Levenberg-Marquardt and Conjugate Gradient Training Algorithms of Neural Network for Parameter Determination of Solar Cell

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1 International Journal of Innovation and Applied Studies ISSN Vol. 9 No. 4 Dec. 204, pp Innovative Space of Scientific Research Journals Levenberg-Marquardt and Conjugate radient raining Algorithms of Neural Networ for Parameter Determination of Solar Cell Fayrouz DKHICHI and Benyounes OUKARFI Department of Electrical Engineering, EEA&I laboratory, Faculty of Sciences and echniques, Hassan II University, Mohammedia, Morocco Copyright 204 ISSR Journals. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. ABSRAC: his present paper deals with the parameter determination of solar cell under different values of irradiance and temperature by using an artificial neural networ. his latter is trained by an optimization algorithm based on gradient descent. In this wor we used two distinguished algorithms from different order of gradient descent: Levenberg-Marquardt and conjugate gradient. he use of these two algorithms is to conduct a comparative study on their performances. he results revealed that the Levenberg-Marquardt algorithm presents the best potential in providing accurate electrical parameters values compared to conjugate gradient algorithm. Moreover, the trends of electrical parameters according to irradiance and temperature show the effect of each of these two meteorological factors on the values of the intrinsic parameters of solar cell. KEYWORDS: Artificial neural networ, Conjugate gradient, Electrical parameters, Levenberg-Marquardt, Solar cell. INRODUCION he continuous exhibition to irradiance (), temperature () and aging leads to prevent the photovoltaic () module to achieve its optimal performances. In order to study the impact of these handicapping factors on the produced electrical energy, we therefore need to now the behavior of the smallest unit of a module which is the solar cell, and thus by determining its intrinsic electrical parameters. Under the lac of a direct mathematical equation between [, ] and the electrical parameters, the Artificial Neural Networ (ANN) seems to be an adequate method to model this implicit non-linear relationship. he feature of this method is presented in its ability to predict result from exploitation of the acquired data. Hence, the information is carried by weights, representing the connections values between neurons. he functioning of ANN requires her learning by a training algorithm, ensuring the minimization of the error generated at the networ output. In this study we used two gradient descent optimization algorithms that allow separately the training of ANN: Conjugate gradient of first order of gradient and the Levenberg-Marquardt of second order. We firstly determined the electrical parameters values of solar cell according to different values of and, then we compared between the potentials of the two algorithms opted for training of ANN. In addition, plotted curves of the electrical parameters values are provided in order to show the impact of and on these parameters during the production of the electrical energy by a solar cell. Corresponding Author: Fayrouz DKHICHI 869

2 Levenberg-Marquardt and Conjugate radient raining Algorithms of Neural Networ for Parameter Determination of Solar Cell 2 SOLAR CELL MODEL 2. SINLE DIODE EQUIVALEN MODEL he solar cell that produces I is modelled by a single diode electrical model [-3]. I I ph R sh R s V Fig.. Equivalent electrical circuit of solar cell he mathematical equation deducted from the circuit of the figure is expressed as follows: Where: I I ph V I s exp R s I n V th V Rs I R R s : Series resistance representing the losses due to various contacts and connections. R sh : Shunt resistance characterizing the lea currents of the diode junction. I ph : Photocurrent. I s : Diode saturation current. n: Diode ideality factor. V th : hermal voltage A q. A: Boltzmann constant J / K. q: Charge of the electron 9 C : emperature of solar cell by Kelvin. sh () 2.2 DEPENDENCE OF ELECRICAL PARAMEERS O IRRADIANCE AND EMPERAURE: An illuminated solar cell generates an I (V ) characteristic for a definite value of and. Indeed, according to the equation (), the solar cell is characterized by five internal electrical parameters R s, R sh, I ph, I s and n, which lead to the production of current I and change their values according to the value of two geometrical factors and, lie it is the case with I ph and I s following these two equations: Where: I ph [ I ph, ref b Isc ( ref )] (2) ref ref: Index indicating the reference value. b Isc : emperature coefficient of the short circuit current. Eg: Energy of band gap of the solar cell material. 3 qe g I, exp s I s ref (3) ref An ref he current I ph varies according to the irradiance (equation (2)) and the current I s varies according to the temperature (equation (3)) while the both resistances R s and R sh change according to the both meteorological factors [4]. he cell handled in this study is a commercial cell characterized by the following parameters at ref = 33 C and ref =000W/m²: Maximum power - P mp (W) Open circuit voltage - V oc (V) Short circuit current - I sc (A) Operating voltage at maximum power - V mp (V) Operating current at maximum power - I mp (A) In order to train our networ, we therefore needed several values of the five parameters and theirs according values of [, ]. Hence, we determined, before we begin the training process, the necessary amount of the parameters values from ISSN : Vol. 9 No. 4, Dec

