Nonlinear adaptive NeuroFuzzy feedback linearization based MPPT control schemes for photovoltaic system in microgrid

Renewable energy resources connected to a single utility grid system require highly nonlinear control algorithms to maintain efficient operation concerning power output and stability under varying operating conditions. This research work presents a comparative analysis of different adaptive Feedback Linearization (FBL) embedded Full Recurrent Adaptive NeuroFuzzy (FRANF) control schemes for maximum power point tracking (MPPT) of PV subsystem tied to a smart microgrid hybrid power system (SMG-HPS). The proposed schemes are differentiated based on structure and mathematical functions used in FRANF embedded in the FBL model. The comparative analysis is carried out based on efficiency and performance indexes obtained using the power error between the reference and the tracked power for three cases; a) step change in solar irradiation and temperature, b) partial shading condition (PSC), and c) daily field data. The proposed schemes offer enhanced convergence compared to existing techniques in terms of complexity and stability. The overall performance of all the proposed schemes is evaluated by a spider chart of multivariate comparable parameters. Adaptive PID is used for the comparison of results produced by proposed control schemes. The performance of Mexican hat wavelet-based FRANF embedded FBL is superior to the other proposed schemes as well as to aPID based MPPT scheme. However, all proposed schemes produce better results as compared to conventional MPPT control in all cases. Matlab/Simulink is used to carry out the simulations.


Introduction
The energy demand of the globe is mainly fulfilled by fossil fuel. Increasing energy demand and limitation of fossil fuel supplies boost the cost of electricity. The environmental hazards due to greenhouse emission and scarcity of fossil fuel supplies diverted the focus towards renewable energy resources. Renewable power is clean, sustainable, green, economical, and durable. The productivity of renewable energy resources depends on meteorological conditions. Thus, a single stand-alone source is unable to supply continuous reliable energy. Therefore several renewable and non-renewable resources are integrated to form a single HPS. Solar energy is a huge reservoir of green energy blessed to this planet [1,2]. To utilize it most reliably a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 conventional and non-conventional techniques were developed to tackle PSC in the near past but they have drawbacks of lager power fluctuation, lower power output, and complexity of control design in some cases [32][33][34][35][36][37].
This paper presents a comparison of the performance of four different adaptive feedback linearization (FBL) techniques incorporated with full recurrent adaptive NeuroFuzzy (FRANF) based controllers for a PV system in a grid-integrated SMG-HPS for three different cases.
Seven-layered full recurrent adaptive NeuroFuzzy structure embedded with four different mathematical functions and wavelets is used to estimate the nonlinear functions of FBL control. FRANF structures are based on Standard Additive Model, Fourier Series, Mexican Hat Wavelet, and Chebyshev Wavelet for the estimation purpose. An online learning algorithm based on the gradient-decent method is applied to update all the parameters of the FRANF structure for each proposed control scheme adaptively. Three different cases are used on the same system to test the performance of proposed controllers. The extraction of maximum power under varying conditions from the PV system in these scenarios is challenging. The comparison of the proposed control schemes is based on the power quality of the SMG-HPS as well as extracted power and performance indexes obtained during the simulation process. The performance of the proposed control schemes is evaluated against the adaptive PID (aPID) control scheme.
The article comprised of four major sections. Section 1 describes the testbed for this research work. Section 2 gives detail of the proposed control schemes for the PV subsystem. Section 3 and 4 provide details of results and conclusion respectively.
Renewable resources like WT, electrolyzer, PV, SOFC, SC, and batteries are connected to DC link and deliver power to the connected converters. These converters are connected to the AC bus that has UG, MT, MH, CS, BM, along with all types of loads attached to it as shown in Fig 1. Specifications of all the sources are described in section 3 Table 1. All the source converters except PV are being controlled by aPID.

