In modern power systems, the rapid integration of distributed renewable energy sources has highlighted the critical role of utility interactive inverters as the interface between generation units and the grid. However, the increasing penetration of these inverters in weak grid conditions—characterized by low short-circuit ratios (SCR) due to long transmission lines and transformer leakage inductances—poses significant stability challenges. Under weak grid scenarios, the point of common coupling (PCC) voltage is influenced by grid impedance and injected currents, which can degrade the accuracy of phase information extracted by phase-locked loops (PLLs) and compromise power quality. Moreover, the coupling interaction among the PLL, current control loop, and grid impedance can reduce the phase margin of the system output impedance, leading to oscillations and even instability in utility interactive inverter operations. Traditional PLL designs often struggle to decouple these interactions, limiting robustness and adaptability. This paper addresses these issues by proposing a novel PLL control structure that reshapes the PLL transfer function through a pre-link mechanism, effectively decoupling the PLL bandwidth from the current loop and enhancing system stability in weak grids. We provide a comprehensive mathematical model, stability analysis based on pole-zero distributions and Nyquist criteria, detailed parameter design, and validation through simulations and experiments. The keyword “utility interactive inverter” is emphasized throughout to underscore its relevance in grid-tied applications.
The topology of a single-phase LCL-filtered utility interactive inverter is commonly used for grid integration, as it offers superior harmonic attenuation compared to simpler filters. The system comprises a DC-link voltage source, an inverter bridge, an LCL filter (with inductors L1 and L2, and capacitor C), and a control scheme that includes a current regulator, a PLL, and active damping components. The grid impedance Zg introduces a feedback path that couples with the PLL and current loop, exacerbating stability issues in weak grids. To analyze this, we first establish a mathematical model for the utility interactive inverter. The control block diagram can be transformed to highlight the interactions, where the PLL introduces a negative impedance term ZPLL(s) and the current loop contributes an output impedance Zout(s). The overall system open-loop transfer function from grid voltage to grid current, denoted as F(s), is derived to assess stability. For a traditional PLL, the transfer function GTPLL(s) is given by:
$$G_{\text{TPLL}}(s) = \frac{k_{\text{p-PLL}} s + k_{\text{i-PLL}}}{s^2 – j2\omega_0 s + U_m (k_{\text{p-PLL}} s + k_{\text{i-PLL}})}$$
where \(k_{\text{p-PLL}}\) and \(k_{\text{i-PLL}}\) are the proportional and integral gains of the PLL controller, \(\omega_0\) is the grid angular frequency (e.g., 100π rad/s for 50 Hz), and \(U_m\) is the amplitude of the PCC voltage. This transfer function shows that the PLL dynamics directly influence the system impedance, leading to coupling with the grid impedance Zg(s). The output impedance of the utility interactive inverter, considering the PLL effect, is expressed as:
$$Z_{\text{out-TPLL}}(s) = \frac{1 + G_c(s)G_1(s)G_2(s)}{G_c(s)G_1(s)G_2(s) – G_{\text{TPLL}}(s)G_c(s)G_1(s)G_2(s)I_2}$$
where \(G_c(s)\) is the current controller transfer function (e.g., a quasi-proportional resonant controller), \(G_1(s)\) and \(G_2(s)\) are filter-related transfer functions, and \(I_2\) is the grid current amplitude reference. The coupling arises because Zg(s) appears in a positive feedback loop, amplifying disturbances and reducing robustness. To quantify this, we analyze the pole-zero map and Nyquist plot of F(s) as Zg and PLL bandwidth vary. For instance, increasing Zg or PLL bandwidth expands the Nyquist curve, potentially encircling the critical point (-1, j0) and violating the Nyquist stability criterion. This indicates that traditional PLL designs suffer from reduced phase margins in weak grids, necessitating a trade-off between dynamic performance and stability.
