The integration of renewable energy sources into the main electrical network is predominantly facilitated by power electronic converters, with the on-grid inverter serving as the critical interface. The primary function of an on-grid inverter is to convert DC power from sources like photovoltaic panels or batteries into AC power that is synchronized in phase, frequency, and voltage with the utility grid. A stable and accurate grid synchronization mechanism is paramount for the safe, efficient, and reliable operation of any on-grid inverter system. Traditionally, this synchronization is achieved using Phase-Locked Loops (PLLs), which are closed-loop control systems designed to track the grid voltage phase angle. While effective under ideal grid conditions, the stability of an on-grid inverter employing a PLL can be severely compromised when connected to a weak grid—a grid scenario characterized by a non-negligible impedance, often quantified by a low Short-Circuit Ratio (SCR).
The core instability issue stems from the negative-resistance characteristic that a PLL introduces at low frequencies. In an on-grid inverter, the PLL is coupled with the current controller through the Point of Common Coupling (PCC) voltage. This coupling modifies the output impedance of the on-grid inverter, often depressing its phase in the low-frequency region (typically below 100-150 Hz). When the grid impedance is significant, the interaction between the grid impedance and the inverter’s output impedance can lead to insufficient phase margin, harmonic resonance, and even system-level instability. This presents a major challenge for the widespread deployment of on-grid inverter systems in areas with weak grid infrastructure.
In response to this challenge, extensive research has been conducted to optimize PLL structures or to find alternative synchronization methods. Among these alternatives, Open-Loop Synchronization (OLS) techniques have garnered significant attention. Unlike a PLL, an OLS scheme has no feedback loop; it estimates the grid phase through a direct computation based on filtered measurements of the grid voltage. This inherently feedforward structure suggests that the OLS itself is unconditionally stable. Consequently, an on-grid inverter utilizing OLS is hypothesized to possess superior robustness against weak grid conditions. However, a significant barrier to adopting and analyzing OLS for on-grid inverters has been the difficulty in obtaining an accurate small-signal linearized model. Most OLS implementations involve a strong nonlinearity—the arctangent function (atan) used for phase extraction. Simplifying or ignoring this nonlinearity leads to models with substantial inaccuracies, particularly in the critical low-frequency band, rendering any subsequent stability analysis unreliable.
This article, therefore, focuses on developing a precise small-disturbance (small-signal) model for a single-phase on-grid inverter that employs a specific and prevalent type of OLS: the Moving Average Filter-based Open-Loop Synchronization (MAF-OLS). A novel linearization method inspired by harmonic linearization theory is proposed to accurately model the atan nonlinearity. This enables the derivation of the full output impedance model for the on-grid inverter. Based on this accurate model, a comprehensive stability analysis is performed, comparing the robustness of the MAF-OLS-based on-grid inverter against a conventional PLL-based design under progressively weaker grid conditions. The findings conclusively demonstrate the exceptional stability of the OLS-based on-grid inverter, even in extremely weak grids.
The Stability Challenge for PLL-Based On-Grid Inverters in Weak Grids
To understand the advantage of an OLS-based on-grid inverter, it is essential to first detail the instability mechanism of its PLL-based counterpart. A typical single-phase LCL-filtered on-grid inverter system is considered. The system parameters, including the LCL filter components (\(L_1\), \(C_f\), \(L_2\)), controller gains, and operating points, are summarized in Table 1. The current controller \(G_c(s)\) is designed for high performance and robustness, incorporating a proportional term and multiple harmonic resonant controllers with phase compensation to ensure adequate stability margins under normal conditions.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | \(P_{rated}\) | 5 kW |
| Grid Voltage (RMS) | \(U_g\) | 220 V |
| DC Link Voltage | \(U_{dc}\) | 380 V |
| Grid Frequency | \(f_0\) | 50 Hz |
| Switching Frequency | \(f_{sw}\) | 15 kHz |
| Inverter-side Inductor | \(L_1\) | 0.75 mH |
| Grid-side Inductor | \(L_2\) | 0.45 mH |
| Filter Capacitor | \(C_f\) | 6.8 μF |
| Proportional Gain | \(k_p\) | 10 |
| Resonant Gain | \(k_r/\omega_r\) | 600 |
The stability of the interconnected system—the on-grid inverter and the weak grid—can be rigorously assessed using the impedance-based stability criterion. The on-grid inverter is modeled as a Norton equivalent circuit: an ideal current source \(I_g(s)\) in parallel with its output impedance \(Z_{out}(s)\). The weak grid is modeled as an ideal voltage source \(U_g(s)\) in series with the grid impedance \(Z_g(s)\). According to the criterion, the system is stable if the minor loop gain \(Z_g(s)/Z_{out}(s)\) satisfies the Nyquist stability criterion, provided the on-grid inverter is stable when connected to an ideal grid (\(Z_g=0\)).
