Robust Integral Backstepping Based Nonlinear MPPT Control for a PV System

Abstract

[...]there are different sources of renewable energy, such as solar, wind, hydropower, geothermal energy, biomass and, biofuel. [...]treating x2 as a virtual control input which can be chosen according to Lyapunov stability theory. [...]considering the time derivative of the Lyapunov candidate V1=12e12 along Equation (14), one comes up with V˙1=e1 e˙1=e1ipv Ci−ux2 Ci−x˙1ref To introduce robustness into the backstepping strategy, x2 becomes as: x2=ipv Ci−x˙1ref+k1 e1+k2signe1Ciu where k1 and k2 must be positive constants. [...]the proposed control law (32) along with virtual control law (16) steers all the error dynamics to zero in finite time with high precision. [...]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.

1. 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 and grid-connected systems. Stand-alone PV systems (used in electric vehicles, domestic and street lights, water pumping, space and military applications) are generally designed for supplying DC and/or AC electrical loads. They operate independently of the utility grid. A grid-connected PV system (hybrid systems, power plants) [2,3] is the one that is connected to the utility grid. It is comprised of of grid-connected equipment, a power conditioning unit, inverter and PV array. Two major issues arise that ought to be addressed in order to make the PV system more efficient. First, how does one extract the maximum electric power from the PV system under ambient environmental conditions? Second, how does one achieve the maximum conversion efficiency under low irradiance levels? By addressing both of these issues, an overall cost of a PV system can be reduced.

The electrical characteristics, such as current-voltage (I-V) and power-voltage (P-V) of PV cells are nonlinear, which entirely depend on environmental conditions [4]. Variation in temperature and irradiance changes the voltage produced, as well as, the generated current by the PV array [5]. So, the generated power also varies. There is merely a single point, called MPP, on the P-V curve, where maximum power occurs. As this point varies, it makes the extraction of maximum power a challenging task. In order to harvest maximum power from PV, it is essential to force it to operate at MPP [6]. So, numerous algorithms on MPPT were developed in the literature and can be distinguished from one another based on various features such as complexity, type of sensor required, range of effectiveness, cost, convergence speed, implementation, hardware requirements and various other respects [7].

Recently, thorough research has been done to make a progress towards efficient and robust MPPT techniques. Generally, these techniques consist of conventional techniques (CTs), soft computing techniques (SCTs), linear and nonlinear control techniques.

Conventional algorithms are mainly variants of two basic algorithms, namely, Incremental conductance (IC) [8] and Perturb and Observe (P&O) [9]. Conventional methods are simple, easy in implementation and have the capability of efficiently tracking MPP at normal environmental conditions. However, during steady-state, CTs have oscillations around MPP resulting in a loss of useful power. Furthermore, CTs have incapability of handling the problems of partial shading [10].

To eliminate the drawback of CTs, in recent years, SCT-based MPPTs attracted vast interest of researchers. Because, SCTs have the capability of handling the problem of partial shading. SCTs can be further categorized into artificial intelligence techniques (AITs) and bio-inspired techniques (BITs). AITs have many advantages such as working with variable inputs, no need for exact mathematical model of the system and handling of nonlinearities. In the class of AITs, fuzzy logic controller (FLC) [11], artificial neural network (ANN) [12] and hybrid techniques, such as genetic algorithms (GAs) combined with FLC [13] and GA combined with ANN [14] are used. The ANN-based MPPT techniques require large amount of computation time and months of training to achieve MPPT. On the other hand, the FLC-based technique needs the knowledge base for creating rules for tracking. Hence, a large memory size is required [10]. Furthermore, with the passage of time usually the electrical characteristics of the PV system vary. Thus, these controllers require periodic tuning [15,16].

To eliminate these drawbacks, the BITs are proposed which ensure the optimal searching ability without performing excessive mathematical calculations. Furthermore, their implementation simplicity and effectiveness in handling the complex nonlinearities make them very attractive for solving the MPPT problems, particularly in partial shading conditions. In the class of BITs, the latest proposed techniques are mentioned as follows: Flashing fireflies algorithm [17], Artificial bee colony [18], Cuckoo search algorithm [19], Ant colony optimization [20], Particle swarm optimization [21] and Evolutionary algorithm [22].

