Robust Integral Backstepping Based Nonlinear MPPT Control for a PV System

: A photovoltaic system generates energy that depends on the environmental conditions such as temperature, irradiance and the variations in the load connected to it. To adapt to the consistently increasing interest of energy, the photovoltaic (PV) system must operate at maximum power point (MPP), however, it has the issue of low efﬁciency because of the varying climatic conditions. To increase its efﬁciency, a maximum power point technique is required to extract maximum power from the PV system. In this paper, a nonlinear fast and efﬁcient maximum power point tracking (MPPT) technique is developed based on the robust integral backstepping (RIB) approach to harvest maximum power from a PV array using non-inverting DC-DC buck-boost converter. The study uses a NeuroFuzzy network to generate the reference voltage for MPPT. Asymptotic stability of the whole system is veriﬁed using Lyapunov stability criteria. The MATLAB/Simulink platform is used to test the proposed controller performance under varying meteorological conditions. The simulation results validate that the proposed controller effectively improves the MPPT in terms of tracking speed and efﬁciency. For further validation of the proposed controller performance, a comparative study is presented with backstepping controller, integral backstepping, robust backstepping and conventional MPPT algorithms (PID and P&O) under rapidly varying environmental conditions. a maximum steady-state error is observed in backstepping controller after faults occurrence. performance indexes of PV system (IAE, ITAE, ISE and ITSE) are depicted. These results validate that the proposed technique performs better than the other MPPT techniques.


Introduction
The energy requirements of the world population have been continuously increasing and are expected to increase at a higher rate in the future. Fossil fuels have been supplying the majority of energy needs over the centuries and they continue to be a major contributor. As the fossil fuel reserves become exhausted and due to increasing environmental issues, people are more concerned with their usage. A solution to these issues is to reduce the dependence on fossil fuels and to use renewable sources of energy as these sources are inexhaustible and cause less pollution when compared to fossil fuels.
In fact, there are different sources of renewable energy, such as solar, wind, hydropower, geothermal energy, biomass and, biofuel. Among these resources, solar energy is one of the most dominant energy sources [1] because of the fact that it is clean, inexhaustible and free.
Daily, the sun gives an unlimited amount of energy that can be directly converted to electricity by using a PV system. The main two classifications of photovoltaic systems are stand-alone systems nonlinear controllers. In the literature of PV system, many MPPT-based nonlinear techniques [23,24] were proposed, in both stand-alone and grid-connected PV systems.
In [25,26], a nonlinear backstepping and integral backstepping controller is proposed to track maximum power point of PV array. However, a considerable steady-state error and overshoot was observed in backstepping and integral backstepping controllers, respectively during MPP tracking. Similarly, robustness of both the controllers have not been evaluated against certain faults or uncertainties occurring in the system. To mitigate these problems, a nonlinear robust integral backstepping-based MPPT control scheme is proposed in this research article, as shown in Figure 1. The proposed control scheme not only minimizes the steady-state errors and overshoots, but also outperforms the backstepping and integral backstepping controllers in terms of providing better efficiencies, better rise time, better peak time, better settling time with and without occurrence of certain faults or uncertainties in the system.
In this research article, a nonlinear robust integral backstepping controller is used for MPPT of PV array, using a non-inverting buck-boost converter. In Section 2, NeuroFuzzy network is presented. Mathematical modeling of the PV system is presented in Section 3. Section 4 presents state-space average modeling of the non-inverting DC-DC buck-boost converter. A nonlinear robust integral backstepping controller is presented in Section 6, whereas the analysis of global asymptotic stability is guaranteed through Lyapunov stability criteria in the corresponding section. The performance of the proposed controller for tracking MPP is analyzed under varying environmental conditions using MATLAB/Simulink. Simulation results are presented in Section 7 to show the effectiveness of the proposed controllers. Finally in Section 8, the conclusion is drawn.

Reference Voltage Generation by NeuroFuzzy Algorithm
NeuroFuzzy network used in this article is based on Takagi-Sugeno fuzzy inference system, which is shown in Figure 2, while a NeuroFuzzy-based nonlinear 3D-plane for PV array is depicted in Figure 3. The NeuroFuzzy network used has two inputs i.e, temperature and irradiance. The input layer is the fuzzification layer which has three Gaussian membership functions for each variable. The output layer has a linear equation for each rule. The NeuroFuzzy algorithm generates reference V pvre f of the peak power voltage to extract maximum power from the PV array under varying environmental conditions.

