A procedure to evaluate the seven parameters of the two-diode model for photovoltaic modules

Accepted Manuscript A procedure to evaluate the seven parameters of the two-diode model for photovoltaic modules

Aldo Orioli, Alessandra Di Gangi PII:

S0960-1481(19)30287-3

DOI:

10.1016/j.renene.2019.02.122

Reference:

RENE 11254

To appear in:

Renewable Energy

Received Date:

12 September 2018

Accepted Date:

22 February 2019

Please cite this article as: Aldo Orioli, Alessandra Di Gangi, A procedure to evaluate the seven parameters of the two-diode model for photovoltaic modules, Renewable Energy (2019), doi: 10.1016/j.renene.2019.02.122

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ACCEPTED MANUSCRIPT

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A procedure to evaluate the seven parameters of the two-diode model for photovoltaic modules

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Aldo Oriolia*, Alessandra Di Gangia

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a

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*Corresponding

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DEIM Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici Università degli Studi di Palermo, Viale delle Scienze Edificio 9, 90128 Palermo – Italy author: email address: [email protected], [email protected]; telephone number: +3909123861905; fax: +39091484425

Abstract The paper presents an analytical procedure to calculate the seven parameters of the two-diode model of photovoltaic (PV) panels for any value of the solar irradiance and cell temperature. Six parameters (the photocurrent, the diode reverse saturation currents, the quality factor of the first diode and the series and shunt resistances), are evaluated by solving the equations related to the properties of the main points of the current-voltage (I-V) characteristics. The further information, necessary to calculate the entire set of seven truly independent parameters, is based on two conditions that have to be simultaneously satisfied: 1) the exclusion of negative values of the model parameters; 2) the minimization of the difference between the diode quality factors, which permits to obtain more precise I-V curves of the PV modules. A method to extend the application of the two-diode model to conditions that are different from the standard rating conditions (SRC) is presented. The comparison with alternative two-diode models showed that accurate representations of the IV characteristics can be achieved using the proposed procedure regardless of the typology of the PV devices (monocrystalline, polycrystalline, HIT, amorphous silicon, CIS, CIGS, tandem).

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Keywords Two-diode model; seven-parameter equivalent model; I-V characteristics; solar energy; PV panels

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1. Introduction The increasing demand of electrical power generated from renewable sources has encouraged the development of new technologies of PV devices in order to reduce the production costs, by adopting less complex manufacturing processes and using cheaper materials. Monocrystalline and polycrystalline PV cells are made by sawing thin wafers from carefully cooled and solidified silicon ingots obtained with a method of crystal growth or from molten silicon. Thin-film solar cells, which are produced by depositing one or more thin films of photovoltaic material on lightweight and flexible substrates, are easier to make and also more resistant than the fragile monocrystalline and polycrystalline silicon PV cells. Such PV devices need a smaller amount of light absorbing material and can be used for numerous innovative applications such as building-integrated photovoltaics, curtain walls, canopies, acoustic barriers, watercrafts, vehicles, portable electronics etc. Although 1

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thin-film solar cells have lower temperature coefficients [1-2], as compared with the crystalline cells, they unfortunately show reduced energy efficiencies and degradation phenomena after long term outdoor exposure [3-6]. Thin-film PV cells present smaller fill factors because their I-V characteristics are smoother than the curves of the crystalline solar cells. Fig. 1 depicts the I-V characteristics at the SRC - irradiance Gref = 1000 W/m2, cell temperature Tref = 25 °C and average solar spectrum at AM 1.5 defined by IEC 60904-3 [7] - of various typologies of PV modules. In order to better appreciate the differences, the curves are plotted using the normalised values of current i and voltage v: I V (1) i v I sc ,ref Voc ,ref where Isc,ref and Voc,ref are the short circuit current and the open circuit voltage of the PV panel at SRC, respectively. Normalized I-V Characteristics at SRC

Normalized current i = I/Isc,ref

1.0

0.8

0.6 Sanyo HIT-240 HDE4 - HIT

0.4

Shell SQ 150-PC - Monocrystalline Kyocera KD245GH-4FB2 - Polycrystalline Exiom EXTF-75 - Amorphous

0.2

Solar Frontier SF130-L CIS Soltecture Linion 90 - CIGS Pramac MCPHP7 125 - Tandem

0.0 0.0

0.2 0.4 0.6 0.8 Normalized voltage v = V/Voc,ref

1.0

52 53

Fig. 1. Normalised I-V characteristics of crystalline and thin-film PV panels at the SRC.

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For the various typologies, the derivatives of the I-V curves in correspondence with the short circuit (1, 0) and open circuit (0, 1) points are very different. Such derivatives indicate how the electrical behaviour of a PV panel is far from that of an ideal electrical source. The ideal current or voltage sources, which are not affected by internal energy dissipation, have I-V characteristics that are straight lines parallel to the voltage axis or the current axis, respectively. On the contrary, the right part of the I-V curve slants to the left, whereas the left part is not horizontal due to the internal series and shunt resistances, respectively. The different slopes of the I-V curves near the short circuit and open circuit points indicate that the high quality silicon slabs of crystalline and CIGS modules depicted in Fig 1 dissipate less energy than the materials used to make amorphous, CIS, and tandem PV panels. The reduced energy efficiency of the thin-film PV modules highlights the importance of designing PV systems using procedures based on reliable physical models in order to get the most accurate prevision of the electrical behaviour of the considered PV devices. Lumped-parameter equivalent circuits (one and two diode models), whose parameters are determined from experimental I-V characteristics, have been conveniently used to perform the simulation of the PV 2

