The widespread adoption of renewable energy sources has positioned the on grid inverter as a critical interface between distributed generation and the utility network. Ensuring high-quality power injection, characterized by low total harmonic distortion (THD) and robust stability under various grid conditions, remains a paramount objective. Modern on grid inverter systems frequently employ LCL filters at their output due to their superior high-frequency harmonic attenuation capability compared to simple L-filters. However, the inherent resonant peak of the LCL filter introduces a significant stability challenge that must be effectively mitigated.
Traditional solutions involve passive damping, which adds resistive losses, or active damping methods requiring additional sensor hardware. This work focuses on an advanced control strategy that addresses both the resonance issue and the computational burden of high-performance controllers. Specifically, I present the design and analysis of a multirate compound repetitive controller (MRC) for a single-phase on grid inverter, utilizing a notch filter for lossless active resonance damping. This approach maintains exceptional harmonic suppression while reducing the digital controller’s sampling rate and associated computational load.
1. System Modeling and Resonance Challenge
The topology of a single-phase LCL-type on grid inverter is considered. The system comprises a full-bridge inverter, an LCL filter (with inductances $L_1$, $L_2$ and capacitance $C$), and the grid. The primary control goal is to force the grid current $i_g$ to accurately track its sinusoidal reference $i_{ref}$.
The plant transfer function from the inverter output voltage $U_{inv}(s)$ to the grid current $I_g(s)$, neglecting parasitic resistances, is given by:
$$ P(s) = \frac{I_g(s)}{U_{inv}(s)} = \frac{1}{L_1 L_2 C s^3 + (L_1 + L_2)s} $$
This third-order system exhibits a resonant peak at the frequency:
$$ f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} $$
At this frequency, the phase drops by 180°, which can easily lead to closed-loop instability if not properly addressed in the controller design for the on grid inverter. The following table summarizes a typical parameter set used for analysis.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| DC-Link Voltage | $E_{dc}$ | 400 | V |
| Grid Voltage (RMS) | $U_g$ | 220 | V |
| Grid Frequency | $f_g$ | 50 | Hz |
| Inverter-side Inductor | $L_1$ | 3.8 | mH |
| Grid-side Inductor | $L_2$ | 2.2 | mH |
| Filter Capacitor | $C$ | 10 | μF |
| Switching Frequency | $f_s$ | 10,000 | Hz |
| Resonant Frequency | $f_r$ | ~1,250 | Hz |
2. Active Damping via Notch Filter
Instead of using lossy resistors or complex observer-based active damping, I employ a notch filter $G_n(s)$ in series with the controller. The notch filter is designed to introduce a pair of complex zeros that cancel the resonant poles of the LCL filter. Its transfer function in the continuous domain is:
$$ G_n(s) = \frac{s^2 + \omega_n^2}{s^2 + 2\xi\omega_n s + \omega_n^2} $$
Here, $\omega_n = 2\pi f_r$ is the notch frequency, and $\xi$ is the damping ratio controlling the notch’s depth and width. The design principle is to set $\omega_n$ equal to the LCL resonant frequency. The damping ratio $\xi$ is a critical trade-off parameter: a smaller $\xi$ creates a deeper but narrower notch, making the system sensitive to parameter variations; a larger $\xi$ offers better robustness at the cost of reduced attenuation at $f_r$. For the on grid inverter application, a value of $\xi=0.7$ provides a good compromise between effective resonance suppression and system robustness. The discrete-time equivalent $G_n(z_m)$, where $T_m$ is the controller sampling period, is obtained using a suitable transformation method like Tustin with prewarping at $\omega_n$.
3. Multirate Compound Repetitive Control Architecture
Repetitive Control (RC), founded on the internal model principle, offers excellent steady-state accuracy and periodic disturbance rejection, making it ideal for harmonic suppression in an on grid inverter. However, a conventional single-rate RC operates at the high switching frequency, leading to significant computational and memory burdens. To alleviate this, I propose a multirate scheme.
In the proposed Multirate Compound Repetitive Control (MCRC) system, the feedback loop (current sampling and PWM generation) retains the high sampling rate $f_s$ (period $T_s$). In contrast, the repetitive controller operates at a lower rate $f_m = f_s / m$, where $m$ is an integer downsampling factor, with period $T_m = mT_s$. A standard Proportional-Integral (PI) controller operates in parallel at the fast rate to ensure basic dynamic response. The overall control law is a plug-in structure, where the multirate RC output compensates for periodic errors.
