The widespread integration of high proportions of renewable energy sources and power electronic devices has introduced new dynamics and challenges to modern power systems. Utility interactive inverters, as the critical interface connecting distributed generation units to the load or the main grid, are responsible for delivering high-quality grid current with low harmonic distortion. In microgrid environments, where system inertia is low, power imbalances can lead to frequency fluctuations. Concurrently, the inherent nonlinear characteristics of power electronic devices exacerbate harmonic pollution. This combination can degrade power quality, potentially leading to sustained oscillations and threatening system stability. Therefore, developing advanced control strategies for utility interactive inverters that maintain excellent harmonic suppression under grid frequency variations is of paramount importance.
Repetitive Control (RC), rooted in the internal model principle, is renowned for its ability to achieve zero steady-state error when tracking periodic signals and eliminating periodic disturbances. This makes it exceptionally effective for suppressing harmonic currents in utility interactive inverters. However, conventional RC suffers from inherent drawbacks: its dynamic response is relatively slow due to the one-fundamental-period delay within its internal model, and its performance deteriorates significantly when the grid frequency deviates from its nominal value, as the delay is no longer an integer multiple of the sampling period.
To address the dynamic performance issue, various solutions have been proposed. Combining RC with Proportional-Integral (PI) control improves transient response but can cause coupling issues. Plug-in or parallel structures with proportional control, forming a Proportional-Integral Multi-Resonant (PIMR) type controller, offer a better compromise. Furthermore, Dual-Mode Repetitive Control (DMRC), featuring independent odd-harmonic and even-harmonic internal models, allows for flexible and targeted harmonic suppression. To tackle the frequency adaptability challenge, methods such as using Finite Impulse Response (FIR) filters to approximate fractional delays have been studied. However, FIR filters often require high orders for accurate approximation in the low-frequency range. In contrast, Infinite Impulse Response (IIR) filters, designed using formulas like the Thiran approximation, can achieve comparable or better performance with lower computational complexity, a feature not yet fully explored in the context of DMRC.
This work proposes a novel Frequency Adaptive Feedforward Dual-Mode Repetitive Control and Proportional Control (FA-FDMRC-PC) scheme for utility interactive inverters. The core innovations are twofold: firstly, a feedforward path is introduced into the DMRC structure to enhance its harmonic suppression capability and dynamic response; secondly, a low-order IIR filter is employed to accurately realize the fractional delay required for synchronization with a fluctuating grid frequency, endowing the controller with inherent frequency adaptability. The proposed strategy ensures robust and high-performance operation of the utility interactive inverter under microgrid conditions.
Control Scheme Architecture and Principle
The proposed control system builds upon the DMRC foundation. The standard DMRC transfer function is given by:
$$G_{DMRC}(z) = k_{o,rc} \frac{z^{-N}}{1 – Q(z)z^{-N}} + k_{e,rc} \frac{z^{-N}}{1 + Q(z)z^{-N}}$$
where $k_{o,rc}$ and $k_{e,rc}$ are the gains for the odd and even harmonic internal models, respectively, $Q(z)$ is a filter (often a low-pass filter or a constant slightly less than 1) to ensure stability, and $N = f_s / f_g$ is the number of samples per fundamental period ($f_s$ is sampling frequency, $f_g$ is grid frequency).
To improve performance, we introduce a feedforward dual-mode repetitive control (FDMRC). The modified structure adds a parallel feedforward branch to each internal model, significantly boosting the open-loop gain at harmonic frequencies. The transfer function of the proposed FDMRC is:
$$G_{FDMRC}(z) = k_{o,ff} \left( 1 – \frac{1 – Q(z)z^{-N}}{1 + Q(z)z^{-N}} \right) + k_{e,ff} \left( 1 + \frac{1 – Q(z)z^{-N}}{1 – Q(z)z^{-N}} \right)$$
where $k_{o,ff}$ and $k_{e,ff}$ are the feedforward gains. A comparison of the Bode plots at a low-order harmonic frequency (e.g., 7th) reveals that FDMRC provides substantially higher gain than both DMRC and conventional RC, leading to superior harmonic rejection.
The complete proposed controller, FA-FDMRC-PC, is formed by placing this FDMRC in parallel with a proportional ($k_p$) controller. The proportional controller significantly improves the transient response of the overall system. The block diagram of the FA-FDMRC-PC applied to an LCL-filter-based utility interactive inverter is shown below. The current error is processed by the composite controller, whose output is the voltage reference for the Pulse Width Modulation (PWM) stage.
