Backstepping Based Super-Twisting Sliding Mode MPPT Control with Differential Flatness Oriented Observer Design for Photovoltaic System

Abstract

The formulation of a maximum power point tracking (MPPT) control strategy plays a vital role in enhancing the inherent low conversion efficiency of a photovoltaic (PV) module. Keeping in view the nonlinear electrical characteristics of the PV module as well as the power electronic interface, in this paper, a hybrid nonlinear sensorless observer based robust backstepping super-twisting sliding mode control (BSTSMC) MPPT strategy is formulated to optimize the electric power extraction from a standalone PV array, connected to a resistive load through a non-inverting DC–DC buck-boost power converter. The reference peak power voltage is generated via the Gaussian process regression (GPR) based probabilistic machine learning approach that is adequately tracked by the proposed MPPT scheme. A generalized super-twisting algorithm (GSTA) based differential flatness approach (DFA) is used to retrieve all the missing system states. The Lyapunov stability theory is used for guaranteeing the stability of the proposed closed-loop MPPT technique. The Matlab/Simulink platform is used for simulation, testing and performance validation of the proposed MPPT strategy under different weather conditions. Its MPPT performance is further compared with the recently proposed benchmark backstepping based MPPT control strategy and the conventional MPPT strategies, namely, sliding mode control (SMC), proportional integral derivative (PID) control and the perturb-and-observe (P&O) algorithm. The proposed technique is found to have a superior tracking performance in terms of offering a fast dynamic response, finite-time convergence, minute chattering, higher tracking accuracy and having more robustness against plant parametric uncertainties, load disturbances and certain time-varying sinusoidal faults occurring in the system.

1. Introduction

Electrical energy production has been a challenging task throughout history. With the industrialization of countries, the energy demand is growing proportionally. Most of the energy production nowadays comes from depleting fossil fuels causing environmental concerns in terms of greenhouse gas emission, global warming and increased pollution. Furthermore, due to the economic and petroleum crisis nowadays, together with increasing efforts for environmental protection, scientific research has now focused on the development of so-called alternative or renewable energy sources [1,2].

Among different forms of the alternative energy sources, electricity generation from solar energy through photovoltaic (PV) cells is regarded as the fast developing technology due to considerable reduction in its equipment cost. It is a naturally and abundantly available clean energy, distributed all over the earth, and can compete with other sources of energy production [2,3]. Despite all the stated attractive attributes of the PV cells, its energy conversion efficiency is still very low. The PV cell has a nonlinear current-voltage and power-voltage (i.e., $I-V$ and $P-V$, respectively) characteristics that vary considerably with the ambient environmental conditions (i.e., temperature and irradiance). Only, under a uniform solar irradiance, the PV cell exhibits a unique operating point, called the maximum power point (MPP), where the maximum voltage and current (i.e., $V_{MPP}$ and $I_{MPP}$, respectively) occurs. This makes the maximum power extraction from the PV cell quite a challenging task under inconsistent atmospheric conditions. Hence, to maximize the efficiency and to extract and transfer the maximum possible power from the PV cell to a load, a sophisticated control strategy is needed, known as the maximum power point tracking (MPPT). The MPPT strategy matches the load resistance with the source (PV cell) resistance, thus forcing the PV cell to operate on the MPP and ensuring the maximum power extraction, despite ambient atmospheric variations or the load. The PV system operation in the MPPT mode also, indirectly, reduces the total number of PV cells required and, hence, its total cost. [4,5,6,7].

The MPPT algorithm is typically integrated into the power electronic converter serving as a hardware interface between the PV cell (source) and the load. This algorithm continuously alters the duty cycle, d, of the power converter switches and adapts the PV system operating point (MPP or $V_{MPP}$) to the varying atmospheric conditions, thus ensuring the optimal power extraction from it [8,9,10,11].

To maximize the power output of a PV system, conventionally, hill climbing strategies are used. These strategies include a number of variants of the two basic algorithms, such as: Perturb-and-observe (P&O) and incremental conductance (IncCond). Both of these stated strategies try to find the MPP of the PV system by introducing oscillations in the output, even if the MPP is reached. Thus, the overall efficiency of the PV system is reduced. However, the IncCond algorithm causes less oscillations than the P&O. Similarly, under rapidly varying atmospheric conditions, the IncCond algorithm performs better than the P&O technique. However, it needs additional control circuitry for proper operation, thus making its implementation more complex [12].

Owing to the nonlinear nature of the electrical characteristics of the PV cell and power converter, different nonlinear MPPT control strategies have been reported in the available scientific literature. In [13], a conventional backstepping based nonlinear MPPT scheme has been proposed for a standalone PV system. However, a significant steady-state error was observed in the output during MPP tracking. This issue has been addressed in [12] through integral backstepping based nonlinear MPPT algorithm, where the output tracking error was reduced to a minute level due to the integral action. Similarly, another nonlinear robust backstepping based MPPT paradigm has been proposed in [14]. This stated strategy not only dealt efficiently with the simultaneous variation of the temperature and irradiance, but also it offered significant robustness against time-varying sinusoidal faults and parametric uncertainties occurring in the system.

In the context of nonlinear control, the backstepping strategy belongs to the recursive control design. It acquires its name from the recursive nature of the controller design, where the design process starts with an inner scalar equation that steps-back towards the external control input after passing through a chain (or sequence of integrators). Its application is based on designing a nonlinear controller recursively by choosing some of the system state variables as the virtual controllers, followed by designing intermediate control laws for these selected virtual controllers. Its attractive attributes are fast dynamic response, external disturbance rejection, robustness to system parametric uncertainties as well as modeled and unmodeled system dynamics. It has the capability of canceling out all the destabilizing effects (i.e., forces or terms) appearing throughout the domain [15,16].

Another well-established nonlinear MPPT control strategy is the conventional sliding mode control (CSMC). It is a robust control strategy based on the variable-structure control (VSC) theory. The key to CSMC implementation is the reduction of the higher order complex closed-loop system to the first order, namely the sliding variable along with its derivative. Consequently, the plant order is reduced, thus the main control design is focused on the reduced plant dynamics. The main attributes of the CSMC include: Simple implementation, good dynamic response, external disturbance rejection and low sensitivity to (internal) plant parametric uncertainties (or variations). However, the main negative aspect of the first-order CSMC is the high-frequency oscillations in the system states, called the chattering phenomenon, resulting from the switching action of the discontinuous control signal as well as other non-idealities (e.g., hysteresis, time-delays, unmodeled system dynamics etc.) [17,18,19]. The chattering leads to a low control accuracy, increased heat losses in the power electronic circuits, and high wear and tear in case of moving mechanical parts [20].

To attenuate chattering, several nonlinear higher order sliding mode control (HOSMC) strategies can be found in the available literature, such as the super-twisting algorithm (STA) [21]. The STA is a second-order SMC strategy where the control signal appears in the first derivative of the sliding variable. Moreover, unlike other second-order SMC strategies, it is applicable to a system (of any order, in general). Some of the main features of the STA are given as follows [18,19,22,23]:

• It offers a finite-time convergence of the output as well its derivative to the origin.
• It can compensate those perturbations/uncertainties that are Lipschitz
• It requires the information of the output only (the sliding variable)
• It introduces an extra integrator (dynamic extension) to the control structure in such a manner that the discontinuous control term is hidden behind this integrator. Thus, it artificially increases the plant relative degree and generates a continuous control signal, thereby attenuating chattering.

Motivation and Significant Contributions

The main motivation of this article is to formulate an MPPT control strategy for a PV system with minute chattering and at the same time offering a high precision performance under different atmospheric conditions and internal as well as external disturbances.

