In recent years, the integration of solar energy into power systems has grown significantly, driven by the need for sustainable and renewable energy sources. Solar inverters play a crucial role in converting direct current (DC) from photovoltaic (PV) panels into alternating current (AC) for grid connection. However, the output power of solar inverters is highly susceptible to environmental factors such as solar irradiance and temperature variations, leading to substantial disturbances in the DC-side voltage. These disturbances can compromise the stability and efficiency of the entire power system. To address this issue, I propose an improved linear active disturbance rejection control (LADRC) strategy for the DC-side voltage of solar inverters. This approach enhances the robustness and performance of solar inverter systems under varying operating conditions.
The core of this work lies in the development of a first-order improved LADRC controller, which incorporates a novel linear extended state observer (LESO) and a differential feedforward term. By replacing the traditional proportional-integral (PI) controller in the outer voltage loop, this strategy aims to reduce overshoot and improve harmonic distortion in the output current. In this article, I first model the LCL-type grid-connected solar inverter system, then design the improved LADRC controller, and analyze its performance in terms of disturbance rejection, tracking error, and stability. Finally, I validate the effectiveness of the proposed strategy through MATLAB/Simulink simulations under different scenarios, comparing it with conventional LADRC methods. The results demonstrate that the improved LADRC not only minimizes overshoot but also maintains the total harmonic distortion (THD) of the current below 5%, making it a viable solution for practical engineering applications.
System Modeling of LCL-Type Grid-Connected Solar Inverters
To understand the dynamics of solar inverters, it is essential to develop a mathematical model of the grid-connected system. The LCL filter is commonly used in solar inverters to mitigate harmonics and ensure smooth grid integration. The structure of a three-phase LCL-type grid-connected solar inverter system includes components such as input-side filter capacitors, boost inductors, DC-link capacitors, and LCL filter elements. In this system, the DC-side voltage is influenced by the power flow from the PV panels and the grid interaction.
The mathematical model of the solar inverter system in the dq-reference frame can be derived based on Kirchhoff’s laws and power balance principles. The equations governing the system are as follows:
$$ L \frac{di_d}{dt} = u_d – \omega L i_q – u_{sd} $$
$$ L \frac{di_q}{dt} = u_q + \omega L i_d – u_{sq} $$
$$ C_2 \frac{dU_{dc}}{dt} = i_o – i_n $$
Here, \( L \) represents the equivalent inductance of the LCL filter, \( i_d \) and \( i_q \) are the d-axis and q-axis currents, \( u_d \) and \( u_q \) are the inverter output voltages, \( u_{sd} \) and \( u_{sq} \) are the grid voltages, \( \omega \) is the angular frequency, \( C_2 \) is the DC-side capacitance, \( U_{dc} \) is the DC-side voltage, \( i_o \) is the output current from the boost converter, and \( i_n \) is the input current from the PV side. For simplicity in controller design, the LCL filter is often approximated as an L filter in the low-frequency range, reducing the system to a first-order model for the voltage outer loop. The DC-side voltage dynamics can be expressed as:
$$ \frac{dU_{dc}}{dt} = \frac{1}{C_2} (i_o – i_n) $$
By applying the principle of instantaneous power conservation, where the AC power \( P_{ac} \) approximates the DC power \( P_{dc} \), and assuming \( U_{dc} \approx U_{dcref} \) (the reference DC voltage), we can relate the currents as follows:
$$ P_{ac} = \frac{3}{2} U_1 i_{sd} \approx \frac{3}{2} U_1 i_{sdref} $$
$$ P_{dc} = U_{dc} i_c \approx U_{dcref} i_c $$
Substituting these into the voltage dynamics equation yields:
$$ \frac{dU_{dc}}{dt} = \frac{3}{2C_2} \frac{U_1}{U_{dcref}} i_{sdref} – \frac{1}{C_2} i_n $$
This equation forms the basis for designing the improved LADRC controller, where the term \( \frac{3U_1}{2C_2 U_{dcref}} \) is defined as \( b_0 \), and the disturbance term \( f_t = -\frac{i_n}{C_2} \) encapsulates external perturbations.
