In distributed generation systems, the stable operation of the utility interactive inverter is paramount for overall system security. The increasing penetration of renewable energy, characterized by intermittency and volatility, leads to significant grid impedance fluctuations. This results in weak grid conditions, typically defined by a low Short-Circuit Ratio (SCR). Under such conditions, the utility interactive inverter must maintain robust performance. Key control loops, namely the Phase-Locked Loop (PLL) for grid synchronization and the grid voltage feedforward (GVF) for improving current quality and reducing steady-state error, can detrimentally interact with the grid impedance. Their destabilizing effects often overlap in frequency, creating a complex stability challenge that complicates the design of a robust utility interactive inverter.
This article investigates this intricate stability problem in LCL-filtered utility interactive inverters. We analyze the mechanisms through which the PLL and GVF induce instability and propose a comprehensive, improved feedforward strategy. The analysis begins with the derivation of the system’s equivalent output admittance, incorporating the effects of both factors.
System Modeling and Admittance Derivation
The control structure of a standard LCL-type utility interactive inverter is considered. Key components include the LCL filter (L1, C, L2), a PI current controller Gi(s), and the grid impedance Lg. Digital control delay is modeled as Gd(s) = e-1.5sTs. The grid voltage feedforward path, with a standard proportional gain Gff(s)=1/KPWM, and a synchronous reference frame PLL are included. The PLL transfer function, considering a delay-based implementation, is given by:
$$ G_{PLL}(s) = \frac{-[K_p(s-j\omega_0) + K_i]}{2s[U_m – j\omega_0] + [K_p(s-j\omega_0) + K_i]} $$
where $K_p$ and $K_i$ are the PLL PI gains, $U_m$ is the grid voltage amplitude, and $\omega_0$ is the fundamental angular frequency. The current reference is thus $i_{ref}(s) = I^* G_{PLL}(s) v_{PCC}(s)$.
By simplifying the system block diagram, the relationship between the grid current $i_g$ and the Point of Common Coupling (PCC) voltage $v_{PCC}$ can be established. The total equivalent output admittance $Y_o(s) = -i_g(s)/v_{PCC}(s)$ is found to be the sum of three distinct components:
$$ Y_o(s) = Y_{con}(s) + Y_{PLL}(s) + Y_{ff}(s) $$
where $Y_{con}(s)$ is the admittance introduced by the current control loop, $Y_{PLL}(s)$ is the admittance introduced by the PLL dynamics, and $Y_{ff}(s)$ is the admittance introduced by the grid voltage feedforward. This model is crucial for analyzing stability using the Nyquist criterion for interconnected systems. For stability, the phase of $Y_o(s)$ at the frequency where its magnitude intersects with the grid admittance $Y_g(s) = 1/(sL_g)$ must be less than 90°.
Stability Analysis and Frequency Band Division
Plotting the Bode diagram of $Y_o(s)$ reveals critical insights. In a weak grid, $Y_o(s)$ can exhibit a phase greater than 90° at low-to-medium frequencies, violating the stability criterion. To dissect the contribution of each factor, the admittances $Y_{con}+Y_{PLL}$ and $Y_{con}+Y_{ff}$ are plotted separately alongside the total $Y_o$.
The analysis leads to a clear frequency band division, as summarized in the table below:
| Frequency Range | Dominant Instability Factor | Typical Frequency Characteristics |
|---|---|---|
| $f < f_{d1}$ | Phase-Locked Loop (PLL) | Phase lead introduced by PLL dynamics near and below its bandwidth. |
| $f_{d1} < f < f_{d2}$ | Joint Action of PLL & GVF | Complex interaction where both factors significantly alter phase. |
| $f > f_{d2}$ | Grid Voltage Feedforward (GVF) | Phase lag primarily due to control delays in the feedforward path. |
This overlap, particularly in the joint-action band, makes the stability analysis and solution design for the utility interactive inverter non-trivial. A solution targeting only one factor may be insufficient if the impedance intersection frequency falls within the overlapping region.
The Role and Limitations of a Weighted Feedforward Coefficient
A common initial approach to mitigate GVF-induced instability is to introduce a weighting coefficient $K_f$ to the feedforward path, modifying it to $G’_{ff}(s)=K_f/K_{PWM}$. This reduces the effective feedforward gain, trading off some harmonic rejection capability for improved phase margin.
