In modern power systems, the integration of distributed generation sources has become increasingly critical, and the grid connected inverter serves as a pivotal interface between these sources and the main grid. Ensuring high-quality power injection into the grid is paramount, and this necessitates advanced control strategies for grid connected inverters to mitigate harmonics and enhance stability. Traditional control methods, such as proportional-integral (PI) and deadbeat controllers, offer simplicity and robustness but often fall short in achieving zero-error tracking of alternating signals. Resonant-based controls, including proportional-resonant (PR) and repetitive control (RC), leverage the internal model principle to provide accurate tracking at specific frequencies. However, these approaches can be limited by poor dynamic performance and sensitivity to grid frequency variations. In this context, we propose a novel control scheme termed second-order higher gain repetitive control (SOHG-RC), which combines the benefits of high-gain repetitive control with enhanced frequency adaptability. This article delves into the design, analysis, and validation of SOHG-RC for LLCL-type grid connected inverters, aiming to improve harmonic suppression and dynamic response under fluctuating grid conditions.
The importance of grid connected inverters in renewable energy systems cannot be overstated. They facilitate the conversion of DC power from sources like photovoltaics or batteries into AC power synchronized with the grid. To achieve clean power injection, filters such as L, LCL, and LLCL types are employed to attenuate switching harmonics caused by pulse-width modulation (PWM). Among these, the LLCL filter offers superior attenuation with reduced inductance values, lowering costs. However, low-frequency harmonics require sophisticated current control strategies. Repetitive control, based on the internal model principle, is effective for periodic signal tracking but suffers from slow dynamics and frequency sensitivity. Prior works have explored variants like fractional-order repetitive control and high-order repetitive control to address these issues. Our contribution lies in integrating a second-order repetitive structure with a high-gain framework, resulting in SOHG-RC that enhances both performance and robustness for grid connected inverters.
The mathematical modeling of a single-phase LLCL-type grid connected inverter forms the foundation for control design. The system comprises a DC source, inverter bridge, LLCL filter, and grid connection. Key parameters include inductances \(L_1\) and \(L_2\), filter capacitance \(C_f\), and damping resistance \(R_d\). The state-space representation or transfer function from inverter output voltage to grid current is derived to facilitate controller synthesis. For the LLCL filter, the continuous-time transfer function \(G_{\text{LLCL}}(s)\) is given by:
$$G_{\text{LLCL}}(s) = \frac{L_f C_f s^2 + R_d C_f s + 1}{C_f (L_1 L_f + L_1 L_2 + L_2 L_f) s^3 + R_d C_f (L_1 + L_2) s^2 + (L_1 + L_2) s}$$
Discretizing this using a zero-order hold (ZOH) with sampling frequency \(f_s\) yields the discrete-time plant \(P(z)\), which is essential for digital control implementation. The grid connected inverter’s dynamics are influenced by component values, and Table 1 summarizes typical parameters used in simulations.
| Parameter | Symbol | Value |
|---|---|---|
| Inverter-side inductance | \(L_1\) | 3 mH |
| Inverter-side resistance | \(R_1\) | 0.4 Ω |
| Grid-side inductance | \(L_2\) | 2.2 mH |
| Grid-side resistance | \(R_2\) | 0.3 Ω |
| Filter capacitance | \(C_f\) | 10 μF |
| Filter inductance | \(L_f\) | 25.33 μH |
| Damping resistance | \(R_d\) | 10 Ω |
| DC voltage | \(E_d\) | 380 V |
| Grid voltage | \(u_g\) | 220 V |
| Grid fundamental frequency | \(f_g\) | 50 Hz |
| Sampling frequency | \(f_s\) | 10 kHz |
| Dead time | \(t_0\) | 3 μs |
To achieve precise current control in a grid connected inverter, we first examine conventional repetitive control (CRC). The CRC transfer function in the discrete domain is:
$$G_{\text{CRC}}(z) = k_r \frac{z^{-N}}{1 – z^{-N}}$$
where \(k_r\) is the control gain, \(N = f_s / f_g\) is the delay number, and \(z\) is the complex variable. CRC provides high gain at harmonic frequencies but has limited bandwidth and slow response. To improve frequency adaptability, second-order repetitive control (SORC) introduces additional delay paths with weights \(w_1\) and \(w_2\):
$$G_{\text{SORC}}(z) = k_r \frac{w_1 z^{-N} + w_2 z^{-2N}}{1 – (w_1 z^{-N} + w_2 z^{-2N})}$$
with \(w_1 + w_2 = 1\). The weights \(w_1\) and \(w_2\) are chosen to enhance gain and bandwidth. For instance, setting \(w_1 = 2\) and \(w_2 = -1\) yields higher gain across a broader frequency range compared to CRC, as shown in frequency response analyses. This makes SORC more robust to grid frequency variations in a grid connected inverter.
