The proliferation of grid-connected photovoltaic (PV) systems represents a cornerstone of the global transition towards sustainable energy. The utility interactive inverter, serving as the critical interface between the PV array and the utility grid, is paramount to this integration. Its primary function is to convert direct current (DC) from the solar panels into high-quality alternating current (AC) that is synchronized with the grid voltage. However, the increasing penetration of distributed generation, often connected via long feeders or transformers with significant leakage inductance, leads to a non-negligible grid impedance. This scenario characterizes a “weak grid,” posing substantial challenges to the stable and reliable operation of utility interactive inverters.
In weak grid conditions, the voltage at the Point of Common Coupling (PCC) becomes susceptible to distortion and fluctuation due to the interaction between the inverter’s output current and the grid impedance. A Phase-Locked Loop (PLL) is universally employed in utility interactive inverters to precisely synchronize the inverter’s output with the grid voltage by tracking the phase angle at the PCC. When the PCC voltage is distorted, the PLL’s output phase deviates from the ideal grid phase. This deviation introduces undesirable coupling dynamics between the PLL and the current control loop, effectively altering the inverter’s output impedance characteristics. Specifically, this interaction can severely degrade the phase margin of the inverter’s output impedance in the low-frequency range, inducing a negative impedance characteristic that threatens system stability. This instability manifests as harmonic resonance, exaggerated grid current distortion, and in severe cases, can lead to protective relay tripping and disconnection of the utility interactive inverter from the grid.
This paper delves into the stability challenges faced by LCL-filter-based utility interactive inverters operating in weak grids. We begin by establishing a comprehensive small-signal model that explicitly accounts for the coupling effects of the PLL. Through impedance-based stability analysis, we elucidate the mechanism by which PCC voltage disturbances deteriorate the system’s phase margin. To mitigate this issue, an improved control strategy based on a compensated small-signal injection is proposed. Unlike conventional small-signal compensation methods which offer limited bandwidth enhancement, the proposed technique incorporates a tailored band-pass filter to provide effective phase compensation across a broader frequency spectrum, particularly targeting the critical low-frequency region where PLL coupling is most detrimental. This method effectively reshapes the output impedance of the utility interactive inverter, restoring sufficient phase margin even under severe weak grid conditions without requiring complex redesign of the PLL’s internal structure. The theoretical derivations are validated through detailed frequency-domain analysis and experimental results from a prototype utility interactive inverter.
Mathematical Modeling of the LCL-Type Utility Interactive Inverter
1.1. Model Without PLL Dynamics
The topology of a three-phase, two-level voltage source utility interactive inverter with an LCL output filter is considered. The system parameters are defined as follows: $L_1$ is the inverter-side filter inductance, $L_2$ is the grid-side filter inductance, $C_f$ is the filter capacitance, $Z_g = sL_g$ represents the grid impedance (modeled as a pure inductance for stability analysis), and $V_{dc}$ is the input DC voltage. The currents $i_1$, $i_c$, and $i_g$ represent the inverter-side current, capacitor current, and grid-injected current, respectively. The voltages $v_{PCC}$ and $v_g$ denote the PCC voltage and the ideal grid voltage.
In the synchronous reference frame (d-q frame) rotating at the fundamental grid frequency $\omega_0$, the current control is typically implemented using PI regulators. For the d-axis, the controller $G_i(s)$ is given by:
$$ G_i(s) = k_p + \frac{k_i}{s} $$
where $k_p$ and $k_i$ are the proportional and integral gains. The system includes computational and PWM delay, modeled as $G_d(s) = e^{-1.5T_s s}$, where $T_s$ is the sampling period. Capacitor current feedback active damping with gain $K_c$ is often used to suppress LCL resonance. The plant transfer functions from the modulating voltage $U_d(s)$ to the inverter-side current $I_{1d}(s)$ and to the grid current $I_{gd}(s)$ can be derived. The control block diagram for the d-axis current loop, ignoring the PLL and grid voltage disturbance, is shown in a simplified form, leading to the open-loop gain $T_A(s)$:
$$ T_A(s) = G_A(s) G_B(s) G_d(s) G_i(s) $$
where $G_A(s)$ and $G_B(s)$ are transfer functions related to the LCL filter and active damping. The output impedance of the utility interactive inverter at the PCC, without considering PLL dynamics, $Z_{inv,0}(s)$, can be expressed as:
$$ Z_{inv,0}(s) = \frac{1 + T_A(s)}{G_A(s)} $$
In a balanced three-phase system and without PLL influence, the d- and q-axis impedances are identical, i.e., $Z_{inv,0}^{dd}(s) = Z_{inv,0}^{qq}(s)$.