The critical component for frequency adaptation is the implementation of the fractional delay $z^{-D/2}$, where $D = f_s / f_g$ becomes a non-integer during frequency excursions. We express this as $z^{-D/2} = z^{-[int(D/2) + d]}$, where $int(\cdot)$ denotes the integer part and $d$ is the fractional part ($0 \le d < 1$). This fractional delay $z^{-d}$ is approximated using a $K$-th order IIR filter based on the Thiran approximation:
$$z^{-d} \approx \frac{a_0 + a_1 z^{-1} + a_2 z^{-2} + \cdots + a_K z^{-K}}{1 + a_1 z^{-1} + a_2 z^{-2} + \cdots + a_K z^{-K}}$$
The coefficients $a_i$ are calculated as:
$$a_i = (-1)^i C_K^i \prod_{k=0}^{K} \frac{d – K + k}{d – K + i + k}, \quad i = 0, 1, 2, …, K$$
$$C_K^i = \frac{K!}{i!(K-i)!}$$
This IIR filter provides an accurate all-pass characteristic with a maximally flat group delay at low frequencies, making it ideal for this application with much lower computational burden than a high-order FIR filter.
Stability Analysis and Controller Design
The stability of the closed-loop system with the FA-FDMRC-PC controller is analyzed using the characteristic equation. The system can be viewed as the FDMRC controlling a modified plant $P_0(z)$ which includes the proportional controller in the loop:
$$P_0(z) = \frac{P(z)}{1 + k_p P(z)}$$
where $P(z)$ is the discrete-time model of the LCL filter including computational delays. The overall characteristic equation leads to two conditions for stability:
Condition 1: All poles of $P_0(z)$ must lie inside the unit circle. This is primarily governed by the design of $k_p$.
Condition 2: The loop gain of the FDMRC around $P_0(z)$ must satisfy the Nyquist criterion. A simplified sufficient condition for stability can be derived as:
$$|Q(z)| – 2 k_{ff} |S(z)P_0(z)| \cos(\theta_S(\omega) + \theta_{P0}(\omega) + m\omega) > -1$$
where $k_{ff} = (k_{o,ff}+k_{e,ff})/2$, $S(z)$ is a compensator (typically a low-pass filter to enhance robustness at high frequencies), and $z^m$ is a phase-lead compensator to adjust the phase of $S(z)P_0(z)$. This inequality guides the selection of the controller parameters $k_{ff}$, $m$, and the design of $S(z)$.
Parameter Design Procedure
The design is illustrated for a single-phase LCL-type utility interactive inverter with the following parameters:
| Parameter | Symbol | Value |
|---|---|---|
| Inverter-side Inductor | $L_1$ | 4.4 mH |
| Grid-side Inductor | $L_2$ | 0.4 mH |
| Filter Capacitor | $C$ | 4.7 µF |
| DC Link Voltage | $V_{dc}$ | 400 V |
| Nominal Grid Frequency | $f_g$ | 50 Hz |
| Sampling/Switching Frequency | $f_s$ | 10 kHz |
1. Proportional Gain ($k_p$): The gain $k_p$ is chosen to ensure $P_0(z)$ is stable and has a favorable frequency response. A value in the range of 8 to 12 typically places all poles inside the unit circle and results in a nearly flat magnitude response for $P_0(z)$ at low frequencies, making it an ideal “plant” for the repetitive controller.
2. Internal Model Filter ($Q(z)$): A simple yet effective choice is $Q(z) = (z + 2 + z^{-1})/4$, which is a low-pass filter. It provides high gain at low frequencies (good for harmonic suppression) and sufficient attenuation at high frequencies (good for stability).
3. Compensator ($S(z)$): A 4th-order Butterworth low-pass filter with a cutoff frequency of 1 kHz is selected. This filter $S(z)$ shapes the loop gain, providing additional attenuation beyond the main harmonic frequencies of interest, thereby enhancing system robustness against high-frequency noise and model uncertainties.
4. Phase-Lead Compensator ($z^m$): The phase $\theta_S(\omega) + \theta_{P0}(\omega)$ is calculated. The lead step $m$ is chosen so that $\theta_S(\omega) + \theta_{P0}(\omega) + m\omega$ is close to zero in the low-frequency band, satisfying the phase condition derived from the stability inequality. For the given system, $m=9$ is found to be optimal.