PV cell model
A simple pn-junction that converts solar irradiation into electric energy is known as PV cell [32]. It is comprised of current source, parallel diode and a series resistor, which is further connected to boost converter [8,32]. PV cells are combined in clusters which are then connected to one-another is series and parallel fashion to obtain the desired power level [9,32,33]. The relationship between output voltage and current of PV cell is given as: [6,8,19,32,33] i pv ¼ n p i p À n p I D exp q AKT c v pv n s þ R s i pv n p where all the symbols are defined in Nomenclature. Photo-current, i p can be determined by solar irradiation as: Beside this, saturation current, i s of PV cell varies with temperature and can be related as: The Eqs (1), (2) and (3) are used to design PV system and also show that the output of a PV array depends on solar irradiance and temperature of the environment [32].
2 Adaptive feedback linearization embedded NeuroFuzzy MPPT for PV system

Control law design
Feedback linearization is a tool that transforms nonlinear system dynamics into linear ones algebraically, either fully or partly, hence linear control techniques can be applied.

PLOS ONE
nonlinearities can be eliminated from any nonlinear system represented in a companion form as [38]: where y 2 < is plant output, n 2 Z is relative degree of system, f(x) and g(x) are unknown nonlinear functions, u MPPT 2 < is control input, and x ¼ ½y; _ y; . . . ; y nÀ 1 Þ� T 2 < n is the statespace vector. The control problem is to find u MPPT that assures y(t) follows the desired trajectory y d (t). If a new input v represents the plant's input then: Using the control law of Eq (5) in Eq (4), nonlinear terms will be canceled and input-output integral form is obtained as: The nonlinear functions in Eq (5) are estimated using NeuroFuzzy systems. b f ðxÞ represents the estimated f(x) and b g ðxÞ represents g(x). The adaptive feedback control law can be rewritten as: The control law of Eq (7) is based on control input v = −K v } − Y D and identified nonlinear functions. where K v is constant and } the tracking error is given as [38]: where^= [λ 1 λ 2 � � � λ n-1 ] is constant weight vector and e = x − x d represents error matrix.
Online adoption of^ensures occupation of poles of s n−1 + λ n−1 s n−2 + � � � + λ 1 in left half of complex plane. To bring the tracking error to zero, the following control law is entertained by identification of b f ðxÞ and b g ðxÞ through FRANF, where ½b g ðxÞ > 0� is obtained by using saturation block and identifier is set to give 0.1 as output if b g ðxÞ ¼ 0 during identification.
To generate appropriate control law, FBL control coefficients^are updated using n LMS algorithm, as: , z = e (n−1) and b is estimation of^.

Full recurrent adaptive NeuroFuzzy identification techniques based on different NeuroFuzzy architectures
A variety of FRANF identifiers are used in order to identify the nonlinear b f ðxÞ and b g ðxÞ functions for PV system in SMG-HPS. The seven-layered FRANF system uses NeuroFuzzy concept for estimation.
Fuzzy logic uses IF-THEN rules for approximation of unknown functions using standard fuzzy model. The unknown functions b f ðxÞ and b g ðxÞ can be identified by the standard fuzzy model using a set of rules as; R m : IF x 1 is A j 1 . . . and x n is A j n THEN y is b j l . Let fuzzy logic controller has q inputs, ρ 1 , ρ 2 , . . ., ρ q . The output of NeuroFuzzy system is given as: where m F l j is the membership function, ρ j and β l are adjustable parameters. It is the point in R at which m b j achieves its maximum value. m is the number of fuzzy rules used to construct the identifier, F l j is the jth fuzzy set corresponding to the lth fuzzy rule, and β l is centroid of the lth fuzzy set corresponding to identifier output, b f ðxÞ and b g ðxÞ. Eq (10) can be written for b f ðxÞ and b g ðxÞ using fuzzy-basis function vector ξ(x), as and ξ(x) is given as A number of mathematical relations and functions are available for designing a fast and robust NeuroFuzzy identifier. The following variants are used to design antecedent and the consequent part of the fuzzy logic system for this research work.

