A Novel Approach for an MPPT Controller Based on the ADALINE Network Trained with the RTRL Algorithm

energies

Article A Novel Approach for an MPPT Controller Based on the ADALINE Network Trained with the RTRL Algorithm

Julie Viloria-Porto *, Carlos Robles-Algarín * and Diego Restrepo-Leal *

Facultad de Ingeniería, Universidad del Magdalena, Santa Marta 470003, Colombia * Correspondence: [email protected] (J.V.-P.); [email protected] (C.R.-A.); [email protected] (D.R.-L.); Tel.: +57-5-421-7940 (C.R.-A.)

Received: 9 November 2018; Accepted: 27 November 2018; Published: 5 December 2018

Abstract: The Real-Time Recurrent Learning Gradient (RTRL) algorithm is characterized by being an online learning method for training dynamic recurrent neural networks, which makes it ideal for working with non-linear control systems. For this reason, this paper presents the design of a novel Maximum Power Point Tracking (MPPT) controller with an artiﬁcial neural network type Adaptive Linear Neuron (ADALINE), with Finite Impulse Response (FIR) architecture, trained with the RTRL algorithm. With this same network architecture, the Least Mean Square (LMS) algorithm was developed to evaluate the results obtained with the RTRL controller and then make comparisons with the Perturb and Observe (P&O) algorithm. This control method receives as input signals the current and voltage of a photovoltaic module under sudden changes in operating conditions. Additionally, the efﬁciency of the controllers was appraised with a fuzzy controller and a Nonlinear Autoregressive Network with Exogenous Inputs (NARX) controller, which were developed in previous investigations. It was concluded that the RTRL controller with adaptive training has better results, a faster response, and fewer bifurcations due to sudden changes in the input signals, being the ideal control method for systems that require a real-time response.

Keywords: RTRL algorithm; MPPT controller; PV module; dynamic neural network controller

1. Introduction Due to the negative impact caused by the use of conventional energy sources, environmental problems such as pollution, and poor quality in the provision of primary energy service, renewable energy is presented as an excellent alternative to reduce dependence on conventional electrical network. The notable increase in the use of renewable energies as an alternative means of generating and storing clean energy is considered worldwide as one of the technological solutions to mitigate climate change. Internationally it is recommended that in places with high solar density, the use of photovoltaic (PV) energy sources be chosen, since this type of energy provides an inexhaustible energy resource [1]. In the global report presented in 2017 by the International Agency for Renewable Energy (IRENA), it is evident that solar energy is a leader in power generation capacity due to its competitive costs in many emerging markets of the world [2]. In order to improve competitiveness, increase demand, and obtain the highest performance, the elements of a PV system must be optimized. There are several techniques to increase the performance of PV systems, among which solar trackers [3–7], hybrid systems [8,9], and controllers for the maximum power point tracking (MPPT) stand out. With the MPPT controllers, better performance of the PV systems is obtained, since they take advantage of all the current generated by the solar panels independently of the voltage. These controllers are more expensive than Pulse-Width Modulation

Energies 2018, 11, 3407; doi:10.3390/en11123407 www.mdpi.com/journal/energies Energies 2018, 11, 3407 2 of 17

(PWM) controllers, but due to their high efficiency they are ideal for working in medium and large solar power plants [10]. In these controllers, the perturb and observe (P&O) algorithm has traditionally been used. This algorithm is characterized by its low efficiency and presenting oscillations around the operating point. The presence of oscillations has contributed to the researchers focusing their efforts on designing MPPT controllers [10] with better performance. In this way, investigations have been carried out with genetic algorithms [11], fuzzy logic [12–17], golden section [18,19], simulated annealing [20], artificial bee colony [21], adaptive control [22], glowworm swarm optimization [23], ant colony optimization [24], artificial neural networks [25–30], and numerical methods [31]. Taking into account the previous context, this paper proposes as a novelty the design of an MPPT neuro-controller based on an adaptive neural network (ADALINE) with Real-Time Recurrent Learning (RTRL) [32]. We decided to explore the application of this method in order to design a robust controller that adapts to sudden climatic changes to which a PV module will be exposed. Specifically, we highlight as a contribution the application of the RTRL algorithm in an MPPT controller, which was developed with a single neuron and without using the bias. In addition, this training algorithm was programmed online with the Simulink function blocks without using the MATLAB Neural Network Toolbox (R2017b Version 9.3). In this way, it was possible to develop an efficient controller with excellent computing times. The results obtained show the advantage of the use of an online learning method in an artificial neural network (ANN), with which the volume of data to be acquired and the times of capture and conditioning of the data are reduced. In addition, the controller is more flexible since it has the ability to learn when there are sudden changes in the operating conditions of the PV system. It is noteworthy that the neurocontroller developed has a single neuron, which is simpler to implement compared to controllers based on nonlinear autoregressive network with exogenous inputs (NARX) [25], since these networks have at least two layers and a feedback between the output layer and the input layer. Additionally, the neuro-controller proposed in this paper is more efficient than those implemented with multilayer perceptron (MLP) [33]. This is one of the most popular architectures of neural networks and usually has at least one hidden layer. In order to evaluate the robustness of the controller implemented with the RTRL algorithm and compare it with other methods, the algorithms least mean square (LMS) [34] (standard and iterative version) and the traditional P&O were implemented. Additionally, the results were compared with previous results of the Magma Ingeniería research group with a fuzzy controller [12,35] and a controller based on a NARX network [25]. For this, modeling and simulation of the PV module, the DC-DC converter, and the MPPT algorithms was carried out in MATLAB/Simulink. It should be noted that this research is part of a set of intelligent control techniques developed in the Magma Ingeniería research group, with the aim of implementing a cost-effective MPPT controller with high efficiency, precision, and low computational cost; that can initially be used in the photovoltaic systems of the Universidad del Magdalena in Santa Marta, Colombia. The modeling of the PV system is presented in Section2. Section3 presents the simulation results with the MPPT controllers. Finally, the conclusions are presented in Section4.

2. Modeling of the PV System

2.1. FV Module To simulate the PV module, the mathematical model proposed in [36,37] was used. From this model, it is possible to ﬁnd the curve ﬁtting parameter (b), directly from the I-V ratio of the PV module shown in Equation (1). Ix h ( V − 1 )i I(V) = 1 − e bVx b (1) ( −1 ) 1 − e b

The variables Vx and Ix must be obtained from Equations (2) and (3). Energies 2018, 11, 3407 3 of 17 EnergiesEnergies 20182018,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 33 ofof 1717

E퐸i푖 푉V푚푎푥max−V푉oc표푐 Ei퐸푖 (( 퐸푖 ln|| 푉푚푎푥−푉표푐||)) V 푉==s 푠 퐸TC푖 푇퐶(T(푇−−T푇 ))++sV푠푉 −−s푠((V푉 − V푉 )푒e (E퐸iN푖푁 ln|푉V푚푎푥max−푉Vmin푚푖푛 |) (2) x 푉푥 = 푠 푇퐶V 푉 (푇 − 푇N푁 ) + 푠푉max푚푎푥 − 푠(푉푚푎푥max − 푉푚푖푛min )푒 퐸푖푁 푉푚푎푥−푉푚푖푛 ((22)) 푥 EiN퐸푖푁 푉 푁 푚푎푥 푚푎푥 푚푖푛 퐸푖푁

