The global energy transition is driving the rapid integration of renewable energy sources like solar and wind into power systems. A significant challenge arises as these resources are often located in remote areas far from load centers, necessitating connection through inherently weak grid infrastructures characterized by high grid impedance. The stability and power quality of on grid inverters, the essential interface between distributed generation and the utility grid, become critically challenged under such conditions. A common metric for grid strength is the Short-Circuit Ratio (SCR), defined as the ratio of the grid’s short-circuit capacity at the Point of Common Coupling (PCC) to the rated capacity of the generating unit. An SCR of 1.5 or lower defines an “ultra-weak” grid, posing stringent requirements for on grid inverter control.
Traditional current-controlled, or grid-following, on grid inverters face two primary issues in ultra-weak grids: 1) Reduced stability margins due to adverse interactions between the inverter’s control loops (especially the Phase-Locked Loop, PLL) and the grid impedance, potentially leading to oscillations; and 2) Amplification of low-frequency current harmonics originating from the inverter’s own switching or background grid distortion, degrading power quality. While grid-forming control strategies (like virtual synchronous generators) offer inherent stability in weak grids, they often suffer from slower dynamic response and may have stability issues in strong grids. Therefore, enhancing the performance of the prevalent grid-following on grid inverter topology remains a crucial research focus.
This article presents a comprehensive analysis of these challenges and proposes a novel output impedance reshaping strategy based on voltage feedforward compensation. First, a detailed small-signal model of an LC-filtered on grid inverter in the dq-frame is established, explicitly accounting for the influence of the PLL and control delays to derive an accurate expression for the inverter’s output impedance. Second, a generalized method for analyzing low-frequency current harmonic amplification is developed using the Nyquist curve of the impedance ratio. Subsequently, the core contribution—a voltage feedforward-based impedance reshaping method—is introduced. An exact compensator transfer function is derived and then simplified for practical digital implementation. Finally, the effectiveness of the proposed harmonic analysis and compensation strategy is validated through Hardware-in-the-Loop (HIL) simulation using parameters from a commercial on grid inverter product.
1. Output Impedance Modeling of an LC-Filtered On Grid Inverter
The topology and control structure of a three-phase two-level voltage source on grid inverter with an LCL filter (simplified to LC for analysis where the grid-side inductor is considered part of the grid impedance $L_g$) is considered. The standard control structure includes an inner inductor current PI control loop, a PLL (typically SRF-PLL), and a proportional grid voltage feedforward path. The primary parameters are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | $P_N$ | 250 kW |
| Grid Voltage (L-L, RMS) | $V_g$ | 800 V |
| DC-Link Voltage | $u_{dc}$ | 1300 V |
| Inverter-side Inductor | $L_1$ | 120 µH |
| Filter Capacitor | $C$ | 34 µF |
| Switching / Control Frequency | $f_s$, $T_s$ | 16 kHz |
| Grid Inductor (SCR=1.5) | $L_g$ | 5.3 mH |
| Grid Resistor | $R_g$ | 0.1 Ω |
| Current PI: Proportional Gain | $k_p$ | 0.754 |
| Current PI: Integral Gain | $k_i$ | 43.8 |
| PLL PI: Proportional Gain | $k_{p,PLL}$ | 0.0962 |
| PLL PI: Integral Gain | $k_{i,PLL}$ | 2.207 |
| Voltage Feedforward Coefficient | $k_f$ | 0.95 |
To analyze stability, the impedance-based approach is employed. The system is modeled in the dq synchronous reference frame. While the full model exhibits coupling between d and q axes due to the PLL, it is widely accepted that the cross-coupling impedance terms are significantly smaller than the direct-axis terms. Furthermore, the PLL’s destabilizing effect is predominantly manifested in the q-axis channel. Therefore, focusing on the q-axis output impedance $Z_o(s)$ provides a conservative and sufficient basis for stability assessment.
The derived control block diagram for the q-axis, incorporating the PLL influence as a feedback path from the PCC voltage to the current reference, leads to the following expression for the q-axis output impedance $Z_o(s)$ of the on grid inverter:
$$Z_o(s) = \frac{sL_1 + G_i(s)G_d(s)}{1 + s^2 L_1 C – k_f G_d(s) + sC G_i(s)G_d(s) – I_d G_{PLL}(s)G_i(s)G_d(s)}$$
where:
- $G_i(s) = k_p + \frac{k_i}{s}$ is the current PI controller transfer function.
- $G_d(s) = k_{PWM} e^{-1.5sT_s}$ models the total digital control and PWM delay (1.5 sampling periods). $k_{PWM}$ is the gain, typically 1.
