Adaptive Sliding Mode Control for Maximum Power Point Tracking in Photovoltaic Systems

Renewable energy has been playing an essential role in electricity production. Solar power is a widely accessible renewable energy source that is both eco-friendly and sustainable. It is a potential and popular choice in the fields of agriculture, industry, services, etc. [1-5]. The problem of improving PV systems' efficiency has attracted many scientists' attention [6-10]. The current-voltage (I-V) and power-voltage (P-V) characteristics of PV systems exhibit nonlinear behavior, varying dynamically with changes in radiation intensity and temperature. There is a single maximum point on these characteristics, which changes with different radiation and temperature values. The objective of the control strategy is to ensure that the PV system continuously operates at the maximum power point (MPP) to optimize energy extraction.

A DC/DC (Direct Current) converter is integrated with the PV system to regulate its output voltage. Common types of DC/DC converters include buck, boost, and buck-boost converters. Switching components such as MOSFETs and IGBTs (Insulated-Gate Bipolar Transistors) are employed to regulate the converters, with control pulses generated using pulse-width modulation (PWM). The controller's task is to create a pulse width for the PWM stage, in which the pulse width has a value from [0,1].

MPPT algorithms are developed in two forms, namely indirect algorithms and direct algorithms. The direct algorithm is commonly used because it only uses information about voltage and current. Meanwhile, in addition to requiring information about voltage and current, the indirect algorithm also involves information about temperature and radiation parameters as control parameters [3, 11]. Therefore, it is difficult for the indirect algorithm to perform well when radiation and temperature change. The P&O and Incremental Conductance (INC) methods are widely adopted direct approaches due to their simplicity and ease of implementation. However, a common drawback of these algorithms is their tendency to oscillate around the MPP under stable radiation conditions and their inability to effectively track the MPP during rapid radiation changes [2, 12-14]. On the other hand, the performance of the PV system is influenced by parameter uncertainties and external disturbances. In reference [15], a sliding mode controller was developed with a surface formulated based on the MPP. However, a major limitation of this approach is the chattering phenomenon occurring in the system. In reference [16], the authors introduced an MPPT controller utilizing quadratic sliding mode control, but the chattering issue persisted. On the contrary, in reference [17], a type 2 fuzzy and sliding mode-based MMPT controller was proposed, eliminating the chattering phenomenon. However, it did not consider the influence of uncertain parameters and external disturbances. Additionally, MPPT techniques utilizing fuzzy logic and neural networks [18, 19] have also been explored and implemented. However, these algorithms are more complex than the conventional MPPT, a simple and low-cost algorithm. The control strategies mentioned above are utilized to generate pulse width values for PWM, which regulate the switching components of the DC/DC converter.

The control strategies mentioned above are utilized to generate pulse width values for PWM, which regulate the switching components of the DC/DC converter. To enhance control performance, MMPT techniques have been developed employing a two-loop control structure. In this structure, the first loop is responsible for setting the reference voltage, while the second loop adjusts the voltage tracking of the PV system based on the reference. For the first loop, widely used algorithms include P&O and INC. The effectiveness of the tracking process is heavily reliant on the tracking controller’s performance in the second loop. The success of the tracking process is primarily determined by how well the controller functions in the second loop. A suitable control method should be capable of accurately tracking the MPP in various scenarios while handling the system's nonlinearities and uncertainties. In reference [20], the authors introduce an MPPT approach that integrates P&O with a sliding mode controller for photovoltaic applications, where the P&O method operates in the first loop and the sliding mode controller manages the reference voltage tracking in the second loop. However, this controller does not address the uncertain parameters, and the chattering phenomenon has not been eliminated. On the contrary, in reference [21], a control scheme based on P&O and sliding mode controller for PV systems was introduced to ensure robustness against the influence of disturbances and uncertainties of system parameters. However, the chattering phenomenon was not eliminated. In reference [22], the authors used the INC algorithm in the first loop and introduced the terminal sliding controller for the second loop. However, the algorithms [21, 22] did not eliminate the chattering phenomenon. In reference [23], the chattering phenomenon of the sliding controller was eliminated by automatic switching factor adjustment. However, it did not consider the uncertain parameters and external disturbances. In addition, in reference [24], an observer combined with the P&O method and a sliding controller was designed for PV systems. Unfortunately, it did not eliminate the chattering phenomenon.

From the analysis provided, this paper presents an adaptive sliding mode control (ASMC) approach to ensure the PV system consistently functions at the MPP. This is achieved using an observer and a fuzzy controller, where the observer estimates the uncertain components and unknown external disturbances, and the fuzzy controller adjusts the switching coefficient to reduce oscillations in the sliding mode control. The highlights and primary contributions of the paper are outlined below: (1) The ASMC employs a two-loop control scheme to enhance control quality, unlike the direct single-loop control schemes used in references [15-19]. (2) In contrast to references [20-24], the ASMC not only ensures the system consistently operates at the MPP but also eliminates the influence of external disturbances, uncertain parameters, and chattering phenomena.

3.1 P&O method

The P&O method is commonly applied to find the MPP of photovoltaic systems, owing to its straightforwardness and effectiveness. It calculates a reference voltage V_ref, after measuring the output of the photovoltaic system. Figure 4 displays the sequence of steps for the P&O method [20], where ΔV_ref is a positive constant represents the increase or decrease of V_ref in each sampling period determined by comparing the power values from the current and previous sampling periods.