For this purpose, a hybrid sensorless observer based nonlinear robust backstepping super-twisting sliding mode control (BSTSMC) MPPT paradigm is proposed in this article for delivering an optimum power from a PV array to a resistive load through a DC–DC converter. The reference peak power voltage is generated via the Gaussian process regression (GPR) based probabilistic machine learning approach that is adequately tracked by the proposed MPPT scheme. A generalized super-twisting algorithm (GSTA) based differential flatness approach (DFA) is used to observe all the missing system states. The Lyapunov stability theory is used for guaranteeing the stability of the proposed closed-loop MPPT technique. The MPPT performance of the proposed control strategy is simulated, tested, validated and compared with the recently proposed benchmark backstepping [13] based MPPT strategy and conventional SMC, PID and P&O based MPPT techniques, in Matlab/Simulink, under simultaneous variation of the temperature, irradiance and load. It is observed that the proposed BSTSMC based MPPT technique offers a superior tracking performance in terms of offering a fast dynamic response, finite-time convergence, minute chattering, higher tracking accuracy and having more robustness against plant parameters perturbations, load disturbances and certain time-varying sinusoidal faults occurring in the system.

As per the available scientific literature, following are the significant contributions made by this research article:

• To the best of the authors knowledge, model based backstepping STA, for MPPT control of the PV system, has never been applied before.
• The authors also claim that the DFA based observer using GSTA has never been implemented for states retrieval before.
• The GPR based voltage generation trajectory also contributes to the scientific literature.

The entire article is organized as follows: Section 1 covers the introduction and background literature review to this article. Section 2 is dedicated to the PV system modeling. Section 3 discusses the PV array reference peak power voltage generation. Section 4 describes the averaged state-space modeling of the DC–DC converter. Section 5 presents the differential flatness based system states observer design. Section 6 is about the proposed MPPT control scheme design. Section 7 discusses the performance validation of the proposed MPPT scheme in Matlab/Simulink. Finally, Section 8 presents concluding remarks to this article.

2. Phtovoltaic Array Mathematical Modeling

A PV cell generates electricity (DC) from sunlight using the photoelectric effect. For getting an increased voltage and current output, PV cells are connected in series and parallel combination, respectively. Series connection of PV cells forms a PV module or panel, series connection of PV modules constitutes a PV string, while parallel connection of PV strings makes a PV array.

Depending on their complexity and accuracy, a PV cell can be represented by several different equivalent circuit models, including: Single-diode, two-diode and three-diode equivalent circuit model. Taking into account its simplicity and reasonable accuracy, the most commonly used equivalent circuit model is the single-diode model of the PV cell [24], as illustrated in Figure 1. Where, $R_s$ and $R_p$ indicate the PV cell equivalent series and shunt resistances, respectively. Normally, $R_s<<R_p$, where $R_s$ exists due to the metallic leads resistances, while $R_p$ due to the leakage current of the $PN$-junction. Furthermore, $I_{ph}$, $I_D$, $I_p$, I and V are the photon-generated current, diode current, current through the equivalent shunt-resistance, cell output current and cell output voltage, respectively. Mathematically, the PV cell output current can be worked out from Figure 1 by applying Kirchhoff’s current law at the junction, as follows:

$$I=I_{ph}-\underset{I_D}{\underbrace{I_0\left[e^{\frac{q}{AkT}\left(V+I{R}_s\right)}-1\right]}}-\underset{I_p}{\underbrace{\frac{V+I{R}_s}{R_p}}}$$

(1)

In Equation (1), $I_D$ indicates the Shockley diode equation, $I_0$ represents the diode leakage (or reverse saturation) current, q equals the electron charge ( $1.6\times {10}^{-19}$ $\mathrm{C}$), k is the Boltzmann constant ($1.38\times {10}^{-23}$ $\mathrm{J}$/$\mathrm{K}$), T represents the PN-junction temperature (in Kelvin) and A denotes the diode ideality factor (or constant), where usually: $1\le A\le 1.50$.

For practical applications, many PV cells are connected in series and parallel combination to obtain higher voltage and current output, respectively. Suppose, $N_p$ and $N_s$ be the number of parallel connected PV modules and series connected PV cells, respectively. Then, the mathematical relation between the PV array output current, $i_{pv}$, and output voltage, $v_{pv}$, can be expressed as follows [25]:

$$\begin{array}{c}\hfill i_{pv}=N_p{I}_{ph}-N_p{I}_0\left[e^{\frac{q}{AkT}\left(\frac{v_{pv}}{N_s}+\frac{i_{pv}{R}_s}{N_p}\right)}-1\right]-\frac{N_p}{R_p}\left(\frac{v_{pv}}{N_s}+\frac{i_{pv}{R}_s}{N_p}\right)\end{array}$$

(2)

In this work, a user-defined PV array with a total power output of $24,880$ $\mathrm{W}$ is employed. Overall, it has 16 PV modules, where 4 modules are connected in a series combination in each string, and then 4 strings are connected in a parallel combination to constitute the complete PV array.

Table 1. Complete parameters of the PV system.

Name	Quantity	Value
PV Array	Series cells/PV module	72
	Parallel cells/PV module	1
	No. of modules/PV string	4
	No. of strings/PV array	4
	No. of modules/PV array	16
	Single module output power	$\left.\begin{array}{c}1555\mathrm{W}\hfill \\ 24.880\mathrm{W}\hfill \\ 102.60\mathrm{V}\hfill \\ 15.16\mathrm{A}\hfill \\ 165.80\mathrm{V}\hfill \\ 17.56\mathrm{A}\hfill \end{array}\right\}@STC$
	PV array output power
	Module voltage at MPP
	Module current at MPP
	Module open-circuit voltage
	Module short-circuit current
DC–DC Converter	Input capacitor, $C_1$	1 $\mathrm{m}$$\mathrm{F}$
	Output capacitor, $C_2$	48 $\mathsf{\mu }$$\mathrm{F}$
	Inductor, L	2 $\mathrm{m}$$\mathrm{H}$
	MOSFET Switching frequency, $f_{sw}$	5 $\mathrm{k}$$\mathrm{Hz}$
	Load resistance, $R_L$	50, 40, 60 $\mathsf{\Omega }$
Controller (BSTSMC)	Constant, ${\kappa }_1$	100
	Constant, ${\kappa }_2$	100,000
	Constant, ${\kappa }_3$	4
	Constant, ${\kappa }_4$	8

describes various electrical and physical parameters of the PV array, under standard test conditions (STC), i.e., 25 ${}^{\circ }\mathrm{C}$ and 1000 $\mathrm{W}$/$\mathrm{m}$${}^2$. Moreover, the PV array electrical characteristics ($I-V$ and $P-V$), are‘depicted in Figure 2.

3. Reference Voltage Generation via Gaussian Process Regression

The MPPT controller must continuously track the PV array output voltage, $v_{pv}$, to its reference, $V_{MPP}$ or $v_{pv}^r$ for delivering the maximum available power to the load. Because, the reference voltage varies with the inconsistent weather conditions, that is, the temperature, (T, ${}^{\circ }\mathrm{C}$), and solar irradiance, (G, $\mathrm{W}$/$\mathrm{m}$${}^2$). Different approaches have been used to estimate/learn the PV array reference peak power voltage, such as regression plane [12,13], Takagi-Sugeno-Kang based adaptive NeuroFuzzy Inference System (ANFIS) [14] and Gaussian process regression (GPR) based learning approaches [26]. In this article, a GPR based probabilistic machine learning procedure is employed for $V_{MPP}$ estimation/learning of the PV array.

Definition 1.

A Gaussian process (GP) can be defined as the collection of random variables, where any finite number of those random variables have a joint Gaussian distribution [27].

