As the global energy transition accelerates, new energy sources such as solar and wind power are increasingly integrated into the power grid through inverter interfaces. Among these, various types of solar inverter, including string inverters, central inverters, and microinverters, have been widely deployed in distributed and utility-scale photovoltaic systems. However, the parallel operation of multiple inverters in weak grid conditions often leads to wideband oscillation issues, which threaten system stability. To address this challenge, I systematically study the impedance-based modeling method for parallel inverter systems, focusing on the wideband aggregation characteristics. This paper presents a comprehensive modeling framework that covers the circuit topology, control system sampling, phase-locked loop (PLL) dynamics, current controller design, and the final derivation of the inverter sequence impedance model. Simulation validations and impedance stability analyses are provided to demonstrate the accuracy of the proposed model and the influence of key parameters on wideband stability.
1. Introduction
Modern power systems are characterized by a high penetration of renewable energy sources, which are inherently intermittent and have low energy density. Traditional synchronous generators cannot efficiently accommodate such sources due to limited ramping capability and insufficient cross-regional transmission capacity. Moreover, the increasing deployment of power electronic converters introduces complex impedance interactions between the new energy generation equipment and the synchronous grid, leading to wideband oscillations. The inverter serves as the critical interface for grid connection, and impedance analysis has become a widely adopted tool to reveal the frequency-dependent characteristics of the inverter and to assess the risk of instability. This work aims to develop a wideband aggregation model for the parallel operation of multiple inverters, considering various types of solar inverter configurations and control parameters.
2. Parallel Inverter Circuit Structure and Operating Principle
A typical grid-connected inverter consists of a main circuit and a control circuit. The main circuit includes a three-phase full-bridge inverter using a two-level voltage source converter topology with a single inductor filter on the AC side. The control circuit comprises a phase-locked loop (PLL), an inner current loop, and an outer voltage/power loop. The controller samples the voltage at the point of common coupling (PCC) and uses the PLL to obtain the phase angle. The AC-side inductor current is transformed into the dq synchronous reference frame via Park and Clark transformations. A proportional-integral (PI) controller determines the modulation voltage components in the dq frame.
For multiple inverters operating in parallel, the system structure follows Kirchhoff’s voltage law. The sampling positions determine the current and voltage values, and the device output current and voltage can be expressed as:
$$
\begin{bmatrix} v_{imva} \\ v_{imvb} \\ v_{imvc} \end{bmatrix}
=
\begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix}
+
L_f \frac{d}{dt}
\begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix}
$$
where $v_{imva}$, $v_{imvb}$, $v_{imvc}$ are the three-phase midpoint voltages of the bridge arms, $v_a$, $v_b$, $v_c$ are the PCC voltages, $i_a$, $i_b$, $i_c$ are the inverter output currents, and $L_f$ is the output filter inductance. By ignoring the commutation time of the switching devices, a switching-cycle model can be established. The key parameters include the DC-link voltage $v_{dc}$, filter inductance $L_f$, and the PWM modulation ratio $K_m$.
3. Control System Sampling Model
During parallel operation, the sampling and structural delays depend on the system hardware configuration. The voltage sampling delay typically includes a zero-order holder, a one-step sampling lag, and a low-pass filter. The equivalent transfer functions for voltage and current sampling are:
$$
G_v(s) = \frac{e^{-sT_v}}{1 + sT_v} \cdot \frac{1}{1 + s/\omega_v}
$$
$$
G_i(s) = \frac{e^{-sT_i}}{1 + sT_i} \cdot \frac{1}{1 + s/\omega_i}
$$
where $T_v$ and $T_i$ are the sampling periods for voltage and current, $\omega_v$ and $\omega_i$ are the cutoff frequencies of the low-pass filters, and $s$ is the Laplace variable. These transfer functions are incorporated into the subsequent impedance model to account for the system’s discrete-time behavior.