However, the weighted coefficient also influences the PLL stability problem indirectly. The GVF path, when active, effectively adds a parallel current feedback loop. This alters the current loop’s open-loop gain $T_A(s)$. The modified gain $T’_A(s)$ shows that the weighted feedforward impacts the current loop bandwidth, as described by:
$$ T’_A(s) = \frac{G_i(s)G_d(s)K_{PWM}H_{i2}}{s^3L_1C(L_2+L_g) + s^2G_i(s)G_d(s)K_{PWM}H_{i1}C(L_2+L_g) + s(L_1+L_2+L_g) – G_i(s)G_d(s)K_{PWM}G’_{ff}(s)L_g} $$
As shown in the analysis, a larger $K_f$ generally increases the current loop bandwidth. This helps to decouple the current loop dynamics from the slower PLL dynamics, thereby slightly improving robustness against PLL-induced oscillations. Therefore, choosing $K_f$ involves a trade-off: a smaller value benefits GVF stability directly but may leave the system more vulnerable to PLL issues; a larger value helps with PLL coupling but may re-introduce GVF-related phase lag. An optimal value must be found within the joint-action frequency region. For a specific case with SCR=5, the optimal $K_f$ was found to be 0.54, providing a maximum phase margin of 13.2°, compared to 7.6° with no feedforward ($K_f=0$).
| Condition | Weighting Coefficient $K_f$ | Phase Margin at SCR=5 | Current Loop Bandwidth (Relative) | Harmonic Rejection |
|---|---|---|---|---|
| No Feedforward | 0 | 7.6° | Lower | Poor |
| Full Feedforward | 1.0 | Negative (Unstable) | Higher | Excellent |
| Optimal Weighted | 0.54 | 13.2° | Moderate | Moderate |
While effective for moderate weak grids, this weighted approach has limited efficacy in very weak grids (e.g., SCR < 5). The phase margin remains too low, and a fundamental compensation of the phase characteristic introduced by the PLL is necessary.
Proposed Improved Feedforward Path with Phase Compensation
To address the limitation of the simple weighted coefficient and specifically target the phase defect caused by the PLL in the joint-action band, an additional, intelligent feedforward path $G_A(s)$ is proposed. This path is designed to introduce a virtual admittance $Y_A(s)$ in parallel, which effectively compensates for the undesirable phase contribution of $Y_{PLL}(s)$.
The goal is to reshape the PLL admittance to $Y_{PLL2}(s) = Y_{PLL}(s) G_{APF\_add}(s)$, where $G_{APF\_add}(s)$ is a compensating filter. To achieve significant phase shift without gain distortion, an All-Pass Filter (APF) is a suitable core component. However, a standard APF would also shift the phase at the fundamental frequency, introducing a steady-state phase error between the grid current and voltage. Therefore, a phase-lead compensation stage, implemented via a High-Pass Filter (HPF), is cascaded with the APF. The combined compensating filter is:
$$ G_{APF\_add}(s) = K_1 \cdot \frac{s}{s+\omega_{x1}} \cdot \frac{-s+\omega_{x2}}{s+\omega_{x2}} $$
where $\omega_{x1}$ is the HPF cutoff frequency, $\omega_{x2}$ is the APF characteristic frequency, and $K_1$ is a gain correction factor to ensure unity gain at the fundamental frequency ($|G_{APF\_add}(j\omega_0)|=1$).
The required feedforward function $G_A(s)$ is then derived by equating its effect to the desired admittance compensation:
$$ G_A(s) = I^* [G_{PLL}(s)G_{APF\_add}(s) – G_{PLL}(s)] $$
The final, complete improved feedforward control law for the utility interactive inverter becomes:
$$ G_{ff\_add}(s) = \frac{K_f}{K_{PWM}} – G_A(s) = \frac{K_f}{K_{PWM}} – I^* G_{PLL}(s) [G_{APF\_add}(s) – 1] $$
Parameter Design for the Compensating Filter
The design of $G_{APF\_add}(s)$ involves determining $\omega_{x1}$ and $\omega_{x2}$. Two constraints guide the design:
- Fundamental Frequency Phase Correction: The filter must compensate the base phase error introduced by the weighted feedforward at $\omega_0$, ensuring $i_g$ and $v_{PCC}$ are in phase. This gives: $\arg[G_{APF\_add}(j\omega_0)] = 180° – \arg[Y_o(j\omega_0)]$.