1.2. Small-Signal Model Including PLL Dynamics
Under weak grid conditions, the PLL dynamics can no longer be ignored. The PLL tracks the phase angle $\theta_{PLL}$ from the q-axis component of the PCC voltage in its own estimated reference frame. A disturbance in $v_{PCC}$ causes a small-signal phase error $\Delta \theta$ between the PLL’s frame (denoted as ‘c’ frame) and the actual grid synchronous frame (denoted as ‘s’ frame). The linearized model of a standard synchronous reference frame PLL is:
$$ \Delta \theta = G_{PLL}(s) \Delta v_{PCCq}^s $$
where $G_{PLL}(s)$ is the PLL’s transfer function. For a PI-type PLL with gains $k_{p,pll}$ and $k_{i,pll}$, and assuming the steady-state d-axis PCC voltage is $V_{PCCd}^s$, it is given by:
$$ G_{PLL}(s) = \frac{H_{PLL}(s)}{s + V_{PCCd}^s H_{PLL}(s)}, \quad H_{PLL}(s) = k_{p,pll} + \frac{k_{i,pll}}{s} $$
This phase error couples into the current control loop. The relationship between variables in the ‘c’ frame (measured/used by the controller) and the ‘s’ frame (actual) for the q-axis is approximated by:
$$ \begin{aligned} \Delta v_{cq}^c &= \Delta v_{q}^s – V_{d}^s G_{PLL}(s) \Delta v_{PCCq}^s \\ \Delta i_{gq}^c &= \Delta i_{gq}^s – I_{gd}^s G_{PLL}(s) \Delta v_{PCCq}^s \end{aligned} $$
where $V_{d}^s$ is the steady-state d-axis component of the inverter output voltage and $I_{gd}^s$ is the steady-state d-axis grid current. Incorporating these coupling terms into the control block diagram reveals the interaction. The q-axis output impedance of the utility interactive inverter, now denoted as $Z_{inv}^{qq}(s)$ to include PLL effects, becomes significantly different from $Z_{inv,0}(s)$. Its expression is more complex:
$$ Z_{inv}^{qq}(s) = \frac{1 + T_A(s)}{G_B(s) + T_A(s)G_{PLL}(s) \left[ 1 + \frac{1}{G_i(s)} \right] } $$
A key observation is the additional term $T_A(s)G_{PLL}(s) \left[ 1 + \frac{1}{G_i(s)} \right]$ in the denominator. This term, which originates from the PLL coupling, is dominant in the low-frequency range (near and below the PLL bandwidth) and can drastically alter the phase of $Z_{inv}^{qq}(s)$. Table 1 summarizes the key parameters and their values used for analysis in this study.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | $P_0$ | 2 kW |
| DC Link Voltage | $V_{dc}$ | 400 V |
| Grid Voltage (Phase) | $V_g$ | 110 V (rms) |
| Grid Frequency | $f_0$ | 50 Hz |
| Inverter-side Inductor | $L_1$ | 2.0 mH |
| Grid-side Inductor | $L_2$ | 0.5 mH |
| Filter Capacitor | $C_f$ | 10 µF |
| Active Damping Gain | $K_c$ | 0.5 |
| Current Controller $k_p$ | $k_p$ | 0.5 |
| Current Controller $k_i$ | $k_i$ | 50 |
| PLL Controller $k_{p,pll}$ | $k_{p,pll}$ | 50 |
| PLL Controller $k_{i,pll}$ | $k_{i,pll}$ | 2000 |
| Sampling/Switching Frequency | $f_s$ | 10 kHz |
Stability Analysis and the Need for Compensation
2.1. Impedance-Based Stability Criterion
The stability of the interconnected system—comprising the utility interactive inverter and the grid—can be effectively assessed using the impedance-based criterion. The inverter is modeled as a Norton equivalent circuit: an ideal current source $i_{o}(s)$ in parallel with its output impedance $Z_{inv}(s)$. The grid is modeled as a Thevenin equivalent: an ideal voltage source $v_g(s)$ in series with the grid impedance $Z_g(s)$. The circuit equation is:
$$ i_g(s) = \frac{i_o(s) – \frac{v_g(s)}{Z_{inv}(s)}}{1 + \frac{Z_g(s)}{Z_{inv}(s)}} $$
The stability of this feedback system is determined by the minor loop gain $L_g(s) = Z_g(s) / Z_{inv}(s)$. According to the Nyquist criterion adapted for impedance ratios, a sufficient condition for stability is that the phase margin of $L_g(s)$ at the crossover frequency $f_{cross}$ (where $|Z_{inv}(j2\pi f_{cross})| = |Z_g(j2\pi f_{cross})|$) is positive. For a purely inductive grid $Z_g(s) = sL_g$, its phase is constant at +90°. Therefore, the phase margin $PM$ is:
$$ PM = 90^\circ + \angle Z_{inv}(j2\pi f_{cross}) $$
For stability, $PM > 0^\circ$. This implies that $ \angle Z_{inv}(j2\pi f_{cross}) > -90^\circ $. A phase more negative than -90° indicates a negative incremental resistance characteristic and potential instability.
