We present a novel dual closed-loop control strategy for single-phase solar inverter systems, combining an improved linear active disturbance rejection controller (LADRC) in the outer voltage loop with a conventional proportional-integral (PI) controller in the inner current loop. Traditional PI-based dual-loop controllers neglect the inherent nonlinearities of the inverter, leading to degraded dynamic and static performance. Our approach enhances the linear state error feedback (LSEF) law by introducing an output error compensation term to mitigate system oscillations and improve robustness. We design a third-order linear extended state observer (LESO) to estimate total disturbances in real time. Through bandwidth parameterization and pole placement, we optimize the observer bandwidth and controller bandwidth. The closed-loop stability is verified via Routh–Hurwitz criterion. Simulation results on a 1.2 kW single-phase solar inverter model demonstrate that the proposed LADRC-PI control significantly reduces output voltage overshoot, total harmonic distortion (THD), and settling time under both resistive and switching loads compared to conventional dual-loop PI control. The strategy exhibits superior steady-state accuracy, faster transient response, and stronger disturbance rejection. This work provides a practical solution for high-performance solar inverter control in modern photovoltaic systems.
1. Introduction
With the growing demand for renewable energy, solar power has become a focus of global development. As the core power conversion equipment, the solar inverter plays a vital role in photovoltaic (PV) systems. However, the actual mathematical model of a solar inverter is complex and highly nonlinear due to internal switching devices, parasitic parameters, and external disturbances such as irradiance fluctuations and load variations. Conventional linear control methods, such as proportional-integral (PI) control, deadbeat control, and sliding mode control, often fail to meet high-performance requirements when applied individually. Dual closed-loop control, typically with an outer voltage loop and an inner current loop, has been widely adopted to improve solar inverter performance. Nonetheless, most existing dual-loop schemes still rely on PI controllers in both loops, which cannot fully account for nonlinear dynamics and lead to overshoot, slow response, and poor robustness.
Active disturbance rejection control (ADRC), originally proposed by Han, offers a powerful alternative by treating internal uncertainties and external disturbances as a total disturbance to be estimated and compensated in real time. Linear ADRC (LADRC) simplifies the tuning process by using bandwidth-parameterized observers and controllers. Recent studies have applied LADRC to solar inverters with promising results, but many implementations still suffer from oscillations or complex parameter tuning. To address these issues, we propose an improved LADRC that incorporates an output error compensation term into the LSEF law, specifically designed for the voltage outer loop of a single-phase solar inverter. The current inner loop retains a PI controller for simplicity. This combined strategy, referred to as improved LADRC-PI dual closed-loop control, aims to enhance stability, response speed, and disturbance rejection. In the following sections, we first derive the mathematical model of the single-phase solar inverter, then design the improved LADRC controller, analyze its performance, and finally validate it through simulations.
2. System Overview and Mathematical Model
The single-phase solar inverter system under study consists of a full-bridge topology, an L-C filter, and a resistive load (or switching loads for dynamic tests). The DC side is supplied by a constant voltage source representing the output of a front-end boost converter with MPPT. The control structure is shown in Figure 1, where the outer voltage loop regulates the output voltage \( V_o \) to follow a sinusoidal reference \( V_{oref} \), and the inner current loop controls the inductor current \( i_L \) to improve transient performance. The PWM signals are generated by comparing the controller output with a triangular carrier.
