The global transition towards sustainable energy systems has dramatically increased the penetration of distributed generation resources. In small-scale applications like rooftop photovoltaic systems, the single-phase utility interactive inverter is a cornerstone technology due to its high conversion efficiency and relatively simple topology. However, the integration of numerous such inverters, often connected via transformers with significant leakage inductance or to remote rural networks, results in a grid environment characterized by non-negligible grid impedance. This “weak grid” condition poses significant stability challenges for grid-following control schemes.
One of the most critical destabilizing factors in weak grids is the interaction between the inverter’s control loops and the grid impedance. The Phase-Locked Loop (PLL), essential for synchronizing the inverter with the grid voltage, is a primary source of instability. Its inherent asymmetrical control structure—where only the q-axis voltage is used for phase correction—introduces a complex phenomenon known as frequency coupling. This effect transforms the system from a straightforward Single-Input Single-Output (SISO) system into a Multi-Input Multi-Output (MIMO) system, where a perturbation at a single frequency can elicit responses at multiple, coupled frequencies (e.g., f_p and f_p ± n*f_0). This frequency coupling complicates impedance-based stability analysis, often leading to optimistic or inaccurate predictions if ignored, and can induce harmonic oscillations that threaten power quality and system reliability.
This article addresses this critical issue by proposing a novel control strategy centered on a Symmetrical Phase-Locked Loop (SPLL) combined with an Enhanced Second-Order Generalized Integrator (ESOGI) based orthogonal signal generator. The primary objectives are twofold: first, to completely eliminate the frequency-coupling effects induced by conventional PLL structures in single-phase utility interactive inverters, thereby simplifying the system model and analysis; and second, to actively reshape the inverter’s output admittance to enhance its stability margin when operating under very weak grid conditions, without compromising dynamic performance.
Modeling and Stability Challenges of Conventional Single-Phase Utility Interactive Inverters
The typical topology of a single-phase LCL-filter based utility interactive inverter and its conventional control structure is considered. The system includes an LCL output filter (L1, C, L2), a grid inductor Lg representing the weak grid, and a control system featuring a Proportional-Resonant (PR) current controller Gc(s), a digital delay model Gd(s), and a standard PLL.
The core instability mechanism stems from the PLL’s influence on the current reference. In a conventional PLL, the q-axis voltage component uq is regulated to zero to track the grid phase angle θ. The current reference is generated as iref = Im cos(θ + φ). A small-signal perturbation in the Point of Common Coupling (PCC) voltage Δupcc affects uq, which in turn perturbs the PLL output Δθ. This perturbation then modulates the amplitude of the current reference. The critical insight is that the steady-state operating point (sin(ω0t), cos(ω0t)) is a periodic time-varying signal. When the linearization is performed and transformed to the frequency domain, this time-varying operation gives rise to frequency-shifted coupling terms.
The frequency-domain relationship between a PCC voltage perturbation and the resulting perturbation in the current reference for a conventional PLL-based utility interactive inverter can be derived as:
$$ \Delta i_{ref}(s) = H_{PLL0}(s)\Delta u_{pcc}(s) + H_{PLLn}(s)\Delta u_{pcc}(s_n) + H_{PLLp}(s)\Delta u_{pcc}(s_p) $$
where s_n = s – j2ω0, s_p = s + j2ω0, and ω0 is the fundamental grid angular frequency. The transfer functions HPLL0(s), HPLLn(s), and HPLLp(s) are defined by the PLL parameters. This equation clearly shows the MIMO nature: a perturbation at a complex frequency s influences the current reference not only at s but also at the coupled frequencies s_n and s_p. These coupling terms, HPLLn(s) and HPLLp(s), are the root cause of the frequency coupling phenomenon.