2.2. Impact of PLL and Weak Grid
Figure 1 shows the Bode plots of the inverter output impedance under different conditions. $Z_{inv,0}$ (without PLL) maintains a phase above -90° for a wide frequency range. However, $Z_{inv}^{qq}$ (with PLL) shows a dramatic phase drop in the low-frequency region (10-100 Hz). With a grid impedance of $L_g = 1$ mH, the phase at crossover dips below -90°, resulting in a negative phase margin. As $L_g$ increases to 8 mH, the crossover frequency moves lower where the phase drop is even more severe, drastically reducing the phase margin and making the utility interactive inverter highly prone to instability. This analysis confirms that the PLL coupling is the root cause of the degraded robustness in weak grids.
2.3. Limitations of Conventional Small-Signal Compensation
A known mitigation technique is to inject a small-signal compensation based on the PCC voltage disturbance. This method adds a feedforward path of the measured PCC voltage disturbance, transformed through the PLL transfer function, to the current reference or modulation signal. While this can improve the phase margin slightly for small $L_g$, its effectiveness is limited. As seen in the analytical plots, for $L_g = 8$ mH, the phase margin remains critically low. Furthermore, the compensation offers little to no improvement in the mid-to-high frequency range, leaving the utility interactive inverter vulnerable to harmonic interactions.
Proposed Compensated Small-Signal Impedance Shaping Method
To overcome the limitations of the conventional approach, we propose an improved compensation strategy. The core idea is to introduce a compensated small-signal injection that more effectively counteracts the specific phase lag introduced by the PLL coupling across the frequency range of interest, especially at lower frequencies.
The proposed control structure modifies the q-axis current control loop. In addition to the standard PI controller and the conventional small-signal feedforward of $\Delta v_{PCCq}$, we introduce a new compensation block $G_c(s)$ in the modulation path. This block processes the PCC voltage disturbance and injects a compensating signal that directly alters the modulation voltage. The modified equation for the q-axis modulating signal in the controller frame becomes:
$$ \Delta v_{mq}^c = G_i(s) (\Delta i_{gq}^{ref} – \Delta i_{gq}^c) – \underbrace{G_{PLL}(s)\Delta v_{PCCq}^s}_{\text{Conventional}} – \underbrace{G_c(s) G_{PLL}(s) \Delta v_{PCCq}^s}_{\text{Proposed Addition}} $$
The design of $G_c(s)$ is crucial. We propose using a second-order band-pass filter (BPF) structure centered near the fundamental frequency or the critical low-frequency region where phase degradation is worst. The transfer function is:
$$ G_c(s) = K \cdot \frac{2\xi\omega_c s}{s^2 + 2\xi\omega_c s + \omega_c^2} $$
where $K$ is the compensation gain, $\xi$ is the damping ratio, and $\omega_c = 2\pi f_c$ is the center frequency of the filter. The purpose of $G_c(s)$ is not to provide damping but to introduce a controlled phase lead precisely where the PLL introduces phase lag in the inverter’s impedance. By carefully tuning $K$, $\xi$, and $f_c$, the phase characteristic of $Z_{inv}^{qq}(s)$ can be reshaped. Table 2 provides a comparison of the key characteristics of the different control strategies.
| Strategy | Principle | Effect on Phase Margin (Low $L_g$) | Effect on Phase Margin (High $L_g$) | Complexity |
|---|---|---|---|---|
| No Compensation | Standard PI + PLL | May be negative | Highly negative (Unstable) | Low |
| Conventional Small-Signal Comp. | Feedforward PCC disturbance | Moderate improvement | Limited improvement | Medium |
| Proposed Compensated Method | BPF-shaped PCC disturbance feedforward | Significant improvement | Major improvement, maintains >0° | Medium-High |
| PLL Redesign (e.g., Lower BW) | Reduce PLL coupling strength | Improvement | Improvement, but degrades synchronization dynamics | High |
| Virtual Impedance | Actively shape output impedance | Good improvement | Good improvement, requires careful design | High |
The effectiveness of the proposed method is analyzed by deriving the new output impedance $Z_{inv, prop}^{qq}(s)$. The inclusion of the $G_c(s)$ term modifies the denominator of the impedance expression, effectively canceling or reducing the adverse term from the PLL. The analytical Bode plots demonstrate that with proper tuning, $Z_{inv, prop}^{qq}(s)$ maintains a phase above -90° for a much wider range of frequencies and grid inductances. For $L_g = 8$ mH, the phase margin is restored to a clearly positive value, ensuring robust stability of the utility interactive inverter.