However, SORC alone suffers from restricted gain \(k_r\) due to stability constraints in plug-in structures. To overcome this, we propose second-order higher gain repetitive control (SOHG-RC), which combines SORC with a proportional-integral multi-resonant (PIMR) framework. The block diagram of SOHG-RC includes a proportional gain \(k_p\), a low-pass filter \(Q(z)\), a phase lead compensator \(z^m\), and a compensator \(S(z)\). The overall controller transfer function \(G_{\text{SOHG-RC}}(z)\) is:
$$G_{\text{SOHG-RC}}(z) = k_p + \frac{k_r (w_1 z^{-N} + w_2 z^{-2N}) Q(z) z^m S(z)}{1 – (w_1 z^{-N} + w_2 z^{-2N}) Q(z)}$$
This structure allows for larger \(k_r\) values, improving dynamic performance while maintaining stability. The design process involves selecting parameters to ensure robustness and effectiveness for the grid connected inverter.
Stability analysis of SOHG-RC is critical for reliable operation. The system’s characteristic equation is derived from the closed-loop transfer function. Let \(P(z)\) be the plant model of the grid connected inverter. The error dynamics yield:
$$E(z) = \frac{1}{1 + [G_{\text{SOHG-RC}}(z) + k_p] P(z)} [I_{\text{ref}}(z) – U_g(z)]$$
where \(I_{\text{ref}}(z)\) is the reference current and \(U_g(z)\) is grid voltage disturbance. Stability requires that all roots of the characteristic polynomial lie within the unit circle. This leads to two conditions: (1) \(1 + k_p P(z) = 0\) must have roots inside the unit circle, and (2) \(\left|1 + G_{\text{SORC}}(z) P_0(z)\right| \neq 0\), where \(P_0(z) = P(z) / [1 + k_p P(z)]\). Expanding condition (2) gives:
$$\left| Q(z) W(z) [1 – z^m k_r S(z) P_0(z)] \right| < 1$$
with \(W(z) = w_1 + w_2 z^{-N}\). In frequency domain, this inequality translates to:
$$0 < k_r < \frac{2 \cos(\theta_s(\omega) + \theta_p(\omega) + m \omega)}{N_s(\omega) N_p(\omega)} + \Delta$$
where \(\theta_s(\omega)\), \(\theta_p(\omega)\) are phase angles of \(S(z)\) and \(P_0(z)\), \(N_s(\omega)\), \(N_p(\omega)\) are magnitudes, and \(\Delta\) is a residual term. The phase lead \(m\) is chosen to satisfy \(\left| \theta_s(\omega) + \theta_p(\omega) + m \omega \right| < 90^\circ\), ensuring stability margins.
Parameter design for SOHG-RC involves several steps. First, the proportional gain \(k_p\) is selected based on root locus analysis of \(P_0(z)\). For the given grid connected inverter parameters, \(k_p = 15\) ensures poles within the unit circle. Second, the low-pass filter \(Q(z)\) is chosen as a zero-phase shift filter to attenuate high-frequency noise:
$$Q(z) = 0.25z + 0.5 + 0.25z^{-1}$$
Third, the compensator \(S(z)\) is a fourth-order Butterworth filter with cutoff at 1 kHz to further dampen high frequencies:
$$S(z) = \frac{0.004824z^4 + 0.0193z^3 + 0.02895z^2 + 0.0193z + 0.004824}{z^4 – 2.37z^3 + 2.314z^2 – 1.055z + 0.1874}$$
Fourth, the phase lead compensator order \(m\) is determined from phase frequency curves. For our grid connected inverter, \(m = 8\) minimizes phase lag in the low-frequency range. Fifth, weights \(w_1\) and \(w_2\) are set to \(w_1 = 1.5\) and \(w_2 = -0.5\) to balance performance and stability. Finally, the control gain \(k_r\) is set to 14, which satisfies the stability inequality. Table 2 summarizes these controller parameters.