To assess stability using the Nyquist criterion, an equivalent SISO impedance model must be constructed. The inverter’s output admittance Y(s) is found to consist of three main components:
$$ Y(s) = Y_0(s) – Y_{nn}(s) – Y_{pp}(s) $$
Here, Y0(s) = Yin(s) – Gp(s)HPLL0(s) is the “self-admittance,” representing the direct effect without considering coupling feedback. Yin(s) is the inherent inverter admittance with no PLL, and Gp(s) is the closed-loop current transfer function. The terms Ynn(s) and Ypp(s) are the “coupling admittances,” which depend on both the inverter’s internal couplings and the grid admittance Yg(s) = 1/(Lgs):
$$ Y_{nn}(s) = \frac{G_p(s)H_{PLLn}(s)G_p(s_n)H_{PLLp}(s_n)}{Y_g(s_n) – (G_p(s_n)H_{PLL0}(s_n)-Y_{in}(s_n))} $$
$$ Y_{pp}(s) = \frac{G_p(s)H_{PLLp}(s)G_p(s_p)H_{PLLn}(s_p)}{Y_g(s_p) – (G_p(s_p)H_{PLL0}(s_p)-Y_{in}(s_p))} $$
The stability is then evaluated by analyzing the loop gain Y(s)/Yg(s). The key problem is that Ynn(s) and Ypp(s) are functions of the grid inductance Lg. As Lg increases (grid weakens), these coupling admittance terms significantly distort the total output admittance Y(s), particularly in the low-to-medium frequency range, often reducing the phase margin and leading to instability at frequencies where a simple model (using only Y0(s)) would predict stability. This discrepancy underscores the necessity of accounting for frequency coupling in the stability assessment of single-phase utility interactive inverters in weak grids.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | Prated | 5 kW |
| Grid Voltage (RMS) | Vg | 220 V |
| Grid Frequency | f0 | 50 Hz |
| DC-Link Voltage | Vdc | 400 V |
| Inverter-side Inductor | L1 | 1.8 mH |
| Grid-side Inductor | L2 | 0.6 mH |
| Filter Capacitor | C | 10 μF |
| Switching/Sampling Frequency | fsw, fs | 10 kHz |
Fundamental Principles of Symmetrical Control for Frequency Coupling Elimination
The analysis reveals that the asymmetry of the conventional PLL—using only q-axis information—is the genesis of the coupling terms HPLLn(s) and HPLLp(s). A logical solution is to implement a symmetrical control structure. The Symmetrical Phase-Locked Loop (SPLL) achieves this by using both d-axis and q-axis voltage information for synchronization. Instead of just forcing uq to zero, the SPLL controls a complex voltage vector to align with a reference frame, effectively regulating both its magnitude and phase.
In the SPLL structure, the transformation and current reference generation are modified. The key equations become:
$$ \begin{bmatrix} u_d \\ u_q \end{bmatrix} = e^{\theta_q} \begin{bmatrix} \cos \theta_d & \sin \theta_d \\ -\sin \theta_d & \cos \theta_d \end{bmatrix} \begin{bmatrix} u_\alpha \\ u_\beta \end{bmatrix} $$
$$ i_{ref} = I_m e^{-\theta_q} \cos(\theta_d + \phi) $$
Performing a small-signal analysis on this symmetrical structure yields a fundamentally different and simpler result. The perturbation in the reference current is now directly related to the perturbation in the PCC voltage by a single transfer function, without any frequency-shifted coupling terms:
$$ \Delta i_{ref}(s) = H_{PLL}^{sym}(s) \Delta u_{pcc}(s) $$
Here, HPLLsym(s) is the symmetrical PLL transfer function. Comparing it to the conventional case, we find HPLLsym(s) = 2HPLL0(s). This is a profound result: the SPLL completely eliminates the frequency-coupling admittances Ynn(s) and Ypp(s). Consequently, the output admittance of the utility interactive inverter simplifies to the SISO form:
$$ Y_{sym}(s) = Y_{in}(s) – G_p(s)H_{PLL}^{sym}(s) = Y_{in}(s) – 2G_p(s)H_{PLL0}(s) $$
This model is no longer dependent on the grid impedance Lg, greatly simplifying stability analysis. The Nyquist criterion can be applied directly to Ysym(s)/Yg(s). However, a new challenge emerges. While the coupling is eliminated, the magnitude of the PLL-induced admittance term is doubled (2Gp(s)HPLL0(s)). This effectively expands the negative conductance region of Ysym(s) in the low-to-medium frequency band, which can degrade the phase margin and potentially cause instability under even moderately weak grid conditions. Therefore, while the SPLL solves the coupling problem, it necessitates an additional mechanism to enhance the stability margin of the utility interactive inverter.