5. Repetitive Control Gains ($k_{o,ff}$, $k_{e,ff}$): Based on the stability condition and Nyquist analysis of the function $H(e^{j\omega}) = Q^2(z)[1 – 2k_{ff}(1/Q(z)) z^m S(z) P_0(z)]$, the maximum allowable gain $k_{ff}$ is determined. To prioritize the suppression of typically larger odd harmonics, the gains are asymmetrically chosen. For example, $k_{o,ff}=4$ and $k_{e,ff}=3$ provide a stable and high-performance setup.
Experimental Verification and Performance Analysis
A single-phase experimental prototype was built to validate the proposed FA-FDMRC-PC strategy for the utility interactive inverter. The grid voltage was intentionally distorted with background harmonics (THD ≈ 5.7%) to simulate a non-ideal microgrid environment.
Steady-State Performance at Nominal Frequency (50 Hz)
At the nominal 50 Hz grid frequency, the proposed FA-FDMRC-PC was compared against a standard RC-PC controller. The key performance metric is the grid current Total Harmonic Distortion (THD). The results are summarized below:
| Control Scheme | Grid Current THD @ 50 Hz | Dominant Harmonics Suppression |
|---|---|---|
| RC-PC | 1.5% | Good |
| FA-FDMRC-PC (Proposed) | 1.0% | Excellent |
The FA-FDMRC-PC demonstrates superior harmonic suppression capability, reducing the current THD by 33% compared to the conventional RC-PC under the same distorted grid conditions.
Dynamic Performance
The dynamic response was tested by subjecting the utility interactive inverter to a step change in the reference current amplitude (from 5 A to 10 A peak). Both controllers managed the transition, but the FA-FDMRC-PC achieved a settled steady state within approximately two fundamental cycles, showcasing its improved transient performance inherited from the combined proportional and feedforward action.
Performance Under Grid Frequency Fluctuations
This is the core test for the frequency adaptability feature. The grid frequency was varied to 49.6 Hz and 50.4 Hz, simulating microgrid frequency excursions. The non-adaptive FDMRC-PC (without the IIR fractional delay) was compared against the proposed FA-FDMRC-PC. The results clearly highlight the necessity and effectiveness of the frequency adaptation mechanism.
| Grid Frequency | Control Scheme | Grid Current THD | Adaptive? |
|---|---|---|---|
| 49.6 Hz | FDMRC-PC (Non-adaptive) | 1.6% | No |
| FA-FDMRC-PC (Proposed) | 0.8% | Yes | |
| 50.4 Hz | FDMRC-PC (Non-adaptive) | 1.9% | No |
| FA-FDMRC-PC (Proposed) | 0.8% | Yes |
The non-adaptive controller shows a marked increase in THD (from 1.0% at 50 Hz to 1.6% and 1.9%) as the frequency shifts, indicating degraded performance. In stark contrast, the proposed FA-FDMRC-PC maintains an exceptionally low and consistent THD of 0.8% across all frequency variations, proving its robust frequency adaptability. The dynamic performance during reference steps also remained consistently fast under these varying grid frequencies.
An additional test involved a grid voltage sag (from 220 V to 200 V). The utility interactive inverter regulated by the FA-FDMRC-PC maintained stable and low-THD current injection, with the system recovering within one cycle, further demonstrating the controller’s robustness to grid disturbances.
Conclusion
This paper has presented a comprehensive solution for enhancing the performance of utility interactive inverters in microgrid applications. The proposed Frequency Adaptive Feedforward Dual-Mode Repetitive Control and Proportional Control (FA-FDMRC-PC) strategy successfully addresses two critical challenges: improving dynamic response and maintaining high-performance harmonic suppression under grid frequency fluctuations.
The integration of a feedforward path into the dual-mode repetitive control structure significantly boosts the harmonic rejection capability. The parallel proportional control ensures rapid transient response to reference changes or disturbances. Most importantly, the incorporation of a Thiran IIR filter to accurately implement the required fractional delay grants the controller inherent frequency adaptability, a crucial feature for stable operation in microgrids with variable frequency.
Rigorous stability analysis provides clear design guidelines for the controller parameters. Experimental results on a single-phase LCL-type utility interactive inverter prototype validate the theoretical developments. The proposed FA-FDMRC-PC scheme not only outperforms conventional RC-PC at nominal frequency but, critically, maintains its superior current quality (THD < 1.0%) and dynamic performance when the grid frequency varies, unlike non-adaptive counterparts whose performance degrades significantly. This makes FA-FDMRC-PC a highly suitable and robust control strategy for ensuring power quality and reliability in modern renewable-energy-rich and power-electronics-dominated microgrid systems.