In recent years, the rapid development of renewable energy technologies has positioned solar power generation as a key player in global energy systems. Solar inverters, which convert DC power from photovoltaic panels to AC power for grid integration, often employ Boost circuits as the primary energy conversion stage. These circuits regulate the output of solar cells and directly impact the overall performance of solar inverters. However, when solar inverters operate on the left side of the PV curve—where solar cells exhibit high internal resistance—the Boost circuit can experience resonance, leading to instability. This issue poses significant challenges for controllers, especially in applications requiring high dynamic performance, such as power reserve control and IV curve scanning for solar inverters.
To address this, I propose a Linear Active Disturbance Rejection Control (LADRC) strategy for solar inverter Boost circuits. LADRC leverages the system’s input-output data to estimate and compensate for total disturbances, including nonlinearities and uncertainties, without relying on an exact model. This approach allows the control bandwidth to exceed the system’s resonant frequency limitations, offering superior performance compared to traditional methods. Additionally, I introduce an improved discretization algorithm for the Linear Extended State Observer (LESO) to enhance stability under low sampling frequencies. Through theoretical analysis and simulations, I demonstrate that this improved method outperforms conventional PI control and traditional LESO discretization in both steady-state and transient operations for solar inverters.
System Modeling of Solar Inverter Boost Circuits
The behavior of solar cells is highly nonlinear, but for analysis, the current-voltage (IV) characteristics can be linearized around an operating point. On the left side of the PV curve, solar cells approximate a current source with high internal resistance, which can lead to resonance in the Boost circuit’s inductor and capacitor. The equivalent circuit of a solar inverter Boost stage includes a solar cell modeled as a variable voltage source \( E \) in series with a variable internal resistance \( R \), followed by an inductor \( L \), capacitor \( C \), and the Boost converter switching network. The DC-link voltage is regulated by an outer loop and is treated as constant for simplification.
The state-space equations derived using Kirchhoff’s laws are:
$$ C \frac{dU}{dt} = \frac{E – U}{R} – i $$
$$ L \frac{di}{dt} = U – (1 – d) U_{bus} $$
where \( U \) is the solar cell output voltage, \( i \) is the inductor current, \( d \) is the duty cycle, and \( U_{bus} \) is the DC-link voltage. Applying Laplace transform, the transfer function from duty cycle to output voltage is:
$$ G(s) = \frac{U(s)}{d(s)} = \frac{-U_{bus}}{LCs^2 + \frac{L}{R}s + 1} $$
As the internal resistance \( R \) increases (e.g., on the left side of the PV curve), the system exhibits a resonant peak, reducing phase margin and potentially causing instability. For instance, with \( L = 1.24 \, \text{mH} \), \( C = 50 \, \mu\text{F} \), and \( U_{bus} = 600 \, \text{V} \), the resonant frequency is approximately \( 4016 \, \text{rad/s} \). The following table summarizes key parameters and their effects:
| Parameter | Value | Effect on Resonance |
|---|---|---|
| Internal Resistance \( R \) | 5–55 Ω | Increase shifts resonance peak higher |
| Inductance \( L \) | 1.24 mH | Higher values lower resonant frequency |
| Capacitance \( C \) | 50 μF | Larger capacitance reduces bandwidth |
The Bode plot of \( G(s) \) shows that as \( R \) increases, the gain at resonance rises, emphasizing the need for a robust controller. Solar inverters must operate reliably across various conditions, including sudden changes in irradiation that shift the operating point to high-resistance regions.
Design of Linear Active Disturbance Rejection Control
LADRC treats model uncertainties and external disturbances as a total disturbance, which is estimated and compensated in real-time. For the second-order Boost circuit system, the equation is rewritten as:
$$ \ddot{U} = -\frac{U_{bus} d}{LC} + w $$
where \( w \) represents the total disturbance, encompassing internal resistance variations, DC-link voltage errors, and grid perturbations. A second-order LESO is designed to estimate the states and disturbance:
$$ \dot{z}_1 = z_2 + \beta_1 (y – z_1) $$
$$ \dot{z}_2 = z_3 + \beta_2 (y – z_1) + b u $$
$$ \dot{z}_3 = \beta_3 (y – z_1) $$
Here, \( z_1 \) estimates \( U \), \( z_2 \) estimates \( \dot{U} \), and \( z_3 \) estimates \( w \). The observer gains \( \beta_1, \beta_2, \beta_3 \) are parameterized by the observer bandwidth \( \omega_o \):
$$ \beta_1 = 3\omega_o, \quad \beta_2 = 3\omega_o^2, \quad \beta_3 = \omega_o^3 $$
The control law uses a PD controller combined with disturbance compensation:
$$ u_0 = k_p (r – z_1) – k_d z_2 $$
$$ u = \frac{u_0 – z_3}{b_0} $$
where \( k_p = \omega_c^2 \), \( k_d = 2\omega_c \), and \( \omega_c \) is the controller bandwidth. The parameter \( b_0 \) is an approximation of the system gain. The closed-loop transfer function demonstrates that LADRC can suppress resonance by providing phase lead at critical frequencies. For solar inverters, this means improved stability even when operating under high internal resistance conditions.