For a PLL-based on-grid inverter, the output impedance model must account for the coupling between the synchronization unit and the current reference. The current reference is generated as \(i_{ref}(s) = I_{ref} G_s(s) u_{pcc}(s)\), where \(G_s(s)\) is the transfer function of the PLL. This coupling modifies the output impedance from \(Z_{out}\) (without PLL effect) to \(Z_{out\_PLL}\). A detailed derivation yields the following expression for the output impedance of the PLL-based on-grid inverter:
$$
Z_{out\_PLL}(s) = \frac{ [L_1 L_2 C_f s^3 + (k_{ad}k_{PWM} L_2 C_f)s^2 + (L_1 + L_2)s] }{ [1 + G_c(s)k_{PWM}(L_1 C_f s^2 + k_{ad}k_{PWM} C_f s) – G_f(s)k_{PWM} – I_{ref}G_s(s)G_c(s)k_{PWM}] }
$$
The critical term is the last one in the denominator: \(- I_{ref}G_s(s)G_c(s)k_{PWM}\). This term, directly contributed by the PLL, is responsible for altering the phase characteristics of \(Z_{out\_PLL}(s)\). For a widely used Single-Phase Second-Order Generalized Integrator PLL (SOGI-PLL) with a bandwidth of approximately 100 Hz, the Bode plot of \(Z_{out\_PLL}(s)\) reveals a significant phase drop in the low-frequency region (around 50-150 Hz) compared to \(Z_{out}(s)\). This phase depression represents the introduced negative-resistance behavior.
The stability margin is evaluated by the Phase Margin (PM) at the frequency \(f_i\) where the magnitudes of \(|Z_g(j2\pi f)|\) and \(|Z_{out\_PLL}(j2\pi f)|\) intersect:
$$ PM = 180^\circ – | \angle Z_g(j2\pi f_i) – \angle Z_{out\_PLL}(j2\pi f_i) | $$
For a purely inductive weak grid (\(Z_g(s) = sL_g\)), the intersection frequency decreases as the grid inductance \(L_g\) (inversely related to SCR) increases. With a high \(L_g\) (e.g., SCR = 2.5), the intersection occurs in the frequency region where \(Z_{out\_PLL}\) has very low or even negative phase, leading to a PM less than \(0^\circ\) and guaranteed instability. This analysis confirms that the PLL is a destabilizing element for an on-grid inverter in weak grids, limiting its operational range.
Open-Loop Synchronization: Structure and Proposed Linearization Method
The Open-Loop Synchronization (OLS) method offers a fundamentally different approach. A common and effective implementation is the MAF-OLS, whose structure is divided into three functional parts, as shown in the conceptual block diagram.
- Orthogonal Signal Generation (OSG): A T/4 delay (where T is the grid period) is applied to the measured PCC voltage \(u_{pcc}\) to generate a quadrature signal pair (\(u_\alpha\), \(u_\beta\)). Contrary to simplistic modeling, the dynamic of this delay must be preserved: \(u_\beta(s) = e^{-sT/4} u_\alpha(s)\).
- Pre-Filtering (MAF): The signals are transformed into the synchronous reference frame (dq-frame) using the nominal grid frequency \(\omega_n\). The Moving Average Filter (MAF), with a window length \(T_\omega\) equal to the grid period (0.02s for 50Hz), is applied in the dq-domain. The MAF’s transfer function is \(G_{MAF}(s) = (1-e^{-T_\omega s})/(T_\omega s)\). It provides exceptional attenuation at harmonic frequencies, effectively cleaning the signals.