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Figure 4. P&O method diagram

3.2 Adaptive sliding mode controller

3.2.1 Control law design

After determining V_ref through the P&O method, the ASMC control law is responsible for determining the control law for V_pv to track V_ref.

First, define tracking errors as follows: $e_1=V_{p x}-V_{r e f}, e_2=\dot{e}_1$, $e_2=I_L-\dot{V}_{\text {ref }}, \dot{e}_2=\dot{I}_L-\ddot{V}_{\text {ref}}$. The sliding surface is defined by the equation:

$\bar{r}=\kappa_1 e_1+e_2$     (9)

where, $\kappa_1>0$. By differentiating Eq. (9), we obtain:

$\dot{\bar{r}}=\kappa_1 e_2+\dot{e}_2$    (10)

Combining Eq. (8) and Eq. (10), we have

$\dot{\bar{r}}=\kappa_1 e_2+F(\zeta)+\bar{F}(\zeta)+G(\zeta) u+\psi(\zeta)-\ddot{V}_{r e f}$     (11)

By the target $\dot{\bar{r}}=0$, we choose the control law as follows:

$u=u_{t d}+u_s$    (12)

where, $u_a$ is the component that pulls the states to the sliding surface defined as $u_a=G^{-1}(\zeta)\left.(\ddot{V}_{r e f}-F(\zeta)-\bar{F}(\zeta)-\right.$ $\left.\kappa_1 e_2\right)-\hat{\psi}(\zeta), \hat{\psi}(\zeta)$ is the estimate of $\psi(\zeta), u_s$ is the component that keeps the states on the sliding surface and moves towards the origin chosen as $u_s=$ $-G^{-1}(\zeta) \kappa_2 \operatorname{sgn}(\bar{r})$, with $\kappa_2>0$ being the switching coefficient. The stability of the system with control law (12) is proven in the following section. The updated law of the observer is defined as

$\dot{\hat{\psi}}(\zeta)=-\xi \bar{r}$     (13)

where, $\xi$ is a positive constant. The approximation error is defined as $\tilde{\psi}(\zeta)=\psi(\zeta)-\hat{\psi}(\zeta)$.

In the control law (12), if $\kappa_2$ is large, the system will quickly move towards the sliding surface, but the chattering phenomenon will oscillate vigorously, and vice versa. Therefore, $\kappa_2$ is selected by the fuzzy controller to reduce the chattering phenomenon.

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Figure 5. Fuzzy functions for input and output

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Figure 6. Control laws

The controller is designed with a single input and output structure, where the input is the sliding surface and the output is the coefficient $\kappa_2$. Figure 5 presents the fuzzy function for input and output, and Figure 6 depicts the control law. The MAX-MIN composition law is applied, and defuzzification is performed using the centroid method.

3.2.2 Proof of stability

The Lyapunov function is chosen as follows:

$L=\frac{1}{2} \bar{r}^2+\frac{1}{2} \xi \tilde{\psi}^2$     (14)

Taking the derivative Eq. (14) along the trajectory of Eq. (8), we get

$\begin{gathered}\dot{L}=\bar{r} \dot{\bar{r}}+\frac{1}{\xi} \tilde{\psi} \dot{\tilde{\psi}} \\ =\bar{r}\left(\kappa_1 e_2+F+\bar{F}+G u+\psi-\ddot{V}_{r e f}\right)+\frac{1}{\xi} \tilde{\psi} \dot{\tilde{\psi}}\end{gathered}$     (15)

Substituting control law Eq. (12) into Eq. (15) one can obtain:

$\dot{L}=\bar{r}\left(-\kappa_2 \operatorname{sgn}(\bar{r})+(\psi-\hat{\psi})\right)+\frac{1}{\xi} \tilde{\psi} \dot{\tilde{\psi}}$     (16)

$=-\kappa_2|\bar{r}|+\tilde{\psi} \bar{r}+\frac{1}{\xi} \tilde{\psi} \dot{\tilde{\psi}}$

Replace updated law Eq. (13) into Eq. (16) one can obtain:

$\dot{L}=-K_2|\bar{r}|$     (17)

It can be seen that $\dot{L} \leq 0$ when $\kappa_2>0$. Therefore, the closed-loop system is stable according to the Lyapunov criterion. It is easy to see that, $L, \bar{r}, \dot{\vec{r}}, \int_0^t \dot{L} d t=-\kappa_2 \int_0^t|\bar{r}| d t$ are all limited. Apply Barbalat Lemma [30], when $t \rightarrow \infty$, we have $\bar{r} \rightarrow 0$, then $e_1 \rightarrow 0, \dot{e}_1 \rightarrow 0$.

This paper has introduced an ASMC scheme for MMPT in PV systems. The proposed controller consists of a P&O method for generating the reference voltage, an adaptive sliding mode approach to ensure the system tracks the reference voltage, with an observer designed to estimate uncertain components and external disturbances, and a fuzzy controller to reduce the chattering phenomenon. The results of the simulation indicate that the ASMC effectively maintains operation at the MPP, not only ensuring the system always operates at MPP but also ensuring sustainability and significantly reducing chattering. It can be seen that ASMC is a comprehensive solution. However, ASMC has only been verified through simulation and requires both PV voltage and current sensors. In future work, we will test and develop the controller to minimize the measurement sensors.