Antecedent part.
The transformation of continuous input variables into linguistic variables is fuzzification. A membership function is always required for the transformation. The importance of the membership function is based on its shape that translates complete information of the plant (uncertainties and nonlinearities) in fuzzy inference system. The membership function chosen for this research work is the Gaussian membership function due to the following properties: • local and nonlinear nature • smooth output • relation between the radial basis functions Neural Networks (NNs) and fuzzy system Gradient-based techniques are highly suitable for use due to the continuous differentiable nature of Gaussian membership function. It is expressed as: where m ij and σ ij are the mean and variance of the ith input and jth membership function.

Variants of consequent part.
The consequent part generates weights based on different mathematical functions like Fourier series function, wavelet networks and polynomial NNs. The operation of consequent part takes place in parallel to antecedent part and produces final output of identifier at defuzzification layer. The variants of consequent part used for this research work are Standard Additive Model (SAM), Fourier series function and wavelet networks (Mexican hat wavelet and Chebyshev wavelet).

Standard Additive Model (SAM).
SAM is an important case of additive fuzzy systems that can estimate any uninterrupted function uniformly over a closed space. In SAM the fuzzy rules are given as: If x i = C Then y i = D such that C and D are one to one mapping of input and output spaces.  where, are the convex coefficients or discrete probability weights, c k is the centroid of the then-part set and V k is the finite positive volume.

Fourier Series Neural Networks (FsNNs).
Fast convergence and accurate modeling capabilities of Fourier series NNs are known with gradient descent algorithm. Mutually orthogonal sine and cosine are the basis functions of FsNNs that guarantee better estimation and convergence.
In FsNN, the jth input signal initiates p j 2 À 1 neurons with equal number of sine and cosine as activation function. Adaptation of output weights occur during learning. Overall output of the FsNN is given as: where, N denotes Kronecker product, n shows the total number of inputs and l ¼ Q n i¼1 ðp i À 1Þ.

Fuzzy Wavelet Neural Networks (NNs).
For a better estimation of nonlinear functions, wavelet NNs were proposed as a substitute to feedforward NNs. Due to enormous neurons, NNs may get struck on the local minima that result in slower convergence of the network. To get rid of it, wavelet functions (WFs) can be used in the structure. Wavelets are waves having a limited duration and zero mean value. The localization characteristics of wavelets and the fast learning abilities of NNs results in better outcomes for complex nonlinear system modeling. The schematic diagram of wavelet NN is given in Fig 6. Two wavelet activation functions, Mexican hat and Chebyshev wavelet are used in this research work.
• Mexican hat wavelet (MHW) is a negative normalized, non-orthogonal second derivative of Gaussian function. MHW function is expressed as; where C i (x i ) is the family of wavelets obtained by single ψ(x i ) function with parameters dilation d ij and t ij respectively.
• Chebyshev wavelet (CW) C rs ðtÞ ¼ cðk; b r; s; tÞ have four arguments; k � N, r = 1, 2, . . ., 2 k−1 , and b r ¼ 2r À 1; s is the degree of the first kind Chebyshev polynomial and t � [0, 1] is the normalized time. They can be expressed on the interval [0, 1] as: where e T s ðtÞ ¼ where N is a fixed positive integer. Orthonormality of the system is given by the coefficients in Eq (22). Here nth degree Chebyshev polynomials orthogonal to wðtÞ ¼ 1= ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 À t 2 p weight function on the interval [−1, 1] is represented by fT s ðtÞ; s�N [ f0gg that satisfy the following recursive formula: The weight functions of CW, e wðtÞ ¼ wð2t À 1Þ generates orthogonal wavelets on dilation and translation as w s ðtÞ ¼ wð2 k t À b rÞ.

