With the rapid proliferation of distributed generation systems, the power grid is increasingly characterized by high impedance due to factors such as dispersed locations, long-distance transmission lines, and numerous transformer installations. This results in a weak grid environment, where the stability of grid-tied inverters can be severely compromised by grid impedance variations. Typically, a grid-tied inverter relies on a phase-locked loop (PLL) to obtain accurate grid synchronization information. Additionally, to suppress background harmonics and amplitude disturbances in grid voltage, grid voltage feedforward (GVF) control is often employed. However, it has been observed that both PLL and GVF can introduce negative resistance characteristics into the inverter’s output impedance, particularly in the q-axis, leading to instability under weak grid conditions. In this paper, we investigate the underlying mechanisms and propose a novel control strategy centered on q-axis impedance reshaping to enhance the stability of grid-tied inverters. Our approach involves designing an impedance controller that actively reshapes the q-axis impedance to exhibit positive resistance behavior in the low-frequency range, thereby mitigating the destabilizing effects introduced by PLL and GVF.
The integration of renewable energy sources, such as solar and wind, has led to a significant increase in the penetration of grid-tied inverters. These inverters play a crucial role in converting DC power from distributed generators into AC power that can be fed into the grid. However, the weak grid scenario, characterized by non-negligible grid impedance, poses significant challenges to the stable operation of grid-tied inverters. The core issue stems from the interaction between the inverter’s control loops and the grid impedance. Specifically, the PLL, which tracks the grid phase, and the GVF, which compensates for grid voltage disturbances, can inadvertently modify the inverter’s output impedance. In the synchronous reference frame (d-q frame), the q-axis output impedance may exhibit negative resistance in certain frequency bands, violating the passivity criterion and leading to oscillations or even instability. This paper delves into a detailed modeling analysis to quantify these effects and presents a systematic solution through impedance reshaping.
To thoroughly understand the stability issues, we first develop a comprehensive small-signal model of a grid-tied inverter incorporating both PLL and GVF. Consider a three-phase grid-tied inverter with an L-type filter, as commonly used in many applications. The main circuit topology includes a DC voltage source \(V_{dc}\), a filter inductance \(L_c\), and the grid represented by a voltage source \(e_{g,abc}\) in series with a grid inductance \(L_g\). The inverter outputs current \(i_g\) to the grid at the point of common coupling (PCC). In the control scheme, the currents are transformed into the d-q synchronous reference frame using the angle \(\theta\) provided by the PLL. The d-axis current reference \(I_{gdref}\) is typically set to regulate active power, while the q-axis current reference \(i_{gqref}\) is often used for reactive power control or set to zero for unity power factor operation. The actual d-axis and q-axis current components are denoted as \(I_{gd}\) and \(i_{gq}\), respectively. The PCC voltages in the d-q frame are \(U_{PCCd}\) and \(u_{PCCq}\).
The current control loop typically employs a proportional-integral (PI) controller in the discrete-time domain. Using Tustin transformation for discretization, the PI controller transfer function is given by:
$$ PI(z) = K_P + K_I \cdot G_{int}(z) $$
where \(K_P\) and \(K_I\) are the proportional and integral gains, respectively, and \(G_{int}(z)\) represents a discrete integrator:
$$ G_{int}(z) = \frac{T_s}{2} \cdot \frac{z+1}{z-1} $$
Here, \(T_s\) is the sampling period of the inverter. The plant model for the L-filter inverter in the discrete domain, considering a zero-order hold, can be approximated as:
$$ G_{INV}(z) = \frac{T_s}{L_c (z-1)} $$
The closed-loop current transfer function, excluding the effects of PLL and GVF, is:
$$ G_{cl}(z) = \frac{i_{gq}(z)}{i_{gqref}(z)} = \frac{z^{-1} PI(z) G_{INV}(z)}{1 + z^{-1} PI(z) G_{INV}(z)} $$
Now, we incorporate the PLL dynamics. The PLL typically uses a PI controller to regulate the q-axis PCC voltage to zero, thereby extracting the grid phase. The small-signal transfer function from perturbations in \(u_{PCCq}\) to perturbations in the PLL output angle \(\theta\) is:
$$ G_{PLL}(z) = \frac{\Delta \theta(z)}{\Delta u_{PCCq}(z)} = \frac{G_{PLL\_PI}(z) G_{int}(z)}{1 + G_{PLL\_PI}(z) G_{int}(z) U_{PCCd}} $$
where \(G_{PLL\_PI}(z) = K_{PLL\_P} + K_{PLL\_I} G_{int}(z)\) is the PI controller for the PLL, with \(K_{PLL\_P}\) and \(K_{PLL\_I}\) as its gains, and \(U_{PCCd}\) is the steady-state d-axis PCC voltage magnitude.