To address these limitations, we propose a novel PLL structure that decouples the PLL bandwidth from the current loop. The key idea is to reshape the PLL transfer function by adding a pre-link block GN1(s) before the traditional PLL, such that the overall PLL transfer function becomes independent of the PLL controller parameters. Based on the inverse of the traditional PLL transfer function, we define a target function GN0(s) and then optimize it to avoid high-frequency noise amplification and implementation difficulties. The proposed pre-link transfer function is:
$$G_{\text{N1}}(s) = \frac{k_2 (s – j\omega_0)}{(s – j\omega_0 + a)} + \frac{k_1}{(s – j\omega_0 + a)}$$
where \(a\), \(k_1\), and \(k_2\) are design parameters. By setting \(k_2 = a/(2U_m)\) and \(k_1 = 2b/k_{\text{p-PLL}}\) with \(b = a/(2U_m)\), we ensure that the gain at \(\omega_0\) is unity and the phase shift is zero. The novel PLL transfer function GNPLL(s) is then:
$$G_{\text{NPLL}}(s) = G_{\text{TPLL}}(s) G_{\text{N1}}(s) = \frac{b}{s – j\omega_0 + a}$$
This simplified form shows that GNPLL(s) is independent of \(k_{\text{p-PLL}}\) and \(k_{\text{i-PLL}}\), effectively decoupling the PLL bandwidth from the system dynamics. Consequently, the output impedance of the utility interactive inverter with the novel PLL becomes:
$$Z_{\text{out-NPLL}}(s) = \frac{1 + G_c(s)G_1(s)G_2(s)}{G_c(s)G_1(s)G_2(s) – G_{\text{NPLL}}(s)G_c(s)G_1(s)G_2(s)I_2}$$
which exhibits improved phase margins and robustness against grid impedance variations. The parameter \(a\) plays a crucial role in balancing robustness and dynamic response. A smaller \(a\) enhances harmonic suppression and phase margin but may slow down the response, while a larger \(a\) improves dynamics at the cost of reduced robustness. Based on analysis, we select \(a = 120\) for a good compromise, ensuring high stability in weak grids without significantly degrading performance.
The stability of the utility interactive inverter with the novel PLL is evaluated using Nyquist criteria. Compared to traditional PLLs, the novel PLL maintains a stable system even under extreme weak grid conditions (e.g., SCR as low as 1.5). The table below summarizes the phase margins for different PLL types and grid impedances, demonstrating the superiority of the novel PLL:
| PLL Type | PLL Bandwidth (Hz) | Phase Margin for Zg=3.5mH (SCR=6) | Phase Margin for Zg=7.1mH (SCR=3) | Phase Margin for Zg=14.1mH (SCR=1.5) |
|---|---|---|---|---|
| Traditional PLL | 130 | 40.4° | 33.0° | 8.7° |
| Traditional PLL | 250 | 34.1° | 4.2° | Unstable |
| Traditional PLL | 500 | 5.9° | Unstable | Unstable |
| Novel PLL | 130 | 41.9° | 37.4° | 32.9° |
| Novel PLL | 250 | 41.9° | 37.4° | 32.9° |
| Novel PLL | 500 | 41.9° | 37.4° | 32.9° |
This table clearly shows that the novel PLL achieves consistent phase margins regardless of PLL bandwidth, whereas traditional PLLs suffer from significant degradation. The robustness of the utility interactive inverter is thereby enhanced, allowing for flexible parameter design without stability compromises.
To validate the proposed novel PLL design, we conducted simulations in MATLAB/Simulink for a 1.5 kW single-phase utility interactive inverter with parameters listed in the following table:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rated Power | 1.5 kW | Grid Voltage | 100 V RMS |
| DC-Link Voltage | 300 V | Grid Frequency | 50 Hz |
| Filter Inductor L1 | 5 mH | QPR Controller kp | 0.1 |
| Filter Inductor L2 | 1 mH | QPR Controller kr | 6.2 |
| Filter Capacitor C | 10 μF | QPR Controller ωc | π rad/s |
| Active Damping kd | 0.26 | Switching Frequency | 10 kHz |
| PWM Gain kPWM | 300 | Sampling Frequency | 20 kHz |
The simulation results compare traditional and novel PLLs under weak grid conditions. For a PLL bandwidth of 250 Hz, the traditional PLL causes severe current distortion at SCR=3 and instability at SCR=1.5, while the novel PLL maintains stable operation with total harmonic distortion (THD) below 2% even at SCR=1.5. Similarly, at 500 Hz bandwidth, the traditional PLL fails at SCR=6, but the novel PLL remains robust. Dynamic performance was tested with an 80° grid voltage phase jump. The novel PLL exhibits smaller overshoot and faster settling in grid current, voltage amplitude, phase, and frequency compared to the traditional PLL, confirming its superior dynamic response.