Proposed FRANF identifier.
The FRANF has seven layers as shown in Fig 7. The antecedent part consists of the first three layers, whereas the rest four layers are consequent part layers. The n number of input signals in the first layer is equivalent to the m number of nodes and these nodes are used for input distribution. Let I k i and O k i represents input and output of ith node in kth layer. The operation function of nodes and the signal propagation in each layer is given as under:

PLOS ONE
• Layer 1 (Input Layer): It takes input variables and these inputs are transmitted to the next layer by the nodes. Feedback connections are the part of this layer that add temporal relationship in the network.
• Layer 2 (Membership Layer): Each node represents one linguistic term and computes membership degree and fuzzy set for all input signals entering into the system. Linguistics terms in each node are computed using Gaussian membership function.
Input is : Output is : Where ij subscript shows the jth term of the ith input O 1 i , where j ¼ 1; . . . ; N; and the superscript (2) represents second layer. Also, i = 1, 2, 3, . . .. . .n; k is number of iterations, m 2 i ðk À 1Þ is the past value of membership function and y 2 ij is the recurrent weight.

• Layer 3 (Rule Layer):
The rule layer is just the product of membership functions. The number of rules, i.e. R 1 , R 2 , . . .R n in this layer are equal to number of nodes and each node corresponds to one fuzzy rule. Min operator is used to compute the output signal's value in each rule. Each O 3 i ðkÞ is the input for the next (consequent) layer.
Input is : Output is : • Layer 4 (Consequent Layer): It determines the difference in the proposed control techniques. The general description of this layer is; Input is : where H i is the output of the hidden layer (previous values of mathematical function), and F i is the feedback weight and the superscript (4) denotes the layer 4. Each mathematical function is multiplied by the weight, w i of the NNs in fourth layer.
Output is : where 9 i ðI 4 i Þ represents one of the used mathematical function discussed in section 2.2.2.

• Layer 5 (Defuzzification Layer 1):
The product of the outputs of antecedent and consequent parts for each input is taken in this layer and then added to each other.
Output is : • Layer 6 (Defuzzification Layer 2): In this layer the summation of rules (output of antecedent) part is calculated.
Output is : • Layer 7 (Output Layer): The final output of the FRANF identifier is estimated in the seventh layer as given below: Output is :
The general equation is given as follows: where, g k is the gradient of cost function at kth iteration, γ > 0 is the learning rate and k is the iteration index. The update equations for antecedent part and consequent part are as follows: 2.2.4.1 Update equations for antecedent part parameters. The update equation for variants of Gaussian membership function is derived from following chain rule [39][40][41][42][43][44].
where, χ shows the variant like mean, variance and feedback weight of Gaussian membership function.
Update equation for mean, m i is; Update equation for variance, σ i is; Update equation for recurrent weight, θ i is; where 9 i represents one of the mathematical functions discussed in section 2.2.2 and $ shows the variants of mathematical function like centroid, dilation, translation and volume etc. Update equation for centroid, c i of SAM is; where, w i is the adaptive weight of consequent part obtained by Eq (51), w i (k) and w l (k) are the weights of SAM. Update equation for volume, V i of SAM is; Update equation for weight, w i (k) of FsNNs is; where, w i is the weight of consequent part from Eq (51), w i (k) is the weight of FsNNs. Chain rule for updating variants of MHW used in this research work is: where here C i represents MHW, Z i defined below is an intermediate state variable and $ shows the variants of MHW like, translation, dilation, feedback weight etc.
Update equation for translation, t i is; Update equation for dilation, d i is; Update equation for feedback weight, F i of MHW is; The update equations of CW are applied according to the equations given in section 2.2.2.3. Weight of consequent part are being updated according to the following chain rule: where w i represents the weight of the consequent layer.
The feedback weight of antecedent part of all proposed controllers is adaptive while feedback weight of consequent part of only Mexican hat wavelet is adaptive. The feedback weight of rest of the consequent parts of other controllers is closed-loop fixed gain.

