In the context of global energy transitions, the rapid adoption of renewable energy sources has become imperative to address climate change, air pollution, and resource scarcity. Among these, solar power generation stands out as a key contributor, with photovoltaic (PV) systems being deployed extensively in various configurations such as agricultural, fishery, and mountainous power plants. As a researcher focused on power electronics and grid integration, I have observed that these installations often utilize multiple solar inverters in parallel within the 20 kW to 50 kW power range to form larger-scale stations. However, the geographical constraints of such sites introduce long-distance transmission lines, which increase impedance and, coupled with the reduced equivalent impedance of high-capacity PV systems, result in weak grid characteristics. This weak grid environment poses significant challenges: parallel-connected solar inverters can experience circulating currents, altered control loop gains due to interactions with the grid, degraded power quality, and even system oscillations leading to instability. Therefore, ensuring efficient, high-power-quality, and stable operation of distributed solar inverter systems in weak grids has become a critical research focus. In this article, I present my insights and proposed methods for grid impedance identification, which is essential for analyzing stability, compensating power quality, and optimizing control loops for solar inverters.
Impedance identification techniques broadly fall into two categories: active and passive methods. For solar inverters, passive methods leverage the inherent switching characteristics or algorithms without injecting external signals. For instance, some approaches extract harmonic components at switching frequencies to compute equivalent grid impedance, but these can be computationally complex and susceptible to interference from other parallel inverters. Active methods, on the other hand, inject specific frequency disturbances—periodic or random—into the grid and analyze the response to derive impedance. While methods like high-frequency harmonic injection minimize impact on fundamental frequencies, they may be affected by capacitive loads on the user side. Alternatively, injecting low-frequency non-characteristic harmonics requires extensive computation and can suffer from reduced accuracy in multi-inverter scenarios. In my work, I have developed a simplified active method based on Q-axis perturbation in the control inner loop, which balances accuracy, simplicity, and reliability for real-time implementation on fixed-point microcontrollers. This method is particularly suited for solar inverter applications in weak grids, as it minimizes impact on power output and avoids additional hardware.
To lay the groundwork, let me first discuss the system structure of a typical PV power plant. A solar inverter system comprises PV arrays that generate DC power under irradiation, followed by power conversion stages. In my analysis, I focus on non-isolated three-phase grid-connected solar inverters with LC filters, which offer cost benefits and simpler control compared to LCL filters, avoiding resonance damping issues. The control strategy for such a solar inverter involves Clark and Park transformations to extract positive and negative sequence components, phase-locked loop (PLL) synchronization, and decoupled DQ-axis control for current regulation. The key equations are as follows:
The Clark transformation (amplitude-invariant) is given by:
$$ T_{\text{Clark}} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $$
The Park transformation is:
$$ T_{\text{Park}} = \begin{bmatrix} \cos \theta_{ge} & \sin \theta_{ge} \\ -\sin \theta_{ge} & \cos \theta_{ge} \end{bmatrix} $$
where \( \theta_{ge} \) is the grid phase angle from PLL. The D-axis control ensures active power regulation:
$$ i_{gd}^* = \left( K_{p1} + \frac{K_{i1}}{s} \right) (U_{dc}^* – U_{dc}) $$
$$ U_{gd}^* = e_{gd}^p – \left( K_{p2} + \frac{K_{i2}}{s} \right) (i_{gd}^* – i_{gd}) – i_{gq} \omega L $$
and the Q-axis control manages reactive power:
$$ i_{gq}^* = \frac{1}{e_{gd}} Q_q^* $$
$$ U_{gq}^* = – \left( K_{p3} + \frac{K_{i3}}{s} \right) (i_{gq}^* – i_{gq}) + i_{gd} \omega L $$
Here, \( i_{gd} \) and \( i_{gq} \) are the DQ-axis currents, \( U_{gd}^* \) and \( U_{gq}^* \) are reference voltages, and \( K_p \), \( K_i \) are PI controller gains. This control framework is fundamental to the operation of a solar inverter and forms the basis for my impedance identification method.
In a multi-solar inverter parallel system, the grid impedance model must account for various components: inherent grid impedance, transformer impedance, and line impedance. For simplification, I consider a subsystem with n solar inverters connected to a point of common coupling (PCC). The impedance on the PCC external side is combined as the utility grid equivalent impedance \( Z_g = R_g + j\omega L_g \), while the internal side includes inverter output impedance and line impedance \( Z_i = R_{ci} + j\omega L_{ci} \). Using Norton’s theorem, each solar inverter can be modeled as a controlled current source \( I_{oi} \) in parallel with an output impedance \( Z_{oi} \). The parallel combination yields a virtual equivalent model for any single solar inverter, as shown in the derived equations.