The GPs put a prior over functions in order to obtain posterior over functions, for some data being observed. When some random function $f\left(x\right)$ follows a GP, it is indicated by a combination of a mean function and a covariance (or kernel) function, as follows:

$$f\left(\mathbf{x}\right)\sim \mathcal{G}\mathcal{P}\left(\mu \left(\mathbf{x}\right),\mathcal{K}\left(\mathbf{x},{\mathbf{x}}^{\prime }\right)\right)$$

where $f\left(\mathbf{x}\right)$ and $\mathcal{GP}$ indicate a real process and a Gaussian process, respectively, $\mathbf{x}$ and ${\mathbf{x}}^{\prime }$ are the arbitrary input variables (normally represented as vectors, because there are many input variables), $\mu \left(\mathbf{x}\right)=\mathbb{E}\left[\right(\mathbf{x}\left)\right]$ represents the mean function, $\mathbb{E}\left[\right(\mathbf{x}\left)\right]$ is the expectation of $\left(\mathbf{x}\right)$ and $\mathcal{K}(\mathbf{x},{\mathbf{x}}^{\prime })=\mathbb{E}\left[(f\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right))\left(f\left({\mathbf{x}}^{\prime }\right)-\mu \left({\mathbf{x}}^{\prime }\right)\right)\right]$ describes the covariance (or kernel) function evaluated at $\mathbf{x}$ and ${\mathbf{x}}^{\prime }$, which is sometimes also known as the kernel trick.

Definition 2.

A function $\mathcal{K}$, capable of mapping a pair of input arguments $\mathbf{x}$ and ${\mathbf{x}}^{\prime }$ into $\mathbb{R}$ (real numbers) is known as a kernel [27].

The covariance function encodes all of the assumptions about the function to be learnt, thus making it a crucial ingredient in GP predictor. Any function could be selected as a valid covariance function, as long as its resulting covariance matrix remains positive semi-definite. Nonetheless, in some learning processes, the input arguments are not necessarily vectors. For such scalar inputs, the most commonly used kernel function is the squared exponential kernel (${\mathcal{K}}_{SE}$). It is expressed as follows:

$$\begin{array}{cc}\hfill & cov\left(f\left(\mathbf{x}\right),f\left({\mathbf{x}}^{\prime }\right)\right)={\mathcal{K}}_{SE}\left(\mathbf{x},{\mathbf{x}}^{\prime }\right)=exp\left(-\frac{r^2}{2{\ell }^2}\right)=exp\left(-\frac{{\left|\mathbf{x}-{\mathbf{x}}^{\prime }\right|}^2}{2{\ell }^2}\right)\hfill \end{array}$$

(3)

where parameters r and ℓ represent the radial basis function and characteristic length-scale, respectively.

It can be seen in Equation (3) that the covariance between the outputs is expressed as a function of the inputs. It means that the covariance, for this particular covariance function, reaches almost unity between the output variables, if the corresponding inputs are close enough. On the other hand, it decreases if the distance between the inputs increases.

In this article, a GPR based $V_{MPP}$ learning process is carried out in Matlab/Simulink using the Regression Learner App. In this process, GPR learns three different variables at a time, that is, it takes the two atmospheric variables (i.e., temperature and solar irradiance) as two input arguments (i.e., known predictors or data) and then maps these variables to their corresponding $V_{MPP}$ (i.e., known response). A trained GPR model, based on the squared-exponential kernel function, is obtained that renders new predicted responses for any new input data.

For GPR based $V_{MPP}$ learning process, the $V_{MPP}$ data is recorded by entering the user-defined PV array specifications, given in Table 1, in Matlab/Simulink. During this process, the temperature is perturbed from 0 ${}^{\circ }\mathrm{C}$ to 75 ${}^{\circ }\mathrm{C}$ in uniform steps of 1 ${}^{\circ }\mathrm{C}$. On the other hand, the solar irradiance is perturbed from 1 $\mathrm{W}$/$\mathrm{m}$${}^2$ to 1000 $\mathrm{W}$/$\mathrm{m}$${}^2$, in uniform steps of 1 $\mathrm{W}$/$\mathrm{m}$${}^2$. As a result, about 76,000 $V_{MPP}$ data points are recorded. The concept of GPR based $V_{MPP}$ learning workflow is depicted in Figure 3. Moreover, the predicted response of the trained GPR model against the true (or actual) response is depicted in Figure 4. As, the predicted response closely matches the true response (i.e., the diagonal line), it indicates that the prediction error is very small and the learning process renders a good trained GPR model. This trained GPR model then generates the reference voltage, $V_{MPP}$, during simulation for any combination of input temperature and irradiance levels that is tracked by the MPPT controller.

4. State-Space Averaged Discrete-Time Bilinear Equivalent Circuit Modeling of the Cascaded Non-Inverting DC-DC Buck-Boost Power Converter

For operating the PV array at its MPP, irrespective of inconsistent atmospheric conditions, the MPPT control algorithm is integrated into the power electronic converter serving as a hardware interface between the PV array (source) and the load. This algorithm constantly adjusts the duty cycle, d, of the power converter switches and adapts the PV array operating point ($V_{MPP}$) to the varying atmospheric conditions, thus ensuring the optimal power extraction from it [8].

Several well-known versions of the DC–DC power converters have been employed as the hardware interface between the PV array and the load, specifically, conventional buck-boost converter, Cuk converter, and single-ended primary inductor (SEPIC) converter. However, all the stated converter variants are prone to high switching stresses, and consequently lower efficiency. Moreover, the output voltage polarity is reversed with respect to the input voltage polarity (or the output voltage polarity is negative with respect to the common ground), specifically in case of the conventional buck-boost converter and the Cuk converter. These stated issues are resolved by using a cascaded non-inverting DC–DC buck-boost (CCNI-BuBo) converter, which is a cascaded combination of a buck converter and a boost converter. Its output voltage is either less than, or higher than in magnitude than its input voltage. Moreover, the output voltage polarity is the same with respect to the input voltage polarity. It has two controllable switches ($S_1$ and $S_2$), an inductor (L) and two capacitors ($C_1$ and $C_2$) in its circuit. It can be operated in three separate modes, that is, the buck mode (when $S_1$: Switching and $S_2$: OFF), the boost mode (when $S_1$: ON and $S_2$: Switching) and the buck-boost mode (when both $S_1$ and $S_2$: Simultaneously switching) [28].

Conventionally, the switching power converters have time-variant nonlinear response. So, to give a better physical insight into a converter operation and properties, different equivalent circuit modeling techniques are used. If the accuracy is not a big concern, a converter can be approximately represented by its continuous, time-invariant and linear equivalent circuit model using the small-signal approximation that are easier to analyze. Since, a switching power converter basically behaves as a sampled system, hence, for a higher level of accuracy it must be represented by its state-space averaged discrete-time bilinear equivalent circuit model that includes the product of the duty cycle, d, and system states [29,30].