| Parameter | Symbol | Value |
|---|---|---|
| Proportional gain | \(k_p\) | 15 |
| Control gain | \(k_r\) | 14 |
| Weight 1 | \(w_1\) | 1.5 |
| Weight 2 | \(w_2\) | -0.5 |
| Phase lead order | \(m\) | 8 |
| Low-pass filter | \(Q(z)\) | \(0.25z + 0.5 + 0.25z^{-1}\) |
| Compensator | \(S(z)\) | Fourth-order Butterworth |
Simulation studies validate the efficacy of SOHG-RC for the grid connected inverter. Using MATLAB/Simulink, we compare SOHG-RC with PIMR-RC under steady-state and dynamic conditions. The grid connected inverter model includes the LLCL filter and PWM switching. Steady-state performance is evaluated at grid frequencies of 49.5 Hz and 50.5 Hz to test frequency adaptability. For each case, grid current waveforms, tracking errors, and total harmonic distortion (THD) are analyzed.
At 49.5 Hz, SOHG-RC reduces current tracking error to ±0.8 A, compared to ±1.4 A with PIMR-RC. THD values are 2.29% for SOHG-RC and 3.53% for PIMR-RC. At 50.5 Hz, SOHG-RC achieves ±0.9 A error and 2.41% THD, while PIMR-RC gives ±1.3 A error and 3.09% THD. Table 3 compiles THD and error metrics across a frequency range from 49.5 Hz to 50.5 Hz, demonstrating SOHG-RC’s superior harmonic suppression and smaller errors.
| Grid Frequency (Hz) | SOHG-RC THD (%) | SOHG-RC Error (A) | PIMR-RC THD (%) | PIMR-RC Error (A) |
|---|---|---|---|---|
| 50.5 | 2.41 | 0.9 | 3.09 | 1.3 |
| 50.4 | 2.20 | 0.8 | 2.63 | 1.1 |
| 50.3 | 1.78 | 0.6 | 2.42 | 0.9 |
| 50.2 | 1.44 | 0.5 | 1.83 | 0.6 |
| 50.1 | 1.24 | 0.4 | 1.38 | 0.5 |
| 50.0 | 1.22 | 0.4 | 1.10 | 0.3 |
| 49.9 | 1.31 | 0.4 | 1.42 | 0.5 |
| 49.8 | 1.42 | 0.5 | 1.99 | 0.7 |
| 49.7 | 1.70 | 0.6 | 2.64 | 1.1 |
| 49.6 | 2.04 | 0.7 | 3.05 | 1.3 |
| 49.5 | 2.29 | 0.8 | 3.53 | 1.4 |
Dynamic performance is tested by applying a step change in reference current from 10 A to 5 A at 0.2 s. With SOHG-RC, the grid connected inverter output settles within two fundamental cycles (40 ms), indicating fast transient response. This showcases the controller’s ability to handle sudden load variations while maintaining stability.
The effectiveness of SOHG-RC stems from its combined structure. The second-order repetitive part enhances frequency robustness, allowing the grid connected inverter to tolerate ±0.5 Hz variations without significant performance degradation. The high-gain aspect, enabled by the proportional term, accelerates error convergence. Moreover, the design ensures computational efficiency, as no online parameter updates are needed, reducing processor burden. These attributes make SOHG-RC suitable for practical grid connected inverter applications where grid conditions may fluctuate.
In conclusion, the proposed second-order higher gain repetitive control offers a significant advancement for grid connected inverters. By integrating second-order repetitive control with a high-gain framework, we achieve improved harmonic suppression, dynamic response, and frequency adaptability. Theoretical analysis confirms stability under parameter variations, and simulations validate superior performance compared to conventional methods. Future work could explore SOHG-RC in three-phase systems or under unbalanced grid conditions. Overall, this control strategy enhances the reliability and power quality of grid connected inverters in modern distributed generation systems.
The role of grid connected inverters in renewable integration underscores the need for robust control solutions. As grid standards tighten, techniques like SOHG-RC will be crucial for compliance. We have detailed the mathematical modeling, controller design, and validation steps, providing a comprehensive guide for implementation. The use of LLCL filters further optimizes cost and performance. With ongoing advancements in power electronics, such control schemes will facilitate smoother grid integration, contributing to a sustainable energy future. This work emphasizes the importance of adaptive control in ensuring that grid connected inverters operate efficiently across varying conditions, ultimately supporting grid stability and power quality.