The tuning guidelines for the compensation block are as follows:
1. Center Frequency ($f_c$): Set near the fundamental frequency (50/60 Hz) or slightly lower, as the PLL-induced phase drop is most critical in this region. It can be aligned with the frequency where the uncompensated impedance phase is minimum.
2. Damping Ratio ($\xi$): Choose a value between 0.5 and 1 to ensure a reasonably wide bandwidth for the compensation effect without excessive gain peaking.
3. Compensation Gain ($K$): This is the most critical parameter. It should be adjusted iteratively based on the desired phase lift. A positive $K$ typically provides phase lead. The value can be derived by analyzing the required compensation to achieve a target phase margin (e.g., 30°) at the expected worst-case crossover frequency.
A simplified design equation can be obtained by evaluating the phase contribution of the compensation term at the target frequency $\omega_t$. The additional phase $\phi_{add}$ provided by the $G_c(s)G_{PLL}(s)$ path relative to the conventional path affects the total loop gain. The gain $K$ can be approximated to satisfy the phase margin condition:
$$ PM_{target} = 90^\circ + \angle Z_{inv,0}(j\omega_t) + \Delta\phi(K, \omega_t) $$
where $\Delta\phi(K, \omega_t)$ is the phase adjustment from the proposed compensator. Since $\Delta\phi$ is a nonlinear function of $K$, an analytical solution is complex, but a graphical or numerical search based on the Bode plot of the modified system loop gain is practical for design.
Experimental Validation
A 2-kW three-phase utility interactive inverter prototype was built to validate the theoretical analysis. The system parameters matched those in Table 1. The controller was implemented on a TMS320F28335 DSP. A programmable AC source combined with series inductors was used to emulate a weak grid with variable impedance $L_g$.
3.1. Performance with Conventional Small-Signal Compensation
First, the system with conventional small-signal compensation was tested.
– With $L_g = 1$ mH, the grid currents were sinusoidal with a Total Harmonic Distortion (THD) of 3.29%, acceptable under IEEE Std. 519.
– With $L_g = 8$ mH, severe instability occurred. The grid currents became highly distorted and oscillatory, with a THD exceeding 14.8%. This confirmed the insufficiency of the conventional method under strongly weak grid conditions, rendering the utility interactive inverter unstable.
3.2. Performance with Proposed Compensated Method
The proposed control strategy with the added compensation block $G_c(s)$ was then implemented. The compensator parameters were tuned to: $K=0.8$, $f_c=45$ Hz, $\xi=0.7$.
– With $L_g = 1$ mH, the current quality remained excellent, with a THD of 3.24%.
– Critically, with $L_g = 8$ mH, the system remained stable. The grid currents were maintained sinusoidal with significantly reduced distortion. The measured THD was 4.12%, well within the acceptable limit. This demonstrates the robust stabilizing effect of the proposed method on the utility interactive inverter.
Table 3 summarizes the key experimental results, highlighting the decisive advantage of the proposed method in maintaining stability and power quality under high grid impedance.
| Grid Impedance $L_g$ | Control Strategy | Stability | Grid Current THD | Observation |
|---|---|---|---|---|
| 1 mH | Conventional Compensation | Stable | 3.29% | Acceptable performance |
| Proposed Compensation | Stable | 3.24% | Acceptable performance | |
| 8 mH | Conventional Compensation | Unstable | >14.8% | Severe distortion, system trips |
| Proposed Compensation | Stable | 4.12% | Stable operation, good power quality |
Conclusion
This paper has presented a comprehensive analysis and a novel solution for the stability enhancement of utility interactive inverters in weak grid environments. The core challenge stems from the adverse coupling between the Phase-Locked Loop (PLL) and distorted PCC voltages, which degrades the phase margin of the inverter’s output impedance and can induce instability.
Through detailed small-signal modeling in the d-q synchronous frame, we quantitatively demonstrated how PLL dynamics introduce a negative impedance characteristic in the low-frequency range. The widely used conventional small-signal compensation method was shown to offer insufficient phase margin improvement, particularly under severe weak grid conditions (high grid inductance).
The proposed improved strategy addresses this limitation by incorporating a tailored band-pass filter into the small-signal compensation path. This active impedance shaping technique effectively counteracts the phase lag introduced by the PLL coupling. The key advantages of this method are:
1. It effectively restores a positive and adequate phase margin for the utility interactive inverter’s output impedance, even with high grid impedance, ensuring robust stability.
2. It maintains good power quality, keeping grid current THD within standards under weak grid conditions.
3. The method is implementable without modifying the internal structure or extensively redesigning the parameters of the standard PLL, preserving its synchronization performance.
Theoretical analysis based on impedance models and experimental results from a 2-kW prototype have conclusively validated the effectiveness of the proposed compensated small-signal impedance shaping method. It provides a viable and effective control solution to enhance the robustness and reliability of utility interactive inverters, facilitating their secure integration into increasingly prevalent weak grid scenarios.