Mathematical Modeling of PV System
The PV cell have p-n junction similar to a diode, which produces the electric power by using photons. A PV cell consists of a series resistance, R s , a shunt resistance, R p , current source, I ph and a diode, D as shown in Figure 4. In order to extract the parameters of the equivalent electrical circuit, it is required to know the PV current-voltage or power-voltage curve in standard conditions of measurement (SCM). As R p has a high value and R s has a low one, so to simplify the study both can be neglected. The PV array electrical characteristics are often determined by the subsequent equations [27].
where I pv is the cell output current in ampere, A ideality factor of diode and V pv is the cell output voltage in V, q is electron charge in C, N s and N p are PV cells connected in series and parallel respectively, K is Boltzmann constant in J/K and T is temperature in K.
Reverse saturation current, I 0 of the cell is given by: where E g = 1.1 eV is energy gap or band gap of semiconductor and T r [K] is the cell reference temperature. So, at T r , the reverse saturation current, I 0r is given by the following equation: where, I scr is the cell's short circuit current at reference temperature, T r and radiation. V oc is the open circuit voltage.
The cell's photocurrent, I ph depends on the irradiation, E e W m 2 and cell temperature, given by the following equation: where K i [A/K] is the temperature coefficient for short circuit current. Power, P pv of the PV array can be calculated by the following equation: The PV array used in this paper consists of 4 strings which are connected in parallel. Every string consists of 4 modules connected in series. So, the total number of PV modules used in this paper are 16. One module has a power of 1555 W. So, the maximum power which can be delivered by this PV system is: 1555 × 16 = 24,880 W.
The PV array I pv − V pv and P pv − V pv characteristics for different values of temperature (T) and solar irradiance (E e ) is shown in Figures 5 and 6, respectively.

Average State-Space Modeling of the Non-Inverting DC-DC Buck-Boost Converter
The non-inverting DC-DC buck-boost converter either steps up or steps down voltage from its input (PV array) to its output (load), in order to force the PV array to operate at the MPP. The converter is periodically controlled by its switching period, T where: T = t on + t o f f ; t on is the ON time and t o f f is the OFF time, over this period. The duty ratio of the converter is defined as: u = t on /T. The input capacitor, C i , is used to limit ripples in the converter input voltage, while the output capacitor, C o , is used to limit ripples in the output voltage. Throughout this paper, the converter is assumed to be operating in continuous conduction mode (CCM). The equivalent circuit of the non-inverting DC-DC buck-boost converter is shown in Figure 1, as a part of the complete model of the system [28].
There are two switching intervals. In the first switching interval, both the switches, S 1 and S 2 are ON while the diodes D 1 and D 2 are OFF. In the second switching interval, both the diodes D 1 and D 2 are ON while the switches, S 1 and S 2 are OFF.
The state-space equations for the first switching interval in vector-matrix form are as follows: On the other hand, the state-space equations for the second switching interval in vector-matrix form are as follows: Now, the average model for non-inverting DC-DC buck-boost converter with a resistive load in vector-matrix form based on inductor volt-second balance and capacitor charge-balance is as follows: Assume x 1 , x 2 , x 3 and u as the average values ofv pv ,ī L ,v c 0 andμ, respectively. Under these assumptions, Equation (9) becomes as: The voltage transformation ratio of non-inverting DC-DC buck-boost converter is given as: The reflected input impedance based on ideal power transfer is given by [25]: where