3 Fayrouz DKHICHI and Benyounes OUKARFI several characteristics I (V ). We considered different characteristics for a margin of solar irradiance values between 00W/m² and 000W/m² and for a cellular temperature values going from 5 C to 45 C. hese I (V ) are obtained from the schematization of a simulation model of the solar cell under MALAB/SIMULINK software V f P Vpv dule (V) cell model I-V characteristic P-V characteristic Solve f(z) z f(z) = 0 Algebraic Constraint Fig. 2. Solar cell model he bloc cell model of Fig. 2 includes a subsystem, modeling the mathematical equation (). 2 3 n n Iph Voc n Iph Voc Vt Vth Vth Iph Voc Is Vt Product Is /u Fcn exp(u) Product Fcn Solve f(z) z f(z) = 0 Algebraic Constraint 2 Vpv Vpv Rs Rs Product2 Product3 Ppv Fig. 3. Internal cell structure inside the bloc Solar cell model he bloc of I ph in figure 3 models the equation (2), while the bloc of V oc and I s in the same figure models successively the both following equations for current I pv = 0 : I ph Voc [ Voc, ref bvoc ( ref )] (4) I s Voc (5) ref V exp oc V th b voc : emperature coefficient of the open circuit voltage. 3 HE USED ARIFICIAL NEURAL NEWORK he determination of the five internal electrical parameters of solar cell for various values of and are ensured by an ANN made in [4]. he architecture includes an entrance layer, a hidden layer and an output layer as shown below: ISSN : Vol. 9 No. 4, Dec

4 Levenberg-Marquardt and Conjugate radient raining Algorithms of Neural Networ for Parameter Determination of Solar Cell Fig. 4. he used topology of ANN he entrance layer contains two inputs [, ], the hidden layer contains twenty hidden neurons and that of the output includes five neurons of output corresponding to the five parameters [R s, R sh, I ph, I s, n] whose we want to predict the values. he entrance and the output of the networ are connected by the following equations: f Activation function «hyperbolic tangent» of the hidden neurons, o f Activation function «linear» of the output h j neurons. s Equations of input Equations of output z ji 2 i x w i ji b ji (6) y f h j ( z ji ) (7) z 20 j y w b (8) y o f s ( z ) (9) 4 RAININ ALORIHMS OF HE NEWORK he training of the ANN is characterized by three stages: learning, validation and test. In order to obtain a good training, the error generated by the output of ANN during the learning stage must be minimized. his error is nown under the name of Mean Squared Error F, which is expressed by the following equation: mean (learning ) F mean p v 2 ( learning ) S learning ( t, m) ylearning ( t, m) (0) p t m Where: p: Number of examples {input, output}. v: Number of networ output. S: Desired values of the networ output. t: Index pointing out the number of examples. m: Index pointing out the number of output. y: Matrix values of the networ output y = [R s, R sh, I ph, I s, n]. he minimization of the error F is ensured by the adjustment of the ANN s weights (w), using one mean (learning ) optimization algorithm among the both: Levenberg-Marquardt or conjugate gradient. 4. LEVENBER-MARQUARD ALORIHM Levenberg-Marquardt algorithm (LM) is well nown for its combination between the steepest descent method and auss- Newton method, which explain its low sensitivity toward initial points and its fast convergence around the optimum [5]. he LM algorithm adjusts the weights (w) of the networ by following the expression: ISSN : Vol. 9 No. 4, Dec

5 Fayrouz DKHICHI and Benyounes OUKARFI Where: J: Jacobian matrix of the error function F. J e w w () J J I mean (learning ) e: Error between the desired and calculated networ output. : Number of iterations. I: Matrix identity. Regulation strategy of the damping factor of Levenberg-Marquardt algorithm is made according to the following principle: If the calculated error F for w mean (learning ) +, decreases then: / 0 Else * 0 and w + = w 4.2 CONJUAE RADIEN ALORIHM Conjugate gradient algorithm (C) is of the same family as the steepest descent algorithm, but it is more efficient. At the first iteration, the direction of gradient descent "d" is defined as follows: d old 2 J e From the second iteration d ( J old e d ) old J e w and d new 2 J e d old w d (2) new 5 RESULS raining of the networ is made with 30 inputs-outputs examples distributed in three sets (learning, validation and test). 5. MEAN SQUARED ERROR LM algorithm allows a good training of the ANN compared with C algorithm; the latter converges slowly as it is shown in figure 5. Fig. 5. Evolution of the mean squared error at learning stage of the ANN trained by LM and C ISSN : Vol. 9 No. 4, Dec

6 Levenberg-Marquardt and Conjugate radient raining Algorithms of Neural Networ for Parameter Determination of Solar Cell 5.2 EVOLUION OF ELECRICAL PARAMEERS ACCORDIN O EMPERAURE Electrical parameters values of solar cell are determined in margin of = [5 C, 45 C]. In order to well observe their evolution, we have chosen to present their curves for two values of (50W/m² and 350W/m²) (figures 6, 0). Fig. 6. Series resistance R s according to temperature Fig. 7. Shunt resistance R sh according to temperature Fig. 8. Photocurrent I ph according to temperature Fig. 9. Saturation current I s according to temperature Fig. 0. Diode ideality factor n according to temperature ISSN : Vol. 9 No. 4, Dec

