As a researcher in power electronics, I have long been fascinated by the challenges of ensuring stable and high-quality power injection from renewable energy sources into the grid. The LCL-type grid-connected inverter is a cornerstone of modern distributed generation systems, responsible for converting DC power into high-fidelity AC current. In my work, I have focused on the critical issue of current harmonic suppression and system stability, particularly under the increasingly common conditions of weak grids with complex and variable impedance. This article presents my proposed solution: a novel repetitive control strategy grounded in the principles of passivity theory. This strategy aims not only to achieve precise current tracking and harmonic rejection but also to guarantee robust stability regardless of grid impedance variations, a feat that traditional methods often struggle with.
The pursuit of low total harmonic distortion (THD), typically below 5%, for the grid-injected current has made repetitive control (RC) a popular choice for LCL-type grid-connected inverters. Its ability to provide infinite gain at fundamental and harmonic frequencies allows it to perfectly track periodic references and reject periodic disturbances, such as those caused by dead-time effects and grid voltage background harmonics. However, in my extensive analysis, I have identified a fundamental conflict. While the RC controller excels at harmonic suppression, its internal model, which inherently contains a delay of one fundamental period, introduces resonant peaks at harmonic frequencies beyond the system’s crossover frequency. These peaks, as I will demonstrate, can render the inverter’s output admittance non-passive—specifically, imparting negative real parts around harmonic frequencies. In a weak grid scenario, where the grid impedance is no longer negligible and can be inductive or even resonant, an interaction between this non-passive inverter admittance and the passive grid admittance can lead to severe instability, threatening the entire interconnection.
This instability manifests as growing oscillations in the grid current. Existing solutions often involve reducing the RC gain to mitigate these negative real parts, but this comes at a significant cost: the dynamic performance of the controller is severely degraded, leading to slower error convergence. My goal was to break this trade-off. I asked: can we design an RC controller that maintains high gain for fast dynamic response and effective harmonic suppression while simultaneously ensuring the overall inverter presents a passive output admittance across a wide frequency range? The answer, as detailed in this article, is yes. My proposed strategy leverages frequency-domain passivity theory as a guiding design principle. The core innovation lies in the development of a specially designed third-order zero-phase-shift low-pass filter within the RC’s internal model. This filter is crafted to eliminate the negative-real-part admittance introduced by the RC without compromising its gain. Furthermore, I employ a capacitor voltage feedforward (CVF) technique, based on a low-pass filter, to cancel the inherent negative real part region of the LCL filter’s admittance under grid-side current control. The synergistic combination of these two techniques results in an inverter output admittance that is passive up to the Nyquist frequency. According to passivity theory, if a system is stable internally and its output admittance is passive, its interconnection with any other passive network (like a real-world grid) is guaranteed to be stable. This forms the bedrock of my proposed control strategy’s robustness.
To lay the groundwork for understanding the proposed method, I first derived a comprehensive admittance model for the LCL-type grid-connected inverter. The system topology, as shown in the figure, consists of the inverter-side inductor $L_1$, the filter capacitor $C$, and the grid-side inductor $L_2$. The grid at the point of common coupling (PCC) is modeled as an ideal voltage source $v_g$ in series with an impedance $Z_g$, which can include inductance $L_g$ and capacitance $C_g$. The control employs grid-side current feedback, where the current controller $G_c(s)$ regulates the current $i_2$ to follow its reference $i_r$. The capacitor voltage $v_c$ is measured for feedforward and phase-locked loop (PLL) synchronization. The PLL’s dynamics are neglected for mid-to-high-frequency stability analysis, as its bandwidth is designed to be lower than the fundamental frequency. Similarly, the DC-link voltage $V_{dc}$ is assumed constant. The system equations in the Laplace domain ($s$-domain) are:
$$
\begin{aligned}
v_i(s) &= Z_1(s) i_1(s) + v_c(s) \\
v_c(s) &= Z_C(s) i_c(s) \\
v_c(s) &= v_{pcc}(s) + Z_2(s) i_2(s) \\
i_1(s) &= i_c(s) + i_2(s)
\end{aligned}
$$
where $Z_1(s)=sL_1$, $Z_C(s)=1/(sC)$, and $Z_2(s)=sL_2$. The inverter output voltage $v_i$ is given by the control law:
$$
v_i(s) = \left\{ [i_r(s) – i_2(s)] G_c(s) + v_c(s) G_v(s) \right\} G_{dc}(s)
$$
Here, $G_{dc}(s)=e^{-sT_{dc}}$ represents the total digital control delay, with $T_{dc}=1.5T_s$ and $T_s$ as the sampling period. $G_v(s)$ is the CVF controller. By treating $v_c$ as a disturbance, the grid-side current $i_2$ can be expressed in Norton equivalent form:
$$
i_2(s) = T_c(s) i_r(s) – Y_c(s) v_c(s)
$$
The current source transfer function $T_c(s)$ and the inverter output admittance $Y_c(s)$ seen from the capacitor voltage node are:
$$
T_c(s) = \frac{G_c(s) G_{dc}(s) / Z_1(s)}{1 + G_c(s) G_{dc}(s) / Z_1(s)}
$$
$$
Y_c(s) = \frac{1/Z_1(s) + [1/Z_1(s) – G_v(s)] G_c(s) G_{dc}(s)}{1 + G_c(s) G_{dc}(s) / Z_1(s)}
$$
Notably, $Y_c(s)$ can be decomposed into two parallel sub-admittances: $Y_{cg}(s)$ stemming from the grid-side current control and $Y_{cv}(s)$ from the CVF:
$$
Y_c(s) = Y_{cg}(s) + Y_{cv}(s) = \frac{1/Z_1(s)}{1 + G_c(s)G_{dc}(s)/Z_1(s)} + \frac{-G_v(s)G_c(s)G_{dc}(s)}{1 + G_c(s)G_{dc}(s)/Z_1(s)}
$$
This decomposition is crucial, as it allows for independent shaping of the overall admittance. The equivalent circuit is a current source $T_c i_r$ in parallel with the admittance $Y_c$, connected to the grid impedance $Z_g$ and voltage source $v_g$ via $L_2$.
For the current controller $G_c(s)$, I selected a Proportional-Repetitive Control (PRC) structure. Compared to the plug-in type, the parallel structure of the PRC can accommodate a larger RC gain $k_{rc}$ without sacrificing stability, leading to faster dynamic response. In the discrete-time domain ($z$-domain), the PRC controller is:
$$
G_c(z) = k_p + k_{rc} \frac{Q(z) z^{-m}}{1 – Q(z) z^{-N}}
$$
where $k_p$ is the proportional gain, $k_{rc}$ is the RC gain, $N=f_s/f_0$ is the number of samples per fundamental period, $Q(z)$ is a low-pass filter (or a constant less than 1) to ensure stability, and $S(z)=z^m$ is a linear phase lead compensator with $m$ steps of advance to counteract the system’s phase lag. The conventional choice for $Q(z)$ is a first-order zero-phase-shift low-pass filter: $Q(z) = \alpha_1 z + \alpha_0 + \alpha_1 z^{-1}$ with $\alpha_0 + 2\alpha_1 = 1$.
The stability of the grid-connected inverter system is assessed using the impedance-based stability criterion. The system is stable if: (1) $T_c(s)$ is stable (internal stability), and (2) the impedance ratio $[Z_2(s)+Z_g(s)]Y_c(s)$ satisfies the Nyquist criterion (external stability). However, this requires prior knowledge of the grid impedance $Z_g(s)$, which is often unknown and variable. A more powerful and general approach is based on passivity theory in the frequency domain. A system with admittance $Y(s)$ is passive if: (1) it is stable, and (2) its real part is non-negative for all frequencies, i.e., $\text{Re}\{Y(j\omega)\} \ge 0$ for all $\omega$, which implies its phase lies within $[-90^\circ, 90^\circ]$. The profound implication is that if the inverter’s output admittance $Y_c(s)$ is passive, then its interconnection with any passive grid impedance (which a real physical grid is) is guaranteed to be stable. This provides a robustness certificate against arbitrary grid impedance variations, which is the ultimate objective for a reliable grid-connected inverter.
Therefore, my design philosophy is to shape $Y_c(s)$ to be passive within the Nyquist frequency range ($0 \le \omega \le \omega_s/2$, where $\omega_s=2\pi f_s$). The first step is to ensure internal stability by proper design of the PRC parameters. Transforming the PRC controller to the $s$-domain for analysis, the internal open-loop transfer function is $k_p G_p(s)$, where $G_p(s)=G_{dc}(s)/Z_1(s)$. The proportional gain $k_p$ is designed to provide a desired phase margin $\phi_m$ at the crossover frequency $\omega_c$:
$$
\omega_c = \frac{\pi/2 – \phi_m}{T_{dc}}, \quad k_p = L_1 \omega_c
$$
The stability condition for the repetitive part leads to an upper bound for $k_{rc}$. After detailed derivation, the condition is $k_{rc} < x_2(\omega)_{\text{min}}$, where $x_2(\omega)$ is a frequency-dependent function. The parameter $m$ influences $x_2(\omega)_{\text{min}}$. For the system parameters in this study, $m=3$ or $4$ maximizes this bound, allowing $k_{rc}$ to be as high as $2k_p$. This is a significant range, enabling high-gain operation.