Improved LESO Discretization Algorithm
In digital implementations, the sampling frequency affects stability. Traditional forward Euler discretization of LESO can lead to instability at low sampling rates. For example, with \( \omega_c = 5000 \, \text{rad/s} \), \( \omega_o = 14000 \, \text{rad/s} \), and a sampling frequency of 19.2 kHz, the system becomes unstable. The discrete LESO using forward Euler is:
$$ z(n+1) = \Phi z(n) + H u(n) + L [y(n) – z_1(n)] $$
where
$$ \Phi = \begin{bmatrix} 1 & h & 0 \\ 0 & 1 & h \\ 0 & 0 & 1 \end{bmatrix}, \quad H = \begin{bmatrix} 0 \\ h b \\ 0 \end{bmatrix}, \quad L = \begin{bmatrix} \beta_1 h \\ \beta_2 h \\ \beta_3 h \end{bmatrix} $$
The characteristic equation in the Z-domain is:
$$ \det\left( zI – (\Phi – LC + H J) \right) = 0 $$
Using the Routh-Hurwitz criterion after bilinear transformation, the stability condition is:
$$ \omega_o^3 h^3 + (3\omega_o^2 – 6\omega_o \omega_c – \omega_c^2) h^2 + (6\omega_o – 4\omega_c) h + 4 > 0 $$
For the given parameters, this condition is violated, indicating instability. To overcome this, I propose an improved discretization based on the implicit Euler method:
$$ A z(n+1) = \Phi z(n) + H u(n+1) + L y(n+1) $$
where
$$ A = \begin{bmatrix} 1 & -h & 0 \\ 0 & 1 & -h \\ 0 & 0 & 1 \end{bmatrix} $$
Approximating \( u(n+1) \approx u(n) \) and \( y(n+1) \approx y(n) \), the update equations become:
$$ z_1(n+1) = z_1(n) + \frac{h [z_2(n) + h z_3(n)] + h^2 b u(n) + t_a e(n)}{1 + t_a} $$
$$ z_2(n+1) = z_2(n) + h z_3(n) + h b u(n) + t_b e(n) $$
$$ z_3(n+1) = z_3(n) + t_c e(n) $$
with \( e(n) = y(n) – z_1(n) \), \( t_a = \beta_1 h + \beta_2 h^2 + \beta_3 h^3 \), \( t_b = \beta_2 h + \beta_3 h^2 \), and \( t_c = \beta_3 h \). The stability condition for this improved discretization is:
$$ \omega_o^3 h^3 + (3\omega_o^2 – 6\omega_o \omega_c – \omega_c^2) h^2 + (6\omega_o – 4\omega_c) h + 4 > 0 $$
which holds for the same parameters, ensuring stability. This allows solar inverters to operate reliably at lower sampling frequencies, reducing hardware requirements.
Simulation Results and Analysis
I validated the proposed method using offline simulations in PSCAD/EMTDC and real-time hardware-in-the-loop (HIL) tests with Typhoon HIL604. The solar inverter parameters are: \( C_1 = 50 \, \mu\text{F} \), \( C_2 = 200 \, \mu\text{F} \), \( C_3 = 8 \, \mu\text{F} \), \( L_1 = 1.24 \, \text{mH} \), \( L_2 = 0.86 \, \text{mH} \), \( L_3 = 0.033 \, \text{mH} \), and switching frequency \( f_s = 19.2 \, \text{kHz} \). The solar cell has a maximum power of 10 kW, open-circuit voltage \( U_{oc} = 643 \, \text{V} \), short-circuit current \( I_{sc} = 19.4 \, \text{A} \), and MPP voltage \( U_{mpp} = 566 \, \text{V} \).
In the simulation, the reference voltage \( U_{ref} \) steps from 550 V to 450 V at 0.2 s and to 350 V at 0.3 s. The internal resistance \( R \) increases as \( U_{ref} \) decreases, exacerbating resonance. The following table compares the performance of PI control, traditional LADRC, and improved LADRC:
| Control Method | Overshoot | Settling Time | Stability at Low \( U_{ref} \) |
|---|---|---|---|
| PI Control | High | Slow | Unstable |
| Traditional LADRC | Moderate | Medium | Divergent |
| Improved LADRC | Low | Fast | Stable |
The improved LADRC achieves fast, non-overshooting tracking of \( U_{ref} \), even under high internal resistance. In HIL tests, the results align with offline simulations, confirming the practicality for solar inverter applications. The control bandwidth extends beyond the resonant frequency, enabling robust operation across the PV curve.
Conclusion
In this paper, I developed a LADRC strategy for solar inverter Boost circuits that addresses resonance issues on the left side of the PV curve. The improved LESO discretization algorithm enhances stability under low sampling frequencies, making it suitable for practical solar inverters. Simulations and HIL tests demonstrate superior steady-state and transient performance compared to PI control, with the control bandwidth overcoming resonant limitations. This approach contributes to the reliable integration of solar power into grids, supporting the growth of renewable energy. Future work will focus on optimizing parameters for varying environmental conditions and extending the method to three-phase solar inverters.