Ei퐸푖 Ix = p 퐸[푖Isc[ + TCi(T − TN)]] (3) 퐼퐼푥 == 푝푝 [퐼퐼푠푐 ++푇퐶푇퐶푖((푇푇 −−푇푇푁))] ((33)) 푥 EiN퐸푖푁 푠푐 푖 푁 퐸푖푁 2 where,where, Ei is thethe solarsolar irradiance,irradiance, EEiNiN is anan irradianceirradiance constantconstant ofof 10001000 W/mW/m2,, Iscsc isis the the short circuit where, Ei is the solar irradiance, EiN is an irradiance constant of 1000 W/m2, Isc is the short circuit current, p is the number of PV modules in parallel, s is the number of PV modules in series, T is the current,current, pp isis thethe numbernumber ofof PVPV modulesmodules inin parallel,parallel, ss isis thethe numbernumber ofof PVPV modulesmodules inin series,series, TT isis thethe operating temperature, TCii and TCvv areare the current and voltage temperature coefﬁcients,coefficients, TN is the operating temperature, TC◦i and TCv are the current and voltage temperature coefficients, TN is the temperature constantconstant ofof 2525 °C,C, VVococ isis thethe openopen circuitcircuit voltage,voltage,V Vmaxmax isis 103%103% ofof VVococ, and Vmin isis the the 85% 85% temperature constant of 25 °C, Voc is the open circuit voltage, Vmax is 103% of Voc, and Vmin is the 85% of Vococ. . of Voc. The electrical parameters were taken from the data sheet of the 65 W PV module (Yingli Solar TheThe electricalelectrical parametersparameters werewere takentaken fromfrom thethe datadata sheetsheet ofof thethe 6565 WW PVPV modulemodule (Yingli(Yingli SolarSolar JSJS JS 65). From these data, the value of b = 0.0737561 was obtained [25]. The mathematical modeling of 65).65). FromFrom thesethese data,data, thethe valuevalue ofof bb == 00..07375610737561 waswas obtainedobtained [[2525].]. TheThe mathematmathematicalical modelingmodeling ofof thethe the PV module was carried out in the MATLAB/Simulink software. PVPV modulemodule waswas carriedcarried outout inin thethe MATLAB/SimulinkMATLAB/Simulink software.software. 2.2. DC-DC Buck Converter 2.2.2.2. DCDC--DCDC BuckBuck ConverterConverter As a control device, a DC-DC Buck converter was designed, in which the output voltage V is As a control device, a DC-DC Buck converter was designed, in which the output voltage Vo is As a control device, a DC-DC Buck converter was designed, in which the output voltage Vo is controlled with a PWM signal. Figure1 shows the design of the DC-DC Buck converter developed in controlledcontrolled withwith aa PWMPWM signasignall.. FigureFigure 11 showsshows thethe designdesign ofof thethe DCDC--DCDC BuckBuck converterconverter developeddeveloped inin MATLAB/Simulink with the SimPowerSystem toolbox. MATLAB/SimulinkMATLAB/Simulink withwith thethe SimPowerSystemSimPowerSystem toolboxtoolbox..

FigureFigure 1.11.. ElectricalElectrical modelmodel ofof the the dc-dcdcdc--dcdc converterconverter designeddesigned inin MATLAB/Simulink.MATLAB/SimulinkMATLAB/Simulink.. 2.3. MPPT Controller 2.3.2.3. MPPTMPPT ControllerController The MPPT controllers guarantee the highest available power in a PV system [30], by modifying TheThe MPPTMPPT controllerscontrollers guaranteeguarantee thethe highesthighest availableavailable powerpower inin aa PVPV systemsystem [30][30],, byby modifyingmodifying the percentage of the duty cycle of the PWM signal, which is the input of the dc-dc converter. In this thethe percentagepercentage ofof thethe dutyduty cyclecycle ofof thethe PWMPWM signal,signal, whichwhich isis thethe inputinput ofof thethe dcdc--dcdc converter.converter. InIn thisthis way, all the controllers were connected individually between the PV module and the dc-dc converter way,way, allall thethe controllerscontrollers werewere connectedconnected individuallyindividually betweenbetween thethe PVPV modulemodule andand thethe dcdc--ddcc converterconverter as shown in Figure2. asas sshownhown inin FigureFigure 22..

Figure 2. Components of the PV system. Figure 2.2. Components of the PV system.

2.3.1.2.3.1. P&O P&O Algorithm Algorithm TheThe P&O P&O algorithm algorithm consists consists of of performing performing a a disturbance, disturbance, modifying modifying the the voltage voltage and and observing observing thethe resulting resulting variation variation in in power power [27 [[2727].]. In In this this algorithm, algorithm, the the duty duty cycle cycle ( (DD)) of of the the DC DC-DCDC--DCDC converter converter is is adjustedadjusted depending depending onon the the value value of of the the power power change change ( (dPdP).). Figure Figure3 33 shows showsshows the thethe ﬂow flowflow diagram diagramdiagram that thatthat was waswas designeddesigned for for this this algorithm. algorithm.

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Figure 3. Flowchart of the P&O algorithm. Figure 3. Flowchart of the P&O algorithm.

2.3.2.2.3.2. RTRL RTRL Algorithm Algorithm TheThe intelligent intelligent control control technique technique for for the the MPPT MPPT proposed proposedin inthis this work,work, isis basedbased on the design o off a neurala neural network network type type ADALINE, ADALINE, with with FIR architecture,FIR architecture, and and trained trained with with the RTRLthe RTRL algorithm algorithm in order in to minimizeorder to minimize the oscillations the oscillations of the P&O of the algorithm. P&O algorithm. FIR architecture FIR architecture is characterized is characterized by being by being easy to easy to design, configure, and implement. design, configure, and implement. To use the ADALINE neural network as an adaptive filter, it was necessary to introduce a line To use the ADALINE neuralFigure network 3. Flowchart as an adaptive of the P&O filter, algorithm. it wasnecessary to introduce a line of of delays (D) in order to modify the impact that the synaptic weights generate on the network delays (D) in order to modify the impact that the synaptic weights generate on the network response. 2.3.2.res ponse.RTRL AlgorithmThe RTRL algorithm is developed for a single neuron as can be seen in Figure 4. For the The RTRL algorithm is developed for a single neuron as can be seen in Figure4. For the voltage and voltage and current signals that were used, it is not necessary to use the bias, since there is no currentdisplacementThe signals intelligent that through were control the used, hyperplane.technique it is not necessaryfor Equation the MPPT to (4 use) representsproposed the bias, tin sincehe this output there work, of is the nois based displacementnetwork. on the design through of thea neural hyperplane. network Equation type ADALINE, (4) represents with the FIR output architecture, of the network. and trained with the RTRL algorithm in order to minimize the oscillations of the P&O algorithm. FIR architecture is characterized by being easy to design, configure, and implement. 2 a(t) = n(t) = ∑ W(d)p(t − d) (4) To use the ADALINE neural network as and =adaptive0 filter, it was necessary to introduce a line of delays (D) in order to modify the impact that the synaptic weights generate on the network whereresponse.p(t) Therepresents RTRL algorithm the inputs is of developed the network for overa single time, neuronW is theas can vector be ofseen synaptic in Figure weights, 4. For a(t)the isvoltage the output and current of the neural signals network, that weren(t) used,is the it weighted is not necessary sum of the to useinputs the and bias, weights since there before is the no activationdisplacement function, through and thed indicates hyperplane. the delayEquation time (4 with) represents respect tothe the output first inputof the data. network.

