The global energy landscape is undergoing a profound transformation, driven by the urgent need to address environmental degradation and the depletion of fossil fuels. Renewable energy sources, particularly photovoltaic and wind power, have emerged as pivotal components of sustainable power systems. At the heart of integrating these distributed energy resources into the main utility network lies the on grid inverter. The performance, efficiency, and, most critically, the stability of these on grid inverters directly dictate the power quality and reliability of the entire grid. As penetration levels increase, on grid inverters are increasingly required to operate in challenging grid conditions characterized by high impedance, commonly referred to as weak or extremely weak grids.
This image exemplifies a modern grid-tied energy system, where the on grid inverter plays the crucial role of converting and synchronizing DC power from solar panels (and batteries) with the AC utility grid. In such applications, maintaining stable and high-quality power injection under varying grid strengths is paramount. A fundamental challenge in the control of LCL-type on grid inverters is the mitigation of grid background harmonics and the enhancement of reference tracking. The grid voltage proportional feedforward strategy is a widely adopted technique to address this. By injecting a signal proportional to the point of common coupling (PCC) voltage into the control loop, this strategy effectively improves the system’s ability to reject grid voltage disturbances and boosts the gain at low frequencies. The standard feedforward path with a gain $$H_f$$ is typically implemented, where $$H_f = 1 / K_{PWM}$$ and $$K_{PWM} = V_{in}/V_{tri}$$ is the inverter bridge gain.
However, this beneficial effect comes at a significant cost when the on grid inverter is connected to a weak grid. The interaction between the feedforward path and the grid impedance, represented as $$Z_g(s) = sL_g$$, can lead to severe degradation of the system’s phase margin. As the grid inductance $$L_g$$ increases, the phase of the on grid inverter‘s equivalent output impedance can drop precipitously, potentially crossing the -90-degree threshold and leading to instability according to the impedance-based stability criterion. This paper delves into this critical issue and proposes a novel, adaptive solution. We begin by establishing the output impedance model of an LCL-type on grid inverter with standard feedforward to analytically demonstrate the phase margin deterioration mechanism. We then introduce an improved feedforward strategy incorporating a second-order low-pass filter (LPF) to reshape the output impedance. Finally, to overcome the limitations of fixed-parameter design, we propose an adaptive parameter tuning method that synergistically combines online grid impedance detection with a genetic optimization algorithm, thereby significantly enhancing the robustness and grid adaptability of the on grid inverter.
Output Impedance Modeling and Stability Analysis for On Grid Inverters
To rigorously analyze the stability problem, we first derive the equivalent output impedance model for a single-phase LCL-type on grid inverter. The system typically employs a current controller, often a quasi-Proportional-Resonant (PR) type, and a capacitor current active damping loop. The PR controller transfer function is given by:
$$G_c(s) = K_P + \frac{2K_r\omega_i s}{s^2 + 2\omega_i s + \omega_o^2}$$
where $$K_P$$ is the proportional gain, $$K_r$$ is the resonant gain, $$\omega_i$$ is the cutoff frequency, and $$\omega_o$$ is the fundamental angular frequency. From the system block diagram, we can define the plant transfer functions. The open-loop transfer function $$T_o(s)$$ of the current control loop without considering the feedforward effect is:
$$T_o(s) = G_c(s) \cdot \frac{K_{PWM}}{L_1 L_2 C s^3 + K_d K_{PWM} L_2 C s^2 + (L_1 + L_2)s}$$
where $$K_d$$ is the active damping coefficient. To isolate the effect of the grid voltage feedforward, we set the current reference to zero and derive the relationship between the grid current $$i_g(s)$$ and the PCC voltage $$u_{pcc}(s)$$. This allows us to define the equivalent output impedance $$Z_{req}(s)$$ of the on grid inverter with standard proportional feedforward as:
$$Z_{req}(s) = \frac{u_{pcc}(s)}{-i_g(s)} = \frac{1 + T_o(s)}{G_2(s) – H_f T_o(s) / G_c(s)}$$
This impedance can be conceptually decomposed into two parallel components: the native inverter output impedance without feedforward $$Z_0(s)$$, and an impedance term $$Z_{ext}(s)$$ introduced solely by the feedforward loop. The relationship is:
$$Z_{req}(s) = Z_0(s) \cdot Z_{ext}(s)$$
where
$$Z_0(s) = \frac{1 + T_o(s)}{G_2(s)} \quad \text{and} \quad Z_{ext}(s) = \frac{1}{1 – H_f G_1(s)}$$
According to the impedance-based stability criterion for a grid-connected system, a sufficient condition for stability is that the phase of the on grid inverter‘s output impedance at the frequency where its magnitude intersects with the grid impedance magnitude must be greater than -90 degrees when the grid impedance is purely inductive:
$$\text{PM} = 180^\circ + \arg(Z_{req}(j2\pi f_c)) > 0^\circ \quad \text{or equivalently} \quad \arg(Z_{req}(j2\pi f_c)) > -90^\circ$$
Analysis of the Bode plot for $$Z_{req}(s)$$ reveals the core problem. While the standard feedforward successfully increases the magnitude of $$Z_{req}(s)$$ at low frequencies (improving disturbance rejection), it drastically reduces the phase in the low-to-mid frequency range. The introduced impedance term $$Z_{ext}(s)$$ has a phase characteristic that pulls down the overall phase of $$Z_{req}(s)$$. As the grid inductance $$L_g$$ increases, the intersection frequency $$f_c$$ shifts lower, often into a region where the phase margin has been severely compromised by the feedforward action. This can lead to the violation of the stability condition, resulting in harmonic oscillation or even instability for the on grid inverter.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Rated Power | P | 6 | kW |
| DC Link Voltage | V_in | 400 | V |
| Grid Voltage (RMS) | V_g | 220 | V |
| Switching Frequency | f_s | 10 | kHz |
| Inverter-side Inductor | L_1 | 3 | mH |
| Grid-side Inductor | L_2 | 1 | mH |
| Filter Capacitor | C | 5 | μF |
| PR Controller Proportional Gain | K_P | 0.063 | – |
| PR Controller Resonant Gain | K_r | 3.14 | – |
| Active Damping Coefficient | K_d | 0.15 | – |
Proposed Adaptive Improved Feedforward Strategy
To mitigate the destabilizing effect of the standard feedforward while preserving its beneficial properties, we propose a modified feedforward structure. The core idea is to insert a second-order low-pass filter (LPF) in series with the proportional feedforward gain. The transfer function of this filter is:
$$H_0(s) = \frac{\omega_n^2}{s^2 + \frac{\omega_n}{Q}s + \omega_n^2}$$
where $$\omega_n$$ is the natural frequency (or cutoff frequency) and $$Q$$ is the quality factor (damping coefficient). This $$H_0(s)$$ block is placed directly in the feedforward path. The new feedforward transfer function becomes $$H_f \cdot H_0(s) / G_c(s)$$ in the system block diagram. Consequently, the improved equivalent output impedance $$Z’_{req}(s)$$ of the on grid inverter is:
$$Z’_{req}(s) = \frac{1 + T_o(s)}{G_2(s) – H_0(s) H_f T_o(s) / G_c(s)} = Z_0(s) \cdot Z’_{ext}(s)$$
where the new feedforward-induced impedance term is:
$$Z’_{ext}(s) = \frac{1}{1 – H_0(s) H_f G_1(s)}$$
The impact of $$H_0(s)$$ is profound. A properly designed second-order LPF provides a phase boost (less negative phase) in a specific frequency range. When applied to the feedforward path, it reshapes $$Z’_{ext}(s)$$, increasing its phase compared to the original $$Z_{ext}(s)$$. This phase boost in $$Z’_{ext}(s)$$ counteracts the phase-lagging effect of the original feedforward, thereby raising the overall phase of $$Z’_{req}(s)$$. The result is a significant recovery of the system’s phase margin at the critical intersection frequency with the grid impedance. Furthermore, by setting $$\omega_n$$ sufficiently low (e.g., around the fundamental frequency or lower), the LPF maintains high gain (near unity) at low frequencies, thus largely preserving the original feedforward’s ability to suppress low-frequency grid voltage disturbances and improve tracking. The design challenge lies in optimally selecting the two parameters $$\omega_n$$ and $$Q$$ to achieve the best possible phase margin recovery for a given grid condition without overly sacrificing low-frequency performance.