Adaptive Feedback Linearization (FBL) embedded with Full Recurrent Adaptive NeuroFuzzy (FRANF) Standard Additive Model (SAM) control.
This controller is based on the FBL scheme which is embedded with a FRANF-SAM identifier to identify unknown nonlinear functions for FBL. The antecedent part of FRANF is modeled using Gaussian membership function while the consequent part is modeled using SAM as discussed in section 2.2.2.1 and section 2.2.3. The overall explicit control law for this control scheme is given in Eq (53).

Adaptive Feedback Linearization (FBL) embedded with Full Recurrent Adaptive NeuroFuzzy (FRANF) Fourier Series (FS) control.
This controller is based on the FBL scheme which is embedded with a FRANF-FS identifier to estimate unknown nonlinear functions for FBL. The antecedent part of FRANF is modeled using Gaussian membership function while the consequent part is modeled using FS as discussed in section 2.2.2.2 and section 2.2.3.
The overall explicit control law for this control scheme is given in Eq (54).

Adaptive Feedback Linearization (FBL) embedded with Full Recurrent Adaptive NeuroFuzzy(FRANF) Mexican Hat Wavelet (MHW) control.
This controller is based on the FBL scheme which is embedded with a FRANF-MHW identifier to identify unknown nonlinear functions for FBL. The antecedent part of FRANF is modeled using Gaussian membership function while the consequent part is modeled using MHW as discussed in section 2.2.2.3 and section 2.2.3. The overall explicit control law for this control scheme is given in Eq (55).

Adaptive Feedback Linearization (FBL) embedded with Full Recurrent Adaptive NeuroFuzzy (FRANF) Chebyshev Wavelet (CW) control.
This controller is based on the FBL scheme which is embedded with a FRANF-CW identifier to identify unknown nonlinear functions for FBL. The antecedent part of FRANF is modeled using Gaussian membership function while the consequent part is modeled using CW as discussed in section 2.2.2.3 and section 2.2.3. The overall explicit control law for this control scheme is given in Eq (56).
Results and discussion The SMG system is developed in Matlab/Simulink R2015a for evaluation of the performance of the proposed controllers. The system is developed by using power generation sources like UG, PV, WT, electrolyzer, SOFC, MT, and the backup sources i.e. SC and batteries whose ratings are given in Table 1. The purpose of using multiple sources is to entertain the dynamic residential load and CS load. Intelligent supervisory control is an essential part of the SMG system. It monitors the power generation and load variations during the simulation time. The supervisory control ensures the power consumption from renewable resources at a priority level during their availability period. It satisfies the load demand by shifting the load to other sources and UG during peak hours and in the absence of renewable power.
Intelligent control schemes like Adaptive FBL embedded FRANF-SAM, Adaptive FBL embedded FRANF-FS, Adaptive FBL embedded FRANF-MHW, Adaptive FBL embedded FRANF-CW are used to extract the maximum power from PV system connected in the microgrid. The performance of all the proposed control schemes is compared to aPID based MPPT control scheme.

Case studies
Three different cases are taken in this research work, e.g., (a) Step change in both solar irradiation and temperature; (b) Partial shading condition; and (c) Daily field data of solar irradiation and temperature in Islamabad, to evaluate the performance of proposed adaptive FBL embedded FRANF controllers under PSC compared to aPID control scheme.