In this research work, the state-space averaged discrete-time bilinear equivalent circuit modeling technique is employed to develop an equivalent circuit model for the CCNI-BuBo converter serving as power electronic interface between the PV array and the resistive load, $R_L$, as illustrated in Figure 5. Different significant parameters of the converter are expressed in Table 1. Let the converter operates in the continuous conduction mode (CCM), then, there are two different switching modes of operation for the CCNI-BuBo converter. That is, Mode 1: Both $S_1$ and $S_2$ are ON, while both $D_1$ and $D_2$ remain OFF. Mode 2: Both $S_1$ and $S_2$ remain OFF, while both $D_1$ and $D_2$ are ON. That is, the converter is operated in the buck-boost mode. Now, the state-space model for operation in Mode 1 of the stated converter, in compact vector-matrix form, can be expressed as follows:

$$\begin{array}{cc}\hfill & \underset{\dot{\mathbf{x}}}{\underbrace{\left[\begin{array}{c}{\dot{v}}_{pv}\\ \\ {\dot{i}}_L\\ \\ {\dot{v}}_{C_2}\end{array}\right]}}=\underset{{\mathbb{A}}_{\mathbf{on}}}{\underbrace{\left[\begin{array}{ccc}0& -\frac{1}{C_1}& 0\\ \\ \frac{1}{L}& 0& 0\\ \\ 0& 0& -\frac{1}{R_L{C}_2}\end{array}\right]}}\underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{v}_{pv}\\ \\ i_L\\ \\ v_{C_2}\end{array}\right]}}+\underset{{\mathbb{B}}_{\mathbf{on}}}{\underbrace{\left[\begin{array}{c}\frac{1}{C_1}\\ \\ 0\\ \\ 0\end{array}\right]}}\underset{\mathbf{w}}{\underbrace{\left[\begin{array}{c}{i}_{pv}\end{array}\right]}}\hfill \end{array}$$

(4)

$$\underset{{\mathbf{y}}_1}{\underbrace{\left[v_{pv}\right]}}=\underset{{\mathbb{C}}_{\mathbf{on}}}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}}\underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{v}_{pv}\\ i_L\\ v_{C_2}\end{array}\right]}}$$

(5)

Similarly, the state-space model for operation in Mode 2 of the stated converter, in compact vector-matrix form, can be expressed as follows:

$$\begin{array}{cc}\hfill & \underset{\dot{\mathbf{x}}}{\underbrace{\left[\begin{array}{c}{\dot{v}}_{pv}\\ \\ {\dot{i}}_L\\ \\ {\dot{v}}_{C_2}\end{array}\right]}}=\underset{{\mathbb{A}}_{\mathbf{off}}}{\underbrace{\left[\begin{array}{ccc}0& 0& 0\\ \\ 0& 0& -\frac{1}{L}\\ \\ 0& \frac{1}{C_2}& -\frac{1}{R_L{C}_2}\end{array}\right]}}\underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{v}_{pv}\\ \\ i_L\\ \\ v_{C_2}\end{array}\right]}}+\underset{{\mathbb{B}}_{\mathbf{off}}}{\underbrace{\left[\begin{array}{c}\frac{1}{C_1}\\ \\ 0\\ \\ 0\end{array}\right]}}\underset{\mathbf{w}}{\underbrace{\left[\begin{array}{c}{i}_{pv}\end{array}\right]}}\hfill \end{array}$$

(6)

$$\underset{{\mathbf{y}}_2}{\underbrace{\left[v_{pv}\right]}}=\underset{{\mathbb{C}}_{\mathbf{off}}}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}}\underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{v}_{pv}\\ i_L\\ v_{C_2}\end{array}\right]}}$$

(7)

In Equations (4)–(7), ${\mathbb{A}}_{\mathbf{on}}$ and ${\mathbb{A}}_{\mathbf{off}}$ are the system matrices, ${\mathbb{B}}_{\mathbf{on}}$ and ${\mathbb{B}}_{\mathbf{off}}$ are the input column vectors and ${\mathbb{C}}_{\mathbf{on}}$ and ${\mathbb{C}}_{\mathbf{off}}$ are the output row vectors for Mode 1 and 2, respectively. Moreover, $\mathbf{x}$ is the state-variable vector indicating the input and output capacitor voltages and inductor current, $\mathbf{w}$ is the input disturbance vector representing the source (PV array) current, and $\mathbf{y}$ is the output vector denoting the PV array output voltage.

Now taking the weighted averages of ${\mathbb{A}}_{\mathbf{on}}$ and ${\mathbb{A}}_{\mathbf{off}}$, ${\mathbb{B}}_{\mathbf{on}}$ and ${\mathbb{B}}_{\mathbf{off}}$, and ${\mathbb{C}}_{\mathbf{on}}$ and ${\mathbb{C}}_{\mathbf{off}}$ with an appropriate duty ratio, d, as follows:

$$\left.\begin{array}{c}\hfill \mathbb{A}=\left(\mathbf{d}\right){\mathbb{A}}_{\mathbf{on}}+(\mathbf{1}-\mathbf{d}){\mathbb{A}}_{\mathbf{off}}\\ \hfill \mathbb{B}=\left(\mathbf{d}\right){\mathbb{B}}_{\mathbf{on}}+(\mathbf{1}-\mathbf{d}){\mathbb{B}}_{\mathbf{off}}\\ \hfill \mathbb{C}=\left(\mathbf{d}\right){\mathbb{C}}_{\mathbf{on}}+(\mathbf{1}-\mathbf{d}){\mathbb{C}}_{\mathbf{off}}\end{array}\right\}$$

(8)

Now, the state-space averaged discrete-time bilinear equivalent circuit model of the CCNI-BuBo converter can be expressed as follows:

$$\begin{array}{cc}\hfill & \underset{\dot{\mathbf{x}}}{\underbrace{\left[\begin{array}{c}{\dot{v}}_{pv}\\ \\ {\dot{i}}_L\\ \\ {\dot{v}}_{C_2}\end{array}\right]}}=\underset{{\mathbb{A}}_{off}={\mathbb{A}}_{\mathbf{c}}}{\underbrace{\left[\begin{array}{ccc}0& 0& 0\\ \\ 0& 0& -\frac{1}{L}\\ \\ 0& \frac{1}{C_2}& -\frac{1}{R_L{C}_2}\end{array}\right]}}\underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{v}_{pv}\\ \\ i_L\\ \\ v_{C_2}\end{array}\right]}}+\underset{{\mathbb{B}}_{cb}}{\underbrace{\left(\underset{{\mathbb{A}}_{\mathbf{on}}}{\underbrace{\left[\begin{array}{ccc}0& -\frac{1}{C_1}& 0\\ \\ \frac{1}{L}& 0& 0\\ \\ 0& 0& -\frac{1}{R_L{C}_2}\end{array}\right]}}-\underset{{\mathbb{A}}_{\mathbf{off}}}{\underbrace{\left[\begin{array}{ccc}0& 0& 0\\ \\ 0& 0& -\frac{1}{L}\\ \\ 0& \frac{1}{C_2}& -\frac{1}{R_L{C}_2}\end{array}\right]}}\right)}}\hfill \\ & \underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{v}_{pv}\\ \\ i_L\\ \\ v_{C_2}\end{array}\right]}}d\left(t\right)+\underset{{\mathbb{B}}_{\mathbf{c}}}{\underbrace{\left(\underset{{\mathbb{B}}_{\mathbf{on}}\mathbf{w}}{\underbrace{\left[\begin{array}{c}\frac{i_{pv}}{C_1}\\ \\ 0\\ \\ 0\end{array}\right]}}-\underset{{\mathbb{B}}_{\mathbf{off}}\mathbf{w}}{\underbrace{\left[\begin{array}{c}\frac{i_{pv}}{C_1}\\ \\ 0\\ \\ 0\end{array}\right]}}\right)}}d\left(t\right)+\underset{{\mathbb{B}}_{\mathbf{off}}}{\underbrace{\left[\begin{array}{c}\frac{1}{C_1}\\ \\ 0\\ \\ 0\end{array}\right]}}\underset{\mathbf{w}}{\underbrace{\left[\begin{array}{c}{i}_{pv}\end{array}\right]}}\hfill \end{array}$$

(9)