Though, bio-inspired techniques have several advantages but too many parameters, theoretical analysis and slow convergence under fast changing environmental conditions obstruct their practical usage.

Typically for PV system, MPPT control is a highly nonlinear problem. So, a steady operation of PV system over a wide range of operating points, can be ensured by designing high performance 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.

2. 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 _Vpvref of the peak power voltage to extract maximum power from the PV array under varying environmental conditions.

3. 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, _Rs , a shunt resistance, _Rp , current source, _Iph 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 _Rp has a high value and _Rs 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].

_Ipv=_{Np Iph}−_{Np I0}expq_VpvNsAKT−1

where _Ipv is the cell output current in ampere , A ideality factor of diode and _Vpv is the cell output voltage in V, q is electron charge in C, _Ns and _Np 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, _I0 of the cell is given by:

_I0=_I0r ^T_Tr3expq_EgKA1_Tr−1T

where _Eg = 1.1 eV is energy gap or band gap of semiconductor and _Tr [K] is the cell reference temperature.

So, at _Tr , the reverse saturation current, _I0r is given by the following equation:

_I0r=_Iscrexpq_VocNsAKT−1

where, _Iscr is the cell’s short circuit current at reference temperature, _Tr and radiation. _Voc is the open circuit voltage.

The cell’s photocurrent, _Iph depends on the irradiation, _EeW^m2 and cell temperature, given by the following equation:

_Iph=_Iscr+_KiT−_TrEe1000

where _Ki [A/K] is the temperature coefficient for short circuit current.

Power, _Ppv of the PV array can be calculated by the following equation:

_Ppv=_{Ipv Vpv}

_Ppv=_{Np Iph Vpv}−_{Np I0 Vpv}expq_VocNsAKT−1

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 _Ipv−_Vpv and _Ppv−_Vpv characteristics for different values of temperature T and solar irradiance _Ee is shown in Figure 5 and Figure 6, respectively.

4. 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=_ton+_toff ; _ton is the ON time and _toff is the OFF time, over this period. The duty ratio of the converter is defined as: u=_ton/T . The input capacitor, _Ci , is used to limit ripples in the converter input voltage, while the output capacitor, _Co , 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, _S1 and _S2 are ON while the diodes _D1 and _D2 are OFF. In the second switching interval, both the diodes _D1 and _D2 are ON while the switches, _S1 and _S2 are OFF.

The state-space equations for the first switching interval in vector-matrix form are as follows:

d_vpvdtd_iLdtd_vc0 dt=0−1_Ci01L0000−1_{RL C0vpviLvc0} +_{ipv C0}00

On the other hand, the state-space equations for the second switching interval in vector-matrix form are as follows:

d_vpvdtd_iLdtd_vc0 dt=00000−1L01_C0−1_{RL C0vpviLvc0} +_{ipv Ci}00

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:

d_v¯pvdtd_i¯Ldtd_v¯c0 dt=0−μ_Ci0μL0μL−1L01_C0−μ_C0−1_{RL C0v¯pvi¯Lv¯c0} +_{ipv Ci}00

Assume _x1 , _x2 , _x3 and u as the average values of _v¯pv , _i¯L , _v¯c0 and μ¯ , respectively. Under these assumptions, Equation (9) becomes as:

_x˙1x˙2x˙3=0−u_Ci0uL0uL−1L01_C0−u_C0−1_{RL C0x1x2x3}+_{ipv Ci}00

The voltage transformation ratio of non-inverting DC-DC buck-boost converter is given as:

_{vc0 vpv}=u1−u

The reflected input impedance based on ideal power transfer is given by [25]:

_Ropt=_Rin

where

_Rin=^1−uu2 _RL

and

_Ropt=_{VMPP2 PMPP}

5. 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, _e1 is defined as the difference between the actual and the desired PV array output voltage, as:

_e1=_x1−_x1ref

where _x1ref refers to _Vpvref . The derivative of Equation (13) along the dynamics reported in Equation (10) becomes

_e˙1=_{ipv Ci}−u_{x2 Ci}−_x˙1ref

Since the objective is to steer the error signal _e1 to zero. Therefore, treating _x2 as a virtual control input which can be chosen according to Lyapunov stability theory. Thus, considering the time derivative of the Lyapunov candidate _V1=12_e12 along Equation (14), one comes up with

_V˙1=_{e1 e˙1}=_{e1ipv Ci}−u_{x2 Ci}−_x˙1ref

To introduce robustness into the backstepping strategy, _x2 becomes as:

_x2=_{ipv Ci}−_x˙1ref+_{k1 e1}+_k2sign_e1Ciu

where _k1 and _k2 must be positive constants. With this choice of _x2 , Equation (15) takes the form

_V˙1=−_{k1 e12}−_{k2 e1}sign_e1

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

_x2ref′=_{ipv Ci}−_x˙1ref+_{k1 e1}+_k2sign_e1Ciu+λϕ

where λ is a constant, and ϕ=_{∫0t e1}dτ . Now, treating _x2ref′ 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:

_e2=_x2−_x2ref′

and

_x2=_e2+_x2ref′

Putting Equation (20) in Equation (15), one may get

_V˙1=_{e1ipv Ci}−u_e2+_x2ref′Ci−_x˙1ref

=_{e1ipv Ci}−u_{x2ref′ Ci}−_x˙1ref−u_{e2 Ci}

Substituting Equation (18) in the above expression, one has

_V˙1=−_{k1 e12}−_{k2 e1}sign_e1−u_{e2 e1Ci}

This inequality can also be written as:

_V˙1=−2_{k1 V1}−2_k2V1−u_{e2 e1Ci}

This differential inequality will be discussed at the end of this section. Now, differentiating Equation (20) with respect to time, it becomes

_e˙2=_x˙2−_x˙2ref′

where the time derivative of _x2ref′ is calculated as follows:

_x˙2ref′=1^u2[u_i˙pv−_{Ci x¨1ref}+_{k1 Ci e˙1}−_ipv−_{Ci x˙1ref}+_{k1 Ci e1}+_{k2 Ci}sign_e1u˙]+λ_e1

Carrying out some algebraic simplification, the expression of _x˙2ref′ becomes

_x˙2ref′=1u_i˙pv−_{Ci x¨1ref}−_{Ci k12 e1}−_{Ci k1 k2}sign_e1−_{k1 e2}−u˙u_x2ref+λ_e1

Using it in Equation (24), one has

_e˙2=_x˙2−[1u_i˙pv−_{Ci x¨1ref}−_{Ci k12 e1}−_{Ci k1 k2}sign_e1−_{k1 e2}−u˙u_x2ref+λ_e1]

A composite Lyapunov function, _Vc , is defined to ensure convergence of the errors _e1 and _e2 to zero and the asymptotic stability of the system, as follows:

_Vc=_V1+12_e22

The time derivative of _Vc along Equation (22) becomes

_V˙c=−_{k1 e12}−_{k2 e1}sign_e1+_e2e˙2−u_{e1 Ci}

For _Vc to be negative definite, let

_e˙2−u_{e1 Ci}=−_{k3 e2}−_k4sign_e2

where _k3 and _k4 are positive constants. Using values of _e˙2 from Equation (27) in Equation (30), it gives

−_{k3 e2}−_k4sign_e2=−u_{e1 Ci}+_x˙2−1u_i˙pv−_{Ci x¨1ref}−_{Ci k12 e1}−_{Ci k1 k2}sign_e1+_{k1 e2}+u˙u_x2ref−λ_e1

Now, making use of _x˙2 from Equation (10) and solving for u˙ , one gets the final expression of the control law as follows:

u˙=1_x2refe2−_k1−_k3u−_{e1Ci k12}−^u2 _Ci−uλ+u_x3L+1_x2refi˙pv−_{Ci x¨1ref}−^u2_x1+_x3L−_{Ci k1 k2}sign_e1+u_x2ref−_k4sign_e2