| Feature | No Feedforward | Standard Proportional Feedforward | Proposed Improved Feedforward |
|---|---|---|---|
| Low-Freq Disturbance Rejection | Poor | Excellent | Good to Excellent |
| Phase Margin in Weak Grid | High (Unaffected by L_g) | Very Low (Degrades with L_g) | High (Recovered by design) |
| Stability in Extremely Weak Grid | Stable | Unstable | Stable |
| Parameter Complexity | None | Single gain (H_f) | Two parameters (ω_n, Q) |
| Adaptability to Grid Changes | N/A | None (Fixed) | Possible with online tuning |
Genetic Algorithm-Based Adaptive Parameter Tuning
A fixed-parameter second-order LPF, while effective for a specific grid impedance, may not be optimal or even sufficient as the grid strength varies. Manual tuning of $$\omega_n$$ and $$Q$$ for every possible condition is impractical. Therefore, we propose an adaptive parameter tuning strategy that combines real-time grid impedance identification with an optimization algorithm. The process is automated and consists of the following steps:
- Online Grid Impedance Detection: The on grid inverter employs a real-time algorithm (e.g., based on injecting a small perturbation signal or analyzing natural harmonics) to estimate the dominant grid inductance $$L_{g,est}$$.
- Optimization Problem Formulation: Using the estimated $$L_g$$ and the derived model for $$Z’_{req}(s, \omega_n, Q)$$, we formulate an optimization problem. The objective is to find the parameters $$\omega_n$$ and $$Q$$ that maximize the phase margin (PM) of the system. A typical formulation aiming for a target PM (e.g., 60°) is:
$$
\begin{aligned}
& \underset{\omega_n, Q}{\text{minimize}}
& & f = | \text{PM}(\omega_n, Q) – 60^\circ | \\
& \text{subject to}
& & 0 < \omega_n < \omega_{max} \\
& & & 0 < Q < 1
\end{aligned}
$$
Here, $$\text{PM}(\omega_n, Q)$$ is calculated by finding the crossover frequency $$f_c$$ where $$|Z’_{req}(j2\pi f_c)| = |Z_g(j2\pi f_c)|$$ and then evaluating $$180^\circ + \arg(Z’_{req}(j2\pi f_c))$$. - Genetic Algorithm Optimization: A Genetic Algorithm (GA) is employed to solve this nonlinear, constrained optimization problem efficiently. GA is a population-based metaheuristic inspired by natural selection. It operates on a set of candidate solutions (chromosomes representing $$[\omega_n, Q]$$ pairs). The fitness of each chromosome is inversely related to the objective function $$f$$. Through iterative operations of selection, crossover, and mutation, the GA evolves the population toward an optimal or near-optimal solution. The key advantage is its ability to explore a wide parameter space without being trapped in local minima, which is ideal for this application.
- Parameter Update: The optimal parameters $$[\omega_n^*, Q^*]$$ found by the GA are then applied to update the coefficients of the second-order LPF $$H_0(s)$$ in the feedforward path of the on grid inverter.
This closed-loop adaptive strategy enables the on grid inverter to continuously and automatically adjust its feedforward compensator in response to changing grid conditions, ensuring robust stability with a consistent phase margin target.
| Step | Action | Description/Output |
|---|---|---|
| 1 | Impedance Identification | Estimate current grid inductance $$L_{g,est}$$ using online detection techniques. |
| 2 | Problem Setup | Formulate the phase margin maximization problem based on $$L_{g,est}$$ and system model $$Z’_{req}(s)$$. |
| 3 | GA Initialization | Generate an initial population of random $$[\omega_n, Q]$$ pairs within defined bounds. |
| 4 | Fitness Evaluation | Calculate the phase margin for each chromosome and evaluate its fitness (closeness to target PM). |
| 5 | Genetic Operations | Perform selection, crossover, and mutation to create a new generation of candidate solutions. |
| 6 | Convergence Check | Repeat steps 4-5 until a maximum generation count is reached or a fitness threshold is met. |
| 7 | Parameter Update | Apply the best-found $$[\omega_n^*, Q^*]$$ to the feedforward filter $$H_0(s)$$ of the on grid inverter. |
Simulation and Validation
The proposed adaptive improved feedforward strategy was rigorously validated through simulation and its principles confirmed. A detailed model of a 6 kW single-phase LCL on grid inverter was constructed using the parameters listed in Table 1. The performance was evaluated under two distinct grid conditions: a weak grid with $$L_g = 2.56 \text{ mH}$$ (Short Circuit Ratio, SCR ≈ 10) and an extremely weak grid with $$L_g = 12.84 \text{ mH}$$ (SCR ≈ 2).