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module behaviour. The one-diode model usually gives appreciable results, especially dealing with the sharp I-V characteristics of the crystalline PV cells. The two-diode model, which uses a larger number of parameters, should result more effective to represent any I-V characteristic regardless of the difference in the curve shapes, which are related to the production technology of the simulated PV devices. Unfortunately, due to the difficulties met in the parameters calculation, which are usually overcome adopting simplifying hypotheses and fixing some values of the parameters, the two-diode model often fails to achieve the high level of the expected performances. In order to meet the requirements of PV system designers and researchers, an analytical procedure to calculate the parameters of a very accurate two-diode model is described in the present paper. Different from the previous two-diode models, the proposed procedure provides a set of seven parameters that are independently calculated on the basis of the specific peculiarities of each analysed PV panel. By this way, it is possible to avoid any arbitrary choice of the model parameters, or adoption of simplifying hypotheses, and the electrical behaviour of any PV device typology can be simulated with a very high level of accuracy.

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2. The two-diode equivalent circuit The equivalent circuit of the two-diode model, which is depicted in Fig. 2, contains seven parameters, which are photocurrent IL, diode reverse saturation currents I01 and I02, diode quality factors n1 and n2, series and shunt resistances Rs and Rsh. I Rs IL

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I01

I02

Rsh

V

Fig.2. Two-diode equivalent circuit for a PV panel.

The two-diode model is traditionally described by the following equation: s  V n TIRs   V n IR  V  IRs (2) 1 I  I L  I 01  e  1  I 02  e 2T  1      Rsh     in which diode quality factors n1 = a1Ncsk/q and n2 = a2Ncsk/q, a1 and a2 are the diode shape factors, Ncs is the number of cells of the panel that are connected in series, q is the electron charge (1.602∙ 10-19 C) and k is the Boltzmann constant (1.381∙10-23 J/K). Following the traditional theory, photocurrent IL depends on the solar irradiance and diode currents I01 and I02 are affected by the cell temperature. The two-diode model parameters are usually obtained from the following data, which are often available in the manufacturer datasheets:  open circuit voltage Voc,ref and short circuit current Isc,ref at the SRC;  voltage Vmp,ref and current Imp,ref at the maximum power point (MPP) at the SRC;  open circuit voltage temperature coefficient μV,oc and short circuit current temperature coefficient μI,sc. 3

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The number of series connected PV cells, and/or the derivative of the I-V curve calculated at the maximum power, short circuit and open circuit points are also required by some models. The equation system, which has to be formulated to calculate the model parameters, cannot be solved using exact mathematical methods, due to the presence of current I in both terms of transcendent Eq. (1). The model parameters have been valuated using numerous procedures such as the genetic algorithms, the particle swarm optimization, the differential evolution, the Chebyshev polynomials, the artificial immune systems, the Lambert W-function, the hybrid evolutionary algorithms, the Taylor series expansion, the fireworks algorithm and the Gauss-Seidel algorithm. Other authors preferred to analytically solve the problem using approximate forms of the equations and numerical solving techniques. Some of the seven parameters were often fixed in order to simplify the solution of the equations involved in the analytical procedures. Chan et al. [8], Enebish et al. [9], Hovinen [10], Gow et al. [11], Haouari-Merbah et al. [12], Hejri et al. [13], Masmoudi et al. [14] and Sallem et al. [15] set a1=1 and a2=2. Salam et al. [16], Ishaque et al. [1718], Marrekchi et al. [19], Duong et al. [20] and Nguyen et al. [21] assumed a1=1, a2 ≥1.2 and I01=I02. Gupta et al. [22] considered a1=1, a2=1, Rsh=∞, a fixed value of Rs and I02 = (T 2/5/3.77) I01, where T is the PV module temperature. Babu et al. [23] presented a simplified two-diode model in which Rs= 0, Rsh= ∞ and I02 was calculated with the expression proposed by Gupta et al. Also Sangeetha et al. [24] used the relation of Gupta et al. Elbaset et al. [25] fixed a1+a2 = 3 for polycrystalline and thin film solar cells, and a1+a2 = 4 for amorphous solar cells. Shannan et al. [26] added a new resistance to the two-diode model and assumed that and I01=I02. Some of the above models, such as those proposed by Chan et al., Enebish et al. and Hovinen, permit to calculate the I-V characteristic only at the SRC; the other procedures were tested on various crystalline (CanadianSolar CS6X-310P, BP Solar MSX-60, GE Energy GEPV-110, Isofoton ISF-240, JA Solar JAC MSF-2, Kyocera KC200GT, Renesola JC260S, SolarWorld SW 280, Shell SQ150-PC, Shell SP-70, Solarex MSX-60) and thin-film (Kaneka U-EA100, Shell ST40, Solarex MST-43LV, UNI-Solar US-64) PV devices, also for conditions different from the SRC. It is easy to verify that none of the above models can be considered a true seven-parameter model because, due to the initial choice of some parameters, the degrees of freedom of the system are always less than seven. Actually, due to the initial selection of a1+a2, the model of Elbaset et al. uses six independent parameters. The models that assume a1=1 and a2 =2, make the two-diode model a five-parameter model. The independent parameters are reduced to four by the Salam et al., Ishaque e al., Marrekchi et al., Duong et al. and Nguyen et al., which also set I01=I02; the same happens with the model of Babu et al., which fixes the values of Rs, Rsh and I02. The model of Gupta et al. has only three independent parameters. As a consequence, none of the above models use a set of seven model parameters in which the value of each parameter is obtained solving the equations formulated on the basis of the specific peculiarities of the simulated PV panel. For this reason the results obtained with such models may not show the superior performances that the two-diode model is supposed to have. Actually, some of these incomplete two-diode models may result even less accurate of the one-diode model [2729].