Figure 4. Block diagram of the ADALINE network with FIR architecture.

2 푎(푡) = 푛(푡) = ∑ 푊(푑)푝(푡 − 푑) (4) 푑=0 where p(t) represents the inputs of the network over time, W is the vector of synaptic weights, a(t) is the output of the neural network, n(t) is the weighted sum of the inputs and weights before the activation function, and d indicates the delay time with respect to the first input data. In developing Equation (4), the network response can be obtained from the first instant of time. FigureFigure 4.4. BlockBlock diagramdiagram ofof thethe ADALINEADALINE networknetwork withwith FIRFIR architecture.architecture. The sum of the quadratic error F(x) represents the standard performance function that was chosen for the training. The gradient was calculated from this performance function [32]. See Equation (5). In developing Equation (4), the network response2 can be obtained from the ﬁrst instant of time. 푄 푄 The sum of the quadratic error F(x) represents푎(푡) = 푛(푡 the) = standard∑ 푊(푑) performance푝(푡 − 푑) function that was chosen for(4) 2 2 the training. The gradient was calculated퐹(푥) = from∑ 푒 this(푡) 푑= performance=0∑(푡(푡) − 푎( function푡)) [32]. See Equation (5). (5) 푡=1 푡=1 where p(t) represents the inputs of the network over time, W is the vector of synaptic weights, a(t) is where x is the vector that contains the parametersQ of Qthe network (synaptic weights), Q is the number the output of the neural network,F(x n(t)) = is thee2( t weighted) = (t( t sum) − a of(t)) the2 inputs and weights before the(5) of samples, e2(t) is the mean square error,∑ and t(t) are∑ the target values of the duty cycle. activation function, and d indicates the delayt=1 time witht=1 respect to the first input data.

In developing Equation (4), the network response can be obtained from the first instant of time. The sum of the quadratic error F(x) represents the standard performance function that was chosen for the training. The gradient was calculated from this performance function [32]. See Equation (5). 푄 푄 퐹(푥) = ∑ 푒2(푡) = ∑(푡(푡) − 푎(푡))2 (5) 푡=1 푡=1 where x is the vector that contains the parameters of the network (synaptic weights), Q is the number of samples, e2(t) is the mean square error, and t(t) are the target values of the duty cycle.

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EnergiesEnergies 2018 2018, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 55 ofof 1717 where x is the vector that contains the parameters of the network (synaptic weights), Q is the number of samples,TheThe execution executione2(t) is the of of mean the the RTRL square RTRL algorithm error, algorithm and t(t) advances advancesare the target very very rapidlyvalues rapidly of over over the duty time, time, cycle. since since it it allows allows simultaneoussimultaneousThe execution calculationcalculation of the RTRL ofof thethe algorithm networknetwork advancesandand thethe calculationcalculation very rapidly ofof over thethe time,explicitexplicit since derivativesderivatives it allows simultaneous ofof EquationEquation (calculation(66),), which which is is ofsolved solved the networkwith with the the help andhelp theof of the the calculation chain chain rule. rule. of The The the development development explicit derivatives o off this this equation equation of Equation has has some some (6), which aspects aspects is similarsolvedsimilar towithto thethe the backpropagationbackpropagation help of the chain algorithmalgorithm rule. The usedused development inin staticstatic multilayermultilayer of this equation netwnetworksorks has [32[32 some].]. uu aspects isis thethe numbernumber similar of toof layersthelayers backpropagation inin thethe network.network. algorithm used in static multilayer networks [32]. u is the number of layers in 푄푄 the network. 푢푢 푇푇 푒푒 휕퐹휕퐹 Q " 휕휕푎u푎 ((푡푡))T 휕휕e 퐹퐹 # ∂F ==∑∑∑∑[[[∂[a (t) ]] ×× ∂ F ]] ((66)) 휕푥휕푥= 휕휕푥푥푇푇 × 휕휕푎푎푢푢((푡푡)) (6) ∑푡=1∑푢∈푈 T u( ) ∂x t=1푡=u∈1U푢∈푈 ∂x ∂a t Finally,Finally, withwith thethe the developmentdevelopment development ofof of EquationEquation Equation ((77) (7)) thethe the synapticsynaptic synaptic weightsweights weights obtainedobtained obtained inin thethe in trainingthetraining training areare updated.areupdated. updated. ∂F ( )( + ) = ( )( ) − 휕퐹휕퐹 www(d(푑푑))t((푡푡++111))==www(d(푑푑))t((푡푡))−−α훼훼 ((7)(77)) ∂w휕휕w(wd(()푑푑)) where α is the dynamic learning rate. wherewhere αα isis thethe dynamicdynamic learninglearning rate.rate. Figure5a shows the ﬂow diagram of the RTRL algorithm used to train the dynamic network. FigureFigure 55aa showsshows thethe flowflow diagramdiagram ofof thethe RTRLRTRL algorithmalgorithm usedused toto traintrain thethe dynamicdynamic network.network. The behavior of the dynamic neural controller is shown in Figure5b. TheThe behaviorbehavior ofof thethe dynamicdynamic neuralneural controllercontroller isis shownshown inin FigureFigure 55b.b.

(a)(a) (b)(b)

FigureFigure 5 5 5... Flow Flow diagrams:diagrams: ((a (aa)) ) RTRLRTRL RTRL algorithm;algorithm; algorithm; (( bb)) RTRLRTRL Neurocontroller. Neurocontroller.

FigureFigure 6 6 shows shows the thethe blockblock diagramdiagram ofof thethe RTRLRTRL neurocontrollerneurocontroller designed designeddesigned in inin MATLAB/Simulink, MATLAB/Simulink,MATLAB/Simulink, inin which which it it is is emphasizedemphasized thatthat itit waswas notnot necessarynecessary toto useuse thethe the NeuralNeural Neural NetworksNetworks Networks ToolboxToolbox Toolbox to to design design the the controlcontrol system.system. TheTheThe block blockblock “Error” ““ErrorError” of” ofof Figure FigureFigure6 is 66 responsible isis responsibleresponsible for ﬁltering forfor filteringfiltering the oscillations thethe oscillationsoscillations that may thatthat occur maymay occurwhenoccur when tryingwhen tryingtrying to track toto thetratrackck MPP. thethe MPP.MPP.

FigureFigure 6 6.6.. BlockBlock diagram diagram of of the the RTRL RTRL Neurocontroller Neurocontroller designeddesigned inin MATLAB/Simulink.MATLAB/Simulink.