Experimental validation was performed on a 1.5 kW utility interactive inverter prototype using an RTU-BOX204 real-time controller. With a PLL bandwidth of 500 Hz, the traditional PLL led to distorted grid current (THD of 24.54%) at SCR=8 and triggered protection at higher impedances. In contrast, the novel PLL enabled stable operation at SCR=4, SCR=2, and even SCR=1.5, with grid current THD as low as 1.70%. The grid voltage at PCC showed some low-frequency distortion due to grid impedance, but the system remained stable. Dynamic tests during current reference step changes from half-load to full-load demonstrated that the novel PLL achieves faster response and smoother transitions than the traditional PLL, highlighting its practical benefits for utility interactive inverter applications.
In conclusion, this paper presents a novel PLL design method for utility interactive inverters in weak grids. By analyzing the coupling effects among the PLL, current loop, and grid impedance, we identified that traditional PLLs suffer from reduced robustness as bandwidth increases. The proposed novel PLL incorporates a pre-link block to reshape the transfer function, making it independent of PLL bandwidth and thus decoupling it from the current loop. This design enhances system stability, allowing for high robustness even in extremely weak grids (SCR down to 1.5) without sacrificing dynamic performance. Parameter design guidelines are provided, focusing on the key parameter \(a\) to balance robustness and response. Simulations and experiments confirm the effectiveness of the novel PLL, showing improved phase margins, lower current distortion, and better dynamic performance compared to traditional approaches. The utility interactive inverter with this novel PLL offers greater adaptability to weak grids, facilitating the reliable integration of renewable energy sources. Future work could explore the coupling mechanisms in three-phase systems or under unbalanced grid conditions to further optimize utility interactive inverter performance.
The mathematical formulations and stability criteria used in this analysis are grounded in control theory. For instance, the Nyquist stability criterion states that a system is stable if the Nyquist plot of the open-loop transfer function does not encircle the point (-1, j0) when the number of right-half-plane poles is zero. For our utility interactive inverter model, the open-loop transfer function F(s) is derived as:
$$F(s) = \frac{Z_g(s)}{Z_{\text{out}}(s)}$$
where \(Z_{\text{out}}(s)\) is the output impedance with PLL effects. The pole-zero analysis involves solving for roots of the characteristic equation, which for the novel PLL system yields improved damping. The Bode plots of GNPLL(s) and Zout-NPLL(s) illustrate the frequency-domain benefits, such as reduced gain at harmonics and increased phase margins. The design equations for the pre-link parameters ensure implementability; for example, the condition \(2bU_m/a = 1\) guarantees unity gain at the fundamental frequency. These analytical tools collectively support the novel PLL’s efficacy in enhancing utility interactive inverter stability.
Throughout this paper, the term “utility interactive inverter” is emphasized to reflect its role in grid-tied systems. The proposed method is applicable to various inverter topologies, but the LCL filter is considered due to its widespread use. The novel PLL design not only addresses stability issues but also simplifies parameter tuning, as the PLL bandwidth can be selected based solely on dynamic requirements without worrying about robustness trade-offs. This advancement contributes to more resilient power electronics interfaces in modern grids, where weak grid conditions are increasingly common. Future research may extend this approach to multi-inverter systems or incorporate adaptive tuning for varying grid impedances, further optimizing the performance of utility interactive inverters.