4. Phase-Locked Loop Frequency-Domain Model
The PLL determines the synchronous reference frame angle $\theta_{PLL}$ from the measured PCC voltage. For the stability analysis of various types of solar inverter systems, the accurate modeling of the PLL is essential. When a small positive-sequence disturbance voltage $v_p$ at frequency $f_p$ and a negative-sequence disturbance voltage $v_n$ at frequency $f_n$ are injected at the PCC, the PLL output angle contains a disturbance component $\Delta\theta$ superimposed on the fundamental angle $\theta_1$. Using harmonic linearization, the relationship between $\Delta\theta$ and the harmonic voltages can be derived.
The time-domain expression of the A-phase voltage at the PCC is:
$$
v_a(t) = V_1 \cos(2\pi f_1 t) + V_p \cos(2\pi f_p t + \varphi_{vp}) + V_n \cos(2\pi f_n t + \varphi_{vn})
$$
where $f_1$ is the fundamental frequency, $V_1$, $V_p$, $V_n$ are the amplitudes, and $\varphi_{vp}$, $\varphi_{vn}$ are the initial phase angles. The PLL control block diagram shows that the dq transformation angle $\theta_{PLL} = \theta_1 + \Delta\theta$. For small disturbances, $\sin\Delta\theta \approx \Delta\theta$, $\cos\Delta\theta \approx 1$. The transfer functions from the harmonic voltages to the angle disturbance are:
$$
G_p(s) = \frac{\Delta\theta}{v_p} = \frac{1}{V_1} \cdot \frac{1}{1 + H_{PLL}(s)/s \cdot V_1}
$$
$$
G_n(s) = \frac{\Delta\theta}{v_n} = \frac{1}{V_1} \cdot \frac{1}{1 + H_{PLL}(s)/s \cdot V_1}
$$
where $H_{PLL}(s) = k_{pp} + k_{pi}/s$ is the PI controller of the PLL, with proportional gain $k_{pp}$ and integral gain $k_{pi}$. This model captures the effect of PLL dynamics on the impedance shaping.
5. Current Controller Model
For the inner current loop, either a PI controller or a quasi-proportional resonant (QPR) controller can be used. In the dq synchronous reference frame, the current controller processes the measured dq currents and outputs the modulation voltage references. The three-phase currents in the time domain, with injected positive- and negative-sequence disturbance currents, are:
$$
i_a(t) = I_1 \cos(2\pi f_1 t + \varphi_{i1}) + I_p \cos(2\pi f_p t + \varphi_{ip}) + I_n \cos(2\pi f_n t + \varphi_{in})
$$
After transformation to the dq frame using $\theta_{PLL} = \theta_1$ (ignoring the small disturbance effect for the current loop in the linearization), the q-axis and d-axis currents are obtained. The modulation signals in the dq frame are then generated by the current controller. By applying the inverse transformation, the three-phase modulation signals can be expressed in terms of the small-signal sequence components. This step is crucial for deriving the overall inverter impedance model.
6. Inverter Sequence Impedance Model
Integrating the modulation signals obtained from the current controller into the switching-cycle model yields the positive-sequence and negative-sequence output impedances of the inverter. The impedance expressions depend on the PLL and current controller parameters, as well as the operating point. For a typical grid-connected inverter with a PI current controller, the positive-sequence impedance $Z_p(s)$ and negative-sequence impedance $Z_n(s)$ are given by:
$$
Z_p(s) = \frac{V_p(s)}{I_p(s)} = sL_f + \frac{K_m V_{dc}}{2} \cdot \frac{1}{1 + G_i(s)G_{PLL}(s) \cdot (I_1 e^{j\varphi_{i1}})} \cdot \text{(control terms)}
$$
$$
Z_n(s) = \frac{V_n(s)}{I_n(s)} = sL_f + \frac{K_m V_{dc}}{2} \cdot \frac{1}{1 + G_i(s)G_{PLL}(s) \cdot (I_1 e^{-j\varphi_{i1}})} \cdot \text{(control terms)}
$$
Due to the frequency-coupling effect introduced by the synchronous reference frame control, the positive and negative impedances exhibit a frequency shift of $\pm f_1$ in the PLL and current controller terms. For example, in the positive-sequence impedance, the frequency of the control signals is shifted by $-f_1$ relative to the natural frequency. This characteristic is common to many types of solar inverter systems.