2.3.3.2.3.3. LMSLMS AlgorithmAlgorithm WidrowWidrow andand MarcianMarcian HoffHoff developeddeveloped thethe ADALINEADALINE neuralneural networknetwork withwith theirtheir ownown learninglearning algoralgorithmithm (LMS:(LMS: LeastLeast MeanMean Square)Square) [32[32].]. ToTo taketake advantageadvantage ofof thisthis learninglearning methodmethod inin digitaldigital signalsignal

Energies 2018, 11, x FOR PEER REVIEW 6 of 17 processing, the two versions of the LMS algorithm (LMS and Iterative LMS) were implemented using Energies 2018 11 the same network, , 3407 architecture. 6 of 17 The design of the ADALINE network with LMS algorithm consists of a vector array of input 2.3.3.data, synaptic LMS Algorithm weights, and target, which are subsequently used in various matrix calculations. The construction of these matrices is key to determining the MSE (Mean Squared Error) and the final synapticWidrow weights and at Marcian the firstHoff instant developed time, since the the ADALINE mathematical neural process network is performed with their only own once. learning See algorithm (LMS: Least Mean Square) [32]. To take advantage of this learning method in digital signal Equations (8)–(10). processing, the two versions of the LMS algorithm (LMS and Iterative LMS) were implemented using 푝(푡) 푊(0) the same network architecture. 푝 = [ ] , 푊 = [ ] (8) 푝(푡 − 1) 푊(1) The design of the ADALINE network with LMS algorithm consists of a vector array of input data, synaptic weights, and target, which are subsequently used in various matrix calculations. 푋 = [푊], 푍 = [푝], 푎(1) = 푋푇푍 (9) The construction of these matrices is key to determining the MSE (Mean Squared Error) and the final synaptic weights at the first instant푐 = 퐸 time,[푡2], ℎ since= 퐸 the[푡푍] mathematical, 푅 = 퐸[푍푍푇] process is performed only once.(10) See Equations (8)–(10). " # " # The vector Z has the input values of thep(t )network, T is theW (transpose0) of the vector, c is a constant, p = , W = (8) h is the cross-correlation between the vectorp(t − of1 )inputs and theW (vector1) associated with the input t, and R is the correlation matrix. The input vectors determine if there is a single solution of the system [32]. However, it must be verified that theX = array[W], ofZ vectors= [p], a does(1) = notXT resultZ in a square matrix. Therefore,(9) for this training method the ADALINEh networki has only oneh delay.i = 2 = [ ] = T The training algorithms andc theE LMSt , controllersh E tZ , Rthat EwereZZ developed are shown in Figures(10) 7 and 8.The In vectorthese figuresZ has the it is input shown values that of these thenetwork, controllersT is were the transposetrained with of the the vector, standardc is aprocedure constant, hdefinedis the cross-correlation through their operating between equations. the vector of inputs and the vector associated with the input t, and R is theAs correlation seen in Figures matrix. 7 The and input 8, the vectors LMS algorithms, determineif in there their is two a single versions solution, have of in the common system [32 the]. However,procedure itused must to beminimize verified the that MSE the with array iterations. of vectors For does this, not th resulte steepest in a descent square matrix.algorithm Therefore, is used, forwith this which training the gradient method thecan ADALINEbe calculated, network and the has performance only one delay. index analyzed. TheDuring training each iteration algorithms k the and synaptic the LMS weights controllers are adjusted, that were and developed the error is are obtained shown. inThis Figures is done7 anduntil8 .the In function these figures converges. it is shown See Equati that theseon (11 controllers). were trained with the standard procedure defined through their operating equations. | 푥푘+1 = 푥푘 − 훼∇F(x) 푥=푥푘 (11)

LMS

(a) (b)

Figure 7. Flow diagrams: ( a)) LMS LMS Algorithm; Algorithm; ( (b)) LMS Controller Controller..

As seen in Figures7 and8, the LMS algorithms, in their two versions, have in common the procedure used to minimize the MSE with iterations. For this, the steepest descent algorithm is used, with which the gradient can be calculated, and the performance index analyzed. During each iteration k the synaptic weights are adjusted, and the error is obtained. This is done until the function converges. See Equation (11).

x = x − α∇F(x)| (11) k+1 k x=xk

Figure9 corresponds to the LMS subsystem and Figure 10 corresponds to the Iterative LMS subsystem, which were designed in MATLAB/Simulink. The architecture of the ADALINE neural EnergiesEnergies 2018 2018, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 77 ofof 1717 Energies 2018, 11, 3407 7 of 17 network was built with operational blocks, such as, products, addition, and subtraction; while the training algorithms were programmed with the Simulink function blocks. Energies 2018, 11, x FOR PEER REVIEW 7 of 17

(b)(b)

(a) (a) FigureFigure 8 8. .Flow Flow diagrams: diagrams: ( (aa)) Iterative Iterative LMS LMS Algorithm; Algorithm; ( (bb)) Iterative Iterative LMS LMS Controller Controller. .

(b) FigureFigure 99 correspondscorresponds toto thethe LMSLMS subsystemsubsystem andand FigureFigure 1010 correspondscorresponds toto thethe IterativeIterative LMSLMS subsystem,subsystem, whichwhich werewere designeddesigned inin MATLAB/Simulink.MATLAB/Simulink. TheThe architecturearchitecture ofof thethe ADALINEADALINE neuralneural (a) networknetwork waswas builtbuilt withwith operationaloperational blocks,blocks, suchsuch as,as, products,products, additionaddition, , andand subtractisubtraction;on; whilewhile thethe trainingtraining algorithms algorithmsFigure 8.8were were. Flow programmed programmed diagrams: ( a) withIterative with the the LMS Simulink Simulink Algorithm; function function ( b) Iterative blocks. blocks. LMS Controller.Controller.

Figure 9 corresponds to the LMS subsystem and Figure 10 corresponds to the Iterative LMS subsystem, which were designed in MATLAB/Simulink. The architecture of the ADALINE neural network was built with operational blocks, such as, products, addition, and subtraction; while the training algorithms were programmed with the Simulink function blocks.

FigureFigure 9 9.9. .Block Block diagram diagram of of the the ADALINE ADALINE network network with with LMS LMS algorithm. algorithm.

Figure 9. Block diagram of the ADALINE network with LMS algorithm.

FigureFigure 10 10 10.. .Block Block diagram diagram of of the the ADALINE ADALINE network network with with Iterative Iterative LMS LMS algorithm. algorithm. 3. Results and Discussion 3.3. Results Results and and Discussion Discussion In order to evaluate the performance of the RTRL Neurocontroller, comparisons with the P&O InIn order order to to evaluate evaluate the the performance performance of of the the RTRL RTRL Neurocontroller Neurocontroller, ,comparisons comparisons with with the the P&O P&O and LMS algorithms were made. In addition, comparisons were established with the previous results andand LMS LMS algorithms algorithms were were made. made. In In addition, addition, comparisons comparisons were were established established with with the the previous previous results results obtained in theFigure Magma 10. Block Ingenier diagramía research of the ADALINE group with network a fuzzy with controller Iterative [LMS12] and algorithm. a controller with an obtainedobtained in in the the Magma Magma Ingeniería Ingeniería research research group group with with a a fuzzy fuzzy controller controller [12] [12] and and a a controller controller with with NARX network [25]. Finally, a set of tables with the results of the comparisons are presented. For each anan NARX NARX network network [25]. [25]. Finally, Finally, a a set set of of tables tables with with the the results results of of the the comparisons comparisons are are presented. presented. For For control3. Results method, and Discussion the maximum efﬁciency is calculated. eacheach control control method, method, the the maximum maximum efficiency efficiency is is calculated. calculated. It is important to highlight that the performance of the algorithms developed in this work will ItItIn is is order important important to evaluate to to highlight highlight the performance thatthat the the performance performance of the RTRL of of Neurocontroller the the algorithms algorithms developed ,developed comparisons in in this thiswith work work the P&Owill will be measured in terms of the power obtained by the controllers, the efﬁciency and the settling time. bebeand measuredmeasured LMS algorithms inin termsterms were of of thethe made. powpow erIner obtainedaddition,obtained bycomparisonsby thethe controllers,controllers, were established thethe efficiencyefficiency with and and the thethe previous settlingsettling results time. time. Additionally, the performance of all controllers will be evaluated according to the computation time Additionally,Additionally,obtained in the the the Magma performance performance Ingeniería of of all all research controllers controllers group will will with be be evaluated evaluated a fuzzy controller according according [12] to to the theand computation computation a controller time timewith necessary to perform the simulations. necessarynecessaryan NARX to tonetwork perform perform [25]. the the Finally,simulations. simulations. a set of tables with the results of the comparisons are presented. For each control method, the maximum efficiency is calculated. 3.1.3.1. Performance PerformanceIt is important of of the the to RTRL RTRL highlight Neurocontroller Neurocontroller that the performance of the algorithms developed in this work will be measured in terms of the power obtained by the controllers, the efficiency and the settling time. Additionally, the performance of all controllers will be evaluated according to the computation time necessary to perform the simulations.