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3. Procedure to calculate the model parameters at the SRC. The new procedure evaluates the model parameters using the information derived from the main points of the I-V characteristic at the SRC, which are the short circuit point, open circuit point, MPP, and the derivatives in correspondence with the same points. On the basis of this information the following equations can be written in the SRC: - Open circuit point: V = Voc,ref I=0

 VnocT,ref   VnocT,ref  V 1 ref 0  I L ,ref  I 01,ref  e  1  I 02,ref  e 2 ref  1  oc ,ref     Rsh ,ref     - Short circuit point: V = 0 I = Isc,ref  I sc ,nrefTRs ,ref   I sc n,refTRs ,ref  I R I sc ,ref  I L ,ref  I 01,ref  e 1 ref  1  I 02,ref  e 2 ref  1  sc ,ref s ,ref     Rsh ,ref     (4) - Maximum power point: V = Vmp,ref I = Imp,ref  Vmp ,ref nITmp ,ref Rs ,ref  1 ref I mp ,ref  I L ,ref  I 01,ref  e  1  I 02,ref     (5) - Derivative at the open circuit point: Voc , ref

157

158

159

dI dV

V Voc , ref I 0



 Vmp ,ref nITmp ,ref Rs ,ref  V  I mp ,ref Rs ,ref 1 ref e  1  mp ,ref   Rsh ,ref  

Voc , ref

I 01 I 1 nT nT e 1 ref  02 e 2 ref  n1Tref n2Tref Rsh ,ref 1  Rs ,ref

Voc , ref Voc , ref  I I 02 1 n1Tref nT 01  e  e 2 ref  n2Tref Rsh ,ref  n1Tref

  



(6)

1 Rso

- Derivative at the short circuit point:

dI dV

V 0 I  I sc , ref



I 01 e n1Tref

I sc , ref Rs , ref n1Tref

 I 1  Rs ,ref  01 e  n1Tref

I  02 e n2Tref

I sc , ref Rs , ref n1Tref



I sc , ref Rs , ref n2Tref

I 02 e n2Tref



1 Rsh ,ref

I sc , ref Rs , ref n2Tref



1 Rsh ,ref (7)

160 161

(3)

- Derivative at the MPP:

5

  



1 Rsho

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dI dV

V Vmp , ref I  I mp , ref



I 01 e n1Tref 1  Rs ,ref

Vmp , ref  I mp , ref Rs , ref n1Tref

I  02 e n2Tref

Vmp , ref  I mp , ref Rs , ref n2Tref



1 Rsh ,ref

Vmp , ref  I mp , ref Rs , ref Vmp , ref  I mp , ref Rs , ref  I I 02 1 n1Tref n2Tref 01  e  e  n2Tref Rsh ,ref  n1Tref (8)

  



I mp ,ref Vmp ,ref

From Eqs. (3) and (7), parameters IL,ref and Rsh,ref can be extracted and substituted in Eqs. (4) and (5) that, due to these substitutions, only contain I01,ref, I02,ref, n1, n2 and Rs,ref. In Appendix A it is showed that Eqs. (4) and (5) can be solved in order to find I01,ref and I02,ref. Parameters I01,ref , I02,ref , IL,ref and Rsh,ref, calculated with Eqs. (A11), (A12), (A1) and (A2), only depend on n1, n2 and Rs,ref. Two of these parameters may be calculated using Eqs. (6) and (8), whereas to evaluate the last parameter a seventh equation should be necessary. Leaving aside for the moment parameter n2, the six parameters n1, I01,ref, I02,ref, IL,ref, Rs,ref, Rsh,ref can be evaluated with a double iterative procedure that uses the normalised values of Voc,ref, Isc,ref, Vmp,ref, Imp,ref, Rso and Rsho, obtained by dividing the voltage data by Voc,ref, the current data by Isc,ref and the resistance data by the ratio of Voc,ref to Isc,ref . The following sequence of steps is adopted: 1) an initial value of n2 is assumed; 2) an initial value of n1 is assumed; 3) an initial value of Rs,ref is assumed; 4) I01,ref and I02,ref are calculated by Eq. (A11) and (A12); 5) Rsh,ref is calculated by Eq. (A2); 6) IL,ref is calculated by Eq. (A1); 7) the first iterative procedure is concluded if Eq. (6) is verified within a fixed tolerance; otherwise, a new value of Rs,ref is assumed and the procedure is repeated starting from step 4); 8) the obtained value of Rs,ref is used to calculate Eq. (8); 9) the second iterative procedure is concluded if Eq. (8) is verified within a fixed tolerance; otherwise, a new value of n1 is assumed and the procedure is repeated starting from step 3). It may happen that the input data set does not permit to solve the six equations system within the required tolerances. Such an occurrence, which has been observed in the iterative solution of Eq. (8), may due to the fact that in correspondence with the values of Vmp,ref and Imp,ref issued by the producer the power-voltage curve has a derivative that is not perfectly equal to zero. In this case the second iteration can be concluded for the value of n1 for which Eq. (8) is solved with the smallest tolerance found during calculations. Due to the lack of a seventh equation, for each initially fixed value of n2, a different set of n1, I01,ref, I02,ref, IL,ref, Rs,ref, Rsh,ref can be evaluated. Even negative values of the model parameters may be obtained, which absolutely contrast with the physical meaning of the two-diode equivalent circuit. Eqs. (A11) and (A12) can lead to negative values of I01,ref and/or I02,ref, which correspond to the presence of diodes that unrealistically behave like generator. Especially for the PV modules with high values of Rsh,ref and/or small values of Rs,ref, it was observed that the I-V characteristics at the SRC calculated with negative values of I01,ref and/or I02,ref can present values of the current greater than Isc,ref, which cannot be considered physically possible. For this reason it was assumed 6

