With the increasing integration of renewable energy sources into power systems, photovoltaic grid-connected inverters have become critical components for converting solar energy into usable AC power. However, the operational stability of these solar inverters is often compromised by three-phase voltage imbalance, which leads to significant steady-state errors in output voltage and current. These errors can cause waveform distortion, reduce power quality, and shorten the lifespan of power electronic devices. To address this issue, I propose an automated control method that specifically considers three-phase voltage imbalance to minimize steady-state errors in solar inverters. My approach involves constructing a steady-state function for the solar inverter, designing an automated controller, and generating control strategies based on predictive algorithms. This method aims to enhance the robustness and performance of solar inverters under unbalanced conditions, ensuring reliable operation in practical applications.
The foundation of my method lies in accurately modeling the steady-state behavior of photovoltaic grid-connected inverters under three-phase voltage imbalance. I begin by collecting operational data from the solar inverter using a data acquisition system. To ensure the reliability of this data, I apply a smoothing process to reduce noise and anomalies. The smoothing function is defined as follows:
$$ \xi_1 = \psi \times \xi $$
where $\xi_1$ represents the actual value of the operational data, $\psi$ is a random parameter, and $\xi$ is the observed value. The weighted function for the data is given by:
$$ E(p) = \begin{cases}
(1 – p)^3 / 3, & \text{if } p < 1 \\
0, & \text{otherwise}
\end{cases} $$
and the smoothing curve $s$ is expressed as:
$$ s = t + k r + \lambda r^2 $$
Here, $t$, $k$, and $\lambda$ represent the fitting degree of the operational data, and $r$ denotes the data points. This preprocessing step ensures that the data used for building the steady-state function is accurate and representative of the solar inverter’s behavior.
Next, I construct the steady-state function for the photovoltaic grid-connected inverter, incorporating the three-phase voltage imbalance. The imbalance degree $k$ is calculated as:
$$ k = \frac{|U_a – U_b| + |U_b – U_c| + |U_c – U_a|}{3 |U_a|} $$
where $U_a$, $U_b$, and $U_c$ are the voltage values of phases A, B, and C, respectively. The steady-state function $M$ for the solar inverter is then defined as:
$$ M = \sum_{i=1}^{3} k I_i^2 / I_1 $$
In this equation, $I_i$ represents the three-phase currents, and $I_1$ is the output current of the solar inverter. This function captures the impact of voltage imbalance on the inverter’s performance and serves as the basis for designing the automated controller. The integration of three-phase voltage imbalance into the model allows for a more realistic representation of the solar inverter’s operational environment, which is crucial for effective error control.
To automate the control of steady-state errors, I design a dedicated controller that leverages the steady-state function. The control objectives are twofold: minimizing the output voltage steady-state error and the output current steady-state error. The structure of the controller involves input variables, control parameters, and feedback mechanisms. The control parameters are computed based on feature and deviation quantities, as shown below:
$$ \beta_{ij} = \frac{\beta_{i|j} + \beta_{j|i}}{2n} $$
where $\beta_{ij}$ is the feature quantity of the control parameter, $n$ is the number of input variables, and $\beta_{i|j}$ is given by:
$$ \beta_{i|j} = \frac{\exp\left(-\frac{\|x_i – x_j\|^2}{2\sigma_i^2}\right)}{\sum_{k \neq i} \exp\left(-\frac{\|x_i – x_k\|^2}{2\sigma_i^2}\right)} $$
The deviation quantity $\eta_{ij}$ is calculated as:
$$ \eta_{ij} = \frac{(1 + y_i – y_{2j})^{-1}}{\sum_{k \neq i} (1 + y_k – y_{2j})^{-1}} $$
Using these, the control parameter $z$ is derived as:
$$ z = \sum_i \sum_j \beta_{ij} \log\left(\frac{\beta_{ij}}{\eta_{ij}}\right) $$
The controller’s prediction result $G$ is then obtained through:
$$ G = \frac{[z \cdot (w \cdot s + h) – y_0 \cdot (s + h)]^2}{2} $$
where $w$ is the current output value, $y_0$ is the input variable value, $s$ is a random constant, and $h$ is the time to the next moment. The automated controller $H$ for the solar inverter steady-state error is finally designed as:
$$ H = G + \frac{2\lambda_1 \beta_{ij} \eta_{ij}}{z^2 + 2\lambda_2 z + h_1^2} $$
with $h_1$ defined as:
$$ h_1 = G + \frac{\lambda_3 z}{z + h_2} $$
Here, $\lambda_1$, $\lambda_2$, and $\lambda_3$ are weight parameters, and $h_2$ is the transfer function of input variables. This controller dynamically adjusts the control parameters to reduce steady-state errors in the solar inverter, ensuring stable operation under three-phase voltage imbalance.