Next, we model the GVF. In the d-q frame, the GVF introduces a feedforward path from the PCC voltages to the modulator. However, due to the PLL, the voltages are transformed between the inverter’s d-q frame and the PLL’s d-q frame. The small-signal relationship is captured by a transformation matrix:
$$ \begin{bmatrix} \Delta u_{PCCd\_PLL}(z) \\ \Delta u_{PCCq\_PLL}(z) \end{bmatrix} = \begin{bmatrix} 1 & -U_{iq} G_{PLL}(z) \\ 0 & G_f(z) [1 – U_{PCCd} G_{PLL}(z)] \end{bmatrix} \begin{bmatrix} \Delta u_{PCCd}(z) \\ \Delta u_{PCCq}(z) \end{bmatrix} = H_V^{PLL} \begin{bmatrix} \Delta u_{PCCd}(z) \\ \Delta u_{PCCq}(z) \end{bmatrix} $$
where \(G_f(z)\) is the original feedforward transfer function, often set to 1 for ideal compensation, and \(U_{iq}\) is the steady-state q-axis inverter voltage. The matrix \(H_V^{PLL}\) encapsulates the coupling introduced by the PLL.
Combining these components, the overall q-axis small-signal model of the grid-tied inverter can be derived. The q-axis output admittance \(Y_{oq}(z)\), defined as the negative ratio of the q-axis current perturbation to the q-axis PCC voltage perturbation, is a key metric for stability analysis. It can be expressed as the sum of three components:
$$ Y_{oq}(z) = -\frac{i_{gq}(z)}{u_{PCCq}(z)} = Y_{INV}(z) + Y_{PLL}(z) + Y_{GVF}(z) $$
where:
$$ Y_{INV}(z) = \frac{G_{cl}(z)}{z^{-1} PI(z)} $$
$$ Y_{PLL}(z) = -G_{cl}(z) G_{PLL}(z) \left( I_{gd} + \frac{U_{id}}{PI(z)} \right) $$
$$ Y_{GVF}(z) = -G_{cl}(z) \frac{G_f(z)}{PI(z)} \left[ 1 – G_{PLL}(z) U_{PCCd} \right] $$
Here, \(Y_{INV}(z)\) represents the inherent inverter admittance without PLL and GVF, \(Y_{PLL}(z)\) is the admittance contribution from the PLL, and \(Y_{GVF}(z)\) is from the GVF. The term \(U_{id}\) is the steady-state d-axis inverter voltage, related to \(I_{gd}\) by:
$$ U_{id} = \sqrt{ U_{PCCd}^2 + (\omega_0 L_c I_{gd})^2 } $$
where \(\omega_0\) is the grid fundamental frequency. The impedance is the inverse of admittance: \(Z_{oq}(z) = 1 / Y_{oq}(z)\).
A critical observation is that both \(Y_{PLL}(z)\) and \(Y_{GVF}(z)\) can exhibit negative real parts in the low-frequency range, effectively acting as negative resistors. This negative resistance behavior is a primary source of instability when the grid-tied inverter is connected to a weak grid with inductive impedance \(L_g\). The system stability can be assessed using the impedance-based criterion, which requires that the ratio of grid impedance to inverter output impedance satisfies the Nyquist stability condition. Negative resistance in the inverter impedance can lead to encirclements, indicating instability.