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that only the sets of calculated parameters with I01,ref > 0 and I02,ref > 0 could be considered acceptable. Moreover, the comparison between the calculated and the issued I-V curves led to observe that the minimum differences were obtained for a value of n2, greater than n1 and as near as possible to n1, for which it was still possible to calculate a data set of positive values of n1, I01,ref, I02,ref, IL,ref, Rs,ref, Rsh,ref. Accordingly, the procedure to calculate the model parameters above described was nested into a third iteration in which the initial value of n2 is gradually reduced in order to find the smallest value of n2 that still permits to calculate positive values of all the model parameters. Fig. 3 depicts the flowchart of the described procedure. INPUT

Data Normalization Set Set Set from Eq. A11 from Eq. A12 from Eq. A2 from Eq. A1 NO

E

C

YES NO

E

C

YES < 0 or

<0

YES

C

NO Data Inverse Normalization

OUTPUT 209 210

Fig.3. Flowchart of the model parameters calculation at the SRC

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The above procedure solves the equations system by means of a routine that can be easily implemented even like VBA macros in Microsoft Excel. The algorithm used to find Rs,ref and n1 is a modified version of the bisection method, which is very simple and robust. The data normalization permits the univocal definition of the numerical parameters involved in the root-finding procedure that are crucial points of the bisection method, such as the first-attempt values, the width of the searching interval, the bisection step and the accuracy level, avoiding the need of adjusting them for each different PV panel. For the same reason, the procedure evaluates n2 using a dimensionless parameter defined as the ratio of n2 to n1.

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4. Procedure to calculate the model parameters for conditions different from the SRC. Effective and reliable models should adequately represent the electrical behaviour of a PV device working in real conditions, in which both solar irradiance G and cell temperature T are usually different from the SRC. In order to reach this aim, the following generalised version of Eq. (2) is used:  V  IRs  G    V  IRs  G   V  IR  G  n1T s (9) I  G , T   I L  G , T   I 01  G , T  e  1  I 02  G , T  e n2T  1  R G      sh 

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As usual, photocurrent IL(G,T) is calculated with the following relation: G I L  G , T    I L ,ref   I ,sc T  Tref  Gref





(10)

and, for Rs(G) and Rsh(G), the expressions proposed by Lo Brano et al. [30-31] may be used: G G (11) Rs  G   Rs ,ref ref Rsh  G   Rsh ,ref ref G G in which Rs,ref and Rsh,ref are the series and shunt resistances calculated at the SRC. Actually, from the comparison of the calculated I-V curves with the characteristics issued by manufacturers it emerged that, for some PV panels, more accurate results can be achieved if the series and/or shunt resistance do not vary with the solar irradiance, i.e. are always equal to the values Rs and Rsh calculated at the SRC. A way to predict the convenience of selecting constant or variable values of the series and shunt resistances, consists in controlling if the shape of the I-V characteristic at the SRC is similar to the other curves depicted for G < 1000 W/m2, at T = Tref. As it is described in Appendix B, the proposed method permits to make the decision of using constant or variable values of the series and shunt resistances on the basis of the results of a comparison between some particular values of the derivative of the I-V characteristics. In order to define the relations describing I01(G,T) and I02(G,T), it is useful to separately consider the dependence on G and T. As first step, an expression for I01(G,Tref) and I02(G,Tref) can be defined, then a second expression for I01(Gref,T) and I02(Gref,T) can be found. Eventually, I01(G,Tref), I02(G,Tref), I01(Gref,T) and I02(Gref,T) will be combined in order to find the general expressions of I01(G,T) and I02(G,T). The procedure to calculate I01(G,Tref), I02(G,Tref) is based on the following expressions: G  200   (12) I 01 G , Tref  I 01,ref 1  RIo1   RIo1  800  





8

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G  200   I 02  G , Tref   I 02,ref 1  RIo2   RIo2  800   where: I I RIo1  01,200 RIo2  02,200 I 01,ref I 02,ref

(13)

(14)

251 252 253 254 255 256 257 258 259 260 261

The equations to calculate I01,200 and I02,200, which are the saturation current evaluated at T = Tref and at the lowest value of the solar irradiation (usually 200 W/m2), are described in the Appendix B. It was observed that, for some PV devices, currents I01,200 or I02,200 may assume negative values. Such an occurrence, which obviously should contrast with physics, seems do not represent an issue for the two-diode model as it, for values of G < Gref, never generates I-V curves with values of the current greater than the value produced at the short circuit point. As second step, to calculate currents I01(Gref,T) and I02(Gref,T) the same Eqs. (A3-A12) used to evaluate I01,ref and I02,ref can be adopted, as the values of n1 and n2 are known and Rs is the value related to the selection criterion above mentioned. Obviously, the values of Isc,ref, Voc,ref, Vmp,ref, Imp,ref in Eqs. (A5-A10) have to be corrected with the temperature: (15) Voc (T )  Voc ,ref  V ,oc (T  Tref )

