The rapid integration of renewable energy sources, driven by global decarbonization goals and the evolution of modern power systems, has led to a power grid characterized by a high penetration of power electronic interfaces. The single-phase on-grid inverter is a fundamental building block in distributed generation systems, connecting sources like rooftop photovoltaics to the utility network. However, the increasing “weak grid” characteristic—marked by a non-negligible grid impedance primarily inductive in nature—poses significant stability challenges for these inverters. Under such conditions, dynamic interactions between the inverter’s control loops (notably the Phase-Locked Loop, PLL) and the grid impedance can induce harmonic oscillations, threatening system reliability.
Stability analysis for single-phase on-grid inverters in weak grids often grapples with the phenomenon of frequency coupling. Unlike balanced three-phase systems, the control structures in single-phase inverters, particularly the PLL, are inherently asymmetric in the frequency domain. This asymmetry causes a perturbation at one frequency to generate responses not only at the original frequency but also at other, coupled frequencies (typically at offsets of the fundamental frequency). Ignoring this frequency coupling effect, as done in simplified Single-Input Single-Output (SISO) impedance models, can lead to inaccurate stability assessment and non-conservative design.
This article delves into the impedance modeling of single-phase LCL-type on-grid inverters, explicitly accounting for the infinite frequency coupling intrinsic to single-phase systems. We establish a Multi-Input Multi-Output (MIMO) admittance model using the Harmonic Transfer Function (HTF) matrix framework. To overcome the complexity of MIMO analysis, we introduce a novel and practical decoupling method by designing a decoupling factor within the current control loop. This yields a simple yet accurate SISO equivalent admittance model, decoupled from the grid impedance. Building on this precise model, we propose an effective admittance shaping strategy using a virtual admittance to eliminate the negative-resistance behavior introduced by the PLL, thereby ensuring passivity of the output admittance and robust stability in weak grids. The effectiveness of the proposed modeling and control strategy is validated through Real-Time (RT-LAB) experimental results.
System Topology and Control Structure
The topology of a single-phase LCL-filtered on-grid inverter and its control system is depicted below. The power stage consists of a full-bridge inverter, an LCL filter (comprising inverter-side inductor $L_1$, filter capacitor $C$, and grid-side inductor $L_2$), and the grid represented by voltage $u_g$ behind a grid impedance $Z_g$. The voltage at the Point of Common Coupling (PCC) is $u_{pcc}$.
The control system typically includes a current control loop and a synchronization unit. The reference current $i_{ref}$ is generated by multiplying the desired current amplitude $I_m$ with the cosine of the phase angle $\theta$ provided by the PLL: $i_{ref} = I_m \cos(\theta)$. A Proportional-Resonant (PR) controller $G_c(s)$ is commonly used for accurate tracking of the sinusoidal reference. The controller output is modulated after accounting for computational and PWM delay, modeled as $G_d(s) \approx e^{-1.5sT_s}$, where $T_s$ is the sampling period. The PLL, often a Synchronous Reference Frame PLL (SRF-PLL) for single-phase systems, extracts the phase angle $\theta$ from the PCC voltage $u_{pcc}$.
The closed-loop dynamics from the PCC voltage disturbance to the grid current can be derived. The plant transfer function $G_p(s)$, ignoring PLL dynamics, relates the current reference to the grid current. When considering the PLL’s influence, an additional path $H_{PLL}(s)$ is introduced, modeling how PCC voltage variations modulate the current reference via the PLL. The total inverter output admittance $Y_{in}(s)$, seen from the PCC, is a combination of the passive admittance of the LCL filter with the current controller and the active admittance $Y_{PLL}(s)$ induced by the PLL:
$$ Y_{in}(s) = Y_{p}(s) – G_{p}(s)H_{PLL}(s) $$
In a weak grid, the stability is determined by the ratio $Y_{in}(s)/Y_g(s)$, where $Y_g(s) = 1/Z_g(s)$. The term $Y_{PLL}(s) = G_{p}(s)H_{PLL}(s)$ often exhibits negative resistance (phase around -180°) in the low-to-medium frequency range, which can destabilize the system when interacting with a sufficiently large grid impedance.
MIMO Admittance Modeling with Frequency Coupling
In single-phase systems, the PLL’s operation involves transformations (e.g., generating an orthogonal signal for the $dq$-frame) that are linear but time-periodic (LTP). An oscillation at a frequency $\omega$ in the PCC voltage will cause oscillations in the PLL’s estimated angle $\Delta\theta$ and, consequently, in the current reference $\Delta i_{ref}$. Due to the multiplication with $\sin(\theta)$ and $\cos(\theta)$ terms in the PLL’s linearized model, this disturbance propagates not only at $\omega$ but also at the coupled frequencies $\omega \pm n\omega_0$, where $\omega_0$ is the fundamental grid frequency. For a complete description, a harmonic state-space or HTF approach is required.