The block diagram of the proposed control system for the on grid inverter can be described conceptually. The error $e(kT_s)=i_{ref}(kT_s)-i_g(kT_s)$ is passed through an anti-aliasing filter $F_1(z)$ and then downsampled by $m$ to become $e(lT_m)$. This signal feeds the multirate repetitive controller $G_{mrc}(z_m)$. Its output $u_r(lT_m)$ is upsampled by $m$ using a zero-order hold (ZOH) and then filtered by an anti-imaging filter $F_2(z)$ to remove spectral replicas. This processed signal is added to the output of the fast PI controller $G_{PI}(z)$ before being modulated.
The transfer function of the multirate repetitive controller is:
$$ G_{mrc}(z_m) = \frac{k_r z_m^k Q(z_m) S(z_m) z_m^{-N_m}}{1 – Q(z_m) z_m^{-N_m}} $$
Where:
- $k_r$ is the RC gain.
- $N_m = f_m / f_g$ is the number of samples per fundamental period at the slow rate.
- $z_m^{-N_m}$ is the periodic delay element.
- $Q(z_m)$ is an optional low-pass filter or constant (typically <1) inside the delay loop to enhance robustness.
- $S(z_m)$ is a phase-lead compensator and/or a low-pass filter to shape the open-loop response for stability.
- $z_m^k$ provides phase lead compensation to counteract the phase lag of the plant and filters.
4. Stability Analysis and Parameter Design
The stability of the overall multirate controlled on grid inverter system can be analyzed using the small gain theorem in the slow-sampled domain ($z_m$). The equivalent plant seen by the repetitive controller $G_{mrc}(z_m)$ is the closed-loop transfer function of the fast-sampled system comprising the PI controller, the notch filter, the actual LCL plant, and the ZOH. Let’s denote this equivalent plant as $P_0(z_m)$:
$$ P_0(z_m) = \frac{G_{PI}(z_m) G_n(z_m) P(z_m)}{1 + G_{PI}(z_m) G_n(z_m) P(z_m)} $$
The PI controller parameters ($K_p$, $K_i$) must first be designed to stabilize this inner loop $P_0(z_m)$. With the notch filter properly tuned, this is readily achievable. The characteristic equation for the entire MCRC system leads to two conditions:
- All roots of $1 + G_{PI}(z_m) G_n(z_m) P(z_m)=0$ lie inside the unit circle (ensured by PI design).
- The following inequality holds for all frequencies $\omega \in [0, \pi/T_m]$:
$$ |Q(e^{j\omega T_m})| \cdot |1 – k_r e^{jk\omega T_m} S(e^{j\omega T_m}) P_0(e^{j\omega T_m})| < 1 $$
Assuming $Q(z_m)$ is a constant or a zero-phase low-pass filter with gain less than 1, the core stability criterion simplifies to ensuring the Nyquist plot of $H(e^{j\omega T_m}) = 1 – k_r e^{jk\omega T_m} S(e^{j\omega T_m}) P_0(e^{j\omega T_m})$ remains inside a unit circle centered at (1,0). This translates to:
$$ |k_r S(e^{j\omega T_m}) P_0(e^{j\omega T_m})| < 2 \cos(\angle(e^{jk\omega T_m} S(e^{j\omega T_m}) P_0(e^{j\omega T_m}))) $$
The design procedure involves iterative shaping:
- Choose the downsampling factor $m$. A factor of $m=2$ or $m=3$ offers a good balance between computational reduction and control performance for a typical on grid inverter. Selecting $m=2$ halves the RC’s operational rate.
- Design the notch filter $G_n(z_m)$. As described, set $\omega_n = 2\pi f_r$ and choose $\xi=0.7$.
- Design the inner PI controller $G_{PI}(z)$ at the fast rate for adequate phase margin of the damped system.
- Select the compensators $S(z_m)$ and $z_m^k$. $S(z_m)$ is often a low-pass filter (e.g., 2nd or 4th order Butterworth) with a cutoff frequency below $f_m/2$ to roll off high-frequency gain. The integer $k$ is chosen to make the phase of $e^{jk\omega T_m} S(e^{j\omega T_m}) P_0(e^{j\omega T_m})$ as close to $0^\circ$ as possible over a wide low-frequency range, maximizing the allowable $k_r$. For the given parameters with $m=2$, $k=4$ was found optimal.
- Tune the RC gain $k_r$. Based on the stability inequality, $k_r$ is set as high as possible without violating the condition, typically between 0.5 and 1.5. For robust performance, $k_r=1.0$ is often a safe and effective choice.
- Set the internal model filter $Q(z_m)$. A simple constant like 0.95 or a zero-phase filter like $Q(z_m)=0.25z_m^{-1}+0.5+0.25z_m$ is used to push the internal model’s poles slightly inside the unit circle, enhancing robustness against frequency variations.