3.1. Performance of the RTRL Neurocontroller

Energies 2018, 11, 3407 8 of 17

3.1. Performance of the RTRL Neurocontroller Energies 2018, 11, x FOR PEER REVIEW 8 of 17 EnergiesThe results 2018,, 11,, x ofx FORFOR the PEERPEER ADALINE REVIEWREVIEW neural network with the RTRL algorithm were compared8 of directly 17 with theThe P&O results and of LMS the ADALINE algorithms neural (LMS network and Iterative with the LMS)RTRL underalgorithm the were same compared conditions directly of solar The results of the ADALINE neural network with the RTRL algorithm were compared directly irradiance,with the temperature, P&O and LMS and algorithms simulation (LMS time, and Iterative in order LMS) to observe under the their same responses conditions and of solarevaluate with the P&O and LMS algorithms (LMS and Iterative LMS) under the same conditions of solar theirradiance, performance temperature of each, ofand these simulation methods. time, Forin order standard to observe test conditionstheir responses with and a solarevaluate irradiance the irradiance,irradiance,performance temperaturetemperature of each of ,, these and simulationmethods. For time, standard in order test to conditionsobserve their with resp a solaronses irradiance and evaluate of 1000 the of 1000 W/m2 and operating temperature of 25 ◦C the signals shown in Figure 11 were obtained. performanceW/m2 and operating of each temperatureof these methods. of 25 °CFor the standard signals testshowntest conditionsconditions in Figure withwith 11 were aa solarsolar obtained. irradianceirradiance ofof 10001000 W/m22 and operating temperature of 25 °C the signals shown in Figure 11 were obtained.

Figure 11. Results of the P&O, LMS, and RTRL controllers for 1000 W/m2 and 25 °C. Figure 11. Results of the P&O, LMS, and RTRL controllers for 1000 W/m2 and 25 ◦C. Figure 11.. Results of the P&O, LMS,, and RTRL controllers for 1000 W/m22 and 25 °C. In orderIn order to verifyto verify the the behavior behavior of of the the oscillations, oscillations, thethe simulationsimulation time time was was increased increased to to 1 1s sand and in in addition,InIn orderorder atoto signal verifyverify was thethe behaviorbehavior designed of of to thethe simulate oscillations,oscillations, the variable thethe simulationsimulation irradiance timetime conditions waswas increasedincreased represented toto 1 s and by addition, a signal was designed to simulate the variable irradiance conditions represented by Figure 12. ininFigure addition, addition, 12. In a aFigure signal signal 1 3 was was it can designed designed be seen tothat, to simulate simulate by increasing the the variable variable the simulation irradiance irradiance time, conditions conditions the oscillations represented represented continue by by In Figure 13 it can be seen that, by increasing the simulation time, the oscillations continue and the Figureand the 1 2P&O.. InIn Figure controller 13 itit does cancan be benot seen stabilize that, byin theincreasing MPP. the simulation time, the oscillations continue P&Oand controller the P&O doescontroller not stabilize does not instabilize the MPP. in the MPP.

Figure 12. Variable irradiance signal used for tests. FigureFigure 12. 12.. Variable Variable irradiance signal signal used used for for tests. tests.

Figure 13. Results of the P&O, LMS, and RTRL controllers under conditions of variable irradiance Figure 13. Results of the P&O, LMS, and RTRL controllers under conditions of variable irradiance FigureFigureand 13.constant 1Results3. Results temperature. of theof the P&O, P&O, LMS, LMS and, and RTRL RTRL controllers controllers under under conditions conditions ofof variablevariable irradianceirradiance and and constant temperature. constant temperature.

Energies 2018, 11, 3407 9 of 17

EnergiesEnergiesEnergies 2018 20182018, 11,, 1111, x,, xFORx FORFOR PEER PEERPEER REVIEW REVIEWREVIEW 99 9 ofof of 1717 17 After evaluating the performance of the controllers for a variable irradiance signal, the signal of FigureAfter 14AfterAfter was evaluating evaluatingevaluating designed the thethe to performance performanceperformance evaluate the of ofof performance the thethe controllers controllerscontrollers for forfor a aa variable variablevariable irradiance temperatureirradiance signal, signal,signal, signal. thethe the signalsignal signal The results ofof of FigureFigureFigure 1 4 11 44was waswas designed designeddesigned to toto evaluate evaluateevaluate the thethe performance performanceperformance forfor aa variablevariable tetemperaturemperature signal. signal.signal. The TheThe resultsresults results obtained are shown in Figure 15, in which it can be noted that the P&O algorithm has oscillations obtainedobtainedobtained are areare shown shownshown in inin Figure FigureFigure 1 11555, ,, in inin which whichwhich ititit cancancan bebebe notednoted thatthatthat thethe P&OP&O algorithmalgorithmalgorithm has hashas oscillations oscillationsoscillations throughout the simulation time. On the other hand, the LMS algorithm does not stabilize in the MPP throughoutthroughoutthroughout the thethe simulation simulationsimulation time. time.time. On OnOn the thethe other otherother hand, hand,hand, thethe LMSLMS algorithmalgorithmalgorithm doesdoesdoes not notnot stabilize stabilizestabilize inin in thethe the MPPMPP MPP and has oscillations of greater amplitude. andandand has hashas oscillations oscillationsoscillations of ofof great greatgreatererer amplitude. amplitude.amplitude.

FigureFigure 1144.. VariableVariable temperaturetemperature signalsignal usedused forfor tests.tests. FigureFigure 14.14. Variable temperature signal signal used used for for tests. tests.