262

I sc (T )  I sc ,ref   I ,sc (T  Tref )

(16)

263

Vmp (T )  Vmp ,ref  V ,mp (T  Tref )

(17)

264

I mp (T )  I mp ,ref   I ,mp (T  Tref )

(18)

265 266

where μV,mp and μI,mp are the MPP voltage and current temperature coefficients. Finally, currents I01(G,T) and I02(G,T) can be calculated with the following relations: G  200   (19) I 01  G , T   I 01 (Gref , T ) 1  RIo1   RIo1  800  

267 268 269 270 271

G  200   (20) I 02  G , T   I 02 (Gref , T ) 1  RIo2   RIo2  800   in which ratios RIo1 and RIo2 are considered invariable with G and T. Fig. 4 depicts the flowchart of the above mentioned procedure.

9

ACCEPTED MANUSCRIPT INPUT

μV,oc μI,sc

from Eq. 10 and and YES

from Eq. B5 from Eq. B6

NO

<

(G ) = (200) =

(G ) from Eq. 11 (200) from Eq. 11

YES

NO

<

(G ) = (200) =

(G ) from Eq. 11 (200) from Eq. 11

from Eq. B9 from Eq. B10 from Eq. 12 from Eq. 13 from Eq. 15 (T ) from Eq. 16 from Eq. 17 from Eq. 18 from Eq. A11 from Eq. A12 from Eq. 19 from Eq. 20

OUTPUT 272 273

( ) Fig.4. Flowchart of the model parameters calculation for conditions different from the SRC

10

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5. Application of the procedure and analysis of the results The model parameters calculated with the new procedure were used for drawing the I-V characteristics of seven different typologies of PV panel (monocrystalline, polycrystalline, HIT, amorphous silicon, CIS, CIGS and tandem junction), whose performance data are listed in Table 1.

279

Table 1. Data for the evaluation of the model parameters at the SRC. PANEL TYPE

280 281

Shell (Monocrystalline) SQ 150-PC Kyocera (Polycrystalline) KD245GH-4FB2 Sanyo (HIT) HIT-240 HDE4 Exiom (Amorphous) EXTF-75 Solar Frontier (CIS) SF130-L Soltecture (CIGS) Linion 90 Pramac (Tandem) MCPH P7 125

283 284 285 286 287 288

Isc,ref [A]

Vmp,ref [V]

Imp,ref [A]

Rsho [Ω]

Rso [Ω]

43.40

4.80

34.00

4.39

444.54

1.08

36.90

8.91

29.50

8.30

120.48

0.49

43.60

7.37

34.31

7.10

3204.64

0.87

134.50

0.99

98.00

0.84

2192.71

23.68

106.00

2.10

75.00

1.80

1640.69

13.59

72.20

1.80

57.50

1.59

1388.89

3.80

131.40

1.54

100.00

1.21

526.32

12.84

Table 2. Data for the evaluation of the model parameters at conditions different from the SRC. PANEL TYPE

282

Voc,ref [V]

Shell (Monocrystalline) SQ 150-PC Kyocera (Polycrystalline) KD245GH-4FB2 Sanyo (HIT) HIT-240 HDE4 Exiom (Amorphous) EXTF-75 Solar Frontier (CIS) SF130-L Soltecture (CIGS) Linion 90 Pramac (Tandem) MCPH P7 125

Voc,200 [V]

Rso,200 [Ω]

μV,oc [V/°C]

μI,sc [A/°C]

μV,max [V/°C]

μI,max [A/°C]

39.70

3.23

-1.49·10-1

1.74·10-3

-1.43·10-1

-1.86·10-3

34.40

2.28

-1.32·10-1

4.08·10-3

-1.40·10-1

1.00·10-4

40.61

4.36

-1.09·10-1

2.12·10-3

-1.08·10-1

6.34·10-4

117.20

80.19

-3.42·10-1

2.49·10-3

-3.11·10-1

4.22·10-4

96.13

32.27

-3.20·10-1

-2.09·10-4

-3.00·10-1

-2.54·10-4

61.00

18.93

-2.64·10-1

1.74·10-4

-2.50·10-1

-5.03·10-4

118.62

45.25

-4.05·10-1

1.09·10-3

-3.89·10-1

8.15·10-4

For the sake of precision the data listed in Tables 1 and 2 were accurately extracted from the graphs provided by manufacturers [32-38]. Table 3 lists the data used to define the condition on the variability of the series and shunt resistances in compliance with the results of the selection method based on Eqs. (B5) and (B6). Table 3. Data for the evaluation of resistances Rs and Rsh at conditions different from the SRC. PANEL TYPE Shell SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4

Rso [Ω]

Rso,200 [Ω]

Rx [Ω]

ΔRso,V

ΔRso,C

Rs

Rsho [Ω]

Rsho,200 [Ω]

ΔRsho,V

ΔRsho,C

Rsh

1.08

3.23

4.33

0.667

0.340

Const.

444.54

862.40

1.577

0.485

Const.

0.49

2.28

1.45

0.079

0.366

Variab.

120.48

573.64

0.050

0.790

Variab.

0.87

4.36

1.74

0.434

0.600

Variab.

3204.64

2374.84

5.747

0.349

Const.

11

ACCEPTED MANUSCRIPT Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

289 290 291 292

23.68

80.19

77.28

0.477

0.036

Const.