To illustrate the parameter dependencies, we summarize the key variables influencing the q-axis output admittance in Table 1.
| Symbol | Description | Typical Value or Range |
|---|---|---|
| \(K_P, K_I\) | Current PI controller gains | Designed for bandwidth ~1/10 of switching frequency |
| \(K_{PLL\_P}, K_{PLL\_I}\) | PLL PI controller gains | Designed for tracking bandwidth (~10-100 Hz) |
| \(L_c\) | Inverter filter inductance | e.g., 2 mH |
| \(L_g\) | Grid inductance | Variable, represents grid strength (0 to several mH) |
| \(U_{PCCd}\) | Steady-state d-axis PCC voltage | Near nominal grid voltage amplitude |
| \(I_{gd}\) | Steady-state d-axis current | Depends on power output |
| \(T_s\) | Sampling period | e.g., 100 µs (10 kHz switching) |
The instability mechanism can be further analyzed by examining the frequency response of \(Z_{oq}(z)\). In the low-frequency region, typically below the PLL bandwidth, the phase of \(Z_{oq}(z)\) may drop below -90°, indicating negative resistance. This is exacerbated when both PLL and GVF are active. Therefore, to ensure stable operation of the grid-tied inverter across various grid conditions, it is imperative to reshape the q-axis impedance.
In this work, we propose a control strategy that actively reshapes the q-axis impedance by modifying the voltage feedforward path. The core idea is to design a new feedforward transfer function \(G_{f\_pro}(z)\) such that the combined admittance contributions from PLL and GFV cancel each other in the low-frequency range. Specifically, we aim to achieve:
$$ Y_{PLL}(z) + Y_{GVF}(z) = 0 $$
Substituting the expressions for \(Y_{PLL}(z)\) and \(Y_{GVF}(z)\), we solve for \(G_{f\_pro}(z)\):
$$ G_{f\_pro}(z) = \frac{G_{PLL}(z) \left[ I_{gd} PI(z) + U_{id} \right]}{G_{PLL}(z) U_{PCCd} – 1} $$
By substituting the detailed forms of \(PI(z)\) and \(G_{PLL}(z)\), we can simplify \(G_{f\_pro}(z)\) into a form that is implementable in a digital controller. After algebraic manipulation, we obtain:
$$ G_{f\_pro}(z) = – \left[ I_{gd} PI(z) + U_{id} \right] G_{PLL\_PI}(z) G_{int}(z) $$
Expanding this expression yields a polynomial in terms of the discrete integrator \(G_{int}(z)\):
$$ G_{f\_pro}(z) = – \left[ \alpha G_{int}(z) + \beta G_{int}^2(z) + \gamma G_{int}^3(z) \right] $$
where the coefficients \(\alpha\), \(\beta\), and \(\gamma\) are functions of the controller parameters and operating points:
$$ \alpha = (K_P I_{gd} + U_{id}) K_{PLL\_P} $$
$$ \beta = (K_P I_{gd} + U_{id}) K_{PLL\_I} + K_I I_{gd} K_{PLL\_P} $$
$$ \gamma = K_I I_{gd} K_{PLL\_I} $$
These coefficients can be categorized into two groups: fixed controller parameters (\(K_P, K_I, K_{PLL\_P}, K_{PLL\_I}\)) and variable operating points (\(I_{gd}, U_{id}\)). Since \(I_{gd}\) and \(U_{id}\) vary with the power output of the grid-tied inverter, the proposed feedforward transfer function \(G_{f\_pro}(z)\) is adaptive. In practice, \(I_{gd}\) can be obtained from the current reference or measured value, and \(U_{id}\) can be calculated using the relation \(U_{id} = \sqrt{ U_{PCCd}^2 + (\omega_0 L_c I_{gd})^2 }\), where \(U_{PCCd}\) is typically measured or estimated.
The implementation of the proposed strategy involves adding a compensation signal derived from \(G_{f\_pro}(z)\) to the q-axis voltage command. Specifically, as shown in the control block diagram, the modified feedforward path processes the q-axis PCC voltage through \(G_{f\_pro}(z)\), and its output is added to the output of the q-axis current PI controller. This effectively injects a compensating voltage that counteracts the negative resistance effects introduced by the PLL and the original GVF. The overall control structure enhances the passivity of the grid-tied inverter by ensuring that the q-axis output impedance exhibits positive resistance characteristics in the low-frequency range.