Since, ${\mathbb{A}}_{\mathbf{on}}\ne {\mathbb{A}}_{\mathbf{off}}$, hence the state-space averaged discrete-time equivalent circuit model of the CCNI-BuBo converter, expressed in Equation (9), is bilinear. It can be simplified as follows:

$$\begin{array}{cc}\hfill & \dot{\mathbf{x}}={\mathbb{A}}_{\mathbf{off}}\mathbf{x}+\underset{{\mathbb{B}}_{\mathbf{cb}}}{\underbrace{\left({\mathbb{A}}_{\mathbf{on}}-{\mathbb{A}}_{\mathbf{off}}\right)}}\mathbf{x}\mathbf{d}+\underset{{\mathbb{B}}_{\mathbf{u}}}{\underbrace{{\mathbb{B}}_{\mathbf{c}}\mathbf{d}\left(\mathbf{t}\right)}}+\underset{{\mathbb{B}}_{\mathbf{w}}}{\underbrace{{\mathbb{B}}_{\mathbf{off}}\mathbf{w}}}\hfill \end{array}$$

(10)

Equation (10) is in the form of a standard bilinear continuous-time system, where $d\left(t\right)=u\left(t\right)$ indicates the input to the system, ${\mathbb{B}}_{\mathbf{cb}}$ is the matrix of the bilinear terms ($\mathbf{x},\mathbf{d}$), ${\mathbb{B}}_{\mathbf{w}}$ is the matrix of input disturbances and ${\mathbb{B}}_{\mathbf{u}}$ is the matrix of control inputs.

Let, $v_{pv}=v_{C_1}=x_1$, $i_L=x_2$, and $v_{C_2}=v_R=x_3$. These notations will be used in the forthcoming sections. Different significant parameters of the CCNI-BuBo DC–DC converter are specified in Table 1.

5. Differential Flatness Based States Observer Design

Usually all the system state variables are available during the implementation of most of the control methodologies. However, some technical and economical constraints may inhibit the availability of the system states. An observer (or differentiator) can be employed for estimating the derivative of a missing/non-measurable system state variable. The main drawback of a (conventional) differentiator is that it increases the high frequency gain. Furthermore, a pure differentiator is not proper (or causal), and in case of a disturbance (e.g., a change or spike or noisy environment), it causes a theoretically infinite control signal. This phenomenon is termed as the chaos in the scientific literature. In general, a differentiator may either be exact or robust alone, but not both simultaneously. It requires a trade-off between exactness and robustness to simultaneously offer both with respect to the input noises and possible measurement errors [31]. For this reason, in this article, both the essential features are integrated into a single differentiator that accurately estimates the system state variables. This stated differentiator is based on a generalized STA (GSTA) that uniformly demonstrates robustness as well as exactness with a finite-time convergence [32].

Such that, for an $i^{th}$ differentiator, an estimation (or observation) error can be defined as follows:

$${\zeta }_1=x_i-{\widehat{x}}_i$$

(11)

where $x_i$ and ${\widehat{x}}_i$ are the actual and the estimated (or observed) values of x, respectively. Moreover, the observed output states of the differentiator, in compact vector-matrix notation, can be expressed as follows:

$$\left[\begin{array}{c}{\widehat{\dot{x}}}_i\\ {\widehat{\dot{x}}}_{i+1}\end{array}\right]=\left[\begin{array}{c}-c_i{\varphi }_i\left({\zeta }_i\right)\\ -c_{i+1}{\varphi }_{i+1}\left({\zeta }_i\right)\end{array}\right]\left[\begin{array}{c}{\widehat{x}}_{i+1}\\ 0\end{array}\right]$$

(12)

where, $i=1,2,3$, ${\widehat{x}}_{i+1}$ is the estimated value of $x_{i+1}$, and $c_i$ and $c_{i+1}$ are the positive design constants. Since, the system under consideration possesses three state variables ($x_1$, $x_2$ and $x_3$), hence, using the stated strategy in Equation (12) ${\widehat{\dot{x}}}_1$, ${\widehat{\dot{x}}}_2$ and ${\widehat{\dot{x}}}_3$ are obtained from a set of three differentiators, respectively.

The terms, ${\varphi }_i\left({\zeta }_i\right)$ and ${\varphi }_{i+1}\left({\zeta }_i\right)$, appearing in Equation (12), are defined as follows:

$${\varphi }_i\left({\zeta }_i\right)={\left|{\zeta }_i\right|}^{\frac{1}{2}}sign\left({\zeta }_i\right)+\lambda {\left|{\zeta }_i\right|}^{\frac{3}{2}}sign\left({\zeta }_i\right)$$

(13)

and

$${\varphi }_{i+1}\left({\zeta }_i\right)=\frac{1}{2}sign\left({\zeta }_i\right)+2\lambda {\zeta }_i+\frac{3}{2}{\lambda }^2{\left|{\zeta }_i\right|}^{2}sign\left({\zeta }_i\right)$$

(14)

where, $\lambda \ge 0$ is a scalar quantity. Putting $\lambda =0$ in Equation (13) and (14) recovers the standard robust exact differentiator, via the SMC technique, as proposed in [31]. The higher-degree terms, that is, ${\left|{\zeta }_i\right|}^{\frac{3}{2}}sign\left({\zeta }_i\right)$ and ${\left|{\zeta }_i\right|}^{2}sign\left({\zeta }_i\right)$, provide the differentiator with a uniform convergence. It means that the convergence time of the differentiator will be bounded by a constant. Moreover, it will be independent of any initial conditions.

Note that the tuning parameters ($c_1$, $c_2$ and $\lambda$) of the three GSTA based differentiators and their convergence analysis, using standard test input signals are given in

Table A1. Tuning parameters of the differentiators (from left to right in Figure 5).

Name	Constant	Value
Differentiator-1	Constant, $c_1$	200
	Constant, $c_2$	120
	Constant, $\lambda$	50
Differentiator-2	Constant, $c_1$	150
	Constant, $c_2$	900
	Constant, $\lambda$	8
Differentiator-3	Constant, $c_1$	15
	Constant, $c_2$	35
	Constant, $\lambda$	170

and Figure A1 and Figure A2, in Appendix A, respectively.

Differential Flatness Approach

The flatness concept has been derived from the differential algebra. In nonlinear control system theory, a flat system is the one that is equivalent to a linear system. In other words, the system dynamics render the ability to support an accurate linearization. This linearization process is supported by a special dynamic feedback mechanism called the endogenous feedback. A system satisfying the flatness property is termed as a differentially-flat system or simply flat system. One major property of a flat system is that the system states and the input variables can be written directly, (without requiring any integration of a differential equation), in terms of a particular set of variables called the flat (or linearized) outputs, along with a finite number of their derivatives [33]. In other words, without needing any integration, all the system state variables and inputs can be extracted from the flat outputs.

Once ${\widehat{\dot{x}}}_1$, ${\widehat{\dot{x}}}_2$ and ${\widehat{\dot{x}}}_3$ are obtained from a set of three differentiators, in the next step, the differential flatness approach (DFA) is applied on these stated differentiators to recover the (actual) missing system states $x_{2f}$ and $x_{3f}$ from ${\widehat{\dot{x}}}_1$, ${\widehat{\dot{x}}}_2$, respectively. The implementation of the DFA is illustrated in Figure 5.

