a grid-side inductor \(L_2\), interfacing with the grid through the PCC. The grid itself is modeled as an ideal voltage source \(u_s\) in series with a grid impedance \(Z_g\), which becomes non-negligible in weak grids.
The grid impedance \(Z_g\) can be expressed as:
$$Z_g = R_g + L_g s = R_g + \beta (L_1 + L_2) s$$
where \(R_g\) is the equivalent grid resistance, \(L_g\) is the equivalent grid inductance, \(\beta\) is a proportionality coefficient indicating the degree of grid weakness (e.g., \(\beta = 0\) for an ideal grid, \(\beta > 0\) for weak grids), and \(s\) is the Laplace operator. The PCC voltage \(u_g\) relates to the ideal grid voltage \(u_s\) and grid current \(i_g\) as:
$$u_g = u_s + Z_g i_g$$
This equation highlights that distortion in \(u_s\) or variations in \(Z_g\) directly inject harmonics into \(u_g\), which then perturb the current control loop of the on-grid inverter.
Voltage feedforward control is a common technique to suppress grid-induced current harmonics. In a typical dual-loop current control structure with proportional PCC voltage feedforward, the feedforward path aims to cancel the disturbance from \(u_g\). However, in weak grids, this feedforward introduces an additional feedback loop through \(Z_g\), potentially degrading stability. The open-loop transfer function of the system with feedforward can be derived, revealing that the phase margin decreases as \(Z_g\) increases, leading to harmonic amplification. To mitigate this, a filtered feedforward approach using a second-order generalized integrator (SOGI) has been proposed. The SOGI filter \(G_h(s)\) allows only the fundamental component to pass:
$$G_h(s) = \frac{u_{g\_f}}{u_g} = \frac{k_0 \omega_0 s}{s^2 + k_0 \omega_0 s + \omega_0^2}$$
where \(\omega_0\) is the fundamental angular frequency, and \(k_0\) is a damping factor. While this enhances stability by attenuating harmonic feedforward, it compromises the suppression of low-order current harmonics, which are prevalent in weak grids. Thus, a more sophisticated strategy is needed for on-grid inverters to balance stability and harmonic rejection.
I propose an adaptive voltage feedforward control algorithm that automatically adjusts the feedforward signal based on real-time harmonic detection. The core idea is to monitor the PCC voltage harmonics using band-pass filters (BPFs) and selectively include or exclude harmonic components from the feedforward path to maintain stability and harmonic suppression. The adaptive control structure integrates multiple BPFs tuned to dominant harmonic frequencies (e.g., 3rd, 5th, 7th). Each BPF \(G_{h\_a}(s)\) for the \(a\)-th harmonic is defined as:
$$G_{h\_a}(s) = \frac{K s}{s^2 + \xi s + (a \omega_0)^2}$$
where \(K\) is the gain, \(\xi\) is the damping coefficient, and \(a\) is the harmonic order. The output of each BPF, denoted \(u_{g\_a}\), represents the extracted harmonic component. An adaptive module, incorporating a detection mechanism (e.g., RMS or FFT-based magnitude tracking), evaluates the trend of \(u_{g\_a}\). If \(u_{g\_a}\) shows a significant increase—indicating that the harmonic is being amplified due to grid impedance changes—the adaptive rule removes that component from the composite feedforward signal \(u_{g\_f}\). Conversely, if the harmonic level is low, the component is retained to aid in harmonic cancellation. This logic ensures that the on-grid inverter adapts to varying grid conditions without requiring explicit estimation of \(Z_g\).
The stability of the adaptive control system can be analyzed using Bode plots. For instance, when the grid impedance suddenly increases (e.g., \(\beta\) jumps from 1 to 3), the phase margin at higher harmonics (like the 7th) may drop critically. With the adaptive algorithm, if \(u_{g\_7}\) rises sharply, it is excluded from \(u_{g\_f}\), thereby restoring phase margin and preventing instability. The table below summarizes the effect of adaptive control on system stability under different \(\beta\) values, illustrating how harmonic components are managed.