2192.71

10525.00

0.042

0.792

Variab.

13.59

32.27

30.46

1.106

0.056

Const.

1640.69

2812.94

1.916

0.417

Const.

3.80

18.93

14.82

0.004

0.217

Variab.

1388.89

5111.11

0.359

0.728

Variab.

12.84

45.25

59.17

0.419

0.308

Const.

526.32

2500.00

0.053

0.789

Variab.

In Table 4 the values of the parameters evaluated with the new procedure are listed. Table 4. Evaluated parameters of the new model at the SRC. PANEL TYPE Shell (Monocrystalline) SQ 150-PC Kyocera (Polycrystalline) KD245GH-4FB2 Sanyo (HIT) HIT-240 HDE4 Exiom (Amorphous) EXTF-75 Solar Frontier (CIS) SF130-L Soltecture (CIGS) Linion 90 Pramac (Tandem) MCPH P7 125

293 294 295 296 297

IL [A]

I01 [A]

I02 [A]

n1 [V/K]

n2 [V/K]

Rsh [Ω]

Rs [Ω]

4.8054

1.8765·10-19

5.9857·10-7

4.5340·10-3

9.1677·10-3

444.1415

0.4981

8.9337

1.8084·10-10

1.9062·10-10

5.0334·10-3

8.0913·10-3

120.1628

0.3192

7.3716

1.9858·10-14

1.6058·10-11

4.3593·10-3

8.0232·10-3

3203.9441

0.6965

0.9945

8.1624·10-12

3.0484·10-5

2.1891·10-2

4.3705·10-2

2206.7297

9.9174

2.1141

6.4952·10-12

1.2504·10-4

1.3495·10-2

4.6583·10-2

1776.1965

11.4410

1.8004

1.1177·10-5

2.8718·10-6

2.0639·10-2

2.0650·10-2

1393.4066

0.2893

1.5499

3.5266·10-5

1.0004·10-6

4.2027·10-2

4.2048·10-2

524.1427

3.3741

Tables 5 and 6 list the values of the saturation currents and the shunt and series resistances for conditions different from the SRC. Table 5. Evaluated parameters of the new model at temperature T = 25°C. PV PANEL

PARAMETER

IRRADIANCE [W/m2] 200

400

600

800

I01 [A]

2.0884·10-14

1.5663·10-14

1.0442·10-14

5.2211·10-15

Shell

I02 [A]

3.7070·10-7

4.2767·10-7

4.8464·10-7

5.4160·10-7

SQ 150-PC

Rsh [Ω]

444.1415

444.1415

444.1415

444.1415

Rs [Ω]

0.4981

0.4981

0.4981

0.4981

2.8834·10-10

2.5251·10-10

2.1668·10-10

I01 [A]

3.2418·10-10

Kyocera

I02 [A]

-7.6544·10-7

-5.7403·10-7

-3.8262·10-7

-1.9122·10-7

KD245GH-4FB2

Rsh [Ω]

600.8139

300.4070

200.2713

150.2035

Rs [Ω]

1.5958

0.7979

0.5319

0.3990

I01 [A]

4.0955·10-14

3.5681·10-14

3.0406·10-14

2.5132·10-14

Sanyo

I02 [A]

-2.2986·10-9

-1.7199·10-9

-1.1413·10-9

-5.6260·10-10

HIT-240 HDE4

Rsh [Ω]

3203.9441

3203.9441

3203.9441

3203.9441

Rs [Ω]

3.4823

1.7411

1.1608

0.8706

I01 [A]

-6.4704·10-11

-4.6487·10-11

-2.8271·10-11

-1.0054·10-11

Exiom

I02 [A]

2.3880·10-5

2.5531·10-5

2.7182·10-5

2.8833·10-5

EXTF-75

Rsh [Ω]

11033.6483

5516.8242

3677.8828

2758.4121

Rs [Ω]

9.9174

9.9174

9.9174

9.9174

I01 [A]

4.9731·10-12

5.3536·10-12

5.7341·10-12

6.1147·10-12

Solar Frontier

12

ACCEPTED MANUSCRIPT SF130-L

I02 [A]

2.4722·10-4

2.1668·10-4

1.8613·10-4

1.5559·10-4

Rsh [Ω]

1776.1965

1776.1965

1776.1965

1776.1965

Rs [Ω]

11.4410

11.4410

11.4410

11.4410

I01 [A]

-7.8821·10-6

-3.1173·10-6

1.6475·10-6

6.4123·10-6

Soltecture

I02 [A]

2.5410·10-5

1.9776·10-5

1.4141·10-5

8.5064·10-6

Linion 90

Rsh [Ω]

6967.0329

3483.5164

2322.3443

1741.7582

Rs [Ω]

1.4465

0.7232

0.4822

0.3616

I01 [A]

4.6230·10-3

3.4760·10-3

2.3291·10-3

1.1822·10-3

Pramac

I02 [A]

-4.6243·10-3

-3.4680·10-3

-2.3116·10-3

-1.1553·10-3

MCPH P7 125

Rsh [Ω]

2620.7134

1310.3567

873.5711

655.1783

Rs [Ω]

3.3741

3.3741

3.3741

3.3741

298 299

Table 6. Evaluated parameters of the new model at solar irradiance G = 1000 W/m2. PV PANEL

TEMPERATURE [°C] 50

55

60

70

75

I01 [A]

-6.5681·10-15

-

2.2826·10-13

-

-

Shell

I02 [A]