If

$$\left\{\begin{array}{c}{v}_{pv}=v_{C_1}=x_1\hfill \\ i_L=x_2\hfill \\ v_{C_2}=v_{R_L}=x_3\hfill \\ d=u\hfill \end{array}\right.$$

Equation (9) can be re-written as follows:

$$\left\{\begin{array}{c}{\displaystyle {\dot{x}}_1=\frac{-u{x}_2}{C_1}+\frac{i_{pv}}{C_1}}\hfill \\ \\ {\displaystyle {\dot{x}}_2=\frac{u{x}_1}{L}+\frac{\left(u-1\right)x_3}{L}}\hfill \\ \\ {\displaystyle {\dot{x}}_3=\frac{\left(1-u\right)x_2}{C_2}-\frac{x_3}{R_L{C}_2}}\hfill \end{array}\right.$$

Then, the DFA can be applied by rewriting the first two equations of the previous set of equations, as follows:

$$x_{2f}=\frac{C_1}{u}\left(-{\widehat{\dot{x}}}_1+\frac{i_{pv}}{C_1}\right)$$

(15)

and

$$x_{3f}=\frac{L}{\left(u-1\right)}\left(-\frac{u{\widehat{x}}_1}{L}+{\widehat{\dot{x}}}_2\right)$$

(16)

Now, these two (actual) missing system states (i.e., $x_{2f}$ and $x_{3f}$), recovered through the DFA, and called the flat or linearized (output) states, are applied as inputs to the differentiators. The Matlab/Simulink implementation of the DFA based states observer design along with the proposed MPPT strategy is illustrated in Figure 6 and Figure 7.

6. Backstepping Based Super-Twisting Sliding Mode MPPT Control Design

This section covers the formulation of the nonlinear BSTSMC based MPPT paradigm for maximizing the power extraction from the PV array. Note that this design procedure uses the observed values of the system state variables (i.e., ${\widehat{x}}_1$, ${\widehat{x}}_2$ and ${\widehat{x}}_3$) instead of the actual (or true) states. The proposed MPPT controller constantly adjusts the duty cycle, d, of the CCNI-BuBo converter switches and adapts the PV system operating point (MPP or $V_{MPP}$) to the varying atmospheric conditions, thus ensuring the optimal power extraction from it.

The proposed BSTSMC law is composed of the backstepping based equivalent control law, $u_{eq}$, and the super-twisting sliding mode based discontinuous control law, $u_{disc}$.

6.1. The Backstepping Based Equivalent Control Law

The backstepping based equivalent control law is designed in the following two step:

Step 1:

The proposed control system design is initiated by defining the PV array output voltage tracking error, ${ϵ}_1$, as follows:

$${ϵ}_1={\widehat{v}}_{pv}-v_{pv}^r={\widehat{x}}_1-x_1^r$$

(17)

In Equation (17), ${\widehat{x}}_1$ is the PV array observed output voltage and $x_1^r=v_{pv}^r$ is the reference (or desired) output voltage, that must be tracked by ${\widehat{x}}_1$. The goal is to drive the error ${ϵ}_1$ to the origin (equilibrium point), O, asymptotically.

Differentiating Equation (17) with respect to time, and substituting ${\dot{v}}_{pv}={\widehat{\dot{x}}}_1$ from Equation (9), it yields:

$${\dot{ϵ}}_1={\widehat{\dot{v}}}_{pv}-{\dot{v}}_{pv}^r={\widehat{\dot{x}}}_1-{\dot{x}}_1^r=-\frac{u{\widehat{x}}_2}{C_1}+\frac{i_{pv}}{C_1}-{\dot{x}}_1^r$$

(18)

For guaranteeing the convergence of the error ${ϵ}_1$ to the equilibrium point, selecting a Lyapunov function candidate, $V_{f1}$, that must satisfy three conditions, namely: (i) $V_{f1}$ must be positive definite (ii) $V_{f1}$ must be radially unbounded, and (iii) $V_{f1}$ must have a negative definite time derivative, in order to guarantee the local asymptotic stability of the system [34,35].

The selected Lyapunov function, $V_{f1}$, along with its time derivative, ${\dot{V}}_{f1}$, are expressed in Equation (19) and (20), respectively, as follows:

$$V_{f1}=\frac{1}{2}{ϵ}_1^2$$

(19)

and

$${\dot{V}}_{f1}={ϵ}_1{\dot{ϵ}}_1={ϵ}_1\left(-\frac{u{\widehat{x}}_2}{C_1}+\frac{i_{pv}}{C_1}-{\dot{x}}_1^r\right)$$

(20)

For ${\dot{V}}_{f1}$ to be negative definite, the following condition must be satisfied:

$$\left(-\frac{u{\widehat{x}}_2}{C_1}+\frac{i_{pv}}{C_1}-{\dot{x}}_1^r\right)=-{\kappa }_1{ϵ}_1$$

(21)

where ${\kappa }_1$ is a positive design constant.

Substituting Equation (21) into (20) yields:

$${\dot{V}}_{f1}={ϵ}_1{\dot{ϵ}}_1={ϵ}_1\left(-\frac{u{\widehat{x}}_2}{C_1}+\frac{i_{pv}}{C_1}-{\dot{x}}_1^r\right)=-{\kappa }_1{ϵ}_1^2$$

(22)

Suppose the second state of the system, that is, the inductor current, ${\widehat{i}}_L={\widehat{x}}_2$, be a virtual control input [15,16]. Then the stabilization function, say $\beta$, that serves as a reference (or desired) current for the inductor current, ${\widehat{x}}_2$, can be obtained by equating Equations (20) and (22) and then deducing ${\widehat{x}}_2=\beta$ as follows:

$$\beta =\frac{1}{u}\left(C_1{\kappa }_1{ϵ}_1+i_{pv}-C_1{\dot{x}}_1^r\right)$$

(23)

Step 2:

To track ${\widehat{x}}_2$ to its reference $\beta$, another error, ${ϵ}_2$, is defined as follows:

$$\left.\begin{array}{c}\hfill {ϵ}_2={\widehat{x}}_2-\beta \\ \hfill \mathrm{or}{\widehat{x}}_2={ϵ}_2+\beta \end{array}\right\}$$

(24)

Substituting ${\widehat{x}}_2$, from Equation (24) into (18) and then simplifying by substituting $\beta$ from Equation (23), it yields:

$${\dot{ϵ}}_1=-\frac{u({ϵ}_2+\beta )}{C_1}+\frac{i_{pv}}{C_1}-{\dot{x}}_1^r=-{\kappa }_1{ϵ}_1-\frac{u{ϵ}_2}{C_1}$$

(25)

Substituting Equation (25) into (20), it yields:

$${\dot{V}}_{f1}=-{\kappa }_1{ϵ}_1^2-\frac{u{ϵ}_1{ϵ}_2}{C_1}$$

(26)

Applying the quotient rule of derivatives, calculating the time derivative of Equation (23) and simplifying by substituting $\beta$ from Equation (23) and $\dot{{ϵ}_1}$ from Equation (25), it gives:

$$\dot{\beta }=\frac{1}{u}\left[-C_1{\kappa }_1^2{ϵ}_1+{\dot{i}}_{pv}-C_1{\ddot{x}}_1^r\right]-{\kappa }_1{ϵ}_2-\frac{\dot{u}\beta }{u}$$

(27)

Taking the time derivative of ${ϵ}_2$ in Equation (24) gives: ${\dot{ϵ}}_2={\widehat{\dot{x}}}_2-\dot{\beta }$, and substituting ${\widehat{\dot{x}}}_2={\widehat{\dot{i}}}_L$ and $\dot{\beta }$ from Equations (9) and (27), respectively, it gives:

$$\begin{array}{cc}\hfill & {\dot{ϵ}}_2=\frac{u{\widehat{x}}_1}{L}+\frac{(u-1){\widehat{x}}_3}{L}-\frac{1}{u}\left(-C_1{\kappa }_1^2{ϵ}_1+{\dot{i}}_{pv}-C_1{\ddot{x}}_1^r\right)+{\kappa }_1{ϵ}_2+\frac{\dot{u}\beta }{u}\hfill \end{array}$$

(28)

Now, selecting another composite Lyapunov function candidate, $V_{f2}$, that will guarantee the convergence of both the errors ${ϵ}_1$ and ${ϵ}_2$, as well as the asymptotic stability of the system to the equilibrium point, under the same assumptions as those made for $V_{f1}$ [34,35].