To validate the effectiveness of the proposed impedance reshaping strategy, we conducted extensive simulations using MATLAB/Simulink. The parameters for the grid-tied inverter system are listed in Table 2.
| Parameter | Value |
|---|---|
| DC voltage \(V_{dc}\) | 700 V |
| Grid voltage (line-to-line RMS) | 380 V |
| Grid frequency \(f_0\) | 50 Hz |
| Filter inductance \(L_c\) | 2 mH |
| Grid inductance \(L_g\) | Variable, up to 5 mH |
| Switching frequency | 10 kHz |
| Sampling period \(T_s\) | 100 µs |
| Current PI: \(K_P\), \(K_I\) | 0.5, 100 |
| PLL PI: \(K_{PLL\_P}\), \(K_{PLL\_I}\) | 50, 2000 |
| Rated power | 10 kW |
We first analyzed the frequency response of the q-axis output impedance \(Z_{oq}(z)\) under three scenarios: (1) without PLL and GVF (only current control), (2) with PLL and conventional GVF (\(G_f(z)=1\)), and (3) with PLL and the proposed adaptive feedforward \(G_{f\_pro}(z)\). The Bode plots of the impedance magnitude and phase are critical for stability assessment. In the conventional case with PLL and GVF, the phase of \(Z_{oq}(z)\) dips significantly below -90° at frequencies below 100 Hz, confirming the presence of negative resistance. This correlates with observed oscillations in time-domain simulations when the grid inductance exceeds a certain threshold.
With the proposed control strategy, the phase of \(Z_{oq}(z)\) remains above -90° across the low-frequency range, indicating that the negative resistance region has been eliminated. The magnitude response also shows improved characteristics, ensuring a sufficient phase margin when interacting with grid impedance. The reshaping effectively decouples the destabilizing interaction, allowing the grid-tied inverter to maintain stable operation even under weak grid conditions with high \(L_g\).
Time-domain simulations were performed to corroborate the frequency-domain analysis. We simulated a step change in the grid inductance from 1 mH (strong grid) to 4 mH (weak grid) at t=0.5 s. With the conventional control, the grid-tied inverter exhibited growing oscillations in the output current, eventually leading to instability. In contrast, with the proposed impedance reshaping strategy, the grid-tied inverter maintained stable current regulation with minimal transients. The total harmonic distortion (THD) of the output current remained well within grid codes (e.g., below 5%) under various operating points, demonstrating the robustness of the approach.
Furthermore, we tested the performance under grid voltage disturbances, such as voltage sags and harmonics. The proposed strategy not only stabilizes the system but also preserves good dynamic response and disturbance rejection capabilities. The adaptive nature of \(G_{f\_pro}(z)\) ensures that the compensation remains effective across different power levels, making it suitable for grid-tied inverters in varying operational scenarios.
The proposed method offers several advantages. First, it directly addresses the root cause of instability by reshaping the q-axis impedance, without requiring modifications to the PLL or current controller structures. Second, it is adaptive to operating conditions, enhancing robustness. Third, it can be implemented digitally with moderate computational overhead, as it involves polynomial functions of the discrete integrator. However, challenges include the need for accurate measurement or estimation of \(I_{gd}\) and \(U_{PCCd}\), and potential sensitivity to parameter variations. Future work could explore robust adaptations and integration with other stability enhancement techniques.
In conclusion, the stability of grid-tied inverters in weak grids is critically influenced by the negative resistance effects introduced by phase-locked loops and grid voltage feedforward control. Through detailed small-signal modeling, we have quantified these effects and demonstrated how they degrade system stability. To counteract this, we have proposed a novel control strategy based on q-axis impedance reshaping. By designing an adaptive feedforward transfer function that cancels the combined admittance contributions of PLL and GVF, the q-axis output impedance is reshaped to exhibit positive resistance in the low-frequency range. Simulation results validate the effectiveness of the approach, showing stable operation under weak grid conditions and improved robustness to grid disturbances. This research contributes to the ongoing efforts to enhance the reliability and performance of grid-tied inverters in modern power systems with high penetration of renewable energy.
The implications of this work extend to the design and operation of distributed generation systems. As grid-tied inverters become ubiquitous, ensuring their stability in varying grid conditions is paramount. The impedance reshaping strategy presented here offers a systematic solution that can be integrated into existing inverter designs. Future research directions may include experimental validation on hardware platforms, extension to inverters with LCL filters, and consideration of unbalanced or distorted grid conditions. Ultimately, advancing such control strategies will support the transition to a more resilient and sustainable power grid.