7.2479·10-6

-

1.7606·10-5

-

-

SQ 150-PC

Rsh [Ω]

444.1415

-

444.1415

-

-

Rs [Ω]

0.4981

-

0.4981

-

-

Kyocera

I01 [A]

8.8267·10-9

-

-

-

2.4566·10-7

KD245GH-

I02 [A]

1.2988·10-6

-

-

-

2.0280·10-5

Rsh [Ω]

120.1628

-

-

-

120.1628

4FB2

300 301 302

PARAMETER

Rs [Ω]

0.3192

-

-

-

0.3192

I01 [A]

1.9253·10-12

-

-

-

9.7324·10-11

Sanyo

I02 [A]

-4.5797·10-8

-

-

-

-7.6115·10-7

HIT-240 HDE4

Rsh [Ω]

3203.9441

-

-

-

3203.9441

Rs [Ω]

0.6965

-

-

-

0.6965

I01 [A]

-

-5.7118·10-9

-

-3.2602·10-8

-

-

4.6639·10-4

-

Exiom

I02 [A]

-

2.0789·10-4

EXTF-75

Rsh [Ω]

-

2206.7297

-

2206.7297

-

Rs [Ω]

-

9.9174

-

9.9174

-

I01 [A]

3.3397·10-10

-

-

-

9.8872·10-9

Solar Frontier

I02 [A]

1.9879·10-4

-

-

-

-6.2719·10-5

SF130-L

Rsh [Ω]

1776.1965

-

-

-

1776.1965

Rs [Ω]

11.4410

-

-

-

11.4410

I01 [A]

9.9857·10-3

-

-

-

9.7015·10-2

Soltecture

I02 [A]

-9.9405·10-3

-

-

-

-9.6932·10-2

Linion 90

Rsh [Ω]

1393.4066

-

-

-

1393.4066

Rs [Ω]

0.2893

-

-

-

0.2893

I01 [A]

2.0014·10-2

6.5940·10-2

Pramac

I02 [A]

-1.9861·10-2

-6.5655·10-2

MCPH P7 125

Rsh [Ω]

524.1427

524.1427

Rs [Ω]

3.3741

3.3741

The saturation currents keep about the same order of magnitude if the solar irradiance is varied from 200 to 800 W/m2; more consistent variations are generally observed when the temperature is 13

ACCEPTED MANUSCRIPT 303 304 305 306 307 308 309

changed. As it will be showed in the following, the negative values of the saturation currents for conditions different from the STC seem do not represent an issue for the precision of the two-diode model. In order to check the performances of the new procedure, a comparison with the I-V curves calculated with the Hejri et al. and Elbaset et al. models and the data issued by the manufacturer’s datasheets was made. Figs. 5-11 depict the calculated I-V curves for the analysed PV modules.

Datasheet New Procedure Elbaset et al. Hejri et al.

5.0

1000 W/m2

4.0

800 W/m2

3.0

600 W/m2

2.0

400 W/m2

1.0

200 W/m2

5.0 4.0

5

10

15

310

50 °C

2.0

60 °C

20 25 Voltage [V]

30

35

40

0

45

5

10

15

20 25 Voltage [V]

30

35

40

45

Fig. 5. Comparison between the issued I-V characteristics of the Shell SQ 150-PC PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and the Hejri et al.

1000 W/m2

9

10

KYOCERA KD245GH-4FB2 G=1000 W/m2 (Polycrystalline)

Datasheet New Procedure Elbaset et al. Hejri et al.

9 8

8

800 W/m2

7 Current [A]

Datasheet New Procedure Elbaset al. Hejri et al.

Kyocera KD245GH-4FB2 T=25°C (Polycrystalline)

6

7 Current [A]

10

600 W/m2

5 4

400 W/m2

25 °C

6 5 4

50 °C

3

3

2

200 W/m2

2

75 °C

1

1

0

0 0

314 315

3.0

0.0 0

313

25 °C

1.0

0.0

311 312

Datasheet New Procedure Elbaset et al. Hejri et al.

Shell SQ 150-PC G=1000 W/m2(Monocrystalline)

Current [A]

Current [A]

Shell SQ 150-PC T=25°C (Monocrystalline)

5

10

15

20 25 Voltage [V]

30

35

0

40

5

10

15

20 25 Voltage [V]

30

35

40

Fig. 6. Comparison between the issued I-V characteristics of the Kyocera KD245 GH-4FB2 PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and Hejri et al.

14

ACCEPTED MANUSCRIPT

8

SANYO HIT-240 HDE4 G=1000 W/m2 (Heterojunction)

Datasheet New Procedure Elbaset et al. Hejri et al.

Sanyo HIT-240 HDE4 T=25°C (Heterojunction) 1000 W/m2

8 7

7 800 W/m2

5

600

6 Current [A]

Current [A]

6

W/m2

4 400 W/m2

3 2

25 °C

5 4 3

50 °C

2

200 W/m2

75 °C

1

1

0

0 0

5

10

15

316 317 318

Datasheet New Procedure Elbaset et al. Hejri et al.

20 25 30 Voltage [V]

35

40

0

45

5

10

15

20 25 30 Voltage [V]

35

40

45

Fig. 7. Comparison between the issued I-V characteristics of the Sanyo HIT-240 HDE4 PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and Hejri et al. Datasheet New Procedure Elbaset et al. Hejri et al.

