The rapid advancement of renewable energy, particularly photovoltaic (PV) generation, has been a defining feature of the global power sector’s evolution. The core interface between a PV array and the utility grid is the on-grid inverter, which performs the critical function of converting direct current (DC) from solar panels into grid-synchronized alternating current (AC). The performance and reliability of the entire distributed generation system hinge on the stable operation of this on-grid inverter. Stability mandates require the inverter’s output current to be precisely in phase with the grid voltage while maintaining high waveform quality, typically with total harmonic distortion (THD) below 5%.
However, the modern power grid is increasingly characterized as “weak” at the point of common coupling (PCC) for many distributed generators. This weakness stems from the significant integration of inverter-based resources like PV and wind farms, which do not contribute to the system’s inherent inertia and short-circuit capacity in the same way as traditional synchronous generators. The resultant increase in grid impedance can severely impact the stability of on-grid inverters. Furthermore, the grid voltage itself is often contaminated with background harmonics. The interaction between the inverter’s control systems—particularly the Phase-Locked Loop (PLL) essential for synchronization—and the weak grid impedance can lead to instability, potentially causing widespread tripping and power outages. Therefore, developing robust stability analysis and enhancement methods for on-grid inverters under weak grid conditions is paramount.
Traditional stability assessment methods for on-grid inverters include impedance-based analysis, state-space eigenvalue analysis, and Nyquist criterion applied to derived transfer functions. This article introduces and explores the damping coefficient of the PLL as a potent and insightful metric for judging system stability. The core premise is that in weak grids, the PLL exhibits pronounced nonlinear characteristics, and its damping is directly influenced by grid strength. Insufficient PLL damping can be a root cause of instability. This work will systematically model the system, derive the PLL damping expression under weak grid conditions, and demonstrate how tailoring PLL parameters can enhance damping and thus secure stable operation for the on-grid inverter.
System Configuration and Modeling of the On-Grid Inverter
The topology of a three-phase LCL-filter-based on-grid inverter is considered. The primary power circuit consists of a DC-link voltage source \(V_{dc}\), a full-bridge inverter, an LCL filter (\(L_1\), \(C\), \(L_2\)), and the grid represented by a voltage source \(u_g\) behind an impedance \(Z_s\). The control strategy employs a synchronous reference frame (dq-frame) controller. The phase angle \(\theta\) for the dq transformation is obtained from a PLL synchronized to the PCC voltage \(u_{PCC}\).
The current control loop uses a standard PI regulator \(G_c(s)\) in the dq-domain:
$$G_c(s) = k_{cp} + \frac{k_{ci}}{s}$$
where \(k_{cp}\) is the proportional gain and \(k_{ci}\) is the integral gain. The reference for the q-axis current \(i_q^*\) is typically set to zero for unity power factor operation, while the d-axis reference \(i_d^*\) dictates the active power output. To mitigate the inherent resonance peak of the LCL filter, active damping via capacitor current feedback is implemented, characterized by a coefficient \(H_{i1}\). The computational and PWM delays are modeled as a total delay \(G_d(s) = e^{-1.5T_s s}\), where \(T_s\) is the sampling period.
The dynamics of the on-grid inverter, excluding the PLL, can be represented by its output impedance \(Z_o(s)\). The interaction between this inverter output impedance and the grid impedance \(Z_g(s)\) determines stability according to the generalized Nyquist criterion. However, this analysis often treats the PLL as ideal or uses linearized approximations that may fail in weak grids.
Nonlinear Modeling of the PLL and Derivation of Grid-Dependent Damping
The PLL is a critical nonlinear component. A standard Synchronous Reference Frame PLL (SRF-PLL) is analyzed. It transforms the PCC voltages \(u_{PCC,abc}\) to the dq-frame using an estimated angle \(\hat{\theta}\). The q-component \(u_{PCC,q}\) is fed to a PI controller, whose output is the estimated grid frequency deviation, integrated to obtain \(\hat{\theta}\). In a strong grid (\(Z_s \approx 0\)), \(u_{PCC} \approx u_g\), and the linearized model of the PLL around the operating point is a second-order system. Its traditional damping ratio \(\zeta_{PLL,strong}\) is given by:
$$\zeta_{PLL,strong} = \frac{k_{pp}}{2} \sqrt{\frac{U_m}{k_{ip}}}$$
where \(k_{pp}\) and \(k_{ip}\) are the PLL PI controller gains, and \(U_m\) is the amplitude of \(u_{PCC}\). This expression ignores grid impedance.
In a weak grid, the PCC voltage is perturbed by the voltage drop across \(Z_s\) due to the injected current \(i_{L2}\): \(u_{PCC} = u_g + i_{L2} Z_s\). This creates a feedback path from the PLL’s output (frequency/angle) through the inverter’s current control and back to its input (\(u_{PCC}\)), introducing additional dynamics. A more accurate small-signal model must account for this.