For impedance identification, I propose a method based on visual vector analysis of voltage relationships. Consider a single solar inverter connected to the grid via an equivalent impedance \( Z_G = Z_N + Z_g \), where \( Z_N \) represents the total impedance from the inverter output to the PCC (excluding grid impedance). Under steady-state, the inverter output voltage \( V_{\text{out}} \), PCC voltage \( V_{\text{pcc}} \), and grid voltage \( V_g \) relate through:
$$ V_{\text{out}} + I_n Z_N = V_{\text{pcc}} $$
$$ V_{\text{pcc}} + I_n Z_g = V_g $$
By introducing a perturbation in the Q-axis current reference \( \Delta I_q \) in the solar inverter control loop, the output current changes from \( I_n \) to \( I_n’ \), leading to corresponding voltage changes. From the vector diagram, the change in PCC voltage along the D-axis, \( \Delta V_{\text{out}} \), is related to \( \Delta I_q \) and grid inductance \( L_g \). Specifically, when power factor is unity, the relationship simplifies to:
$$ \Delta V_{\text{out}} = \Delta I_q \omega L_g $$
In practice, \( \Delta V_{\text{out}} \) can be obtained as the change in the positive-sequence D-axis component of the inverter output voltage, denoted \( \Delta e_{gd}^p \), which is readily available from the PLL in the solar inverter control system. Thus, the grid inductance is calculated as:
$$ L_g = \frac{\Delta e_{gd}^p}{\omega \Delta I_q} $$
This formula provides a straightforward way to estimate \( L_g \) without requiring direct measurement of PCC voltage, reducing hardware costs and complexity. The resistance \( R_g \) can be derived similarly from the real part of impedance, but for weak grids, inductive dominance makes \( L_g \) more critical. My method involves iteratively applying this perturbation and computation to enhance accuracy, as outlined in the flowchart below.
The implementation process for this impedance identification in a solar inverter system is summarized in the following steps:
- Operate the solar inverter at steady-state with normal grid connection.
- Introduce a small step change in the Q-axis current reference \( Q_q^* \) to generate \( \Delta I_q \).
- Measure the resulting change in the D-axis voltage component \( \Delta e_{gd}^p \) from the PLL output.
- Compute \( L_g \) using the above formula, with \( \omega = 2\pi \times 50 \) rad/s for 50 Hz grids.
- Repeat steps 2-4 multiple times to average out noise and improve precision.
- Optionally, adjust DC bus voltage and power stabilization controls to maintain system stability during perturbation.
This approach is efficient because it leverages existing solar inverter sensors and control loops, minimizing additional footprint. Moreover, the perturbation is small and transient, ensuring minimal impact on power quality and overall solar inverter performance.
To validate my method, I conducted comparative simulations in Matlab/Simulink against a conventional non-characteristic harmonic injection technique. The harmonic injection method involves injecting a 75 Hz disturbance and extracting 25 Hz components via DFT, which is computationally intensive. My simulation model for the solar inverter system included the control strategy, grid impedance, and perturbation logic. The results for grid inductance estimation are summarized in the table below.
| Method | Theoretical \( L_g \) (mH) | Identified \( L_g^* \) (mH) | Error (%) |
|---|---|---|---|
| Non-characteristic Harmonic Injection | 5.0 | 5.06 | 1.2 |
| Q-axis Perturbation (Proposed) | 5.0 | 5.03 | 0.6 |
| Non-characteristic Harmonic Injection | 2.0 | 2.03 | 1.5 |
| Q-axis Perturbation (Proposed) | 2.0 | 2.02 | 1.0 |
| Non-characteristic Harmonic Injection | 1.0 | 1.03 | 3.0 |
| Q-axis Perturbation (Proposed) | 1.0 | 1.02 | 2.0 |
The table shows that my proposed method achieves comparable or better accuracy than harmonic injection, with errors below 2% in most cases. This confirms the feasibility of the Q-axis perturbation approach for solar inverter applications. Additionally, I tested various perturbation magnitudes \( \Delta I_q \) to assess robustness, as shown in another simulation set.
| \( \Delta I_q \) (A) | Theoretical \( L_g \) (mH) | Identified \( L_g^* \) (mH) | Error (%) |
|---|---|---|---|
| -5 | 7.0 | 7.06 | 0.86 |
| -5 | 5.0 | 5.03 | 0.60 |
| -5 | 2.0 | 2.02 | 1.00 |
| -5 | 1.0 | 1.02 | 2.00 |
| -5 | 0.5 | 0.52 | 4.00 |
| -2 | 2.0 | 2.02 | 1.00 |
| -1 | 2.0 | 2.03 | 1.50 |
Smaller perturbations (e.g., \( \Delta I_q = -1 \) A) yield slightly higher errors due to measurement noise, but overall, the method remains effective across a range of conditions. This robustness is crucial for real-world solar inverter deployments where grid impedance may vary.
Beyond simulations, I built an experimental prototype to test the method on a physical solar inverter system. The setup consisted of a 10 kW non-isolated three-phase solar inverter with LC filter, connected to a programmable AC source via a series inductor of 3 mH to emulate grid impedance. The solar inverter control was implemented on a fixed-point DSP microcontroller, with the impedance identification algorithm integrated into the software. During testing, I collected data on \( \Delta e_{gd}^p \) and \( \Delta I_q \) under different operating points. The results demonstrated consistent identification of grid inductance with an average error of around 4%, which is acceptable for practical solar inverter applications. For instance, in one test run, the identified inductance averaged 2.89 mH against a theoretical 3 mH, representing a 3.7% deviation.