Robust Integral Backstepping Controller Design
In order to effectively track the reference voltage generated by a NeuroFuzzy algorithm and to extract maximum power from the PV array, a nonlinear robust integral backstepping MPPT controller is proposed. To proceed with the design, an error, e 1 is defined as the difference between the actual and the desired PV array output voltage, as: where x 1re f refers to V pvre f . The derivative of Equation (13) along the dynamics reported in Equation (10) becomesė Since the objective is to steer the error signal e 1 to zero. Therefore, treating x 2 as a virtual control input which can be chosen according to Lyapunov stability theory. Thus, considering the time derivative of the Lyapunov candidate V 1 = 1 2 e 2 1 along Equation (14), one comes up witḣ To introduce robustness into the backstepping strategy, x 2 becomes as: where k 1 and k 2 must be positive constants. With this choice of x 2 , Equation (15) takes the forṁ Since the main objective is to provide a robust performance with almost zero steady-state error. Therefore, an integral action term λφ, is added (i.e., integral backstepping strategy is adapted) to Equation (16). Consequently, one gets where λ is a constant, and φ = t 0 e 1 dτ. Now, treating x 2re f as a new reference for the next step which will be tracked by the second state of the system. The tracking error is defined as follows: and Putting Equation (20) in Equation (15), one may geṫ Substituting Equation (18) in the above expression, one haṡ This inequality can also be written as: This differential inequality will be discussed at the end of this section. Now, differentiating Equation (20) with respect to time, it becomeṡ where the time derivative of x 2re f is calculated as follows: Carrying out some algebraic simplification, the expression ofẋ 2re f becomeṡ Using it in Equation (24), one haṡ A composite Lyapunov function, V c , is defined to ensure convergence of the errors e 1 and e 2 to zero and the asymptotic stability of the system, as follows: The time derivative of V c along Equation (22) becomeṡ For V c to be negative definite, leṫ where k 3 and k 4 are positive constants. Using values ofė 2 from Equation (27) in Equation (30), it gives Now, making use ofẋ 2 from Equation (10) and solving foru, one gets the final expression of the control law as follows: This choice of the control law guides Equation (29) to the forthcoming forṁ This expression can also be written as follows: wherek 1 = min(k 1 , k 3 ) andk 2 = min(k 2 , k 4 ). The differential Equation (34) looks very similar to the fast terminal attractor [29]. This confirms that V c → 0 in finite time. In other words, e 2 = x 2 − x 2re f → 0 in finite time which confirms high precisions in tracking as well as in regulation problems. As e 2 → 0 in finite time, the last term in the differential Equation (23) vanishes. Consequently, a terminal attractor in terms of V 1 is obtained which, once again, confirms the fast finite time convergence of e 1 to zero. Hence, the proposed control law (32) along with virtual control law (16) steers all the error dynamics to zero in finite time with high precision. Now, the authors aim to present the stability of the zero dynamics. Since, a two step integral backstepping law is designed, so the dynamicṡ are straight a way the internal dynamics of this PV system. According to the nonlinear theory [30], the zero dynamics can be obtained by substituting the applied control input u and the control driven states x 1 and x 2 equal to zero. Thus one hasẋ Since the typical parameters R L and C 0 are positive, therefore, Equation (36) has poles in the left half plane at − 1 R L C 0 . This validates that the zero dynamics are exponentially stable and confirms the minimum phase nature of the under study PV system. Now, in the forthcoming section the simulation results will demonstrate the effectiveness of the proposed law in sound details.

Simulation Results and Discussion
Matlab/Simulink (SimPowerSystems toolbox) is used to simulate the PV array model, the average non-inverting DC-DC buck-boost converter model and the proposed MPPT technique. The information about PV array, non-inverting DC-DC buck-boost converter and controllers used in this study is given in Table 1.
The proposed MPPT technique is evaluated from three different aspects i.e., (1) robustness to climatic changes, (2) faults and (3) uncertainties. The irradiance and temperature profiles are depicted in Figure 7.

Robustness to Climatic Changes
In this case, temperature, irradiance as well as load is varied to validate the robustness of the proposed MPPT technique. In the first time interval t = [0, 0.1] (s), temperature is maintained at 25 • C, irradiance at 650 W/m 2 and load at 30 Ω, and the maximum PV power is 18.4 kW. In the second time interval t = [0.1, 0.2] (s), temperature is changed to 65 • C, irradiance to 1000 W/m 2 and load to 40 Ω, and maximum PV power is 19.02 kW. Finally, in the time interval t = [0.2, 0.3] (s), the temperature is settled back to 25 • C, irradiance to 650 W/m 2 and load to 50 Ω, and maximum PV power is 18.4 kW.
Reference, V re f of the peak power voltage generated by NeuroFuzzy network, is successfully tracked by all the three controllers. However, it can be observed that the proposed controller reaches steady-state at all levels at 0.01 s, again which is faster compared to the other MPPT techniques, as shown in Figure 8.
PV array output power along with MPP curves is shown in Figure 9. It can be observed that MPP is successfully achieved by the proposed controller within 0.01 s, with almost negligible ripples, compared to other MPPT techniques. Similarly, maximum power is transmitted to load by the proposed robust integral backstepping controller with efficiency of 98.04%, which is maximum compared to other MPPT techniques.
In this manner, the validation of the robustness of the proposed robust integral backstepping controller under varying temperature, irradiance and load is guaranteed.