First, the system with the standard proportional feedforward was tested. Under the weak grid condition ($$L_g=2.56 \text{ mH}$$), the on grid inverter operated stably, producing sinusoidal grid current with low total harmonic distortion (THD). However, under the extremely weak grid condition ($$L_g=12.84 \text{ mH}$$), the standard feedforward strategy failed. Severe oscillation and distortion appeared in both the grid current and voltage waveforms, with the current THD exceeding 5%, clearly indicating instability. This observation aligns perfectly with the theoretical instability predicted by the impedance model’s phase margin analysis.
Subsequently, the proposed strategy with a fixed, manually tuned second-order LPF (e.g., $$\omega_n = 1000 \text{ rad/s}, Q=0.1$$) was applied. The results showed a remarkable recovery of stability. Even with $$L_g=12.84 \text{ mH}$$, the on grid inverter resumed stable operation, generating clean current and voltage waveforms with THD well within acceptable limits. The output impedance Bode plot confirmed a significant phase boost in the critical frequency region, restoring a healthy phase margin of over 58 degrees.
Finally, the complete adaptive strategy was implemented. For the $$L_g=12.84 \text{ mH}$$ case, the GA was tasked with optimizing the LPF parameters to achieve a target phase margin of 60 degrees. The algorithm successfully converged, yielding optimal parameters $$\omega_n^* \approx 423.44 \text{ rad/s}$$ and $$Q^* \approx 0.6085$$. When these parameters were applied, the measured phase margin was 60.7°, demonstrating the effectiveness of the automated tuning process. Furthermore, the strategy’s robustness was tested by simulating a step change in grid impedance. When the grid inductance suddenly increased from 2.56 mH to 12.84 mH, the on grid inverter equipped with the adaptive feedforward maintained stable and high-quality power injection without any transient instability, whereas the system with standard feedforward collapsed into oscillation. Load step-change tests (from half-load to full-load and vice-versa) under weak grid conditions also confirmed excellent dynamic performance and harmonic suppression capability of the proposed adaptive on grid inverter control scheme.
Conclusion
In this paper, we have thoroughly investigated the stability challenge posed by standard grid voltage proportional feedforward in LCL-type on grid inverters operating in extremely weak grids. Through impedance-based modeling and analysis, we explicitly demonstrated how this feedforward strategy, while beneficial for disturbance rejection, actively degrades the system phase margin as grid impedance increases, leading to potential instability. To resolve this fundamental conflict, we proposed a novel adaptive improved feedforward control strategy. The core of the solution is the insertion of a second-order low-pass filter into the feedforward path. This filter is designed to provide a controlled phase lead that counteracts the phase-lagging effect of the original feedforward loop, thereby effectively recovering the system’s phase margin and ensuring stability.
Recognizing the limitation of fixed-filter parameters in a dynamically changing grid environment, we advanced the strategy by incorporating an online adaptation mechanism. This mechanism synergistically combines real-time grid impedance identification with a Genetic Algorithm-based optimizer. The GA automatically and efficiently tunes the two key parameters of the second-order filter (natural frequency $$\omega_n$$ and quality factor $$Q$$) to maintain a consistent, user-defined target phase margin regardless of the grid strength. This renders the on grid inverter inherently robust and self-adaptive.
Comprehensive simulation studies validate the theoretical developments. The results unequivocally show that the proposed adaptive strategy enables stable, high-performance operation of the on grid inverter under a wide range of grid conditions, including extremely weak grids with very low short-circuit ratios where traditional methods fail. It successfully maintains low harmonic distortion in the output current during steady-state and exhibits robust dynamic performance during grid impedance variations and load transients. Therefore, this work provides a practical and effective solution to enhance the grid resilience and interoperability of modern on grid inverters, facilitating the higher and safer integration of renewable energy sources into power systems of the future.