This image illustrates a typical solar inverter system with battery storage, similar to those used in modern PV installations. The integration of impedance identification capabilities into such solar inverters can enhance grid adaptability and reliability.
The advantages of my proposed method are multifaceted. First, it requires no extra sensors or hardware modifications to the solar inverter, leveraging existing voltage and current measurements. Second, the computational burden is low, involving simple arithmetic operations suitable for real-time execution on microcontrollers. Third, the perturbation is small and brief, minimizing disruption to power output and grid power quality. Fourth, the method is inherently compatible with multi-solar inverter parallel systems, as each inverter can independently perform identification without cross-interference, provided perturbations are coordinated or randomized. However, challenges remain, such as the influence of DC bus voltage fluctuations and power oscillations during perturbation. In my implementation, I addressed these by incorporating voltage stabilization loops and ensuring that the solar inverter operates in a steady power region before identification.
Looking ahead, there are several directions for improving this impedance identification technique for solar inverters. One area is extending the method to identify resistive components \( R_g \) more accurately, which may be significant in highly resistive grids. This could involve additional perturbations or analysis of the real part of the impedance equation. Another direction is adaptive control integration, where the identified impedance parameters are used to dynamically adjust solar inverter control gains, optimizing performance under varying grid conditions. For example, in weak grids with high inductance, the solar inverter can increase damping margins or modify current loop bandwidth to prevent resonance. Furthermore, machine learning algorithms could be employed to enhance identification accuracy by learning from historical data, though this would increase computational requirements.
In conclusion, grid impedance identification is a vital function for solar inverters operating in weak grid environments. My proposed method, based on Q-axis perturbation and visual vector analysis, offers a simplified, effective solution that balances accuracy, simplicity, and practicality. Through simulations and experimental validation, I have demonstrated its feasibility for real-time implementation in solar inverter systems. This approach contributes to the broader goal of improving solar inverter reliability and grid integration, supporting the global transition to sustainable energy. As solar inverter technologies evolve, incorporating such adaptive features will be key to maximizing the potential of photovoltaic power generation in diverse grid conditions.
To further elaborate on the mathematical foundations, let me derive the complete impedance model for a multi-solar inverter system. Assuming n identical solar inverters with equal line impedances \( Z_i \) and output currents \( I_\epsilon \), the voltage at the PCC can be expressed as:
$$ V_g = V_n + I_\epsilon (Z_n + n Z_g) $$
where \( V_n \) is the inverter output voltage. For a single solar inverter perspective, the equivalent impedance seen from its output terminals is \( Z_{\text{eq}} = Z_n + n Z_g \). In practice, variations in parameters require decentralized identification. My method avoids this complexity by focusing on the local response of each solar inverter. The core equations for grid inductance identification, as implemented in the solar inverter control software, are:
$$ \Delta V_{\text{out}} = \Delta e_{gd}^p $$
$$ L_g = \frac{\Delta e_{gd}^p}{\omega \Delta I_q} $$
where \( \Delta I_q \) is the perturbation in Q-axis current, and \( \omega \) is the grid angular frequency. For a 50 Hz system, \( \omega = 100\pi \) rad/s. The perturbation magnitude is typically set to 1-5% of the rated current to ensure minimal impact. In my experiments, I used \( \Delta I_q = -2 \) A for a 10 kW solar inverter with a rated current of 15 A, resulting in negligible power quality degradation.
Additionally, I evaluated the method’s performance under noisy conditions by adding measurement noise to the voltage and current signals in simulation. The results indicated that averaging over multiple iterations (e.g., 10 cycles) can reduce error to below 2%, making it suitable for industrial solar inverter applications. The table below summarizes key parameters used in the simulations.
| Parameter | Value | Description |
|---|---|---|
| Solar Inverter Power Rating | 10 kW | Rated output power |
| Grid Voltage | 400 V (line-to-line) | Three-phase system |
| Grid Frequency | 50 Hz | Fundamental frequency |
| LC Filter Inductance (L) | 2 mH | Inverter-side inductor |
| LC Filter Capacitance (C) | 10 µF | Filter capacitor |
| Switching Frequency | 10 kHz | PWM frequency |
| Control Loop Sampling Time | 100 µs | Digital control period |
| Perturbation Duration | 100 ms | Time for \( \Delta I_q \) application |
These parameters reflect typical values for commercial solar inverters, ensuring the relevance of my findings. In summary, the impedance identification method I propose enhances the functionality of solar inverters, enabling them to adapt to weak grid conditions and improve overall system stability. As the deployment of solar inverters continues to grow, such advancements will play a crucial role in ensuring reliable and efficient renewable energy integration.