- Phase Compensation at Crossover Frequency: To maximize robustness at a target weak grid condition (e.g., SCR=5), the filter should provide specific phase compensation (e.g., -100°) at the anticipated admittance crossover frequency $f_z$.
Constraint 1 establishes a relationship between $\omega_{x1}$ and $\omega_{x2}$. Constraint 2 is then used to solve for the specific value of $\omega_{x2}$. A larger $\omega_{x2}$ is preferred to minimize harmonic amplification from the HPF stage. For the example system with SCR=5 ($f_z$=322.6 Hz), solving yields $\omega_{x2}$ = 1536 rad/s, $\omega_{x1}$ = 221.3 rad/s, and $K_1$ = 1.22. This design results in a phase margin of approximately 46°, a substantial improvement over the 13.2° offered by the weighted feedforward alone.
Simulation and Experimental Verification
A 1 kW prototype utility interactive inverter was built to validate the theoretical analysis. The system parameters are listed below:
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC Bus Voltage $V_{in}$ | 180 V | Grid Voltage $V_g$ | 110 V |
| Rated Power $P_o$ | 1 kW | Switching Frequency $f_{sw}$ | 15 kHz |
| Inverter-side Inductor $L_1$ | 550 μH | Grid-side Inductor $L_2$ | 110 μH |
| Filter Capacitor $C$ | 5 μF | Current PI: $K_{pi}, K_{ii}$ | 0.831, 2050 |
| PLL PI: $K_p, K_i$ | 5.57, 2400 | Sampling Frequency | 30 kHz |
Experiments were conducted under varying grid impedances ($L_g$).
Case 1: Moderate Weak Grid ($L_g$ = 1.4 mH). With standard GVF, the system was oscillatory. Applying the weighted feedforward ($K_f$=0.54) stabilized the system but introduced a 10.8° steady-state phase error. The proposed $G_{ff\_add}(s)$ stabilized the system while eliminating this phase error.
Case 2: Very Weak Grid ($L_g$ = 7.8 mH, SCR≈5). The weighted feedforward alone yielded poor waveform quality with low phase margin. The proposed improved feedforward path $G_{ff\_add}(s)$ significantly enhanced stability, resulting in clean sinusoidal currents with a high phase margin, as predicted.
Case 3: Extreme Weak Grid ($L_g$ = 10 mH). The system with the proposed method remained stable, demonstrating the extended robustness range of the utility interactive inverter.
Simulation results further corroborate the experimental findings. The FFT analysis of grid current under weak grid conditions confirms that the proposed method maintains low harmonic distortion (THD below 3%) even at high grid impedance, whereas conventional control leads to significant low-order harmonics. Dynamic simulation in strong grid also shows that the improved feedforward does not compromise the system’s transient response.
| Grid Impedance $L_g$ | Control Method | Stability | Phase Margin | Current THD | Steady-State Phase Error |
|---|---|---|---|---|---|
| 1.4 mH | Standard GVF ($K_f$=1) | Unstable/Oscillatory | -0.9° | High | N/A |
| 1.4 mH | Weighted GVF ($K_f$=0.54) | Stable | 48.1° | Acceptable | 10.8° |
| 1.4 mH | Proposed $G_{ff\_add}(s)$ | Stable | >45° | Low (<3%) | ~0° |
| 7.8 mH | Weighted GVF ($K_f$=0.54) | Marginally Stable | 13.2° | Poor | Present |
| 7.8 mH | Proposed $G_{ff\_add}(s)$ | Stable | 45.8° | Low (<3%) | ~0° |
| 10 mH | Proposed $G_{ff\_add}(s)$ | Stable | 37.6° | Low | ~0° |
Conclusion
This work addresses the complex stability problem in utility interactive inverters operating in weak grids, where the PLL and grid voltage feedforward interact detrimentally. The analysis clearly delineates the frequency bands dominated by each factor and their overlap. A simple weighted feedforward coefficient offers a trade-off solution but has limited effectiveness in very weak grids. The proposed improved feedforward path $G_{ff\_add}(s)$ provides a comprehensive solution. It combines a weighted coefficient for basic GVF gain management with an intelligent phase-compensating path designed to counteract the specific phase distortion introduced by the PLL dynamics. The method requires no grid impedance measurement and is straightforward to implement. Both simulation and experimental results on a 1 kW prototype confirm that the enhanced utility interactive inverter maintains high robustness, low harmonic distortion, and good dynamic performance across a wide range of grid impedance variations, significantly extending its viable operating range in weak grid environments.