In the modern era of renewable energy integration, the grid-connected inverter serves as a critical interface between distributed generation sources, such as photovoltaic (PV) arrays, and the utility grid. Ensuring high-quality power injection with low total harmonic distortion (THD) and robust performance under grid disturbances is paramount for the stability and efficiency of power systems. Among various topologies, the LCL-type grid-connected inverter is widely adopted due to its superior harmonic attenuation capabilities. However, conventional control strategies, such as repetitive control (RC), often exhibit degraded performance when the grid frequency deviates from its nominal value, leading to increased harmonic content and reduced tracking accuracy. This work addresses these challenges by proposing a frequency adaptive repetitive control (FARC) system combined with deadbeat (DB) control for LCL-type grid-connected inverters. The core innovation lies in employing a finite impulse response (FIR) filter based on the Lagrange interpolation algorithm to approximate fractional-order delays, enabling the system to adapt dynamically to grid frequency fluctuations. Additionally, DB control is integrated to enhance dynamic response speed. Through comprehensive modeling, parameter design, and simulation validation, this approach demonstrates significant improvements in harmonic suppression, tracking precision, and overall system robustness. The following sections detail the system structure, mathematical formulation, control design, and performance evaluation, emphasizing the adaptability and effectiveness of the proposed strategy for grid-connected inverter applications.
The architecture of a typical LCL-type grid-connected inverter system is illustrated above. It consists of a DC-DC converter interfaced with a PV array on the DC side and a three-phase voltage source inverter (VSI) connected to the grid via an LCL filter on the AC side. The LCL filter, comprising inductors L1 (inverter-side), L2 (grid-side), and a capacitor C, effectively attenuates switching harmonics. The primary control objective is to regulate the grid-side current i2 to follow a sinusoidal reference synchronized with the grid voltage ug, while minimizing THD and ensuring stability under varying grid conditions. For analysis, parasitic resistances are neglected due to their relatively small values. The system variables include the DC-link voltage Udc, inverter output voltage uin, capacitor voltage uc, and grid voltage ug. The control strategy generates pulse-width modulation (PWM) signals for the IGBT switches based on current feedback, aiming to achieve high-performance operation for the grid-connected inverter.
To simplify the control design and avoid resonance peaks that limit bandwidth, the average weighted current method is employed. This method reduces the third-order LCL system to a first-order model by feeding back a weighted sum of the inverter-side current i1 and grid-side current i2. Defining the weighted coefficient β as β = L2/(L1 + L2), the weighted average current i is given by:
$$ i = \beta i_2 + (1-\beta) i_1 $$
The transfer function from the inverter output voltage uin to the weighted current i is derived as:
$$ G_p(s) = \frac{i(s)}{u_{in}(s)} = \frac{1}{s(L_1 + L_2)} $$
This simplified model facilitates the design of current controllers, such as DB and FARC, for the grid-connected inverter. The parameters of a typical grid-connected inverter system are summarized in Table 1, which will be referenced throughout the design process.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (RMS) | Ug | 220 V |
| DC-Link Voltage | Udc | 700 V |
| Rated Current | Irated | 15 A |
| Rated Power | Prated | 10 kW |
| Sampling Frequency | fs | 10 kHz |
| Inverter-Side Inductance | L1 | 1 mH |
| Grid-Side Inductance | L2 | 3 mH |
| Filter Capacitance | C | 5 μF |
Conventional repetitive control (RC) is renowned for its ability to eliminate periodic errors in grid-connected inverters. The internal model of an RC in discrete-time is expressed as:
$$ G_{rc}(z) = \frac{z^{-N}}{1 – Q(z)z^{-N}} C(z) $$
where N = fs / f0 is the ratio of sampling frequency to fundamental grid frequency (typically 50 Hz), Q(z) is a low-pass filter to enhance stability, and C(z) is a compensator. When the grid frequency f deviates from f0, N becomes fractional, denoted as N = ND + M, where ND is the integer part and M is the fractional part (0 ≤ M < 1). This deviation causes the resonant peaks of the RC to shift away from harmonic frequencies, degrading harmonic suppression. The magnitude response at harmonic frequency hf0Δ (where f0Δ is the actual grid frequency) is given by:
$$ |G_{rc}(j2\pi h f_0^\Delta)| = \frac{1}{\sqrt{2 – 2\cos(2\pi h \gamma)}} $$
with γ representing the frequency deviation coefficient. To maintain high performance, a frequency adaptive repetitive control (FARC) is proposed, incorporating a fractional delay approximation using Lagrange interpolation. The fractional delay z-M is approximated by an FIR filter:
$$ z^{-M} \approx \sum_{k=0}^{n} b_k z^{-k} = \sum_{k=0}^{n} \left( \prod_{\substack{i=0 \\ i \neq k}}^{n} \frac{M – i}{k – i} \right) z^{-k} $$
where n is the interpolation order. For balance between computational complexity and accuracy, n = 2 is selected. The coefficients bk are calculated in real-time based on M, enabling adaptive adjustment to grid frequency variations. The FARC controller transfer function becomes:
$$ G_{farc}(z) = K_{rc} \frac{z^{-N_D} \left( \sum_{k=0}^{2} b_k z^{-k} \right) Q(z)}{1 – z^{-N_D} \left( \sum_{k=0}^{2} b_k z^{-k} \right) Q(z)} C(z) $$
Here, Krc is a gain factor, and Q(z) is designed as a zero-phase low-pass filter to improve stability without sacrificing low-frequency tracking. A common choice is:
$$ Q(z) = 0.25z + 0.5 + 0.25z^{-1} $$
This filter ensures |Q(z)| ≈ 1 at low frequencies and |Q(z)| < 1 at high frequencies, striking a balance between robustness and precision for the grid-connected inverter.