The following table summarizes the key controller parameters for the case study with $m=2$.
| Component | Parameter | Designed Value / Transfer Function |
|---|---|---|
| Downsampling Factor | $m$ | 2 |
| Notch Filter (Discrete) | $G_n(z_m)$ | $\frac{0.5915z_m^2 – 0.1944z_m + 0.5915}{z_m^2 – 0.1944z_m + 0.1831}$ |
| PI Controller (Fast Rate) | $G_{PI}(z)$ | $K_p + K_i \frac{T_s z}{z-1}$ with $K_p=17$, $K_i=17000$ |
| Compensator $S(z_m)$ | 4th Order Butterworth LPF | Cutoff at 1000 Hz |
| Phase Lead | $k$ | 4 |
| Repetitive Control Gain | $k_r$ | 1.0 |
| Internal Model Filter | $Q(z_m)$ | $0.25z_m^{-1}+0.5+0.25z_m$ |
5. Performance Evaluation and Comparative Analysis
The effectiveness of the proposed strategy for the on grid inverter is evaluated in the frequency domain. The open-loop transfer function of the repetitive control loop, $G_{ol}(z_m) = G_{mrc}(z_m)P_0(z_m)$, reveals the harmonic suppression characteristics.
$$ G_{ol}(e^{j\omega T_m}) = \frac{k_r e^{jk\omega T_m} Q(e^{j\omega T_m}) S(e^{j\omega T_m}) P_0(e^{j\omega T_m})}{1 – Q(e^{j\omega T_m}) e^{-jN_m\omega T_m}} $$
At harmonics of the fundamental frequency ($\omega = h \cdot 2\pi f_g$), the denominator term $1 – Q(e^{j\omega T_m}) e^{-jN_m\omega T_m}$ becomes very small (ideally zero if $Q=1$), resulting in extremely high open-loop gain. This ensures perfect tracking and disturbance rejection at these frequencies in steady-state. The notch filter $G_n$ ensures the resonance peak in $P_0(z_m)$ is flattened, providing ample stability margin.
The key advantage of the multirate approach becomes clear when comparing the open-loop gain plots of the Single-Rate RC (SRC, $m=1$) and the Multirate RC (MRC, $m=2$). Both systems show high gain at low-frequency harmonics. However, the MRC system exhibits a faster roll-off at higher frequencies (above approximately 1 kHz) due to the lower sampling rate of the RC, which inherently provides a lower Nyquist frequency. This faster roll-off enhances the stability robustness of the on grid inverter against unmodeled high-frequency dynamics. Crucially, the MRC achieves this while maintaining near-identical harmonic suppression performance in the critical low-frequency range (up to the 25th harmonic, 1250 Hz).
From an implementation perspective, the multirate repetitive controller for the on grid inverter offers significant benefits:
- Reduced Computational Load: The update rate of the complex RC algorithm is reduced by a factor of $m$. The number of delay-line elements stored in memory is also reduced from $N = f_s/f_g$ to $N_m = f_s/(m f_g)$.
- Lower Memory Consumption: Proportional reduction in the buffer size needed for the $z^{-N_m}$ delay line.
- Maintained Performance: The plug-in structure ensures the fast PI controller handles transient response, while the slower RC finely corrects periodic steady-state errors.
The integration of the notch filter provides a lossless and sensorless solution to the LCL resonance problem, making the entire control strategy for the on grid inverter both efficient and cost-effective.
6. Conclusion and Future Perspectives
In this work, I have presented a comprehensive design methodology for a multirate compound repetitive controller applied to an LCL-filter-based on grid inverter. The proposed scheme synergistically combines two advanced techniques: notch-filter-based active damping and multirate repetitive control. The notch filter elegantly suppresses the LCL resonance peak without introducing losses or requiring additional voltage/current sensors. The multirate implementation of the repetitive controller significantly alleviates the computational burden and memory requirements on the digital signal processor, which is a critical practical advantage for high-switching-frequency on grid inverter systems.
The stability analysis provides clear guidelines for parameter selection, including the downsampling factor $m$, the notch filter damping $\xi$, the PI gains, and the RC compensators $k$, $k_r$, and $S(z_m)$. The resulting controller ensures high gain at fundamental and low-order harmonic frequencies for excellent steady-state tracking and disturbance rejection, while providing ample stability margins through active resonance damping and the natural high-frequency roll-off of the multirate structure.
Future research directions could explore the adaptation of the notch filter frequency for on grid inverters connected to weak grids with variable resonance frequency. Furthermore, extending the multirate compound repetitive control framework to three-phase systems and integrating it with grid-synchronization techniques under unbalanced and distorted grid conditions would be valuable advancements for real-world renewable energy integration.