| Grid Condition (\(\beta\)) | Included Harmonics in \(u_{g\_f}\) | Phase Margin at 350 Hz | Stability Outcome |
|---|---|---|---|
| 0 (Ideal) | 1st, 3rd, 5th, 7th | > 45° | Stable |
| 1 (Moderate Weak Grid) | 1st, 3rd, 5th, 7th | ~30° | Stable |
| 2 (Weak Grid) | 1st, 3rd, 5th | ~25° | Stable with Adaptive Action |
| 3 (Very Weak Grid) | 1st, 3rd, 5th | ~20° | Stable after 7th Harmonic Removal |
Control delay, arising from computational lag, PWM updating, and sensor dynamics, further complicates the performance of on-grid inverters. In weak grids, delay can cause phase shifts that destabilize the system, particularly when combined with voltage feedforward. To address this, I incorporate a repetitive prediction (RP) scheme to compensate for the delay. The RP predictor estimates the future grid voltage based on past cycles, leveraging the periodic nature of grid signals. The transfer function of the RP predictor is:
$$G_{RP}(z) = \frac{1 – Q z^{-N} + m z^{-N+k}}{1 – Q z^{-N} + m z^{-N}}$$
where \(Q\) is a low-pass filter coefficient (typically \(Q < 1\)), \(N\) is the number of sampling periods per grid cycle, \(k\) is the prediction step to account for delay, and \(m\) is a gain factor (e.g., \(m = 0.96\)). This predictor provides nearly unity gain and minimal phase error at fundamental and harmonic frequencies, ensuring accurate feedforward without amplifying background harmonics. Compared to traditional numerical interpolation prediction (NIP), RP exhibits superior robustness in distorted grids. The effectiveness of RP can be quantified by analyzing its frequency response. For a system with switching frequency \(f_s = 18 \text{kHz}\) and \(N=360\), the gain at harmonics remains close to 0 dB, and phase error is below 2°, which is critical for maintaining stability in on-grid inverters.
The combined strategy—adaptive feedforward with RP delay compensation—enhances the harmonic suppression capability of on-grid inverters. To validate this, I conducted simulation experiments in MATLAB/Simulink for a single-phase on-grid inverter with parameters: \(L_1 = 2 \text{mH}\), \(L_2 = 1 \text{mH}\), \(C_f = 10 \mu\text{F}\), DC link voltage \(u_{dc} = 400 \text{V}\), grid voltage \(u_s = 220 \text{V RMS}\) at 50 Hz, with added voltage harmonics (5% 3rd, 3% 5th, 2% 7th, 1% 11th) to emulate a weak grid. The control parameters were set as: current controller \(G_c(s)\) as a proportional-resonant type, \(k_0=1.414\) for SOGI, \(K=1\) and \(\xi=0.1\) for BPFs, and RP with \(Q=0.98\), \(k=2\). The performance was evaluated in terms of total harmonic distortion (THD) of grid current \(i_g\) and PCC voltage \(u_g\) under varying \(\beta\).
The simulation results demonstrate the superiority of the proposed strategy. Below is a table comparing THD values for different control methods: traditional proportional feedforward, SOGI-based feedforward, and the proposed adaptive RP-based feedforward. The on-grid inverter operates at full power (1 kW) with the same grid distortion profile.
| Control Strategy | Grid Impedance (\(\beta\)) | \(i_g\) THD (%) | \(u_g\) THD (%) | Stability (Oscillation Observed) |
|---|---|---|---|---|
| Traditional Proportional Feedforward | 0 | 5.41 | 6.65 | No |
| Traditional Proportional Feedforward | 2 | 8.50 | 9.20 | Yes (Severe) |
| SOGI-Based Feedforward | 0 | 5.41 | 6.65 | No |
| SOGI-Based Feedforward | 2 | 6.50 | 8.20 | Moderate |
| Proposed Adaptive RP-Based Feedforward | 0 | 2.66 | 6.65 | No |
| Proposed Adaptive RP-Based Feedforward | 2 | 3.08 | 7.70 | No |
| Proposed Adaptive RP-Based Feedforward | 3 | 5.08 | 7.89 | No (Stable after Adaptation) |
The data shows that the proposed strategy reduces \(i_g\) THD by over 50% compared to traditional methods in ideal grids (\(\beta=0\)), and maintains low THD even in very weak grids (\(\beta=3\)), whereas other methods exhibit instability or higher harmonics. The adaptive mechanism successfully removed the 7th harmonic component at \(\beta=3\), preventing resonance. Furthermore, the RP compensation ensured that control delay did not degrade performance; in contrast, NIP prediction amplified harmonics by up to 15% in distorted grids. The mathematical basis for this improvement can be summarized through the enhanced output impedance of the on-grid inverter. With adaptive feedforward and RP, the output impedance \(Z_o(s)\) at harmonic frequencies is increased, which improves disturbance rejection. The modified impedance can be approximated as:
$$Z_o(s) \approx \frac{1 + G_c(s) G_{PWM}(s) H(s)}{G_{PWM}(s)} + Z_g \cdot \frac{G_{RP}(s) \cdot A(s)}{1 + G_c(s) G_{PWM}(s) H(s)}$$
where \(G_{PWM}(s)\) is the PWM gain, \(H(s)\) is the feedback filter, and \(A(s)\) represents the adaptive feedforward transfer function. This formulation highlights how the strategy reshapes impedance to suppress harmonics.