The critical challenge arises when examining the real part of the admittance $Y_{cg}(s)$ contributed by the current control loop. Without RC ($k_{rc}=0$), $Y_{cg}$ has an inherent negative real part region due to the LCL resonance, approximately between $\sqrt{1/(L_1 C)}$ and $\omega_s/6$. The RC controller modifies this. My analysis reveals that the real part of $Y_{cg}$ with PRC can be expressed as:
$$
\text{Re}\{Y_{cg}(j\omega)\} = \frac{k_p (1 – L_1 C \omega^2) \cos(\omega T_{dc}) \cdot f(\omega)}{a^2(\omega) + b^2(\omega)}
$$
where $a(\omega)$ and $b(\omega)$ are functions related to the system, and $f(\omega)$ is a function that captures the effect of the RC. Specifically, $f(\omega)$ can become negative around harmonic frequencies, particularly near $\omega_s/6$ and $\omega_s/2$, due to the periodic nature of the cosine terms in its expression. When $f(\omega)<0$, it can flip the sign of $\text{Re}\{Y_{cg}\}$, introducing new negative-real-part regions on top of the inherent one. This is the root cause of RC-induced instability in weak grids.
Reducing $k_{rc}$ can make $f(\omega)$ positive at some harmonics, but it is ineffective near $\omega_s/6$ and $\omega_s/2$ because the term causing sign change becomes exceedingly large there. Moreover, reducing $k_{rc}$ degrades dynamic performance. My solution is to redesign the filter $Q(z)$ within the RC’s internal model. The conventional first-order $Q(z)$ has poor high-frequency attenuation and only one notch (zero-gain frequency). I developed a third-order zero-phase-shift low-pass filter:
$$
Q(z) = \alpha_3 z^3 + \alpha_2 z^2 + \alpha_1 z + \alpha_0 + \alpha_1 z^{-1} + \alpha_2 z^{-2} + \alpha_3 z^{-3}
$$
with the constraint $2\alpha_3 + 2\alpha_2 + 2\alpha_1 + \alpha_0 = 1$. The key design insight is to place zeros of its magnitude response precisely at the problematic frequencies $\omega_s/6$ and $\omega_s/2$. This imposes conditions on the coefficients:
$$
\begin{aligned}
Q(e^{j\omega T_s})|_{\omega=\omega_s/6} &= 0 \quad \Rightarrow \quad \alpha_1 = 0.25 – 0.5\alpha_0 \\
Q(e^{j\omega T_s})|_{\omega=\omega_s/2} &= 0 \quad \Rightarrow \quad \alpha_2 = -0.5\alpha_0 \\
& \quad \Rightarrow \quad \alpha_3 = 0.25 + 0.5\alpha_0
\end{aligned}
$$
Thus, the filter has only one free parameter, $\alpha_0$, which controls the position of a third zero between $\omega_s/6$ and $\omega_s/2$, enhancing high-frequency roll-off. When this third-order $Q(z)$ is used, $f(\omega)$ remains positive for all $\omega$, even with a large $k_{rc}$. Consequently, the RC-induced negative real parts in $Y_{cg}$ are completely eliminated, as confirmed by the admittance plots. This allows the PRC to operate with high gain without threatening stability.