7 Fayrouz DKHICHI and Benyounes OUKARFI 5.3 EVOLUION OF ELECRICAL PARAMEERS ACCORDIN O IRRADIANCE Electrical parameters values of solar cell are determined in margin of = [00W/m², 000W/m²]. Since the curves are crowded, we have chosen two values of (5 C and 35 C) to observe well the evolution of these parameters (figures,5). Fig.. Series resistance R s according to irradiance Fig. 2. Shunt resistance R sh according to irradiance Fig. 3. Photocurrent I ph according to irradiance Fig. 4. Saturation current I s according to irradiance Fig. 5. Diode ideality factor n according to irradiance ISSN : Vol. 9 No. 4, Dec

8 Levenberg-Marquardt and Conjugate radient raining Algorithms of Neural Networ for Parameter Determination of Solar Cell 6 DISCUSSIONS As it is shown in fig. 5, the LM algorithm converges quicly and presents a minimal value of F compared to the mean (learning ) error generated by the C algorithm. his ability of LM to reach lower value of the mean squared error better than C is explained by the features that LM inherits from auss-newton. his algorithm becomes fast and effective when the networ weights are well refined by the steepest descent algorithm. In other hand the C which behaves lie steepest descent, it had a slow convergence around the optimal networ weighs, which explain the behavior of the algorithm that does not achieve a minimal error lie it is the case with LM. he obtained curves of the electrical parameters of the solar cell according to temperature and irradiance (fig. 6 to fig. 5), shows clearly that the networ trained by LM gives values near to desired values than those obtained by ANN trained by C. he temperature and the irradiance apply different effects on the five parameters R s, R sh, I ph, I s and n. As it is clearly seen in fig. 6 and in fig., the values of R s increase when and/or increase too, while in the two fig. 7 and fig. 2, the values of R sh decrease with the variation of, then they remain almost the same although the variation of. he photocurrent I ph as its name indicates, it is more affected by the change of than by, so this is obvious in fig. 8 and fig. 3. he values of I s decrease with and increase with (fig. 9 and fig. 4), this evolution is also observed in the values of the diode ideality factor n (fig. 0 and fig. 5). Meteorological Conditions able. Percentage of error of electrical parameters obtained par the ANN trained par LM and C algorithms. Conventional model [6] Proposed ANN trained by LM Proposed ANN trained by C [W/m²] [ C] Rs [%] Rsh [%] Iph [%] Is [%] n [%] Rs [%] Rsh [%] Iph [%] Is [%] n [%] Rs [%] Rsh [%] Iph [%] Is [%] n [%] In a previous wor [6], we identified the electrical parameters of the mathematical model equation () by a conventional method. We therefore adjusted numerically the sum of squared error in order to determine these parameters by using one characteristic I (V ). his characteristic is taen for a determined value of and. he determination of the corresponding values of the electrical parameters maes the identification issue a little bit slow and heavy to implement. Consequently the ANN method resolves this problem and presents an adapted determination platform, which establish a relationship between [, ] and [R s, R sh, I ph, I s, n]. As it is shown in able, we observe that the ANN trained by LM and C presents the error percentages of electrical parameters minimum compared to that supplied by the conventional method, except for the percentages of error concerning the parameters R sh et I ph supplied by the C algorithm which had important values compared to those presented by the conventional method. When we compare between ANN trained by LM and that trained by C, we noteworthy that the first one presents the most minimal errors. 7 CONCLUSIONS Levenberg-Marquardt algorithm provides the best performances in the training of the artificial neural networ compared to the conjugate gradient algorithm. he determined values of the five electrical parameters of solar cell obtained by the networ trained by the Levenberg-Marquardt algorithm are so close to the desired values, due to the ability of this algorithm to combine between the characteristics of steepest descent algorithm and the auss-newton algorithm which ensure a maximal minimization of the mean squared error. ISSN : Vol. 9 No. 4, Dec

9 Fayrouz DKHICHI and Benyounes OUKARFI REFERENCES [] A. Jain, A. Kapoor, Exact analytical solutions of the parameters of real solar cells using Lambert W-function, Solar Energy Materials & Solar Cells, vol. 8, pp , [2] H. Qin, J. W. Kimball, Parameter Determination of Photovoltaic Cells from Field esting Data using Particle Swarm Optimization, IEEE, 20. [3]. Iegami,. Maezono, F. Naanishi, Y. Yamagata, K. Ebihara, Estimation of equivalent circuit parameters of module and its application to optimal operation of system, Solar Energy Materials & Solar Cells, vol. 67 pp , 20. [4] E. Karatepe, M. Boztepe, M. Cola, Neural networ based solar cell model, Energy Conversion and Management, vol. 47, pp , [5] B.. Kermani, S. S. Schiffmanb, H.. Nagle, Performance of the Levenberg Marquardt neural networ training method in electronic nose applications, Sensors and Actuators, vol. 0, pp. 3 22, [6] B. Ouarfi, F. Dhichi, A. Faar, Parametric identification by minimizing the squared residuals (Application to a photovoltaic cell), Proc. th IEEE Conf. on Renewable and sustainable energy conference, Ouarzazate, IEEE Explore, (March ). ISSN : Vol. 9 No. 4, Dec