The circuit equations are derived using Kirchhoff’s laws:
\[
V_{ab} = L \frac{di_L}{dt} + R i_L + V_o
\]
\[
i_L = C \frac{dV_o}{dt} + i_o
\]
Applying Laplace transformation yields the transfer function from the bridge midpoint voltage \( V_{ab} \) to the output voltage \( V_o \):
\[
\frac{V_o(s)}{V_{ab}(s)} = \frac{1}{LC s^2 + RC s + 1}
\]
Considering the inverter as a unity-gain transformer, we have \( V_{ab}(s) = V_{oref}(s) \) when the PWM gain is properly normalized. Thus, the plant transfer function \( G_{PV}(s) \) is:
\[
G_{PV}(s) = \frac{1}{LC s^2 + RC s + 1}
\]
In state-space form, let \( x_1 = V_o \), \( x_2 = \dot{V_o} \), and \( x_3 = f \) (total disturbance). The system can be expressed as:
\[
\begin{bmatrix}
\dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3
\end{bmatrix} =
\begin{bmatrix}
0 & 1 & 0 \\
-a_0 & -a_1 & 1 \\
0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
+
\begin{bmatrix}
0 \\ b_0 \\ 0
\end{bmatrix} u
+
\begin{bmatrix}
0 \\ 0 \\ 1
\end{bmatrix} \dot{f}
\]
\[
y = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \mathbf{x}
\]
where \( a_0 = 1/(LC) \), \( a_1 = R/L \), \( b_0 = 1/(LC) \), and \( u = V_{ab} \). This model serves as the foundation for the LADRC design.
3. Improved LADRC Design
3.1 Linear Extended State Observer (LESO)
A third-order LESO is constructed as:
\[
\dot{\hat{\mathbf{x}}} = \mathbf{N}_m \hat{\mathbf{x}} + \mathbf{P}_m u + \boldsymbol{\beta}_m (y – \hat{y})
\]
\[
\hat{y} = \mathbf{C}_m \hat{\mathbf{x}}
\]
Using pole placement, we set all observer poles at \( -\omega_o \). The observer gains are:
\[
\boldsymbol{\beta}_m = \begin{bmatrix} 3\omega_o & 3\omega_o^2 & \omega_o^3 \end{bmatrix}^T
\]
The observer bandwidth \( \omega_o \) determines the tracking speed and noise sensitivity. Through Bode analysis of the estimation error transfer function \( z_{e3}/f \), we select \( \omega_o = 14,000 \) rad/s as a trade-off between low-frequency accuracy and high-frequency noise immunity.
3.2 Improved Linear State Error Feedback (LSEF)
Conventional LADRC uses a PD-type control law. To avoid oscillations, we add an output error compensation term \( T = -\beta_2 (V_{oref} – z_1) \) to the LSEF. The improved LSEF is:
\[
u_c = k_p (V_{oref} – z_1) – k_d z_2 + T
\]
\[
u = \frac{u_c – z_3}{b_0}
\]
where \( k_p = \omega_c^2 \), \( k_d = 2\omega_c \), and \( \omega_c \) is the controller bandwidth. The final control variable \( u \) is fed to the PWM modulator. The output error compensation reduces overshoot and suppresses resonance.
3.3 Parameter Tuning
The system parameters for the 1.2 kW solar inverter are listed in Table 1.
| Parameter | Value |
|---|---|
| DC-link voltage \( V_{dc} \) | 400 V |
| Filter inductance \( L \) | 4.06 mH |
| Filter capacitance \( C \) | 6.23 μF |
| Switching frequency | 10 kHz |
| Line equivalent resistance \( R \) | 0.1 Ω |
| Grid frequency | 50 Hz |
The inner current loop uses a PI controller with parameters \( k_{pi} = 24 \), \( k_{ii} = 7 \), obtained from frequency-domain design. For the outer voltage LADRC, we set \( b_0 = 1/(LC) = 1.54 \times 10^{-3} \). The controller bandwidth \( \omega_c \) is chosen as 6000 rad/s after analyzing the closed-loop Bode plot of \( y(s)/V_{oref}(s) \). The final controller parameters are summarized in Table 2.
| Parameter | Value |
|---|---|
| Observer bandwidth \( \omega_o \) | 14,000 rad/s |
| Controller bandwidth \( \omega_c \) | 6,000 rad/s |
| Compensation factor \( b_0 \) | 1.54×10⁻³ |
| Voltage loop PI gains (for comparison) | \( k_{pv}=0.02 \), \( k_{iv}=21 \) |
| Current loop PI gains | \( k_{pi}=24 \), \( k_{ii}=7 \) |
4. Performance Analysis
4.1 Disturbance Rejection
The disturbance rejection capability is evaluated by examining the transfer function from total disturbance \( f \) to the estimation error \( z_{e3} \). The Bode plot (omitted here for text brevity) shows that at low frequencies the observer attenuates disturbances well; increasing \( \omega_o \) improves low-frequency rejection but reduces high-frequency noise immunity. The chosen \( \omega_o = 14,000 \) rad/s provides a good balance.