However, the inherent negative real part of $Y_{cg}$ from the LCL filter remains. To achieve full passivity, I employ a capacitor voltage feedforward (CVF) path with a specific design. The CVF controller is chosen as:
$$
G_v(s) = K_f (1 – a + a e^{-sT_s})
$$
where $K_f$ is the feedforward gain and $a$ is a low-pass filter coefficient. This structure allows the sub-admittance $Y_{cv}(s)$ to have a positive real part that can exactly cancel the negative real part of $Y_{cg}(s)$ in the low-to-mid frequency range. By properly selecting $K_f$ and $a$, the overall inverter output admittance $Y_c(s) = Y_{cg}(s) + Y_{cv}(s)$ can be shaped to have a non-negative real part for all frequencies up to the Nyquist frequency. The following table summarizes the key system parameters used for design and validation.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link Voltage | $V_{dc}$ | 400 V |
| Grid Voltage (RMS) | $V_g$ | 220 V |
| Grid Fundamental Frequency | $f_0$ | 50 Hz |
| Sampling Frequency | $f_s$ | 16 kHz |
| Inverter-side Inductor | $L_1$ | 600 µH |
| Filter Capacitor | $C$ | 10 µF |
| Grid-side Inductor | $L_2$ | 150 µH |
| Base Grid Inductance | $L_g$ | 0-200 µH |
| Proportional Gain | $k_p$ | 6 |
| RC Gain | $k_{rc}$ | 1 or 4 |
| Phase Lead Steps | $m$ | 3 |
| 1st-order $Q(z)$ coeff. | $(\alpha_0, \alpha_1)$ | (0.2, 0.4) |
| 3rd-order $Q(z)$ coeff. | $(\alpha_0, \alpha_1, \alpha_2, \alpha_3)$ | (0.2, 0.15, -0.1, 0.35) |
| CVF Gain | $K_f$ | 1.0 |
| CVF Filter Coeff. | $a$ | 0.8 |
To validate the theory, I performed extensive simulations and experiments. The stability was first assessed theoretically using Bode plots of the inverter output admittance $Y_c(j\omega)$ and the equivalent grid admittance $Y_{ge}(j\omega)=1/(j\omega(L_2+L_g))$. The phase margin at their intersection indicates stability. The results are summarized below:
| Case | Grid $L_g$ | RC Filter $Q(z)$ | $k_{rc}$ | $\text{Re}\{Y_c\}$ Neg. Regions? | Phase Margin | Predicted Stability |
|---|---|---|---|---|---|---|
| 1 | 0 µH | 1st-order | 4 | Yes (but no intersection) | > 0° | Stable |
| 2 | 200 µH | 1st-order | 1 | Reduced | > 0° | Stable |
| 3 | 200 µH | 1st-order | 4 | Yes | < 0° | Unstable |
| 4 | 200 µH | 3rd-order | 4 | No | > 0° | Stable |
Simulation results in MATLAB/Simulink perfectly corroborated these predictions. With $L_g=0$ µH, both the first-order and third-order $Q(z)$ filters yielded stable operation with low grid current THD. With $L_g=200$ µH and the first-order $Q(z)$, the system was stable with $k_{rc}=1$ but became unstable with oscillatory divergent grid current when $k_{rc}=4$. In stark contrast, with the proposed third-order $Q(z)$, the system remained perfectly stable even with $k_{rc}=4$ under the weak grid condition. This demonstrates that the proposed filter eliminates the stability threat without sacrificing the high-gain, fast-response benefit of the RC. Furthermore, the dynamic performance was tested under grid voltage sags and swells. The grid current showed no oscillation and recovered to steady-state within two fundamental cycles, confirming excellent transient response.
For a fair comparison, I also tested a conventional plug-in RC controller with the same first-order $Q(z)$ and only CVF for admittance shaping. While stable under strong grid ($L_g=0$), it became unstable under the weak grid ($L_g=200$ µH), highlighting the superior robustness of the proposed proportional-repetitive control structure combined with the third-order filter.
The experimental validation was conducted on a 5-kW three-phase LCL-type grid-connected inverter prototype. The control algorithm was implemented on a TMS320F28335 DSP. The experimental results were consistent with simulations. With $L_g=200$ µH, using the first-order $Q(z)$ and $k_{rc}=4$ led to growing oscillations in both inverter-side ($i_1$) and grid-side ($i_2$) currents, indicating instability. Reducing $k_{rc}$ to 1 restored stability. However, when the proposed third-order $Q(z)$ was deployed with $k_{rc}=4$, the inverter operated stably with clean, sinusoidal currents. This conclusive experimental evidence validates the practical effectiveness of the proposed passivity-based repetitive control strategy for LCL-type grid-connected inverters.
In conclusion, I have presented a comprehensive design methodology for a repetitive controller that ensures both high performance and robust stability for LCL-type grid-connected inverters. The core of the strategy is the application of passivity theory as a design constraint. The novel third-order zero-phase-shift low-pass filter within the RC’s internal model is the key innovation that selectively eliminates the negative-real-part admittance introduced by the high-gain RC at harmonic frequencies. This allows the RC to retain its high gain for fast dynamic response and precise harmonic suppression. The capacitor voltage feedforward technique then complements this by canceling the inherent negative real part of the LCL filter’s admittance. The synergy of these techniques results in an inverter output admittance that is passive across the entire Nyquist frequency range. Consequently, the grid-connected inverter system is guaranteed to be stable when connected to any passive grid impedance, providing unprecedented robustness for weak grid integration. This work, therefore, offers a significant step forward in the design of reliable and high-performance power converters for renewable energy systems, making the grid-connected inverter a more resilient and adaptable component in the modern electrical grid.