- $G_{PLL}(s) = \frac{k_{p,PLL} + k_{i,PLL}/s}{s + U_m(k_{p,PLL} + k_{i,PLL}/s)}$ is the linearized PLL transfer function, with $U_m$ being the voltage amplitude.
- $I_d$ is the steady-state d-axis current.
The grid impedance is $Z_g(s) = R_g + sL_g$. According to the Generalized Nyquist Criterion (GNC) or its simplified application for decoupled axes, the stability of the interconnected system can be assessed by the loop gain $Z_g(s)/Z_o(s)$. The phase margin (PM) at the crossover frequency $f_c$ where $|Z_g(j2\pi f_c)| = |Z_o(j2\pi f_c)|$ is a key metric:
$$\theta_{PM} = 180^\circ – \left[ \angle Z_g(j2\pi f_c) – \angle Z_o(j2\pi f_c) \right]$$
A larger positive PM indicates better stability. In weak grids, the phase of $Z_o(s)$ is often degraded, especially at low frequencies due to the PLL’s negative impedance effect, reducing $\theta_{PM}$ and risking instability.
2. Analysis of Low-Frequency Current Harmonic Amplification
Beyond stability, power quality is a major concern. The output current $i_g(s)$ of the on grid inverter can be expressed in terms of the equivalent circuit as:
$$i_g(s) = \underbrace{\left[ I_s(s) – \frac{V_g(s)}{Z_o(s)} \right]}_{M_1(s)} \cdot \underbrace{\frac{1}{1 + Z_g(s)/Z_o(s)}}_{M_2(s)}$$
where $I_s(s)$ is the equivalent current source from the controller. The term $M_2(s)$ dictates how grid impedance modifies the current. If $|M_2(j2\pi f)| > 1$ at a given frequency $f$, the corresponding current harmonic component is amplified compared to a strong grid case ($Z_g \approx 0$).
We define $N(s) = Z_g(s)/Z_o(s)$. The condition for harmonic amplification can be visualized graphically on the complex plane. Plotting the Nyquist curve of $N(j2\pi f)$ for $f \in [0, \infty)$ reveals that any part of the curve lying inside a circle centered at $(-1, j0)$ with a radius of 1 corresponds to frequencies where $|M_2(j2\pi f)| > 1$. The boundary of this “harmonic amplification region” is the circle $|1+N(j\omega)|=1$. The frequency $f_h$ where the $N(j2\pi f)$ curve intersects this circle marks the upper limit of the amplification band. The degree of amplification at any frequency within this band is given by $1 / |1+N(j2\pi f)|$. This method provides a more complete picture than the simplified rule-of-thumb requiring a 60-degree phase margin to avoid amplification only at $f_c$.
3. Proposed Voltage Feedforward-Based Impedance Reshaping Strategy
To simultaneously improve stability margin and suppress low-frequency harmonic amplification, the goal is to reshape the output impedance $Z_o(s)$ by increasing its phase at the crossover frequency $f_c$ without significantly altering its magnitude at that point. This is effectively a phase compensation or lead-lag compensation problem applied to the inverter’s output impedance characteristic.
The proposed method modifies the conventional proportional voltage feedforward path. Instead of a simple gain $k_f$, a dynamic compensator $G_f(s)$ is introduced. The design aims to make the new output impedance $Z_o'(s)$ relate to the original $Z_o(s)$ as:
$$Z_o'(s) = Z_o(s) \cdot \frac{1 + k_1 s}{1 + k_1 k_2 s} \cdot k_3$$
The term $\frac{1 + k_1 s}{1 + k_1 k_2 s} \cdot k_3$ is a lead-lag compensator. By proper parameter selection, it can provide positive phase boost at the target frequency $f_c$ while maintaining a gain of 0 dB (i.e., $k_3$ adjusts the DC gain).