This newly selected Lyapunov function, $V_{f2}$, along with its time derivative, ${\dot{V}}_{f2}$, are expressed in Equations (29) and (30), respectively, as follows:

$$V_{f2}=V_{f1}+\frac{1}{2}{ϵ}_2^2$$

(29)

and

$$\begin{array}{cc}\hfill & {\dot{V}}_{f2}={\dot{V}}_{f1}+{ϵ}_2{\dot{ϵ}}_2=-{\kappa }_1{ϵ}_1^2+{ϵ}_2[\frac{u{\widehat{x}}_1}{L}+\frac{(u-1){\widehat{x}}_3}{L}-\frac{1}{u}\left(-C_1{\kappa }_1^2{ϵ}_1+{\dot{i}}_{pv}-C_1{\ddot{x}}_1^{ref}\right)+\hfill \\ & {\kappa }_1{ϵ}_2+\frac{\dot{u}\beta }{u}-\frac{u{ϵ}_1}{C_1}]\hfill \end{array}$$

(30)

For ${\dot{V}}_{f2}$ to be negative definite, the following condition must be satisfied:

$${\dot{V}}_{f2}=-{\kappa }_1{ϵ}_1^2-{\kappa }_2{ϵ}_2^2$$

(31)

where both ${\kappa }_1$ and ${\kappa }_2$ are positive design constants.

Now, comparing Equations (30) and (31) and working out $\dot{u}$, it gives:

$$\begin{array}{cc}\hfill & \dot{u}=-\frac{u}{\beta }\left[{\kappa }_2{ϵ}_2+\frac{u{\widehat{x}}_1}{L}+\frac{(u-1){\widehat{x}}_3}{L}-\frac{1}{u}\left(-C_1{\kappa }_1^2{ϵ}_1+{\dot{i}}_{pv}-C_1{\ddot{x}}_1^{ref}\right)+{\kappa }_1{ϵ}_2-\frac{u{ϵ}_1}{C_1}\right]\hfill \end{array}$$

(32)

Simplifying Equation (32) it yields the backstepping based equivalent control law, $u_{eq}$, as follows:

$$\begin{array}{cc}\hfill & {\dot{u}}_{eq}=\frac{1}{\beta }\left[-{ϵ}_2\left({\kappa }_1+{\kappa }_2\right)u-{ϵ}_1\left(C_1{\kappa }_1^2-\frac{u^2}{C_1}\right)-\frac{u^2{\widehat{x}}_1}{L}-\frac{u(u-1){\widehat{x}}_3}{L}+{\dot{i}}_{pv}-C_1{\ddot{x}}_1^r\right]\hfill \end{array}$$

(33)

where $\beta \ne 0$.

6.2. The Super-Twisting Sliding Mode Based Discontinuous Control Law

Now, the super-twisting sliding mode based discontinuous control law, $u_{disc}$, is designed as follows [18]:

$$\begin{array}{cc}\hfill & {\dot{u}}_{disc}=-{\kappa }_3\sqrt{\left|{ϵ}_1\right|}sign\left({ϵ}_1\right)-{\kappa }_4\int sign\left({ϵ}_1\right)dt\hfill \end{array}$$

(34)

where ${\kappa }_3$ and ${\kappa }_4$ are positive design constants.

6.3. The Proposed Mppt Control Law

Finally, the proposed BSTSMC based MPPT law, $u_T$, is given as follows:

$$\begin{array}{cc}\hfill & {\dot{u}}_T={\dot{u}}_{eq}+{\dot{u}}_{disc}=-{\kappa }_3\sqrt{\left|{ϵ}_1\right|}sign\left({ϵ}_1\right)-{\kappa }_4\int sign\left({ϵ}_1\right)dt+\frac{1}{\beta }[-{ϵ}_2\left({\kappa }_1+{\kappa }_2\right)u-{ϵ}_1\left(C_1{\kappa }_1^2-\frac{u^2}{C_1}\right)\hfill \\ & -\frac{u^2{\widehat{x}}_1}{L}-\frac{u(u-1){\widehat{x}}_3}{L}+{\dot{i}}_{pv}-C_1{\ddot{x}}_1^r]\hfill \end{array}$$

(35)

Different design constants of the BSTSMC law (i.e., ${\kappa }_1$, ${\kappa }_2$, ${\kappa }_3$ and ${\kappa }_4$) are expressed in Table 1. In Equation (35), ($0<u_T<1$). As ${\displaystyle \underset{t\to \infty }{lim}{v}_{pv}=V_{MPP}}$. In the same way, as ${\displaystyle \underset{t\to \infty }{lim}{P}_{pv}=P_{MPP}}$. Note that the proposed BSTSMC based MPPT law, $u_T$, constantly adjusts the duty cycle, d, of the CCNI-BuBo converter switches, $S_1$ and $S_2$, as shown in Figure 5, and adapts the PV system operating point (MPP or $V_{MPP}$) to the varying atmospheric conditions, thus maximizing its power output.

The implementation of the BSTSMC based MPPT law, $u_T$ is illustrated in Figure 8, where each step includes the dynamics of the previous subsystem(s). The Matlab/Simulink implementation of the proposed MPPT strategy along with the DFA based states observer strategy is illustrated in Figure 6 and Figure 7.

Note that the stability analysis of the zero dynamics state, $x_3$, is given in Figure A3 and Figure A4 in Appendix B.

7. MPPT Performance Evaluation in Matlab/Simulink

This section covers the performance validation of the proposed MPPT controller in Matlab/Simulink platform under five different case studies, described below:

Case 1: Performance evaluation test under simultaneous variation of the temperature, irradiance and load

Case 2: Performance evaluation test for robustness against faults under simultaneous variation of the temperature, irradiance and load

Case 3: Performance evaluation test for robustness against plant parametric uncertainties under simultaneous variation of the temperature, irradiance and load

Case 4: Performance comparison with conventional MPPT schemes under simultaneous variation of the temperature, irradiance and load

Case 5: Performance comparison with conventional MPPT schemes for robustness against faults under simultaneous variation of the temperature, irradiance and load

The total simulation time, t, is chosen to be short (i.e., $0.3$ $\mathrm{s}$) for each case study to indicate the fast response time of the proposed MPPT controller. The overall simulation time is further subdivided into three equal time intervals of $0.1$ $\mathrm{s}$ each. Moving from one sub-interval of time to another, the temperature, irradiance and load resistance are varied in a quick succession as follows:

• Sub-interval 1 (0 → $0.1$ $\mathrm{s}$): ( 25 ${}^{\circ }\mathrm{C}$, 1000 $\mathrm{W}$/$\mathrm{m}$${}^2$, 50 $\mathsf{\Omega }$)
• Sub-interval 2 (0.1 → $0.2$ $\mathrm{s}$: ( 65 ${}^{\circ }\mathrm{C}$, 850 $\mathrm{W}$/$\mathrm{m}$${}^2$, 40 $\mathsf{\Omega }$)
• Sub-interval 3 (0.2 → $0.3$ $\mathrm{s}$): ( 25 ${}^{\circ }\mathrm{C}$, 650 $\mathrm{W}$/$\mathrm{m}$${}^2$, 60 $\mathsf{\Omega }$)

The temperature and irradiance profiles are illustrated in Figure 9.

The simulation with and without states observer was run on a computing machine with Intel(R) Core (TM) i5-6200 CPU @2.30 GHz (4 CPUs), ∼2.40 GHz, 8 GB RAM, 6th generation. On this system, a $0.3$ $\mathrm{s}$ simulation took 20 $\mathrm{s}$ without flatness-based observer, and $20.00123$ $\mathrm{s}$ with flatness-based observer. It shows that the observer is computationally not costly and hence offers a negligible computational cost. Therefore, its implementation is highly justified in this application.