Figure 15. Results of the P&O, LMS, and RTRL controllers under conditions of variable temperature FigureFigure 15. 1Results5. Results of of the the P&O, P&O, LMS, LMS, andand RTRLRTRL controllers under under conditions conditions of variable of variable temperature temperature Figureandand constantconstant 15. Results irradiance.irradiance. of the P&O, LMS, and RTRL controllers under conditions of variable temperature and constant irradiance. and constant irradiance. The Neurocontroller with the Iterative LMS algorithm showed good results for both irradiance TheThe Neurocontroller Neurocontroller with with the the Iterative Iterative LMS algorithm algorithm showed showed good good results results for forboth both irradiance irradiance andandThe constantconstant Neurocontroller temperaturetemperature with asas wellwell the asIterativeas forfor variablevariable LMS conditions,algorithmconditions, showed andand itsits perfor perforgood manceresultsmance was wasfor both similarsimilar irradiance toto thatthat and constant temperature as well as for variable conditions, and its performance was similar to that andobtainedobtained constant withwith temperature thethe RTRLRTRL Neurocontroller. Neurocontroller.as well as for variable conditions, and its performance was similar to that obtained with the RTRL Neurocontroller. obtained with the RTRL Neurocontroller. 3.2.3.2. ComparisonComparison ofof thethe RTRLRTRL NeurocontrollerNeurocontroller withwith thethe P&OP&O ControllerController 3.2. Comparison of the RTRL Neurocontroller with the P&O Controller 3.2. ComparisonTheThe conditionsconditions of the RTRL presentedpresented Neurocontroller inin FigureFigure 1616 with denotedenote the P&O aa moremore Controller complexcomplex scenarioscenario withwith suddensudden changeschanges inin The conditions presented in Figure 16 denote a more complex scenario with sudden changes in thethe The solarsolar conditions irradianceirradiance presented andand operatingoperating in Figure temperature.temperature. 16 denote TheThe a RTRL RTRLmore andandcomplex P&OP&O controllerscontrollersscenario with werewere sudden exposedexposed changes toto thesethese in the solar irradiance and operating temperature. The RTRL and P&O controllers were exposed to these theconditionsconditions solar irradiance andand theirtheir and responsesresponses operating werewere temperature. comparedcompared The simultaneously.simultaneously. RTRL and P&O controllers were exposed to these conditionsconditions and and their their responses responses werewere comparedcompared simultaneously.

Figure 16. Variable irradiance and temperature signals used to evaluate the performance of the RTRL Figure 16. Variable irradiance and temperature signals used to evaluate the performance of the RTRL andand P&OP&O controllerscontrollers.. Figure 16. Variable irradiance and temperature signals used to evaluate the performance of the RTRL Figure 16. Variable irradiance and temperature signals used to evaluate the performance of the RTRL andand P&O P&O controllers. controllers.

Energies 2018, 11, x FOR PEER REVIEW 10 of 17

The results shown in Figure 17 indicate that the P&O controller has power losses when there are Energiesabrupt2018 and, 11 ,proportional 3407 changes in the incident solar irradiance in the PV module. 10 of 17 Energies 2018, 11, x FOR PEER REVIEW 10 of 17

TheThe results results shown shown in in Figure Figure 17 17 indicate indicate that the the P&O P&O controller controller has has power power losses losses when when there there are are abruptabruptEnergies and 2018and proportional, 11proportional, x FOR PEER changes REVIEWchanges in in the the incidentincident solar irradiance irradiance in in the the PV PV module. module. 10 of 17

The results shown in Figure 17 indicate that the P&O controller has power losses when there are abrupt and proportional changes in the incident solar irradiance in the PV module.

Figure 17. Results obtained with the P&O and RTRL controllers for variable signals of irradiance and temperature.

In order to evaluate the performance of the controllers for a longer simulation time, the test Figure 17. Results obtained with the P&O and RTRL controllers for variable signals of irradiance and signalFigure of Figure 17. Results 18 was obtained designed with for the a simulation P&O and RTRLtime of controllers 25 s. for variable signals of irradiance temperature. andFigure temperature. 19 shows the results obtained, equivalent to 6 min of computational processing. These results demonstrate the robustness of the RTRL algorithm to support sudden changes and different In orderInFigure order to 17. evaluate toResults evaluate obtained the the performance with performance the P&O of and theof theRTRL controllers controllers controllers for forfor alonger variablea longer simulation signals simulation of irradiance time, time, the andthe test test signal operatingtemperature. conditions in the PV module. On the other hand, the P&O algorithm improved its of Figuresignal 18of Figurewas designed 18 was designed for a simulation for a simulation time of time 25 s.of 25 s. performanceFigure 19in theshows simulation the results time obtained, greater thanequivalent 20 s, however, to 6 min itof still computational shows power processing. losses. These resultsIn demonstrateorder to evaluate the robustness the performance of the RTRL of the algorithm controllers to supportfor a longer sudden simulation changes time,and different the test operatingsignal of Figure conditions 18 was in designed the PV for module. a simulation On the time other of 25 hand, s. the P&O algorithm improved its performanceFigure 19 in shows the simulation the results time obtained, greater thanequivalent 20 s, however, to 6 min it of still computational shows power processing. losses. These results demonstrate the robustness of the RTRL algorithm to support sudden changes and different operating conditions in the PV module. On the other hand, the P&O algorithm improved its performance in the simulation time greater than 20 s, however, it still shows power losses.

FigureFigure 18. 18.Variable Variable irradianceirradiance and temperature signals signals for for a asimulation simulation time time of of25 25s. s.

Figure 19 shows the results obtained, equivalent to 6 min of computational processing. These results demonstrateFigure 18. Variable the irradiance robustness and oftemperature the RTRL signals algorithm for a simulation to support time of sudden 25 s. changes and different operating conditions in the PV module. On the other hand, the P&O algorithm improved its performance inFigure the simulation 18. Variable time irradiance greater and than temperature 20 s, however, signals for it a stillsimulation shows time power of 25 losses.s.

Figure 19. Results obtained with the P&O and RTRL controllers for variable signals of irradiance and temperature with a simulation time of 25 s. Figure 19. Results obtained with the P&O and RTRL controllers for variable signals of irradiance and temperature with a simulation time of 25 s.

FigureFigure 19. 19.Results Results obtained obtained with with the the P&OP&O and RTRL controllers controllers for for variable variable signals signals of irradiance of irradiance and and temperature with a simulation time of 25 s. temperature with a simulation time of 25 s.

Energies 2018, 11, 3407 11 of 17 Energies 2018, 11, x FOR PEER REVIEW 11 of 17 Energies 2018, 11, x FOR PEER REVIEW 11 of 17 3.3. Comparison of the RTRL Neurocontroller with the Fuzzy Controller 3.3. Comparison of the RTRL Neurocontroller with the Fuzzy Controller 3.3. Comparison of the RTRL Neurocontroller with the Fuzzy Controller In this section we present the results obtained when making a comparison between the RTRL In this section we present the results obtained when making a comparison between the RTRL neurocontrollerIn this section and we the present fuzzy controller the results developed obtained by when researchers making working a comparison in the same between research the RTRL group neurocontroller and the fuzzy controller developed by researchers working in the same research neurocontrolleras the authors of and this paper.the fuzzy The controller controllers developed were evaluated by researchers with the same working conditions in the of same irradiance research and group as the authors of this paper. The controllers were evaluated with the same conditions of grouptemperature as the to authors compare of their this behavior paper. The and controllers determine whichwere evaluated has a better with response. the same It should conditions be noted of irradiance and temperature to compare their behavior and determine which has a better response. It irradiancethat to carry and out temperature these tests, to the compar authorse their of the behavior fuzzy controller and determine provided which us withhas a their better modeling response. and It should be noted that to carry out these tests, the authors of the fuzzy controller provided us with shouldsimulation be noted ﬁles. that to carry out these tests, the authors of the fuzzy controller provided us with their modeling and simulation files. their Inmodeling these simulations and simulation we noticed files. that the fuzzy controller showed good results that can be seen in In these simulations we noticed that the fuzzy controller showed good results that can be seen FigureIn 20these. For simulations a simulation we time noticed of 1 that s, the the fuzzy fuzzy controller controlle lastedr showed approximately good results 14 that min, can while be seen the in Figure 20. For a simulation time of 1 s, the fuzzy controller lasted approximately 14 min, while the inRTRL Figure for 20. the For same a simulation simulation time time of lasted 1 s, the only fuzzy 19 s.controller For these lasted tests, approximately the variable irradiance 14 min, while signal the of RTRL for the same simulation time lasted only 19 s. For these tests, the variable irradiance signal of RTRLFigure for 12 thewas same used. simulation time lasted only 19 s. For these tests, the variable irradiance signal of Figure 12 was used. Figure 12 was used.

