The proliferation of renewable energy sources has led to a significant increase in the penetration of power electronic converter-based generation. The utility interactive inverter, as the critical interface, faces mounting stability challenges when connected to weak grids characterized by high grid impedance. A primary source of instability in such grid-following inverters stems from the dynamic interaction between the Phase-Locked Loop (PLL) system and the grid impedance, often manifesting as low-frequency oscillations.
Traditional solutions to mitigate this interaction frequently involve reducing the bandwidth of the commonly used Type-II PLL (employing a PI controller). While this can enhance stability margins, it invariably degrades the dynamic performance of the PLL, slowing down the synchronization process and consequently impacting the current control loop’s response. Furthermore, fixed low-bandwidth designs struggle to adapt to the wide variation of grid impedance encountered in practice. This paper addresses this core challenge. We propose a fundamental shift from the Type-II PLL structure to a Type-I PLL structure. This change substantially improves the stability robustness of the utility interactive inverter under weak grid conditions without compromising dynamic performance. Recognizing a inherent limitation of the basic Type-I PLL—a steady-state phase error under grid frequency deviations—we further introduce an adaptive frequency feedforward mechanism, creating a Quasi-Type-I PLL. This enhanced version eliminates the phase tracking error, thereby improving both the stability and dynamic performance of the grid-connected system in weak grids.
System Modeling and Stability Challenge
The stability of a utility interactive inverter in a weak grid can be effectively analyzed using impedance-based methods. The interaction between the inverter output impedance and the grid impedance determines system stability. A key component shaping the inverter’s output impedance, especially in the low-frequency range, is the PLL. The standard Synchronous Reference Frame PLL (SRF-PLL) uses a PI controller, making it a Type-II system with two open-loop poles at the origin. Its closed-loop transfer function can be expressed as:
$$
H_{\text{PLL}}(s) = \frac{\theta_{\text{PLL}}(s)}{\theta_g(s)} = \frac{U_{\text{PCC}}(K_{P\_PLL} + \frac{K_{I\_PLL}}{s})}{s + U_{\text{PCC}}(K_{P\_PLL} + \frac{K_{I\_PLL}}{s})} = \frac{2\zeta\omega_n s + \omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}
$$
where $\theta_g$ is the grid phase, $\theta_{\text{PLL}}$ is the estimated phase, $U_{\text{PCC}}$ is the PCC voltage magnitude, $K_{P\_PLL}$ and $K_{I\_PLL}$ are the PI controller parameters, $\zeta$ is the damping ratio, and $\omega_n$ is the natural frequency. The PLL bandwidth $f_{\text{bw}}$ is related to these parameters.
A simplified sequence-domain model of the utility interactive inverter, incorporating the PLL dynamics, can be used to derive the equivalent single-input single-output (SISO) admittance $Y_{\text{SISO}}$ of the inverter. The stability is then assessed by analyzing the ratio $Y_{\text{SISO}} / Y_g$, where $Y_g$ is the grid admittance. The system approaches instability when this ratio exhibits a poor phase margin. The following table lists the core parameters of a typical three-phase LCL-filtered utility interactive inverter used for analysis.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC-Link Voltage ($U_{dc}$) | 700 V | Switching/Sampling Frequency | 20 kHz |
| Grid Phase Voltage ($U_g$) | 311 V / 50 Hz | Current Loop PI Controller | $10 + 600/s$ |
| Inverter-side Inductor ($L_1$) | 2 mH | Rated Apparent Power ($S$) | 20 kVA |
| Grid-side Inductor ($L_2$) | 0.75 mH | Reference Current ($I_{dref}+jI_{qref}$) | 42.87 A |
| Filter Capacitor ($C_f$) | 10 μF | Damping Resistor ($R_d$) | 3 Ω |
The Proposed Type-I PLL Approach
The conventional approach of reducing the Type-II PLL’s bandwidth to improve stability comes at the cost of slower response. The settling time $t_s$ for a Type-II PLL (second-order) is approximately $t_{s,\text{II}} \approx 3 / (\zeta \omega_n)$, whereas for a Type-I PLL (first-order) it is $t_{s,\text{I}} \approx 3 / (K_{P\_PLL} U_{\text{PCC}})$. For comparable controller gains, the Type-I PLL offers a faster settling time.
More importantly, from a stability perspective, replacing the PI controller with a simple P controller transforms the PLL into a Type-I system. The closed-loop transfer function becomes:
$$
H’_{\text{PLL}}(s) = \frac{\theta_{\text{PLL}}(s)}{\theta_g(s)} = \frac{U_{\text{PCC}} K_{P\_PLL}}{s + U_{\text{PCC}} K_{P\_PLL}} = \frac{1}{Ts + 1}
$$
where $T=1/(U_{\text{PCC}} K_{P\_PLL})$ is the time constant. This structural change significantly alters the admittance shaping effect of the PLL. Analysis shows that the utility interactive inverter employing a Type-I PLL presents a smaller magnitude and phase shift in its equivalent admittance at low frequencies compared to one using a Type-II PLL with similar dynamic performance. This effectively expands the “passive” region of the inverter’s impedance, greatly enhancing its robustness to grid impedance variations. For instance, with a grid inductance of $L_g=8$ mH (SCR≈2.9), a Type-II PLL with $f_{\text{bw}}=200$ Hz leads to instability (phase margin -0.4°), while a Type-I PLL with $f_{\text{bw}}=153$ Hz maintains a healthy phase margin of 43.1°.