To achieve this exact reshaping, the required feedforward compensator $G_f(s)$ is derived from the model:
$$
\begin{aligned}
G_f(s) = & \frac{s^2 L_1 C + sC G_i G_d – I_d G_{PLL} G_i G_d + 1}{G_d} \\
& – \frac{s^2 L_1 C + sC G_i G_d – k_f G_d – I_d G_{PLL} G_i G_d + 1}{G_d} \cdot \frac{(1 + k_1 k_2 s)}{(1 + k_1 s)k_3}
\end{aligned}
$$
However, this exact $G_f(s)$ is complex, containing differentiators and the inverse of the delay $G_d(s)$, making it impractical for implementation. A significant contribution is the simplification of this compensator for real-world application. By neglecting terms that primarily affect very high or very low frequencies (like the capacitor $C$ and PLL $G_{PLL}$ dynamics for the mid-frequency range of interest around $f_c$), and approximating the delay inverse, a practical feedforward compensator $G_f'(s)$ is obtained:
$$
G_f'(s) = \left[1 – \left(1 – \frac{k_f}{1 + 1.5sT_s}\right) \cdot \frac{1 + k_1 k_2 s}{1 + k_1 s} k_3 \right] \cdot \frac{1 + 1.5sT_s}{1 + 1.5\alpha s T_s}
$$
where $\alpha$ is a filter tuning parameter between 0 and 1 to manage high-frequency noise. The parameters $k_1$, $k_2$, and $k_3$ are designed to achieve a desired phase boost $\phi$ at $f_c$:
$$
k_1 = \frac{\tan \phi + \sqrt{1 + \tan^2 \phi}}{2\pi f_c}, \quad
k_2 = \frac{1}{(2\pi f_c k_1)^2}, \quad
k_3 = \frac{1 + (2\pi f_c k_1 k_2)^2}{1 + (2\pi f_c k_1)^2}
$$
This simplified $G_f'(s)$ consists of cascaded lead-lag and low-pass filters, which are straightforward to implement in a digital controller, making the impedance reshaping strategy highly feasible for practical on grid inverter products.
4. Validation via Hardware-in-the-Loop Simulation
A HIL simulation platform was constructed using a real-time simulator (e.g., MT6040) to emulate the power circuit and grid, while the actual DSP control board from a commercial on grid inverter executed the control algorithms, including the proposed feedforward compensator. Parameters from Table 1 were used. Two grid conditions were tested: a strong grid with SCR = 8 ($L_g=1$ mH) and an ultra-weak grid with SCR = 1.5.
Stability and Waveform Improvement: Under full load, the on grid inverter output voltage and current were measured. With SCR=1.5 and conventional feedforward ($k_f=0.95$), significant distortion was observed in both voltage and current waveforms due to harmonic amplification and reduced stability margin, with current THD reaching 1.41%. After implementing the simplified impedance reshaping compensator ($G_f'(s)$) designed to boost the phase by 30° at the original crossover frequency $f_c=162$ Hz, the waveform quality improved markedly. The current THD reduced to 1.11%, and voltage THD reduced from 4.75% to 3.70%, demonstrating enhanced stability and power quality for the on grid inverter.
Low-Frequency Harmonic Suppression: The envelope of the PCC voltage over a longer time scale clearly showed increased low-frequency ripple for the uncompensated SCR=1.5 case compared to the SCR=8 case, visually confirming low-frequency harmonic amplification. This ripple was substantially suppressed after applying the proposed impedance reshaping strategy. A detailed spectral analysis of the output current up to 250 Hz validated this: the uncompensated weak grid case showed pronounced amplification of harmonic components in the 100-200 Hz range compared to the strong grid baseline. The compensated case showed significantly reduced harmonic magnitudes across this low-frequency band, aligning perfectly with the theoretical prediction that the $N(s)$ Nyquist curve was pushed away from the (-1, j0) point.
5. Conclusion
This work addresses the critical challenges of stability and power quality for grid-following on grid inverters operating in ultra-weak grids. A precise output impedance model incorporating PLL effects and digital delays was established, forming the basis for a clear stability assessment using phase margin. A generalized graphical method based on the Nyquist plot of $Z_g/Z_o$ was introduced, providing a powerful tool for analyzing the frequency-dependent amplification of current harmonics, which is more comprehensive than rules based solely on phase margin at crossover.
The core of the solution is a novel output impedance reshaping strategy implemented via the voltage feedforward path. An exact compensator transfer function was derived and then intelligently simplified into a practical, implementable form consisting of lead-lag and filtering blocks. This compensator is designed to directly increase the phase of the on grid inverter‘s output impedance at the critical frequency, thereby improving the system phase margin and moving the impedance ratio $N(s)$ away from the critical (-1, j0) point in the complex plane.
HIL simulation results, using parameters from a real 250 kW commercial on grid inverter, conclusively validated the proposed theory and method. The impedance reshaping strategy successfully stabilized the system and significantly suppressed the amplification of low-frequency current harmonics in an SCR=1.5 ultra-weak grid scenario, leading to improved voltage and current waveforms. This work provides a practical and effective reference for enhancing the robustness and power quality of on grid inverters in future renewable-rich, weak-grid power systems.