| Control Feature | Conventional PLL-based Inverter | SPLL-based Inverter |
|---|---|---|
| Control Symmetry | Asymmetric (q-axis only) | Symmetric (d & q-axis) |
| Frequency Coupling | Present. MIMO Model Required. | Eliminated. Pure SISO Model. |
| Output Admittance Model | Y(s) = Y0(s) – Ynn(s) – Ypp(s) (Lg-dependent) | Ysym(s) = Yin(s) – 2Gp(s)HPLL0(s) (Lg-independent) |
| Primary Stability Challenge | Coupling-induced model distortion and oscillation. | Expanded negative admittance region due to larger PLL gain. |
| Analysis Complexity | High (GNC or equivalent SISO derivation) | Low (Standard Nyquist) |
Enhanced Stability via ESOGI-Based Orthogonal Signal Generation
To address the stability limitation of the basic SPLL, we focus on the orthogonal signal generator (OSG), a crucial block in single-phase systems that creates the β-component needed for the αβ transformation. The common T/4 delay and standard Second-Order Generalized Integrator (SOGI) OSG have limitations. The T/4 delay offers no frequency response shaping, while the standard SOGI can provide some phase lead compensation. The transfer functions for the standard SOGI are:
$$ G_{\alpha}^{SOGI}(s) = \frac{k\omega_0 s}{s^2 + k\omega_0 s + \omega_0^2}, \quad G_{\beta}^{SOGI}(s) = \frac{k\omega_0^2}{s^2 + k\omega_0 s + \omega_0^2} $$
Where k is the damping factor. The SOGI essentially acts as a band-pass filter GBPF(s) = GαSOGI(s). When used with an SPLL, it modifies the PLL’s contribution to the output admittance: YPLLSOGI(jω) = YPLLT/4(jω) GBPF(jω). This provides a phase compensation of arg(GBPF(jω)) = arctan((ω_0^2 – ω^2)/(k ω ω_0)). Reducing k increases this compensation, reducing the negative admittance region and improving stability, but at the cost of degraded dynamic response (slower tracking).
To break this trade-off, we propose an Enhanced SOGI (ESOGI) structure. The core idea is to apply an additional, fixed 90-degree phase shift to the orthogonal signals before they are processed by the SPLL. In the frequency domain, this is equivalent to multiplying the αβ voltage vector by -j:
$$ \begin{bmatrix} u_{\alpha_e} \\ u_{\beta_e} \end{bmatrix} = -j \begin{bmatrix} u_{\alpha} \\ u_{\beta} \end{bmatrix} $$
This intentional phase shift fundamentally alters the admittance shaping capability. Analyzing the modified control loop, the new PLL-related transfer function HPLLESOGI(s) for the utility interactive inverter becomes:
$$ H_{PLL}^{ESOGI}(s) = -j\frac{I_m}{2} \left[ T_{PLL}(s-j\omega_0)\left(G_\alpha(s) + jG_\beta(s)\right) – T_{PLL}(s+j\omega_0)\left(G_\alpha(s) – jG_\beta(s)\right) \right] $$
where Gα(s) and Gβ(s) are the OSG transfer functions from upcc to uα and uβ. The resulting output admittance is YESOGI(s) = Yin(s) – Gp(s)HPLLESOGI(s). The critical effect of this modification is that it significantly reduces the phase lag (or can introduce phase lead) contributed by the PLL path in the crucial frequency range above the fundamental frequency. This effectively shrinks or even eliminates the negative conductance region of the utility interactive inverter’s output admittance for frequencies above the fundamental, dramatically improving the phase margin when interacting with a capacitive grid impedance.
Since the ESOGI operation rotates the voltage vector, it introduces a steady-state phase error in the PLL output. This is easily corrected by adding a fixed +90-degree compensation angle (φ0) to the final PLL output angle used for current reference generation and dq transformations, ensuring correct synchronization with the grid.
Comprehensive Stability Analysis and Experimental Verification
The combined strategy of SPLL and ESOGI provides a comprehensive solution for the single-phase utility interactive inverter. The SPLL eliminates the complex frequency-coupling phenomenon, and the ESOGI actively reshapes the output admittance to enhance robustness.
A comparative stability analysis using the impedance-based Nyquist criterion clearly demonstrates the advantage. Consider a weak grid scenario with a high grid inductance of Lg = 20 mH (Short Circuit Ratio, SCR ≈ 1.4). For a standard SPLL with a T/4 delay OSG, the inverter’s output admittance Ysym(s) shows a large negative conductance region, leading to an unstable condition. Using a standard SOGI-OSG with k=1.414 improves the margin but may still be unstable under such weak grid conditions; achieving stability would require reducing k to a very low value (e.g., 0.8), which severely hampers dynamic performance.