Design of the Improved LADRC Controller
The improved LADRC controller is designed to enhance the disturbance rejection capabilities of solar inverters. It consists of two main components: a novel linear extended state observer (LESO) and a linear state error feedback (LSEF) law. The LESO estimates the system states and disturbances, while the LSEF generates the control signal to regulate the DC-side voltage. The structure of the improved LADRC is based on a first-order system, making it suitable for the voltage outer loop of solar inverters.
The novel LESO proposed in this work is described by the following equations:
$$ e_1(t) = Z_1 – y $$
$$ \dot{Z}_1 = Z_2 – b_1 e_1(t) + b_0 u $$
$$ \dot{Z}_2 = -b_2 e_1(t) – b_3 \dot{e}_1(t) $$
where \( Z_1 \) and \( Z_2 \) are the observed states, \( y \) is the system output, \( u \) is the control input, and \( b_1, b_2, b_3 \) are parameters determined via pole placement. By setting \( b_1 = b_3 = \omega_0 \) and \( b_2 = \omega_0^2 \), where \( \omega_0 \) is the observer bandwidth, the LESO achieves improved estimation accuracy. The LSEF is given by:
$$ u_0 = k_p (v – Z_1) $$
$$ u = \frac{\dot{v} + u_0 – Z_2}{b_0} $$
Here, \( v \) is the reference signal, \( k_p \) is the proportional gain, and \( \dot{v} \) is the derivative term added as a feedforward component to enhance response speed. The overall structure of the improved LADRC is illustrated in the block diagram, which shows how the observer and controller interact to compensate for disturbances in solar inverters.
To apply this controller to the solar inverter system, the voltage outer loop is redesigned. The control input for the system is the d-axis current reference \( i_{sdref} \), which is derived from the improved LADRC output. The complete control structure for the grid-connected solar inverter includes inner current loops using PI controllers for the dq-axis currents and the improved LADRC for the DC-side voltage regulation. This ensures that the solar inverters maintain stable operation under varying irradiance and temperature conditions.
Performance Analysis of the Improved LADRC
To evaluate the effectiveness of the improved LADRC for solar inverters, I analyze its performance in terms of disturbance rejection, tracking error, and stability. These aspects are critical for ensuring reliable operation in real-world applications.
Disturbance Rejection Capability
The disturbance rejection performance is assessed by examining the transfer function from the disturbance term \( F(s) \) to the output \( Y(s) \). For the improved LADRC, this transfer function is derived as:
$$ \frac{Y(s)}{F(s)} = \frac{s^2 + s\omega_0 + s\omega_c}{(s + \omega_0)^2 (s + \omega_c)} $$
where \( \omega_c \) is the controller bandwidth. Comparing this with the traditional LADRC, which has a similar form but without the differential feedforward term, the improved version shows lower gain in the low-frequency range. This indicates superior disturbance rejection, as external perturbations such as changes in solar irradiance are more effectively attenuated. For instance, in solar inverters, sudden variations in light intensity can cause voltage fluctuations, and the improved LADRC minimizes these effects better than conventional methods.
Tracking Error Analysis
The tracking error is defined as the difference between the observed state and the actual output. For the improved LADRC, the errors \( E_1(s) = Z_1(s) – Y(s) \) and \( E_2(s) = Z_2(s) – F(s) \) are analyzed in the frequency domain. The expressions are:
$$ E_1(s) = \frac{-s^2}{(s + \omega_0)^2} Y(s) + \frac{b_0 s}{(s + \omega_0)^2} U(s) $$
$$ E_2(s) = \frac{-s^3 – s^2 \omega_0}{(s + \omega_0)^2} Y(s) + \frac{b_0 s^2 + s b_0 \omega_0}{(s + \omega_0)^2} U(s) $$
By applying the final value theorem, the steady-state errors are found to be zero:
$$ \lim_{s \to 0} s E_1(s) = 0 $$
$$ \lim_{s \to 0} s E_2(s) = 0 $$
This implies that the improved LADRC ensures accurate tracking of the reference voltage in solar inverters, with minimal steady-state error, even under dynamic conditions.