Let the grid voltage be \(V_g \angle \theta_g\), the PCC voltage be \(V_P \angle \theta_p\), and the grid-side current be \(I_{L2} \angle \theta_1\). The grid impedance is \(Z_s = R_g + j\omega_1 L_g\), with a phase angle \(\phi_s = \arctan(\omega_1 L_g / R_g)\). The relationship is:
$$V_P \angle \theta_p = V_g \angle \theta_g + I_{L2} Z_s(\omega_1) \angle (\theta_1 + \phi_s(\omega_1))$$
The q-axis component of \(V_P\) after transformation by the PLL angle \(\hat{\theta} \approx \theta_1\) (assuming a well-tuned current loop tracks the dq-frame) is:
$$V_{Pq} = V_g \sin(\theta_g – \theta_1) + I_{L2} Z_s(\omega_1) \sin(\phi_s(\omega_1))$$
Linearizing around the nominal frequency \(\omega_n\) and nominal angles, and considering perturbations \(\Delta \theta_g\), \(\Delta \theta_1\), and \(\Delta \omega = \Delta \dot{\theta}_1\), we derive a revised linear model for the PLL. The key transfer function from grid angle perturbation to inverter output current angle perturbation \(\Delta \theta_1 / \Delta \theta_g\) takes the form:
$$\frac{\Delta \theta_1(s)}{\Delta \theta_g(s)} = \frac{N(s)}{M_1 s^2 + M_2 s + M_3}$$
where the coefficients \(N, M_1, M_2, M_3\) incorporate terms dependent on \(V_g, I_{L2}, Z_s, k_{pp}, k_{ip}\), and the operating point. The denominator represents the characteristic equation of the PLL-inverter system in the weak grid.
By comparing this to a standard second-order system \(s^2 + 2\zeta\omega_n s + \omega_n^2\), the effective damping ratio \(\zeta_{PLL,weak}\) of the PLL under weak grid conditions can be extracted as:
$$\zeta_{PLL,weak} = \frac{M_2}{2 \sqrt{M_1 M_3}}$$
A detailed derivation yields the following insightful expression:
$$\zeta_{PLL,weak} = \frac{ k_{pp} V_g \sqrt{1 – \frac{1}{SCR^2}} – \frac{k_{ip} V_g}{\omega_n SCR} }{ 2 \sqrt{ k_{ip} V_g \left(1 – \frac{k_{pp} V_g}{\omega_n SCR}\right) \sqrt{1 – \frac{1}{SCR^2}} } }$$
where \(SCR\) is the Short-Circuit Ratio, a standard metric for grid strength defined at the PCC:
$$SCR = \frac{V_g^2}{|Z_s| S_{rated}} = \frac{V_g}{Z_s I_{L2,rated}} \quad \text{(for simplified analysis with } Z_s \approx \omega_n L_g \text{)}$$
Here, \(S_{rated}\) is the rated apparent power of the on-grid inverter. An \(SCR \geq 3\) denotes a strong grid, \(2 \leq SCR < 3\) a weak grid, and \(SCR < 2\) a very weak grid. Stability typically requires \(SCR > 1\).
Analysis of Damping Behavior and Stability Implications
The derived formula for \(\zeta_{PLL,weak}\) reveals the profound impact of grid conditions on the on-grid inverter’s synchronization stability. The following table summarizes the key parameters used for a case study analysis.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| DC-Link Voltage | \(V_{dc}\) | 800 | V |
| Grid Voltage (Phase) | \(V_g\) | 155 | V |
| Inverter-side Inductor | \(L_1\) | 200 | µH |
| Grid-side Inductor | \(L_2\) | 100 | µH |
| Filter Capacitor | \(C\) | 5 | µF |
| Switching Frequency | \(f_{sw}\) | 10 | kHz |
| Current Ctrl. Prop. Gain | \(k_{cp}\) | 200 | – |
| Current Ctrl. Int. Gain | \(k_{ci}\) | 30 | – |
| PLL Prop. Gain (Range) | \(k_{pp}\) | 0.5 – 3.0 | – |
| PLL Int. Gain | \(k_{ip}\) | 30 | – |
| Base Grid Inductance | \(L_g\) | Variable (e.g., 18) | mH |
| Rated Current | \(I_{L2,rated}\) | 25 | A |
Using the formula, we can plot \(\zeta_{PLL,weak}\) against the SCR for different PLL proportional gains \(k_{pp}\), holding \(k_{ip}\) constant. This relationship is crucial for understanding on-grid inverter stability.
1. Strong Grid (High SCR) Behavior: For \(SCR \geq 3\), the term \(1/SCR^2\) becomes very small. The expression simplifies, and \(\zeta_{PLL,weak}\) becomes largely independent of \(SCR\), approaching its strong-grid value. The damping is primarily determined by the PLL’s own parameters \(k_{pp}\) and \(k_{ip}\), as well as \(V_g\).
2. Weak Grid (Low SCR) Behavior: As SCR decreases below 3, the damping ratio \(\zeta_{PLL,weak}\) drops sharply. This is the critical region for on-grid inverter operation. The denominator terms containing \(SCR\) in the denominator cause this rapid decline. A low damping ratio (\(\zeta \ll 1\)) indicates a poorly damped or oscillatory system, making the on-grid inverter highly susceptible to disturbances and harmonic instability.