The control strategy is implemented through a generated control algorithm that adjusts the control parameters in real-time. The control signal $u(t)$ is given by:
$$ u(t) = k_1 \cdot e(t) + k_2 \cdot \sum e(t) + k_3 \cdot \frac{de(t)}{dt} $$
where $e(t)$ is the current error value, $\sum e(t)$ is the cumulative steady-state error of the solar inverter, and $k_1$, $k_2$, and $k_3$ are weight, cumulative, and differential parameters, respectively. This algorithm continuously monitors the error and revises the control parameters to maintain optimal performance. If the control results do not meet the predefined standards, the parameters are readjusted, and the process is repeated until the desired outcome is achieved. This iterative approach ensures that the solar inverter operates efficiently even under varying conditions of three-phase voltage imbalance.
To validate the effectiveness of my method, I conducted experimental tests in a controlled laboratory environment. The setup included a data acquisition system, a controller, an oscilloscope, a three-phase voltage imbalance generator, and a power source. The key parameters for the experiments are summarized in the table below:
| Parameter | Value |
|---|---|
| Solar Inverter Rated Power | 1000 W |
| Solar Inverter Rated Voltage | 22 V |
| Solar Inverter Rated Current | 42.3 A |
| Three-Phase Voltage Imbalance Generator Rated Voltage | 22 V |
| Three-Phase Voltage Imbalance Generator Rated Current | 42.3 A |
| Three-Phase Imbalance Degree | 1:1:1 |
| Imbalance Phase Difference | 90 degrees |
| Output Power Frequency | 50 Hz |
| Controller Sampling Frequency | 15 kHz |
| Control Algorithm Weight Parameter | 0.5 |
| Control Algorithm Cumulative Parameter | 0.01 |
| Control Algorithm Differential Parameter | 0.001 |
During the experiments, I recorded the initial steady-state errors of the solar inverters and applied my control method to observe the changes. The results demonstrated a significant reduction in steady-state errors, confirming the efficacy of my approach. For instance, the initial steady-state error of 0.42 V was reduced to 0.12 V after applying the control method. This improvement highlights the ability of my method to maintain stability in solar inverters under three-phase voltage imbalance.
Furthermore, I evaluated the robustness of my method by varying the degree of three-phase voltage imbalance. The results, as shown in the table below, indicate that my method maintains high robustness with minimal fluctuations, outperforming other existing methods. The steady-state error values remained low across different imbalance conditions, proving the reliability of my automated control strategy for solar inverters.
| Three-Phase Imbalance Degree | Robustness Value (My Method) | Robustness Value (Method 2) | Robustness Value (Method 3) |
|---|---|---|---|
| 0.05 | 0.95 | 0.85 | 0.80 |
| 0.10 | 0.93 | 0.82 | 0.78 |
| 0.15 | 0.91 | 0.79 | 0.75 |
| 0.20 | 0.90 | 0.76 | 0.72 |
The experimental data, including voltage and current variations, were collected and analyzed. The voltage changes exhibited larger fluctuations, while the current changes were more regular. My control method effectively stabilized these variations, reducing the steady-state errors in both parameters. The use of the steady-state function and automated controller allowed for precise adjustments, ensuring that the solar inverters operated within acceptable error limits. The integration of three-phase voltage imbalance considerations into the control logic was key to achieving this performance.
In conclusion, my automated control method for steady-state error in photovoltaic grid-connected inverters, which accounts for three-phase voltage imbalance, offers a robust and efficient solution for improving the stability and performance of solar inverters. By leveraging data smoothing, steady-state modeling, and predictive control algorithms, I have developed a method that minimizes errors and adapts to changing conditions. The experimental results validate the superiority of my approach over existing methods, demonstrating its potential for widespread application in renewable energy systems. Future work could focus on optimizing the control parameters further and integrating additional sensor data to enhance precision. This research contributes to the advancement of solar inverter technology, supporting the reliable integration of photovoltaic systems into the power grid.