Enhancement to Quasi-Type-I PLL
A known drawback of the basic Type-I PLL is its inability to track phase without error when the grid frequency $\omega_g$ deviates from the feedforward frequency $\omega_f$ used in the Park transform. The steady-state phase error is given by:
$$
\theta_{ss} = \sin^{-1}\left( \frac{\Delta \omega_g}{K_{P\_PLL} U_{\text{PCC}}} \right), \quad \Delta \omega_g = \omega_f – \omega_g
$$
While this error is negligible for small frequency deviations (e.g., ±0.5 Hz), it can become significant in systems with larger frequency tolerance (e.g., marine grids with ±10% variation).
To overcome this limitation, we propose the Quasi-Type-I PLL. Its structure incorporates an additional, very slow, frequency tracking loop based on a P controller that adaptively adjusts the frequency feedforward $\omega_f$. The core synchronization remains a fast Type-I PLL. The small-signal control block diagram is shown below, where $K_{P1}$ and $K_{P2}$ are the P controller gains for the frequency loop and the main PLL loop, respectively, and $G_{LPF}(s)$ is a low-pass filter.
The closed-loop transfer function of the Quasi-Type-I PLL is derived as:
$$
M_{\text{PLL\_adp}}(s) = \frac{\theta_{\text{PLL}}(s)}{\theta_g(s)} = \frac{U_{\text{PCC}} K_{P2} (1 + \frac{K_{P1} G_{LPF}(s)}{s})}{s + U_{\text{PCC}} K_{P2} (1 + \frac{K_{P1} G_{LPF}(s)}{s})}
$$
The frequency loop gain $K_{P1}$ is chosen to be a small fraction (e.g., 0.1) of the main PLL gain $K_{P2}$ ($K_{P1} = K_{adj} K_{P2}$). This ensures the adaptive loop slowly corrects the feedforward frequency without adversely affecting the dynamic response or stability of the main Type-I synchronization loop. With proper parameter selection, the Quasi-Type-I PLL eliminates the steady-state phase error while retaining the superior stability properties of the Type-I PLL. The parameters for the Quasi-Type-I PLL are summarized below.
| PLL Type | Bandwidth (fbw) | Controller $G(s)$ | Settling Time (ts) |
|---|---|---|---|
| Type-II | 200 Hz | $2.08 + 673.66/s$ | 9.27 ms |
| Type-II | 75 Hz | $0.35 + 18.71/s$ | 55.6 ms |
| Type-I / Quasi-Type-I | 153 Hz | $K_{P2} = 2.0814$ ($K_{P1}=0.2081$) | 4.63 ms |
Stability Analysis and Verification
The stability improvement offered by the Quasi-Type-I PLL is pronounced. In a weak grid scenario ($L_g=8$ mH, SCR=2.9), the system with a Type-II PLL is unstable. The Quasi-Type-I PLL provides a phase margin of 43.6°. More critically, in a very weak grid ($L_g=17$ mH, SCR=1.36), reducing the Type-II PLL bandwidth to 75 Hz still results in an unstable system. In contrast, the utility interactive inverter equipped with the Quasi-Type-I PLL (with an effective bandwidth of 153 Hz) maintains a positive, albeit small, phase margin of 1.6°, allowing for stable operation. This demonstrates that the proposed method offers a more effective way to enhance stability than simply reducing the bandwidth of a traditional Type-II PLL.
Time-domain simulations and experimental results from a 720 VA prototype validate the theoretical analysis. Under very weak grid conditions (SCR=1.36), the utility interactive inverter with a Type-II PLL (200 Hz or 75 Hz) exhibits severe instability with high current THD (>20%). When controlled with the proposed Quasi-Type-I PLL, the system operates stably with a grid current THD well below 5%. Furthermore, the experiment confirms the elimination of the phase-locking error when a frequency mismatch is intentionally introduced, showcasing the effectiveness of the adaptive feedforward loop.
Conclusion
This paper has presented and analyzed Type-I and Quasi-Type-I PLL structures as effective solutions for enhancing the stability of utility interactive inverters in weak grids. The core findings are:
- Reducing the bandwidth of a conventional Type-II PLL can improve stability but is often insufficient for very weak grids and significantly degrades dynamic performance.
- Adopting a Type-I PLL structure inherently provides greater stability robustness against grid impedance variations while offering faster dynamic response compared to a Type-II PLL with similar tuning.
- The basic Type-I PLL’s phase error under frequency deviations is negligible for standard power grids but can be addressed for broader applications by the proposed Quasi-Type-I PLL, which incorporates a slow, adaptive frequency feedforward loop.
- The Quasi-Type-I PLL effectively eliminates the steady-state phase error and provides superior stability in both weak and very weak grid conditions, outperforming the traditional bandwidth-reduction method for Type-II PLLs.
The proposed methods are simple to implement, require no complex online measurements, and offer a significant practical advantage for stabilizing utility interactive inverters, especially during commissioning and operation in challenging weak grid environments.