In contrast, the proposed SPLL with ESOGI-OSG (with k=3, maintaining good dynamics) produces an output admittance YESOGI(s) with a drastically reduced negative conductance region. The phase plot shows sufficient phase margin, confirming stable operation even at SCR=1.4. The ESOGI’s admittance shaping effectively decouples the PLL’s destabilizing effect from the mid-high frequency range, where the inherent inverter admittance Yin(s) is naturally more stable.
| Control Method | Grid Inductance L_g | OSG Parameter (k) | Stability Prediction | Key Observation |
|---|---|---|---|---|
| Conventional PLL | 11 mH (SCR~2.8) | T/4 Delay | Unstable (Oscillation ~120Hz with couplings at 20Hz & 220Hz) | Frequency coupling leads to instability not predicted by simple model. |
| Basic SPLL | 11 mH (SCR~2.8) | T/4 Delay | Unstable (Oscillation ~135Hz, no coupling) | Coupling eliminated, but enlarged negative admittance causes instability. |
| SPLL + Standard SOGI | 16 mH (SCR~1.75) | k = 3 | Unstable | Moderate k-value insufficient for very weak grid. |
| SPLL + Standard SOGI | 16 mH (SCR~1.75) | k = 1.414 | Stable | Lower k stabilizes but slows dynamic response. |
| SPLL + ESOGI (Proposed) | 20 mH (SCR~1.4) | k = 3 | Stable (Phase Margin ~18.4°) | Stable in very weak grid with good dynamic k-value. |
The proposed control strategy was validated using a hardware-in-the-loop (HIL) experimental platform, featuring a real-time simulator (RT-LAB) and a DSP (TMS320F28335) controller implementing the algorithms. A 5kW, 220V/50Hz single-phase utility interactive inverter model was used.
1. Decoupling Performance Verification: A 130 Hz, 5V disturbance was injected into the PCC voltage. With a conventional PLL, the grid current spectrum showed responses not only at 130 Hz but also at the coupled frequencies of 30 Hz and 230 Hz. With the proposed SPLL+ESOGI control, the current response was only at 130 Hz, experimentally confirming the complete elimination of frequency coupling.
2. Stability Enhancement Verification: The grid inductance was progressively increased.
– With a conventional PLL, the system became unstable at L_g = 11 mH, exhibiting oscillations at 120 Hz and coupled harmonics.
– With a basic SPLL (T/4 delay), the system also became unstable at L_g = 11 mH, oscillating at 135 Hz but with no coupled harmonics, proving coupling elimination.
– With SPLL and standard SOGI (k=1.414), stability was maintained at L_g = 16 mH.
– Crucially, with the proposed SPLL and ESOGI (k=3), the utility interactive inverter remained stable with low current distortion even at an extremely weak grid condition of L_g = 20 mH (SCR=1.4).
3. Dynamic Performance: The dynamic response was tested by stepping the load from 50% to 100% under L_g = 20 mH. The utility interactive inverter with the proposed controller settled to the new steady state within half a grid cycle, demonstrating that the enhanced stability does not come at the expense of dynamic performance.
Conclusion
This work has addressed the critical challenges of frequency coupling and stability degradation in single-phase utility interactive inverters operating in weak grids. The analysis demonstrated that the conventional PLL’s asymmetric control introduces complex frequency-coupling admittances that distort the inverter’s output impedance model and can lead to unexpected harmonic oscillations.
The proposed integrated solution, combining a Symmetrical Phase-Locked Loop (SPLL) with an Enhanced Second-Order Generalized Integrator (ESOGI) based orthogonal signal generator, provides a comprehensive remedy. The SPLL fundamentally eliminates the frequency-coupling effect by employing symmetrical dq-axis control, simplifying the system to a pure SISO model that is independent of grid impedance. Building upon this decoupled foundation, the ESOGI structure performs active admittance shaping. By introducing a strategic phase shift in the signal processing chain, it effectively compresses the negative conductance region in the inverter’s output admittance, thereby significantly boosting the phase margin in weak grid scenarios.
This approach allows the utility interactive inverter to maintain robust stability under very weak grid conditions (e.g., SCR down to 1.4) without needing to severely compromise the bandwidth of the PLL or the orthogonal signal generator. Consequently, both good dynamic performance and strong robustness are achieved simultaneously. The theoretical framework, supported by detailed impedance modeling and stability analysis, was conclusively validated through RT-LAB hardware-in-the-loop experiments. This control strategy offers a valuable and practical design method for enhancing the reliability and power quality of single-phase distributed generation systems in increasingly prevalent weak grid environments.