Step change in both solar irradiation and temperature.
Step-changing solar irradiation and temperature profile are simulated for 24 seconds, where each second represents one hour. The solar irradiation is gradually increased in many steps to its maximum and then gradually decreased to zero. Figs 8 and 9 shows the step profile of solar irradiation and ambient temperature used for this case study. Fig 10 shows the output power comparison of all intelligent proposed control schemes with aPID. Every step change is accurately determined by the proposed control schemes and the each proposed control scheme generates more output power compared to aPID. Furthermore, it is also clear that the power generated by three proposed controllers, i.e., adaptive FBL embedded FRANF-CW, adaptive FBL embedded FRANF-FS, and adaptive FBL embedded FRANF-SAM, seems to be very near to each other. However, the power generated by adaptive FBL embedded FRANF-MHW is maximum than other proposed schemes. Power error (P error ) can be described as: where P ref is the reference power and P PV is tracked output power obtained from the PV system under the action of applied controllers. The maximum and average power error of all the control schemes is given in Table 2.  Table 3 shows the values of various indexes of all the proposed controllers compared to aPID showing least to most performing from top to bottom. The switching in the converter circuits arises harmonics in the load voltage and current that are not synchronized with the frequency of the system. It wastes power as heat and should be minimized. Voltage fluctuations and flickers are caused by higher frequency harmonics. Fig 15 shows a comparison of the percentage change in total harmonic distortion (THD) for load current due to individual control scheme. The result shows that the percentage change in THD due to the Adaptive FBL embedded FRANF-MHW control scheme is the smallest of all and proves its better performance among other proposed controllers. The percentage change in frequency is also shown in Fig 16 for all proposed controllers. It can be observed that the percentage change in frequency due to the adaptive FBL embedded FRANF-MHW control scheme is almost flat and nearly zero.  3.1.2 Partial shading condition. PSCs arise due to moving clouds, airplanes, dust, and shadows of the building. They cause a sudden drop in irradiation level as well as temperature ranging from a short interval of time to many hours. To evaluate the performance of proposed controllers for this type of phenomenon, PSCs are introduced in solar and temperature profiles by introducing multiplying factors at certain intervals of time as shown in Fig 17. This suddenly changes the magnitude of irradiation level, as well as temperature, thus produce fluctuations in irradiation and temperature curve as shown in Figs 18 and 19. Fig 20 shows the output power tracked by all the proposed control schemes compared with aPID. The sudden drops in irradiation and temperature level are successfully adopted by all the proposed control schemes, whereas, aPID produces the least accurate results. It is obvious to note that the output power produced using FRANF-CW, FRANF-SAM, and FRANF-FS is better than each other but the performance of FRANF-MHW is superior over all the other proposed control schemes. The maximum and average power errors of all the control schemes calculated according to Eq (57) are given in Table 4.  Table 5. Fig 25 shows a comparison of the percentage change in total harmonic distortion (THD) for load current for all control schemes. It is obvious to note that the percentage change in THD due to the adaptive FBL embedded FRANF-MHW control scheme is the smallest

PLOS ONE
compared to the rest of the proposed controllers. The percentage change in frequency is also shown in Fig 26 for all proposed controllers. It can be observed that the percentage change in frequency due to adaptive FBL embedded FRANF-MHW control scheme is the least among all the proposed schemes.

PLOS ONE
changes in the irradiance level under PSCs that causes a control error. This initiates the controller which adjusts the duty cycle to minimize the power error. Fig 30 also shows that the maximum power error generated by the least performing controller adaptive FBL embedded FRANF-SAM is not exceeding 31.3 kW at its highest peak, whereas the maximum power error generated by the best performing controller i.e. Adaptive FBL embedded FRANF-MHW is below 21.9 kW at its highest peak. However, aPID produced a maximum power error of 58kW at its highest peak. The maximum peak and average power error of all the control schemes starting from the least performance to the best is shown in Table 6.
The output power comparison of all proposed controllers and reference power is shown in Fig 31. For every sharp change in reference power, there is also a sharp spike in tracked power at the same instant of time. This proves the ability of proposed controllers in dealing with sharp sudden changes in the reference signal. The only difference in the performance of the proposed controllers is the net output generated power at the same instant of time. It can be seen clearly that adaptive FBL embedded FRANF-SAM and adaptive FBL embedded