Robustness to Faults under Climatic Changes
In this case, multiple faults are introduced under varying irradiance, temperature and load condition. In the time interval t = [0.06, 0.08] (s), a fault, The PV output voltage deviates from the V re f , due to occurrence of faults in the system, as shown in Figure 14. However, the proposed controller deviates from 329 V to 415 V, which is minimum deviation from V re f compared to integral backstepping from 329 V to 470 V and backstepping from 329 V to 476 V, in the time interval t = [0.16, 0.18] (s). Also, in the time interval t = [0.06, 0.08] (s), a fault deviates the backstepping controller from V re f = 450 V to 412 V and integral backstepping controller from 450 V to 412 V, while the proposed controller shows robustness against fault. It can be observed that the proposed controller reaches steady-state faster than other controllers. Besides, a maximum steady-state error is observed in backstepping controller after faults occurrence. Figure 15 depicts the PV array output power. It is clear that proposed controller performs the best, and reaches steady-state quickly in 0.002 s after faults, with almost negligible ripples.
The performance indexes of PV system (IAE, ITAE, ISE and ITSE), as shown in Figures 16-19 validate the effectiveness of the proposed control scheme. Also, maximum power is transmitted to load by the proposed robust integral backstepping controller with efficiency of more than 95%, outperforming the other MPPT techniques.

Robustness to Uncertainties under Climatic Changes
In this case, multiple uncertainties are introduced under varying irradiance, temperature and load condition. In the time interval t = [0.06, 0.08] (s), uncertainty of ∆L = 200 mH, is added to inductor L, as L t = L + ∆L. A second uncertainty of ∆C = 0.48 µF, is added to output capacitor C 0 , in the time interval t = [0.16, 0.18] (s), as C t = C 0 + ∆C.
From Figure 20, it can be observed that backstepping controller deviates about 6 V and integral backstepping controller 15 V from the V re f , while there is no deviation in the proposed controller. Besides, a maximum steady-state error is observed in backstepping and integral backstepping.
In Figure 21, it is clear that the proposed controller has negligible ripples and no deviation from MPP. Similarly

Comparison with Conventional Techniques
The performance of the proposed robust integral backstepping controller is further validated by comparing its performance with conventional P&O and PID MPPT controllers.

Robustness to Climatic Changes
First conventional techniques (P&O and PID) are compared with the proposed robust integral backstepping technique under the same test (robustness to climatic changes).
The proposed MPPT technique outperforms the conventional PID and P&O techniques. Under varying temperatures, irradiance and load, the PV array output power is shown in Figure 26. It can be seen that the proposed MPPT technique is not only robust, but also the ripples are negligible. Moreover, one can observe that the proposed technique presents a very fast system reaction against meteorological conditions. The PID controller performs better than the P&O technique but results in oscillations around MPP.
Also the proposed robust integral backstepping controller transmitted maximum power to load with efficiency of 98.7%, which is greater than the efficiencies of PID and P&O MPPT techniques.

Robustness to Faults under Climatic Changes
Again the same test (robustness to faults under climatic changes) was performed to compare the performance of conventional PID and P&O with the proposed robust integral backstepping controller. Figure 27 shows the PV output power under faults condition. It can be seen that due to the occurrence of faults in the system, PID controller deviates from MPP to negative power, while the P&O technique does not track MPP at all. However the proposed controller shows robustness against faults with almost negligible ripples. Moreover, the proposed robust integral backstepping controller transmits maximum power to the load with efficiency of 98.6%.

Robustness to Uncertainties under Climatic Changes
In this case, the performance of the PID and P&O are compared with the proposed robust integral backstepping under the same test (robustness to uncertainties under climatic changes).
It is clear from Figure 28 that the proposed technique has much smaller ripples in the output power due to uncertainties as compared to those produced in PID and P&O controllers.
Also, maximum power is transmitted to load by the proposed robust integral backstepping with efficiency of 98.6%, outperforming the conventional PID and P&O MPPT techniques.
It can be noted that the desirable performance of the proposed robust integral backstepping in various conditions has been maintained and tracking operations have been carried out as well.

Conclusions
In this article, a nonlinear robust integral backstepping MPPT technique is proposed. A NeuroFuzzy network is used to generate the reference voltage for MPPT. Asymptotic stability of the system is guaranteed using Lyapunov stability criteria. The robustness of the proposed technique is investigated by simulation under climatic changes, uncertainties and faults. Results clearly showed that the proposed robust integral backstepping controller tracks the MPP quickly and accurately compared to PID, P&O, backstepping and integral backstepping. The simulation results validate the high performance of the proposed robust integral backstepping controller in terms of steady-state error, tracking accuracy, overshoot, rise time, settling time and efficiency. The simulation results verify the superiority of the proposed robust integral backstepping controller.