3. Parameter Tuning for Stability: The formula suggests a path to mitigate this. For a given weak-grid SCR and fixed \(k_{ip}\), increasing the proportional gain \(k_{pp}\) can increase the numerator and positively influence the damping ratio. There is, however, an upper limit related to the \( (1 – k_{pp} V_g / (\omega_n SCR)) \) term in the denominator’s root; \(k_{pp}\) must be chosen to keep this term positive. The analysis provides a clear guideline: in weak grids, to prevent PLL-induced instability in the on-grid inverter, the PLL’s proportional gain \(k_{pp}\) may need to be increased (within stability limits) to boost the synchronization loop’s damping.
The following table illustrates the calculated damping for a specific weak-grid scenario (\(L_g = 18\) mH, \(I_{L2}=25\) A, \(SCR \approx 1.1\)) with different \(k_{pp}\) values.
| PLL Prop. Gain \(k_{pp}\) | Calculated Damping \(\zeta_{PLL,weak}\) | Stability Prediction | Expected Current THD |
|---|---|---|---|
| 0.5 | Very Low (\(\approx\) 0.2) | Unstable | Very High, Distorted |
| 1.2 | 0.884 | Marginally Stable/Unstable | High (>5%) |
| 1.3 | 0.958 | Stable | Acceptable (~4-5%) |
| 3.0 | >1.2 | Very Stable | Low (<2%) |
Simulation Verification
Time-domain simulations in MATLAB/Simulink were conducted for the on-grid inverter system with parameters from Table 1 to validate the damping-based stability theory. The grid inductance was set to \(L_g = 18\) mH, creating a weak grid with \(SCR \approx 1.1\) at full power (25 A). The PLL integral gain was fixed at \(k_{ip}=30\).
Case 1: Low PLL Damping (\(k_{pp}=0.5\)): With a very low calculated damping, the system is predicted unstable. The simulation confirms this. Upon connection or during operation, the PCC voltage and grid current exhibit growing oscillations, leading to severe distortion and eventual collapse of the controlled current. The on-grid inverter fails to maintain stable synchronization.
Case 2: Marginal Damping (\(k_{pp}=1.2\)): The calculated damping is about 0.884. The simulation shows a critically stable behavior. When the on-grid inverter’s reference current is stepped from 10 A (higher SCR) to 25 A (SCR=1.1), the system becomes unstable. The grid current becomes highly distorted, with a measured THD exceeding 9%, which is unacceptable for grid codes. This demonstrates the threshold of stability.
Case 3: Adequate Damping (\(k_{pp}=1.3\)): Increasing \(k_{pp}\) to 1.3 raises the damping to approximately 0.958. In this simulation, the on-grid inverter maintains stability even after the current step to 25 A. The current waveform is sinusoidal with a THD of about 4.64%, meeting the sub-5% requirement. The system successfully rides through the transient caused by the power change.
Case 4: High Damping (\(k_{pp}=3.0\)): With \(k_{pp}=3.0\), the damping ratio is significantly increased. The on-grid inverter shows excellent dynamic performance. The response to the current step is fast and smooth, with minimal overshoot and a very clean current waveform having a THD of only 1.58%. This confirms that increasing the PLL’s proportional gain—thereby increasing its damping—robustly stabilizes the on-grid inverter in a weak grid.
The simulation results perfectly correlate with the stability predictions made using the derived PLL damping coefficient \(\zeta_{PLL,weak}\). This validates the premise that the PLL damping is a reliable and insightful metric for assessing and ensuring the stability of an on-grid inverter operating in weak grid environments.
Conclusions
This analysis has established the damping coefficient of the Phase-Locked Loop as a fundamental and effective metric for evaluating the stability of LCL-filter-based on-grid inverters in weak grids. The key conclusions are:
1. The stability of an on-grid inverter is significantly influenced by the damping of its synchronization PLL. A higher PLL damping coefficient provides greater robustness against grid impedance interactions and better suppresses harmonic-induced instability, leading to lower grid current THD.
2. Under weak grid conditions (SCR < 3), the effective damping of the PLL is drastically reduced compared to its strong-grid value, primarily due to the feedback effect of the grid impedance. This reduction is a primary catalyst for synchronization instability in on-grid inverters.
3. The PLL damping can be actively managed through controller tuning. For a fixed integral gain, increasing the proportional gain \(k_{pp}\) of the PLL’s PI controller effectively increases the damping ratio \(\zeta_{PLL,weak}\), thereby enhancing the stability margin of the on-grid inverter. This provides a clear and practical design guideline for engineers deploying on-grid inverters in areas with weak grid infrastructure.
This damping-centric perspective offers a complementary and physically intuitive approach to traditional impedance or eigenvalue-based stability analysis for on-grid inverters. It directly links a key control loop characteristic (PLL damping) to the overall system stability, facilitating more informed design and tuning decisions to ensure the reliable integration of renewable energy sources into the modern power grid.