In-depth analysis of the adaptive algorithm reveals its decision-making process. The harmonic detection module uses a sliding window FFT to compute the magnitude of \(u_{g\_a}\) every cycle. If the magnitude exceeds a threshold (e.g., 1.5 times the nominal value for that harmonic), the component is excluded. The threshold is set based on pre-defined stability margins, ensuring a balance between harmonic suppression and robustness. For instance, when \(\beta\) increases from 1 to 2, the 7th harmonic magnitude in \(u_g\) might rise from 0.5% to 2.5%, triggering the adaptive rule. This process is autonomous and requires no external grid impedance sensors, making it practical for real-world on-grid inverter deployments. The computational burden is minimal, as BPFs can be implemented efficiently in digital signal processors.
The RP predictor’s performance was further verified under dynamic grid conditions. In a test where \(u_s\) suddenly introduced a 5th harmonic step change from 3% to 7%, the RP predictor adjusted within two cycles, maintaining feedforward accuracy, while NIP caused a transient overshoot of 10% in \(i_g\). The prediction error \(e_{RP}\) can be expressed as:
$$e_{RP}(n) = u_g(n) – \hat{u}_g(n)$$
where \(\hat{u}_g(n)\) is the predicted voltage. With RP, \(e_{RP}\) converges to near zero within a few cycles, thanks to the internal model principle. This robustness is crucial for on-grid inverters in weak grids, where voltage distortions are common.
To further illustrate the benefits, consider the harmonic spectrum of \(i_g\) under different controls. With the proposed strategy, the amplitudes of low-order harmonics (3rd, 5th, 7th) are reduced by 60-70% compared to SOGI feedforward. The reduction is quantified by the harmonic attenuation factor \(H_{att}(a)\) for the \(a\)-th harmonic:
$$H_{att}(a) = 20 \log_{10} \left( \frac{I_{g, \text{prop}}(a)}{I_{g, \text{SOGI}}(a)} \right) \text{ dB}$$
where \(I_{g}(a)\) is the RMS current at harmonic \(a\). For \(a=5\), \(H_{att}(5)\) averages -12 dB, indicating significant suppression. This translates to better compliance with grid codes (e.g., IEEE 1547) for on-grid inverters.
Another aspect is the transient response during grid impedance variations. The adaptive algorithm responds within 10-20 ms, ensuring continuity of operation. In contrast, fixed feedforward methods can lead to sustained oscillations, as seen in simulation waveforms where \(i_g\) and \(u_g\) exhibit ringing at frequencies near the LCL resonance. The proposed strategy dampens these oscillations by dynamically adjusting the feedforward path, akin to an impedance-matching mechanism. This is particularly important for on-grid inverters in renewable energy systems, where grid conditions fluctuate rapidly.
The integration of adaptive control and RP prediction also enhances the stability margins of the on-grid inverter. The gain margin (GM) and phase margin (PM) were calculated from the open-loop transfer function. For \(\beta=2\), the proposed strategy achieves PM > 30° and GM > 6 dB across all harmonics, whereas traditional feedforward yields PM < 15° at higher harmonics. These margins ensure reliability even under parameter uncertainties, such as variations in \(L_1\) or \(C_f\) due to aging or temperature.
In summary, the optimized harmonic suppression strategy for on-grid inverters presented here addresses two critical issues: the adverse effects of PCC voltage feedforward in weak grids and the destabilizing impact of control delay. The adaptive feedforward algorithm, powered by BPF-based harmonic detection, enables intelligent feedforward tuning without grid impedance estimation, thereby maintaining stability while suppressing low-order harmonics. The RP predictor compensates for control delay with high accuracy, preventing harmonic amplification. Simulation results confirm that this combined approach reduces grid current THD by over 50% in typical weak grid scenarios, outperforming conventional methods. Future work could extend this strategy to three-phase on-grid inverters, incorporate machine learning for adaptive threshold tuning, and validate it in hardware prototypes. For now, this strategy offers a robust solution for improving the power quality and resilience of on-grid inverters in evolving power networks.
From my perspective, the key takeaway is that harmonic suppression in on-grid inverters requires a holistic view of grid interactions, control dynamics, and real-time adaptation. By leveraging adaptive algorithms and predictive techniques, we can design on-grid inverters that not only meet regulatory standards but also contribute to grid stability. The repeated emphasis on on-grid inverters in this discussion underscores their pivotal role in modern energy systems; as penetration increases, such advanced control strategies will become indispensable for ensuring efficient and clean power integration.