- Phase Estimation (atan): The filtered q-axis component \(u_q’\) and d-axis component \(u_d’\) are used to compute the phase angle via the arctangent function: \(\theta_o = \tan^{-1}(u_q’ / u_d’)\). This phase is then used to generate the sinusoidal current reference \(i_{ref} = I_{ref} \cos(\theta_o)\).
The key modeling challenge lies in linearizing the cascaded system, particularly the atan block. The proposed method treats the system by injecting a small-signal disturbance of frequency \(f_1\) and amplitude \(A\) (much smaller than the grid voltage amplitude \(U_m\)) into the input \(u_\alpha\). The corresponding small-signal component in the output \(i_{ref}\) is then derived.
Assuming the MAF outputs are nearly ideal orthogonal signals after filtering, the inputs to the atan block can be expressed with the disturbance:
$$ u’_\alpha \approx U_m \cos(\omega_n t) + A \cos(2\pi f_1 t + \phi_1) $$
$$ u’_\beta \approx U_m \sin(\omega_n t) + A \sin(2\pi f_1 t + \phi_1) $$
Using trigonometric identities and the small-signal approximation (\(A/U_m \ll 1\)), the disturbance-induced phase error \(\Delta\theta\) is found to be approximately:
$$ \Delta\theta \approx \frac{A}{U_m} \sin(2\pi f_1 t + \phi_1 – \omega_n t) $$
Substituting into \(i_{ref} = I_{ref} \cos(\omega_n t + \Delta\theta)\) and applying trigonometric expansions yields the fundamental result: the disturbance at frequency \(f_1\) in \(u_\alpha\) produces a disturbance in \(i_{ref}\) primarily at the same frequency \(f_1\), with a linear gain relationship. Crucially, due to frequency shifting in the cosine product, the effective gain differs by a factor of 1/2 for disturbances around the base frequency. The linearized transfer function for the atan and reference generation block (Part III) is:
$$ G_{III}(s): \quad \frac{i_{ref}(s)}{u’_\alpha(s)} = \frac{I_{ref}}{2U_m} $$
Combining this with the linear models for the OSG and MAF-in-dq-frame (Parts I & II), the complete small-signal transfer function of the MAF-OLS from PCC voltage to current reference is derived:
$$ G_{MAF-OLS}(s) = \frac{i_{ref}(s)}{u_{pcc}(s)} = \frac{I_{ref}}{2U_m} \cdot \frac{(1 – e^{-T_\omega s})(e^{-sT/4} – e^{sT/4})}{T_\omega s} \cdot \frac{s}{s^2 + \omega_n^2} $$
This can be simplified to:
$$ G_{MAF-OLS}(s) = \frac{I_{ref}}{U_m} \cdot \frac{(1 – e^{-T_\omega s})}{T_\omega s} \cdot \frac{\omega_n}{s^2 + \omega_n^2} \cdot \sin(sT/4) $$
For small \(s\), \(\sin(sT/4) \approx sT/4\), confirming the proper behavior. This precise model \(G_{MAF-OLS}(s)\) can now be substituted into the general output impedance formula in place of \(G_s(s)\) to obtain \(Z_{out\_OLS}(s)\) for the on-grid inverter using open-loop synchronization.
Stability Analysis: OLS vs. PLL for the On-Grid Inverter
Using the derived model \(Z_{out\_OLS}(s)\), a direct comparison with \(Z_{out\_PLL}(s)\) is performed. The Bode plots of both output impedances are analyzed alongside the magnitudes of various grid impedances \(Z_g = sL_g\), representing different grid strengths (SCR values).
The results are striking. The PLL-based on-grid inverter’s output impedance \(Z_{out\_PLL}\) shows a deep phase depression beginning below 100 Hz and extending over a wide bandwidth. As the grid inductance increases (SCR decreases to 2.5), the magnitude curves of \(Z_g\) and \(Z_{out\_PLL}\) intersect precisely within this low-phase region, resulting in a negative phase margin and predicting instability.