7.1. Performance Evaluation Test under Simultaneous Variation of the Temperature, Irradiance and Load

This test is carried out to evaluate the performance of the proposed MPPT technique under simultaneous variation of the atmospheric conditions and the load.

In Figure 10, the PV array output voltages are compared for the three MPPT candidates. It can be observed that the proposed BSTSMC based MPPT candidate offers a superior tracking performance with very a fast rise time, fast settling time and minute chattering. The conventional SMC strategy exhibits considerable chattering, while the backstepping strategy continuously renders steady-state error during MPP tracking. For change in the atmospheric conditions as well the load, after every 0.1 s, the proposed scheme performs the best by converging the PV array output voltage to its reference, $V_{MPP}$, earlier than the other two MPPT candidates. Similarly, Figure 11 illustrates the the PV array output powers comparison for each candidate MPPT strategy. Again, the proposed MPPT technique has the best tracking performance in terms of having faster rise time, faster convergence and minute chattering.

The actual and the observed system states are compared in Figure 12, from which it can be concluded that the GSTA based DFA is accurately retrieving the system states.

7.2. Performance Evaluation Test for Robustness against Faults under Simultaneous Variation of the Temperature, Irradiance and Load

In this test, the robustness of the proposed MPPT candidate is evaluated against multiple sinusoidal faults occurring in the CCNI-BuBo converter, under simultaneous variation of the environmental conditions and the load.

For this purpose, time-varying sinusoidal faults are injected into the DC–DC converter input and output voltages (i.e., ${\widehat{x}}_1$ and ${\widehat{x}}_3$, respectively). Under fault injections, ${\widehat{x}}_{1,new}={\widehat{x}}_1+\delta {\widehat{x}}_1={\widehat{x}}_1+\frac{0.5u}{C_1}sin\left(t\right)$, and ${\widehat{x}}_{3,new}={\widehat{x}}_3+\delta {\widehat{x}}_3={\widehat{x}}_3+\frac{55u}{C_1}sin\left(t\right)$. Furthermore, $\delta {\widehat{x}}_1$ remains active from 0.16–$0.18$ $\mathrm{s}$ only, while $\delta {\widehat{x}}_3$ from 0.06–$0.08$ $\mathrm{s}$ only.

For each MPPT candidate, the PV array output voltages, under faults, are compared in Figure 13. It is evident that at the onset of faults, both the backstepping and the conventional SMC schemes deviate from the $V_{MPP}$, thereby losing tracking. Rather, the SMC technique doesn’t track the $V_{MPP}$, during the sub-interval 3 (0.2 → $0.3$ $\mathrm{s}$) with ( 25 ${}^{\circ }\mathrm{C}$, 650 $\mathrm{W}$/$\mathrm{m}$${}^2$, 60 $\mathsf{\Omega }$). However, the proposed MPPT technique remains almost unaffected during faults and still tends to adequately keep tracking the $V_{MPP}$. This confirms the robustness of the proposed MPPT strategy against the injection of time-varying sinusoidal fault voltages in the PV array output voltage (i.e., converter input voltage) and converter output. Similarly, Figure 14, provides a comparison of the PV array output powers, under faults, for each MPPT candidate. Again, the proposed MPPT strategy remains almost unaffected in case of faults, while the other two MPPT strategies deviate from the $P_{MPP}$, thus losing tracking. Hence, it can be concluded that the proposed MPPT strategy is more robust than the other two MPP candidates. Figure 15 depicts a comparison of the actual and the observed system states under faults. It is evident that the GSTA based DFA is accurately observing the system states.

7.3. Performance Evaluation Test for Robustness against Plant Parametric Uncertainties under Simultaneous Variation of the Temperature, Irradiance and Load

This test covers the the sensitivity analysis of the proposed MPPT scheme against the plant (DC–DC converter) parametric uncertainties under simultaneous variation of the atmospheric conditions and the load.

For this purpose, parametric uncertainties are introduced into the inductor (L) and output capacitor ($C_2$) of the DC–DC converter. Such that L is increased by 25 times, while $C_2$ is decreased by 100 times with the end result: $L_{new}$ = 50 $\mathrm{m}$$\mathrm{H}$ and $C_{2,new}$ = $0.48$ $\mathsf{\mu }$$\mathrm{F}$. Furthermore, $L_{new}$ remains effective from 0.06–$0.08$ $\mathrm{s}$ only, while $C_{2,new}$ remains effective from 0.16–$0.18$ $\mathrm{s}$ only.

In Figure 16, the PV array output voltages are compared, under uncertainties, for each MPPT candidate. It can be seen that both the capacitive and inductive uncertainties greatly deteriorate the MPPT performance of the backstepping as well the conventional SMC techniques. However, the proposed MPPT technique shows more robustness by remaining almost unaffected during plant parametric uncertainties. Similarly, Figure 17 shows the PV array output powers comparison for each MPPT scheme, under plant parametric uncertainties. Again, it can be observed that the backstepping technique has the worst performance, while the proposed MPPT scheme has the best performance under parametric uncertainties. The actual and the observed system states are compared in Figure 18 under plant parametric uncertainties. It is evident that the GSTA based DFA is accurately estimating the system states.

7.4. Performance Comparison with Conventional MPPT Schemes under Simultaneous Variation of the Temperature, Irradiance and Load

In this test, the effectiveness and MPPT performance of the proposed MPPT scheme is compared with the conventional PID and P&O based MPPT schemes, under simultaneous variation of the atmospheric conditions and the load.

The irradiance, temperature and load profiles are kept the same as previous. The PV array output voltages and powers comparison, shown in Figure 19 and Figure 20, respectively, demonstrate that both the conventional PID and P&O based MPPT schemes exhibit a lot of oscillations around the $V_{MPP}$ during their steady-states. This is practically undesirable. On the contrary, the proposed MPPT scheme offers the best MPPT performance, thus, completely outmatching the conventional MPPT techniques.

7.5. Performance Comparison with Conventional MPPT Schemes for Robustness against Faults under Simultaneous Variation of the Temperature, Irradiance and Load

This test is carried out to further compare the effectiveness of the proposed MPPT strategy with the conventional PID and P&O based MPPT techniques, under faults and simultaneous variation of the atmospheric conditions and the load.

The load and atmospheric conditions profiles are the same as previous. Furthermore, the faults are the same as injected in Case 2. It is clear from the PV array output voltages and powers, depicted in Figure 21 and Figure 22, respectively, that the proposed BSTSMC based MPPT strategy has the best MPPT performance and is much robust against faults than both of the conventional MPPT schemes.

8. Conclusions

To optimize the electric power extraction from a standalone PV array under inconsistent ambient weather conditions that is delivering power to a resistive load through a CCNI-BuBo converter, in this article, a hybrid nonlinear sensorless observer based robust BSTSMC MPPT strategy is proposed. The reference peak power voltage is generated via the GPR based probabilistic machine learning approach that is adequately tracked by the proposed MPPT scheme. All the missing system states are retrieved through the GSTA based DFA approach. The Lyapunov stability theory is used to guarantee the closed-loop system stability. Matlab/Simulink software platform is used for simulation, testing and performance validation of the proposed MPPT strategy under simultaneous variation of the temperature, irradiance and load. When the MPPT performance of the proposed MPPT scheme is compared with the recently proposed benchmark backstepping based MPPT control strategy [13] and other conventional SMC, PID and P&O based MPPT schemes, the proposed technique is found to have a superior performance in terms of offering a fast dynamic response, finite-time convergence, minute chattering, higher tracking accuracy and having more robustness against plant parameter perturbations, load disturbances and certain time-varying sinusoidal faults occurring in the system.