Figure 20. Results obtained with the RTRL and Fuzzy controllers for a signal of variable irradiance FigureFigure 20. Results obtained withwith thethe RTRL RTRL and and Fuzzy Fuzzy controllers controllers for for a signal a signal of variableof variable irradiance irradiance and and constant temperature. andconstant constant temperature. temperature. The two controllers presented a good response time, compared to the P&O algorithm, but the TheThe two two controllers controllers presented presented a a good good response response time, time, compared compared to to the the P&O P&O algorithm, algorithm, but but the the controller trained with the RTRL algorithm stands out for minimizing oscillations and for stabilizing controllercontroller traine trainedd with with the the RTRL RTRL algorithm algorithm stands stands out out for for minimizing minimizing oscillations oscillations and and for for stabilizing stabilizing faster than the fuzzy controller. In Figure 21 we note that using the irradiance and temperature signals fasterfaster than than the the fuzzy fuzzy controller. controller. In In Figure Figure 2121 wewe note note that that using using the the irradiance irradiance and and temperature temperature signals signals of Figure 16, the fuzzy controller tracks the MPP during the first 0.1538 s of simulation. However, ofof Figure Figure 16,16, the the fuzzy fuzzy controller controller tracks tracks the the MPP MPP during during the the first ﬁrst 0 0.1538.1538 s s of simulation. However However,, when there is a positive slope and several disturbances, the controller does not stabilize in the MPP when there is a positive slope and several disturbances, the the controller controller does does not not stabilize stabilize in in the the MPP MPP and remains oscillating between 5 W for approximately 0.5 s of simulation. After 0.656 s, the andand remains remains oscillating oscillating between between 5 W 5 for W approximately for approximately 0.5 s of 0.5 simulation. s of simulation. After 0.656 After s, the 0.656 controller s, the controller responds again and tracks the MPP. controllerresponds againresponds and again tracks and the tracks MPP. the MPP.

Figure 21. Results obtained with the RTRL and Fuzzy controllers for variable signals of irradiance FigureFigure 21. ResultsResults obtained obtained with with the the RTRL RTRL and and Fuzzy controllers for for variable signals signals of of irradiance and temperature. andand temperature. temperature. These results show that the RTRL controller has an excellent performance, since it minimizes These results show that the RTRL controller has an excellent performance, since it minimizes power losses due to oscillations. power losses due to oscillations.

Energies 2018, 11, 3407 12 of 17

These results show that the RTRL controller has an excellent performance, since it minimizes power losses due to oscillations. Energies 2018, 11, x FOR PEER REVIEW 12 of 17 3.4.Energies Comparison 2018, 11, ofx FOR the PEER RTRL REVIEW Neurocontroller with the NARX Neurocontroller 12 of 17 3.4.In Comparison this section of the we RTRL present Neurocontroller the results with obtained the NARX when Neurocontroller making a comparison between the RTRL NeurocontrollerIn this section and we the present NARX the Neurocontroller results obtained [when25] developed making a comparison by researchers between working the RTRL in the sameNeurocontroller research group and asthe the NARX authors Neurocontroller of this work. [25] In developed [25] a neural by resea controllerrchers working of inverse in modelthe same was implemented,research group which as the consists authors of a of nonlinear this work. autoregressive In [25] a neural network controller with exogenous of inverse inputs model to was track theimplemented, MPP of a PV which module. consists For theof a designnonlinear of theautoregressive network, the network researchers with exogenous used the Backpropagationinputs to track trainingthe MPP algorithm of a PV module. with Levenberg-Marquardt For the design of the optimization, network, the whichresearchers was designedused the Backpropagation with the MATLAB Neuraltraining Network algorithm Toolbox. with Levenberg-Marquardt optimization, which was designed with the MATLAB NeuralThe NARXNetwork network Toolbox. has the same inputs (current and voltage) and the same output (duty cycle) as the RTRLThe NARX neurocontroller network has proposed the same inputs in this (current work. The and difference voltage) and between the same these output two (du controllersty cycle) is thatas thethe dynamicRTRL neurocontroller neural network proposed with RTRLin this algorithmwork. The difference only has abetween neuron these with two Pureline controllers activation is that the dynamic neural network with RTRL algorithm only has a neuron with Pureline activation function,that the whereas dynamic the neural NARX network network with has RTRL in its algorithm hidden layer only 10 has neurons a neuron with with Tansig Pureline transfer activation function function, whereas the NARX network has in its hidden layer 10 neurons with Tansig transfer function (Sigmoidfunction, Tangent) whereas and the NARX a neuron network in its outputhas in its layer hidden with layer Pureline 10 neurons transfer with function Tansig transfer for the function duty cycle. (Sigmoid Tangent) and a neuron in its output layer with Pureline transfer function for the duty cycle. (SigmoidFigures Tangent) 22 and and23 showa neuron the in results its output obtained layer with with Pureline the simulations transfer function of the for RTRL the duty and cycle. NARX Figures 22 and 23 show the results obtained with the simulations of the RTRL and NARX controllers.Figure Boths 22 controllers and 23 show track the the results ideal output obtained power with of the the simulations PV module of and the have RTRL fewer and oscillations NARX controllers. Both controllers track the ideal output power of the PV module and have fewer thanoscillations the P&O than controller. the P&O However, controller. at each However, change at in each operating change conditions,in operating the conditions, NARX neurocontroller the NARX hasoscillations strong oscillations than the andP&O stabilizes controller. after However, approximately at each 0.0325change s. in In operating contrast, theconditions, RTRL neurocontroller the NARX neurocontroller has strong oscillations and stabilizes after approximately 0.0325 s. In contrast, the stabilizes at approximately 0.005295 s and without oscillations. RTRL neurocontroller stabilizes at approximately 0.005295 s and without oscillations.

Figure 22. Results obtained with the RTRL and NARX controllers for a signal of variable irradiance FigureFigure 22. 22Results. Results obtained obtained with with thethe RTRLRTRL and NARX contr controllersollers for for a asignal signal of of variable variable irradiance irradiance and constant temperature. andand constant constant temperature. temperature.

Figure 23. Results obtained with the RTRL and NARX controllers for variable signals of irradiance Figure 23. Results obtained with the RTRL and NARX controllers for variable signals of irradiance and temperature. and temperature. It is important to note that for the simulation of 1 s, the NARX neurocontroller in the MATLAB/Simulink software lasted approximately 7 h of computational processing, while the RTRL controller for the same simulation time lasted only 22 s.

Energies 2018, 11, 3407 13 of 17

It is important to note that for the simulation of 1 s, the NARX neurocontroller in the MATLAB/ Simulink software lasted approximately 7 h of computational processing, while the RTRL controller for the same simulation time lasted only 22 s.