PLOS ONE
FRANF-CW are producing low output power as compared to adaptive FBL embedded FRANF-FS and Adaptive FBL embedded FRANF-MHW. This is because of the different consequent structures used in FRANF. The P PV−FB−MHW acquires the PV system output power with steady-state error = 0.3 kW, undershoot = -9.19% and overshoot = 4.39%. The P PV−FB−FS extracts the PV system output power with steady-state error = 11.5 kW, undershoot = -20.95% and overshoot = 2.48%. The P PV−FB−SAM obtained the PV system output power with steady state-error = 18.1 kW, undershoot = -26.99% and overshoot = 1.16%. The P PV−FB−CW obtained the PV system output power with steady state-error = 18.4 kW, undershoot = -28.33% and overshoot = 1.49%.
The dynamic efficiency η PV of various controllers on the basis of their tracked powers and power error are calculated as: where; P PV = V PV × I PV , t o = 0 h and t f = 24 h are initial and final intervals respectively. The dynamic efficiency, η PV of the proposed controllers are shown in Fig 32. The peak η PV of aPID is 75.56%, adaptive FBL embedded FRANF-SAM is 87.1%, adaptive FBL embedded FRANF-CW is 86.4%, adaptive FBL embedded FRANF-FS is 93.1% and adaptive FBL embedded FRANF-MHW is 95.9%. The efficiency plot of all the controllers shows a small variation. This is due to continuous change in the reference signal according to the solar irradiation at that instant of time and shows the high sensitivity of the proposed controllers. The peak and average efficiency of all controllers starting from the least to the most efficient are shown in Table 7. Performance indexes including Integral Square Error (ISE), Integral Time Square Error (ITSE), Integral Absolute Error (IAE), and Integral Time Absolute Error (ITAE), calculated based on P error in Eq (57) as shown in Figs 33, 34, 35 and 36. The comparison of performance indexes plots shows that the accumulative error in all schemes increases with time. Again it is clear that the index of adaptive FBL embedded FRANF-MHW is least among all proposed controllers. Table 8 shows the values of various indexes of all the proposed controllers in contrast with aPID showing least to most performing from top to bottom. Fig 37 shows a comparison of the percentage change in total harmonic distortion (THD) for load current due to individual control scheme. The percentage change in THD complies with the IEEE standard 1547 [45]. The result shows that the percentage change in THD due to the adaptive FBL embedded FRANF-MHW control scheme is the smallest of all and proves its better performance among other proposed controllers. The percentage change in frequency is also shown in Fig 38 for all proposed controllers. It can be observed that the percentage change in frequency due to the adaptive FBL embedded FRANF-MHW control scheme is almost flat and nearly zero. The percentage change in frequency for all the proposed controllers is within the acceptable range according to the IEEE standard 1547 [45].
Another factor of comparison is the percentage change in voltage (V RMS ) produced in the AC-bus during the conversion process. Fig 39 shows  proposed scheme. It can be seen that the percentage change in V RMS due to adaptive FBL embedded FRANF-MHW control scheme has the least value among all proposed control scheme and thus ensures its superior performance. The overall performance of all the described controllers is compared based on power quality, average and peak power error in MPPT, efficiency, performance indexes, percentage THD change, percentage frequency change, and percentage V RMS in load current. For daily field data, a spider chart is plotted for all comparable performance parameters in

Conclusions
In this article, four intelligent control schemes are presented for the MPPT problem of a gridconnected PV subsystem in an SMG-HPS for three different solar and temperature profiles. Results of all the control schemes are compared against one another and aPID for various parameters obtained through simulations. The performance of all the proposed control schemes is within the acceptable range and simply cannot be rejected at any ground. However, the overall analysis shows the performance of the adaptive FBL embedded FRANF-MHW is superior to all the other proposed control schemes. This is due to the use of a continuous signal wavelet that is the MHW in the consequent part of the identifier as well as the adaptive recurrent weights in antecedent and consequent parts of the same identifier. On the other hand, the CW is a discrete composite mathematical function and the recurrent weight of the consequent part of this and other proposed identifier schemes are fixed gains that are not being updated during the simulation. Future studies include testing of adaptive FBL embedded FRANF hybrid wavelet control compared to those presented in this article.