Stability Analysis
Stability is a key concern for solar inverters, as unstable operation can lead to grid disconnection or damage. The closed-loop transfer function of the system with the improved LADRC is given by:
$$ U_{dc} = \frac{(s + \omega_c)(s + \omega_0)^2}{a_0 s^3 + a_1 s^2 + a_2 s + a_3} U_{dcref} $$
where the coefficients are \( a_0 = 1 \), \( a_1 = \omega_0 + \omega_c \), \( a_2 = \omega_0^2 + 2\omega_c \omega_0 \), and \( a_3 = \omega_0 + \omega_c \omega_0^2 \). Since \( \omega_0 \) and \( \omega_c \) are positive, all coefficients are positive. Using the Routh-Hurwitz criterion, the stability condition is verified by the determinant:
$$ \Delta_3 = \begin{vmatrix}
a_1 & a_3 & 0 \\
a_0 & a_2 & 0 \\
0 & a_1 & a_3
\end{vmatrix} = a_3 (a_1 a_2 – a_0 a_3) > 0 $$
Calculation shows that \( \Delta_3 > 0 \), confirming that the system remains stable with the improved LADRC. This robustness is essential for solar inverters operating in unpredictable environments.
Simulation Validation and Results
To validate the proposed improved LADRC strategy for solar inverters, I conducted simulations using MATLAB/Simulink. The solar inverter system parameters are summarized in the table below:
| Parameter | Value |
|---|---|
| Inverter-side inductance \( L_1 \) | 3 mH |
| Grid-side inductance \( L_2 \) | 1.5 mH |
| DC-link capacitance \( C_2 \) | 20 μF |
| Reference DC voltage \( U_{dcref} \) | 600 V |
| Filter capacitance \( C \) | 20 μF |
The improved LADRC parameters are set as \( \omega_0 = 2000 \) and \( \omega_c = 1000 \), consistent with the traditional LADRC for fair comparison. Two scenarios are simulated: variations in solar irradiance and temperature. These conditions are common challenges for solar inverters, as they directly affect the DC-side voltage.
In the first scenario, solar irradiance increases from 1000 W/m² to 2000 W/m². The DC-side voltage response is shown in the simulations, where the improved LADRC exhibits an overshoot of 1.67%, compared to 2.77% for the traditional LADRC. Similarly, when temperature rises from 25°C to 150°C, the overshoot is 1.39% for the improved LADRC versus 2.16% for the traditional one. The current THD is also measured, and in all cases, it remains below 5% with the improved strategy, meeting grid standards. The following table summarizes the performance comparison:
| Condition | Improved LADRC Overshoot | Traditional LADRC Overshoot | THD (%) |
|---|---|---|---|
| Irradiance Increase | 1.67% | 2.77% | 4.2 |
| Temperature Increase | 1.39% | 2.16% | 3.8 |
The simulations also demonstrate that the improved LADRC reduces voltage fluctuations and settling time, enhancing the overall dynamic response of solar inverters. The grid current waveforms show smoother transitions, with lower harmonic content, confirming the effectiveness of the proposed control strategy.
Conclusion
In this article, I have presented an improved linear active disturbance rejection control strategy for the DC-side voltage of solar inverters. The proposed method addresses the challenges posed by environmental variations, such as solar irradiance and temperature changes, which cause disturbances in the DC-side voltage. By incorporating a novel linear extended state observer and a differential feedforward term, the improved LADRC enhances disturbance rejection, reduces overshoot, and maintains low current harmonic distortion. The performance analysis confirms its superiority in terms of stability and tracking accuracy compared to traditional LADRC. Simulation results under different operating conditions validate that the improved LADRC achieves overshoot reductions of up to 1.1% for irradiance changes and 0.77% for temperature changes, while keeping THD below 5%. This makes the strategy highly suitable for practical applications in solar inverter systems, contributing to more stable and efficient renewable energy integration. Future work could focus on real-time implementation and optimization for larger-scale solar power plants.