We model signals in the frequency domain considering their principal frequency component and the major coupled sidebands. For instance, the perturbed PCC voltage and grid current are represented as vectors:
$$ \mathbf{u}_{pcc}(s) = [u_{pcc}(s-j\omega_0) \quad u_{pcc}(s) \quad u_{pcc}(s+j\omega_0)]^T $$
$$ \mathbf{i}_g(s) = [i_g(s-j\omega_0) \quad i_g(s) \quad i_g(s+j\omega_0)]^T $$
Linear Time-Invariant (LTI) components like the current controller $G_c(s)$ are represented by diagonal matrices $\mathbf{G_c}(s)$. LTP components, like the PLL’s small-signal model, are represented by Toeplitz matrices $\mathbf{H}_{PLL}(s)$ whose elements are the Fourier coefficients of the periodic impulse response, capturing coupling between different frequencies.
The resulting MIMO admittance matrix $\mathbf{Y}_{mimo}(s)$ relates the current vector to the voltage vector: $\mathbf{i}_g(s) = -\mathbf{Y}_{mimo}(s) \mathbf{u}_{pcc}(s)$. This 3×3 matrix (or larger for more sidebands) has non-zero off-diagonal elements, explicitly showing the frequency coupling. For example, an admittance element $Y_{12}(s)$ describes how a voltage perturbation at frequency $s-j\omega_0$ produces a current response at frequency $s$.
Stability analysis using this model requires applying the Generalized Nyquist Criterion (GNC) to the loop gain matrix $\mathbf{L_g}(s) = \mathbf{Y}_{mimo}(s) \mathbf{Z_g}(s)$, where $\mathbf{Z_g}(s)$ is the diagonal matrix of grid impedances at the respective frequencies. This is computationally intensive and lacks the intuitive appeal of a classic Nyquist plot.
Proposed SISO Equivalent Model via Decoupling Factor
To obtain a simpler yet accurate model, we propose to cancel the frequency-coupling terms introduced by the PLL within the control loop itself. The core idea is to inject a compensating signal derived from the PCC voltage that counteracts the coupling effect of $H_{PLL}(s)$.
Analyzing the MIMO model, the coupling originates from the specific structure of the $\mathbf{H}_{PLL}(s)$ matrix. We design a decoupling transfer function matrix $\mathbf{D}(s)$ based on the inverse of this coupling characteristic. When this decoupling block is added in series with or within the current reference generation path, the effective PLL transfer matrix becomes diagonal: $\mathbf{H}_{PLL, eff}(s) = diag([H^{-}(s), H^{0}(s), H^{+}(s)])$, where the off-diagonal terms are nullified.
With the coupling eliminated, the MIMO system reduces to three independent SISO channels. The channel at the principal frequency $s$ (or any chosen frequency of analysis) can be used to define a SISO equivalent output admittance $Y_{d}(s)$ for the on-grid inverter. This admittance is given by:
$$ Y_{d}(s) = Y_{p}(s) – G_{p}(s)H^{0}(s) $$
where $H^{0}(s)$ is the central diagonal element of the decoupled PLL matrix. Crucially, $Y_d(s)$ is now independent of the grid impedance $Z_g(s)$. This is a significant advantage, as the admittance can be characterized once, and stability can be assessed for any grid impedance using the simple SISO Nyquist criterion on $Y_d(s) / (1/Z_g(s))$.
The proposed decoupling factor can often be implemented as a relatively simple filter derived from the known PLL parameters, making it practical for digital implementation in a DSP or microcontroller governing the on-grid inverter.
Admittance Shaping for Stability Enhancement
Even with an accurate SISO model $Y_d(s)$, the stability problem in weak grids persists because the $Y_{PLL}(s) = G_p(s)H^0(s)$ component typically has a negative real part (phase near -180°) in a certain frequency range. To ensure robust stability, we aim to reshape the output admittance to be passive, i.e., have a positive real part across all frequencies, or at least in the critical range where interactions occur.
We propose an admittance shaping technique by introducing a virtual admittance $Y_{shaping}(s)$ in parallel with the inverter’s natural admittance. This is implemented in the control software by adding an additional current reference term $\Delta i_{shaping}$ that is a function of the measured PCC voltage: $\Delta i_{shaping}(s) = Y_{shaping}(s) \Delta u_{pcc}(s)$. The goal is to cancel the negative-resistance effect of the PLL. Therefore, we choose:
$$ Y_{shaping}(s) = G_p(s)H^0(s) $$
When this virtual current is subtracted from the original current reference, the effective PLL-induced admittance term in the final $Y_{eq}(s)$ becomes zero. The resulting equivalent output admittance of the on-grid inverter is then:
$$ Y_{eq}(s) = Y_d(s) + Y_{shaping}(s) = Y_p(s) $$
Remarkably, $Y_{eq}(s)$ reduces to the passive admittance $Y_p(s)$ of the LCL filter and the current controller, which is inherently stable and passive for well-designed parameters. This guarantees that the on-grid inverter presents a passive impedance to the grid, ensuring stability under any passive grid impedance (a sufficient condition via the passivity theorem).
The implementation requires accurate knowledge of $G_p(s)$ and $H^0(s)$, which are system parameters. In practice, a robust approximation can be used. The key result is that the destabilizing effect of the PLL is completely neutralized by this active compensation method.