3.5. Performance of the Control Algorithms for the MPPT In order to analyze the performance of all the control algorithms, 5 test scenarios were designed with different values for the solar irradiance and the operating temperature of the PV module (Table1). For case 2, the average value of the solar irradiance signal of Figure 12 was used. In case 3, the average value of the temperature signal of Figure 14 was used. For cases 4 and 5, the average values of the temperature and solar irradiance signals of Figures 16 and 18 were used.

Table 1. Test cases to evaluate the performance of the control algorithms.

Test Signals Case 2 ◦ Ei (W/m ) T ( C) 1 1000 25 2 588.38 25 3 1000 41.79 4 545.12 28.85 5 631.65 28.95

Table2 shows the maximum power values obtained during the simulation of the 5 cases, in which it can be seen that the Iterative LMS and RTRL controller showed similar power values. In each case, the power data presented for each controller represents an average value.

Table 2. Maximum power obtained with each control algorithm.

Case Pref (W) P&O (W) LMS (W) Iterative LMS (W) RTRL (W) Fuzzy (W) NARX (W) 1 64.87 56.05 60.16 61.14 61.33 57.28 58.88 2 36.13 22.37 20.5 35.37 35.47 33.03 34.57 3 61.4 56.3 39.61 55.33 57.09 57.47 4 33.25 2.79 28.86 31.96 32.36 16.87 31.61 5 38.86 12.84 36.29 36.43 37.19 32.54

Generally, the controllers showed a good performance under standard test conditions (Case 1) and for conditions of variable irradiance and constant temperature (Case 2), except for the P&O algorithm that stands out for having oscillations and low efficiency. Under conditions of constant irradiance and variable temperature (Case 3) the P&O algorithm showed good results and obtained an efficiency of 91.69%, despite not tracking the MPP completely. For the most demanding test conditions in which variable signals of irradiance and temperature were used (Cases 4 and 5), exposing the PV module to sudden changes and very steep slopes, the RTRL neurocontroller showed the maximum possible efficiency in most of the tests (See Table3).

Table 3. Maximum efﬁciency obtained for the simulated control algorithms.

Case η (P&O) η (LMS) η (Iterative LMS) η (RTRL) η (Fuzzy) η (NARX) 1 86.4% 92.74% 94.25% 94.54% 88.3% 90.77% 2 61.92% 56.74% 97.9% 98.17% 91.42% 95.68% 3 91.69% 64.51% 90.11% 92.98% 93.6% 4 8.39% 86.8% 96.12% 97.32% 50.74% 95.07% 5 33.04% 93.39% 93.75% 95.7% 83.74%

The efﬁciency (η) was found as the ratio of the average values of the power reached by each control method (Po) and the average values of the power obtained from the PV module (Pref)[38]. Energies 2018, 11, 3407 14 of 17

The results of Table4 show the efﬁciency of the ADALINE network, in terms of computing time, for each of the three training algorithms (RTRL, LMS and Iterative LMS). These results are due to the fact that the ADALINE network only uses one (1) layer for its operation and that, in addition, for each algorithm only one (1) training cycle was needed.

Table 4. Simulation and computation times obtained for the controllers.

Case Simulation Time (Seconds) MPPT Controller Computation Time (hh:mm:ss) P&O 00:00:02 LMS 00:00:01 Iterative LMS 00:00:02 1 0.025 RTRL 00:00:01 Fuzzy 00:00:27 NARX 00:02:18 P&O 00:00:18 LMS 00:00:19 Iterative LMS 00:00:19 2 1 RTRL 00:00:21 Fuzzy 00:15:16 NARX 06:07:46 P&O 00:00:35 LMS 00:00:56 Iterative LMS 00:00:41 3 3 RTRL 00:00:45 Fuzzy 00:36:48 NARX P&O 00:00:19 LMS 00:00:18 Iterative LMS 00:00:20 4 1 RTRL 00:00:22 Fuzzy 00:12:02 NARX 07:29:00 P&O 00:08:00 LMS 00:07:03 Iterative LMS 00:06:32 5 25 RTRL 00:07:33 Fuzzy 05:41:14 NARX

This information is consistent with the results obtained in [39], where the performance of ﬁve training algorithms was evaluated (ADALINE, Backpropagation, Levenberg-Marquardt, Simulated Annealing, Genetic Algorithm). In this work, the authors showed that the ADALINE network showed the best performance, which was measured in terms of the number of training cycles and the MSE. Finally, Table5 shows the settling times, peak power, and rise time, which were obtained for a simulation time of 0.025 s, irradiance of 1000 W/m2, and temperature of 25 ◦C. It is highlighted that the best settling times were obtained with the three algorithms based on the ADALINE network. In this way, the efﬁciency of the algorithms developed in this work is also shown, in terms of the step-response for dynamic systems. Energies 2018, 11, 3407 15 of 17

Table 5. Step-response characteristics for a simulation time of 0.025 s.

MPPT Controller Rise Time (ms) Settling Time (ms) Peak (W) Peak Time (ms) P&O 0.42 16.36 64.67 2.55 LMS 0.46 3.16 65.15 2.45 Iterative LMS 0.58 3.32 64.71 2.55 RTRL 0.59 3.34 65.17 2.50 Fuzzy 0.81 8.65 64.98 2.70 NARX 0.34 4.37 64.70 1.35

4. Conclusions From the development of a dynamic neural controller for the tracking of the maximum power point, it can be concluded that before designing the control system, the non-linear behavior of the PV module must be taken into account and that the output signals will be a function of the climatic conditions that are used as inputs of the module. Therefore, the simulations showed the performance of the controllers under variations in the solar irradiance and operating temperature. With the design of the ADALINE network with its different training methods, it was possible to develop different control methods to solve the problem presented by the P&O algorithm. However, it was established that only the RTRL algorithm showed an optimal response. The mathematical development of the ADALINE network and the RTRL algorithm was a very efficient process, since only 6 previous data were used for the training. This situation contrasts with the 50.931 power data that was used in the training of the NARX controller. In addition, it was not necessary to use the MATLAB Neural Network Toolbox to perform the controller training. Dynamic neural networks have a high computational cost compared with other neural network control systems. However, the simulations of the control methods developed in this work required much less time than the control methods used in the comparisons. The small database used for the training was the most appropriate, since a single neuron was designed. During the literature review it was possible to confirm that the other neural control methods need more than one layer and more than one neuron to obtain good results. The results of the RTRL neurocontroller simulations showed that mathematically it is possible to correct the oscillations that occur around the MPP when the P&O algorithm is used. The RTRL neurocontroller was adapted to the climatic conditions to which the PV module was exposed and presented a rapid response and a much lower error than the other training methods. As future work, the experimental validation of the MPPT controller proposed in this paper will be carried out. The dynamic neural network trained with the RTRL algorithm will be implemented, in order to evaluate its efficiency through different experimental tests. Among the factors that will be evaluated are the efficiency, the program memory, and the data memory required by the proposed method and the execution time in the developed hardware. So far, the ADALINE neural network and the RTRL algorithm are designed in C language in mid-range and high-end PIC (Programmable Intelligent Computer) microcontrollers.

Author Contributions: J.V.-P. performed the design and modeling of the MPPT algorithms. C.R.-A. designed the experiments and wrote the paper. D.R.-L. supported the design of the ADALINE network with the RTRL and LMS algorithms. Funding: This research was funded by Vicerrectoría de Investigación of the Universidad del Magdalena. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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