4.2 Tracking Performance
The closed-loop transfer function from reference to output for the improved LADRC-PI scheme is derived as:
\[
\frac{y(s)}{V_{oref}(s)} = \frac{k_p (s^3 + \beta_1 s^2 + \beta_2 s + \beta_3) G_{PV}(s) (k_{pi} s + k_{ii})}{b_0 s^2 (s^3 + M_1 s^2 + M_2 s + M_3) G_{PV}(s) + (M_4 s^2 + M_5 s + M_6)}
\]
where \( M_1 = k_d + \beta_1 \), \( M_2 = k_p + k_d \beta_1 + \beta_2 \), \( M_3 = k_p \beta_1 + k_d \beta_2 + \beta_3 \), \( M_4 = k_p \beta_2 + k_d \beta_3 \), \( M_5 = k_p \beta_3 \). The Bode comparison with conventional dual-loop PI control (omitted) reveals that the proposed method has a larger crossover frequency, smaller phase lag, and slower magnitude attenuation, indicating faster response and better tracking accuracy.
4.3 Stability Analysis
Using the Routh–Hurwitz criterion on the characteristic equation:
\[
s^5 + \varphi_4 s^4 + \varphi_3 s^3 + \varphi_2 s^2 + \varphi_1 s + \varphi_0 = 0
\]
After substituting the system parameters and controller gains, all coefficients are positive and the first column of the Routh array contains no sign changes, confirming closed-loop stability.
5. Simulation Results
We built a MATLAB/Simulink model of a 1.2 kW single-phase solar inverter with the parameters in Table 1. The reference voltage is \( V_{oref} = 300\sin(100\pi t) \) V. Two control schemes are compared: conventional dual-loop PI (both loops PI) and the proposed improved LADRC-PI. Steady-state and dynamic tests are performed.
5.1 Steady-State Performance under Resistive Load
Table 3 summarizes the steady-state metrics.
| Metric | Dual-loop PI | Improved LADRC-PI |
|---|---|---|
| Output voltage amplitude (V) | 309 | 300.85 |
| Overshoot at peak (%) | 4.23 | 0.28 |
| Maximum waveform overshoot (%) | 6.30 | 0.67 |
| Voltage dip at peak (V) | 5.3 | ~0 |
| THD (%) | 2.54 | 0.28 |
| Fundamental amplitude (V) | 309.8 | 300.7 |
The improved LADRC-PI dramatically reduces overshoot and THD. The output voltage closely tracks the reference with minimal distortion.
5.2 Dynamic Performance under Load Switching
We simulate a load profile: 0–0.3 s resistive, 0.3–0.6 s inductive, 0.6–1 s resistive-inductive. The dual-loop PI control exhibits a settling time of about 1.6 ms when switching to inductive load and 1 ms when switching to resistive-inductive load, with noticeable waveform distortion during transitions. In contrast, the improved LADRC-PI control shows almost instantaneous settling (less than 0.1 ms) and no visible distortion. This demonstrates superior transient capability and robustness.
6. Conclusion
We have developed an improved LADRC-PI dual closed-loop control strategy for single-phase solar inverters. By adding an output error compensation term to the LSEF, the proposed controller effectively eliminates system oscillations and enhances robustness. The LESO with optimized bandwidth accurately estimates total disturbances, while the PI inner loop ensures fast current regulation. Theoretical analyses and simulations confirm that the improved LADRC-PI scheme outperforms conventional dual-loop PI control in terms of lower overshoot, reduced THD, faster dynamic response, and better disturbance rejection. This work provides a practical and effective solution for high-quality solar inverter control in photovoltaic power systems.