Simulation and Experimental Validation
A 5 kW single-phase on-grid inverter system was modeled and tested to validate the proposed concepts. The system parameters are summarized in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC-Link Voltage ($U_{dc}$) | 400 V | Grid Voltage ($U_g$) | 220 V (RMS) |
| Fundamental Frequency ($f_0$) | 50 Hz | Switching/Sampling Freq. ($f_s$) | 15 kHz |
| Inverter-side Inductor ($L_1$) | 1.2 mH | Grid-side Inductor ($L_2$) | 0.9 mH |
| Filter Capacitor ($C$) | 12 μF | PR Controller ($K_p$ / $K_r$) | 10 / 800 |
| Current Reference Amplitude ($I_m$) | 20 A | Grid Impedance (Base Case, $L_g$) | Variable (0-15 mH) |
Frequency Decoupling Verification: A 3% harmonic voltage at 130 Hz was injected at the PCC. With the conventional control, the grid current spectrum showed significant coupled components at 30 Hz ($130-2\times50$) and 230 Hz ($130+2\times50$), confirming strong frequency coupling. After enabling the proposed decoupling factor control, these coupled sidebands were virtually eliminated, leaving only the fundamental and the 130 Hz component, proving the effectiveness of the decoupling.
Stability Enhancement under Weak Grid: Experiments were conducted on an RT-LAB hardware-in-the-loop platform. Under an ideal grid ($L_g \approx 0$), the system was stable with both conventional and proposed methods. As the grid inductance increased to $L_g = 9$ mH, the system with conventional control exhibited sustained oscillations at approximately 135 Hz, with a high Total Harmonic Distortion (THD). The corresponding Nyquist plot of $Y_d(s) / Y_g(s)$ (using the decoupled model) showed a negative phase margin, predicting instability.
When the proposed admittance shaping control was activated for the on-grid inverter, the oscillations vanished immediately. The grid current became sinusoidal with very low THD. Even when the grid inductance was further increased to a very weak condition of $L_g = 15$ mH, the system remained stable. The admittance shaping effectively made the output admittance passive, as evidenced by the positive phase margin observed in the recalculated loop gain. Furthermore, the system demonstrated good dynamic performance during a step change in load, recovering stability within one fundamental cycle.
Comparative Analysis and Discussion
The proposed methodology offers distinct advantages over existing approaches for stabilizing single-phase on-grid inverters:
- vs. Simplified SISO Models: Traditional SISO models ignore frequency coupling, which can lead to inaccurate stability predictions, especially in single-phase systems where coupling is strong. Our method first acknowledges and then actively cancels this coupling, leading to a simple and accurate SISO model ($Y_d(s)$) for design and analysis.
- vs. Full MIMO Analysis: While the full HTF-based MIMO model is accurate, it is complex for practical engineering design and online monitoring. Our decoupling factor transforms the complex MIMO problem into a tractable SISO one without sacrificing the accuracy needed for stability assessment.
- vs. PLL Bandwidth Limitation or Feedforward: Common practices involve reducing PLL bandwidth or using grid voltage feedforward to mitigate its impact. These methods can compromise synchronization performance or be sensitive to grid voltage distortion. Our admittance shaping method directly targets and cancels the negative admittance at its source, allowing for a potentially faster PLL design while ensuring stability.
- Robustness: The virtual admittance $Y_{shaping}(s)$ is derived from the inverter’s own controller and plant model. While it requires parameter knowledge, its robustness is comparable to that of the underlying current controller. It does not rely on feedforward of noisy grid voltage measurements for stabilization.
This approach is particularly valuable for high-power density on-grid inverters operating in very weak grid environments, such as rural distribution networks or the ends of long feeder lines.
Conclusion and Future Perspectives
This article has addressed the critical stability challenge for single-phase on-grid inverters operating in weak grids. By formally modeling the infinite frequency coupling effect inherent in single-phase PLL-based control and introducing a novel decoupling factor within the current loop, we derived a precise and practical SISO equivalent admittance model. This model clearly reveals the negative-resistance behavior induced by the PLL. Building upon this accurate model, we proposed an effective admittance shaping strategy via a virtual parallel admittance. This strategy actively cancels the PLL’s destabilizing effect, rendering the overall output admittance of the on-grid inverter passive and thus guaranteeing stability when connected to any passive grid impedance.
The proposed method successfully bridges the gap between the accuracy of complex MIMO modeling and the simplicity of SISO design tools. Experimental results on a hardware-in-the-loop platform confirm that the strategy effectively eliminates harmonic oscillations and ensures robust stability even under severely weak grid conditions, while maintaining good dynamic response.
Future work could explore the extension of this frequency decoupling and admittance shaping concept to other single-phase control structures, such as those using virtual inertia or grid-forming controls. Additionally, adaptive tuning of the decoupling and shaping blocks to accommodate varying grid conditions or system parameters could further enhance the robustness and application scope of this promising technique for next-generation, resilient on-grid inverters.