To address control delays and enhance dynamic response, deadbeat (DB) control is integrated. DB control predicts the required control action to achieve zero error within one sampling period. Considering the simplified plant Gp(s) and accounting for computational delay, the DB control law in discrete-time is:
$$ u_{in}(k) = \frac{L_1 + L_2}{T_s} \left[ i^*(k) – i(k) \right] + u_g(k) $$
where Ts is the sampling period, i*(k) is the reference current, i(k) is the measured current, and ug(k) is the grid voltage. Incorporating a tuning coefficient K to handle parameter variations, the DB controller transfer function with delay compensation is:
$$ G_{DB}(z) = K \frac{T_s}{L_1 + L_2} z^{-1} $$
The overall control structure for the grid-connected inverter combines FARC and DB in parallel, as shown in the block diagram. The reference current i* is generated from grid voltage phase θ using a phase-locked loop (PLL). The combined controller output drives the PWM modulator to produce switching signals for the inverter.
The closed-loop transfer function of the system, derived from the block diagram, is:
$$ G_i(z) = \frac{G_{farc}(z) G_{DB}(z) G_p(z)}{1 + G_{farc}(z) G_{DB}(z) G_p(z)} = 1 – \frac{1 – Q(z) z^{-N}}{1 – \left[ 1 – K \cdot K_{rc} \frac{L_1 + L_2}{T_s} C(z) G_p(z) \right] Q(z) z^{-N}} $$
Stability and performance depend on proper selection of parameters K, Krc, and the compensator C(z). The compensator typically includes a phase lead element to counteract delays: C(z) = zp, where p is the number of lead steps. Based on frequency response analysis, p = 3 is chosen to maximize bandwidth while maintaining stability. The tuning coefficient K for DB control is set to 0.5 after evaluating Bode plots for different values, as summarized in Table 2.
| K Value | Bandwidth (Hz) | Phase Margin (degrees) | Stability |
|---|---|---|---|
| 0.3 | 850 | 45 | Stable |
| 0.5 | 1200 | 40 | Stable |
| 0.7 | 1500 | 30 | Marginally Stable |
The parameter design ensures that the grid-connected inverter operates robustly across frequency variations. For instance, when the grid frequency shifts to 49.6 Hz (a common fluctuation), the FARC system adapts by updating M in the fractional delay filter. The performance is evaluated through simulations in MATLAB/Simulink, comparing the proposed FARC-DB system with traditional RC-DB approaches.
Simulation results under grid frequency disturbance demonstrate the superiority of the adaptive system. With a frequency drop to 49.6 Hz, the traditional grid-connected inverter exhibits increased current THD and slower error convergence, while the FARC-based grid-connected inverter maintains low THD and fast tracking. Key metrics are compared in Table 3.
| Metric | Traditional RC-DB | Proposed FARC-DB |
|---|---|---|
| THD at 49.6 Hz (%) | 3.21 | 1.44 |
| Current Error Convergence Time (s) | 0.25 | 0.1 |
| Maximum Current Error (A) | 0.8 | 0.3 |
| Dynamic Response Time (ms) | 5 | <1 |
The dynamic performance is further tested by applying a step change in reference current from 15 A to 10 A at t = 0.1 s. The grid-connected inverter with FARC-DB control responds within 1 ms, showcasing rapid tracking capability. The current waveform remains sinusoidal with minimal overshoot, affirming the effectiveness of the combined control strategy.
Mathematical analysis of harmonic suppression reinforces these findings. The FARC system’s magnitude response at harmonic frequencies is nearly infinite even under frequency deviations, due to the fractional delay approximation. For the h-th harmonic, the adapted resonant frequency f_res is given by:
$$ f_{res} = \frac{f_s}{N_D + M} $$
where M is continuously adjusted based on real-time frequency estimation from a PLL. This ensures that the grid-connected inverter consistently targets actual harmonic components, reducing THD. The error dynamics can be modeled as:
$$ e(k+1) = e(k) – K_{rc} C(z) G_p(z) Q(z) z^{-N} e(k) $$
which converges to zero asymptotically when |Q(z)| < 1 and the phase of C(z) compensates for plant delays.
In practical implementations, computational efficiency is crucial for real-time control of grid-connected inverters. The Lagrange interpolation filter requires only a few multiplications and additions per sample, making it suitable for digital signal processors (DSPs) or FPGAs. The update of M involves simple arithmetic based on the measured grid frequency, which can be obtained from standard PLL algorithms. This lightweight adaptation mechanism ensures that the grid-connected inverter can handle frequency swings up to ±2 Hz without significant performance degradation.
Furthermore, the robustness of the grid-connected inverter against parameter variations is enhanced by the DB control’s tuning coefficient K. Sensitivity analysis shows that with K = 0.5, the system remains stable even if the actual inductance values deviate by ±20% from nominal. This is validated by Monte Carlo simulations, where L1 and L2 are varied randomly within this range. The results indicate that THD stays below 2% in 95% of cases, meeting grid codes such as IEEE 1547.
To summarize, the proposed FARC-DB control system offers a comprehensive solution for LCL-type grid-connected inverters operating under grid frequency fluctuations. The integration of adaptive fractional delay approximation and fast deadbeat response addresses key limitations of conventional methods. Future work may explore hardware-in-the-loop (HIL) testing and field trials to validate performance in real-world grid environments. Extensions to multi-inverter systems and microgrid applications are also promising directions.
In conclusion, this design significantly advances the state-of-the-art in grid-connected inverter control by achieving high precision, low THD, and robust adaptability. The use of Lagrange interpolation for fractional delays, combined with DB control, provides a scalable framework for renewable energy integration. As grid-connected inverters become ubiquitous in smart grids, such adaptive strategies will be essential for maintaining power quality and stability in the face of increasing grid volatility.