In contrast, the OLS-based on-grid inverter’s output impedance \(Z_{out\_OLS}\) exhibits a vastly different characteristic. The negative-resistance effect is not eliminated but is confined to an extremely narrow frequency band (approximately 40-60 Hz). More importantly, the phase of \(Z_{out\_OLS}\) recovers quickly outside this narrow band and remains high across the rest of the low-frequency spectrum. Consequently, even for very large grid inductances corresponding to an extremely weak grid (e.g., SCR = 1.07), the intersection with \(Z_g\) occurs at a frequency where \(Z_{out\_OLS}\) maintains a healthy positive phase, ensuring a sufficient phase margin for stability. This analysis robustly demonstrates that an on-grid inverter equipped with MAF-OLS possesses intrinsic and superior robustness against weak grid conditions compared to a standard PLL-based design.
Simulation and Experimental Validation
The accuracy of the proposed small-signal model and the stability conclusions are verified through detailed time-domain simulations and experimental tests on a 5 kW single-phase on-grid inverter prototype.
First, the frequency response of \(Z_{out\_OLS}\) is validated via simulation. A perturbation at frequency \(f_1\) is injected into the grid voltage, and the corresponding component in the grid current \(i_g\) at \(f_1\) is measured. The ratio \(u_{pcc}(f_1)/i_g(f_1)\) gives the empirical output impedance. As shown in comparative Bode plots, the proposed model (Model I) matches the simulation sweep accurately across the entire frequency range, especially in the critical sub-150 Hz region where a simplified model (Model II, which neglects atan nonlinearity) shows significant deviation.
Time-domain stability tests under weak grid conditions provide conclusive evidence. The performance of the on-grid inverter using MAF-OLS is compared against one using SOGI-PLL for increasing grid inductance \(L_g\).
| Grid Impedance \(L_g\) | SCR | On-Grid Inverter with MAF-OLS | On-Grid Inverter with SOGI-PLL |
|---|---|---|---|
| 6 mH | 5.0 | Stable, Low THD | Stable |
| 12 mH | 2.5 | Stable, Low THD | Unstable (Oscillations) |
| 20 mH | 1.5 | Stable | Severely Unstable |
| 28 mH | 1.07 | Stable | Severely Unstable |
The simulation and experimental waveforms consistently show that the SOGI-PLL-based on-grid inverter becomes unstable when \(L_g\) reaches 12 mH (SCR=2.5), exhibiting severe oscillations in both PCC voltage and grid current. The MAF-OLS-based on-grid inverter, however, remains perfectly stable and maintains a high-quality grid current even under the extreme condition of \(L_g = 28\) mH (SCR=1.07). The experimental current THD increases somewhat due to non-ideal conditions (e.g., grid background harmonics, measurement noise), but the system stability is never compromised. These results irrefutably validate the model and underscore the exceptional weak-grid adaptability of the open-loop synchronization approach for on-grid inverters.
Conclusion and Future Perspectives
This work addresses a critical challenge in the operation of single-phase on-grid inverters in weak grids: the instability induced by conventional Phase-Locked Loops. By proposing a novel and accurate harmonic linearization method, a precise small-signal impedance model is developed for an on-grid inverter utilizing a Moving Average Filter-based Open-Loop Synchronization (MAF-OLS) scheme. The model successfully captures the effect of the nonlinear arctangent function, which had been a major obstacle in previous analyses.
The stability analysis based on this model reveals a fundamental advantage of the OLS structure. While it does introduce a minor negative-resistance characteristic, its effect is confined to a very narrow frequency band, unlike the broad low-frequency phase degradation caused by a PLL. Consequently, an on-grid inverter with MAF-OLS maintains a high phase margin over a wide range of frequencies, ensuring robust stability even when connected to very weak grids with a low Short-Circuit Ratio. This finding is of paramount importance for deploying on-grid inverter systems in remote areas or on distribution networks with high impedance.
The superior performance of the OLS-based on-grid inverter is conclusively demonstrated through comprehensive simulations and experimental tests, which show stable operation under grid conditions where a PLL-based on-grid inverter fails catastrophically. Future work may focus on enhancing the dynamic performance and frequency adaptability of OLS schemes for on-grid inverters under large grid frequency deviations, as well as investigating their behavior in three-phase systems and under unbalanced grid faults. Nevertheless, this research firmly establishes open-loop synchronization as a highly robust and reliable alternative for grid synchronization in next-generation on-grid inverter applications.