7. Wideband Impedance Characteristics Analysis
Using the derived sequence impedance model, I performed Bode plots in MATLAB to analyze the impact of various parameters on the wideband impedance characteristics. The inverter parameters used in the simulation are listed in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Grid voltage $V_1$ | 690 V | Filter inductance $L_f$ | 0.15 mH |
| PLL proportional gain $k_{pp}$ | 0.09 | PLL integral gain $k_{pi}$ | 6.667e3 |
| Current controller proportional gain $K_p$ | 32 | Current controller integral gain $K_i$ | 0.5 |
First, the effect of system power on impedance was investigated. Operating at 0.6 pu, 0.85 pu, and 1.0 pu, the results show that in the low-frequency range, the impedance magnitude decreases as power increases, while the phase remains nearly unchanged. At high frequencies approaching the switching frequency, power variation has minimal effect. This behavior is consistent with the theoretical expression where increased active current increases the denominator of the impedance formula.
Second, the influence of grid voltage was studied. The grid voltage was varied between 0.9$V_1$, 1.0$V_1$, and 1.1$V_1$. Under a QPR controller, the system remains stable across the range. Under a PI controller, the stability margin reduces when voltage drops to 0.9$V_1$, indicated by the positive-sequence impedance ratio curve approaching the critical point (-1, j0). This suggests that voltage fluctuations affect the positive-sequence impedance significantly, and the QPR controller provides better adaptability for different types of solar inverter systems.
8. Impedance Stability Analysis
To assess stability, the parallel inverter system is modeled as a series interconnection of an ideal voltage source and an output impedance, with the grid represented by its equivalent impedance. The Nyquist criterion for the ratio $Z_g(s)/Z_{inv}(s)$ determines system stability. I performed simulations for both PI and QPR current controllers, with control parameters listed in Table 2.
| QPR Parameters | Value | PI Parameters | Value |
|---|---|---|---|
| Proportional resonant gain $K_{pr}$ | 3 | Proportional gain $K_p$ | 0.3 |
| Resonant frequency gain $K_{ir}$ | 1000 | Integral gain $K_i$ | 350 |
| Differential gain $K_d$ | 0.047 | — | — |
The analysis focused on the influence of grid voltage variation. Under the QPR controller, the impedance ratio curves for positive and negative sequences do not encircle the critical point (-1, j0) for grid voltages from 0.9 pu to 1.1 pu, indicating robust stability. Under the PI controller, when grid voltage drops to 0.9 pu, the positive-sequence impedance ratio curve passes near the critical point on the right side, implying reduced stability margin. This confirms that the PI-controlled inverter is more sensitive to grid voltage fluctuations than the QPR-controlled inverter. The choice of current controller thus plays a crucial role in the wideband stability of various types of solar inverter systems.
9. Conclusion
In this work, I have developed a wideband aggregation modeling method for the parallel operation of new energy grid-connected inverters. Using harmonic linearization and frequency-domain analysis, I derived the sequence impedance model that captures the effects of PLL, current controller, and sampling delays. The model was validated through simulation, showing that system power and grid voltage have significant impacts on the impedance magnitude and stability margin. The comparison between PI and QPR controllers reveals that QPR provides better tolerance to voltage variations, making it suitable for weak grid applications. The proposed model can be extended to large-scale parallel inverter systems, offering a valuable tool for assessing the wideband stability of various types of solar inverter installations.
The results highlight the importance of considering the frequency-dependent characteristics of inverters in modern power systems. By understanding the impedance interactions, engineers can design more robust control strategies and optimize the integration of renewable energy sources. Future work will focus on the aggregation of multiple inverters with different ratings and topologies, further enhancing the applicability of the model to real-world power systems.