This choice of the control law guides Equation (29) to the forthcoming form

_V˙c=−_{k1 e12}−_{k2 e1}sign_e1−_{k3 e22}−_{k4 e2}sign_e2

This expression can also be written as follows:

_V˙c=−2_{k¯1 Vc}−2_{k¯2 Vc12}

where _k¯1=min(_k1,_k3) and _k¯2=min(_k2,_k4) . The differential Equation (34) looks very similar to the fast terminal attractor [29]. This confirms that _Vc→0 in finite time. In other words, _e2=_x2−_x2ref′→0 in finite time which confirms high precisions in tracking as well as in regulation problems. As _e2→0 in finite time, the last term in the differential Equation (23) vanishes. Consequently, a terminal attractor in terms of _V1 is obtained which, once again, confirms the fast finite time convergence of _e1 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 dynamics

_x˙3=1_C0−u_C0x2−1_{RL C0x3}

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 _x1 and _x2 equal to zero. Thus one has

_x˙3=−1_{RL C0x3}

Since the typical parameters _RL and _C0 are positive, therefore, Equation (36) has poles in the left half plane at −1_{RL C0} . 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.

6. 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.

6.1. 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, _Vref 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.

Figure 10, Figure 11, Figure 12 and Figure 13 depicts performance indexes IAE (Integral Absolute Error), ITAE (Integral Time Absolute Error), ISE (Integral Square Error) and ITSE (Integral Time Square Error) of the PV system. The superiority of the proposed technique has been verified based on these performance indexes.

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.

6.2. 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, _x3f=30usin(t)/_Ci , is added to output voltage capacitor _x3 , as Δ_x3=_x3+_x3f . A second fault _x2f=0.5usin(t)/_Ci , is added to inductor current _x2 , in the time interval t=[0.16,0.18] (s), as Δ_x2=_x2+_x2f .

The PV output voltage deviates from the _Vref , 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 _Vref 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 _Vref=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 Figure 16, Figure 17, Figure 18 and Figure 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.

6.3. 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 ΔL = 200 mH, is added to inductor L, as _Lt=L+ΔL . A second uncertainty of ΔC = 0.48 μ F, is added to output capacitor _C0 , in the time interval t=[0.16,0.18] (s), as _Ct=_C0+ΔC .

From Figure 20, it can be observed that backstepping controller deviates about 6 V and integral backstepping controller 15 V from the _Vref , 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, in Figure 22, Figure 23, Figure 24 and Figure 25 the 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.

Also, maximum power is transmitted to load by the proposed robust integral backstepping controller with efficiency of 98.07% , which is maximum compared to the other MPPT techniques.

7. 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.

7.1. 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.

7.2. 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% .

7.3. 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.

8. 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.

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Type	Name of Parameters	Symbols	Magnitude
PV Module	Maximum power	Pmax	1555 W
	Number of Cells per module	Ns	72
	Voltage at maximum power	Vmp	102.6 V
	Open circuit voltage	Voc	165.8 V
	Short circuit current	Isc	17.56 A
	Current at maximum power	Imp	151.16 A
Converter	Input Capacitor	Ci	1×10−3 F
	Inductor	L	20×10−3H
	Output capacitor	C0	48×10−6F
	Load resistance	RL	50Ω
	IGBT switching frequency	fs	5000 Hz
RIB Controller	Constant	k1	60
	Constant	k2	9000
	Constant	k3	2000
	Constant	k4	10
	Constant	λ	29
PID Controller	Gain	kp	1.74
	Gain	kd	1
	Gain	ki	2.33

Author Contributions

K.A. and L.K. conceived and designed the experiments; K.A. performed the experiments.; writing-review and editing, S.U., S.A., S.M., F.W.K., and Naghmash; supervision, L.K. and Q.K.; K.A. wrote the paper.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflicts of interest.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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