Exiom EXTF-75 T=25°C (Amorphous) 1.0

1000 W/m2

0.8

800 W/m2

0.6

600 W/m2

0.4

W/m2

1.2

Datasheet New Procedure Elbaset et al. Hejri et al.

Exiom EXTF-75 G=1000 W/m2 (Amorphous)

400

Current [A]

Current [A]

1.0

55 °C

70 °C

0.0 0

320 321

0.6

0.2

0.0

319

25 °C

0.4

200 W/m2

0.2

0.8

25

50

75 Voltage [V]

100

0

125

25

50

75 Voltage [V]

100

125

Fig. 8. Comparison between the issued I-V characteristics of the Exiom EXTF-75 PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and Hejri et al.

15

ACCEPTED MANUSCRIPT

2.4

Datasheet New Procedure Elbaset et al. Hejri et al.

Solar Frontier SF130-L T=25°C (CIS) 1000 W/m2

2.4 2.0

800

W/m2

600

W/m2

1.6

Current [A]

Current [A]

2.0

1.2 400 W/m2

0.8

1.6

25 °C

1.2 50 °C

0.8

200 W/m2

0.4

75 °C

0.4 0.0

0.0 0

20

322 323 324

Datasheet New Procedure Elbaset et al. Hejri et al.

Solar Frontier SF130-L G=1000 W/m2 (CIS)

40

60 Voltage [V]

80

0

100

20

40 60 Voltage [V]

80

100

Fig. 9. Comparison between the issued I-V characteristics of the Solar Frontier SF130-L PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and Hejri et al.

2.0

Soltecture Linion 90 F T=25°C (CIGS) 1000 W/m2

Datasheet New Procedure Elbaset et al. Hejri et al.

2.0

Datasheet New Procedure Elbaset et al. Hejri et al.

Soltecture Linion 90 F G=1000 W/m2 (CIGS)

1.6

1.6

800 W/m2

Current [A]

Current [A]

25 °C

1.2

600 W/m2

0.8

400 W/m2

0.4

200 W/m2

50 °C

75 °C

0.0 0

326 327

0.8 0.4

0.0

325

1.2

15

30 45 Voltage [V]

60

0

75

10

20

30 40 Voltage [V]

50

60

70

Fig. 10. Comparison between the issued I-V characteristics of the Soltecture Linion 90 F PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and Hejri et al.

16

ACCEPTED MANUSCRIPT Datasheet New Procedure Elbaset et al. Hejri et al.

Pramac MCPH P7 125W T=25°C (Tandem) 1.6

1000 W/m2

1.6 1.4

1.4 800 W/m2

1.0

600

1.2 Current [A]

Current [A]

1.2

W/m2

0.8 400 W/m2

0.6 0.4

25 °C

1.0 0.8 55 °C

0.6 0.4

200 W/m2

70 °C

0.2

0.2

0.0

0.0 0

328

Datasheet New Procedure Elbaset et al. Hejri et al.

Pramac MCPH P7 125W G=1000 W/m2 (Tandem)

25

50

75 Voltage [V]

100

0

125

25

50

75 Voltage [V]

100

125

329 330

Fig. 11. Comparison between the issued I-V characteristics of the Pramac MCPH P7 125W PV panel and the curves calculated with the new procedure and the models of Elbaset et al. and Hejri et al.

331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355

The above figures permit to make some qualitative observations about the capability of the analysed models to represent the I-V characteristics of the PV modules whose performance data are listed in Tables 1 and 2. The Elbaset et al. model provides a very good representation of the I-V curves of the Shell PV panel; only a small imprecision is showed for low values of the solar irradiance. The performance curves of the Kyocera PV module at constant cell temperature are very accurately calculated; the model is imprecise when is used at G=1000 W/m2 and T=75°C. With the Sanyo, Exiom, Solar Frontier and Soltecture PV panels the model is accurate for the SRC values of the solar irradiance. In the region between the MPP and the open circuit point a lack of precision is observed as the values of G begin to be far from the SRC. The characteristics of the Pramac PV module at the SRC are perfectly calculated by the Elbaset et al. model; unfortunately the model becomes imprecise for condition far from the SRC. The Hejri et al. model perfectly calculates the I-V curves of the Shell PV panel. A good precision is achieved with the Kyocera and the Solar Frontier PV modules. With the Sanyo PV panel, the model results less precise for values of the solar irradiance and cell temperature far from the SRC. The Hejri et al. model is accurate in the evaluation of the curves at constant temperature of the Exiom PV module; a lower precision is showed for values of the cell temperature higher than 25 °C. The representation of the characteristics of the Soltecture and Pramac PV panels is very good at the SRC. The model becomes less accurate when is used for conditions far from the SRC. It can be easily noted that the accuracy achieved by the Elbaset et al. and Hejri et al. models is significantly influenced by the typology of the considered PV device. Conversely, for each value of the solar irradiance and/or cell temperature, the curves calculated with the new procedure are very similar to the issued characteristics of all the analysed PV panels. A more precise assessment of the performances of the studied models can be executed by means of the numerical information contained in the following tables.

17

ACCEPTED MANUSCRIPT 356 357 358 359 360 361 362 363 364 365 366 367 368

To evaluate the quality of the calculated I-V curves the mean absolute difference of current to the rated current at the MPP and of the mean absolute difference of power to the rated power at the MPP, calculated with the following expressions:  100  1 N (49) ADI    I calc , j  I iss , j  I mp  N j 1  N  100  1 (50) ADP    Viss , j I calc , j  Viss , j I iss , j  Vmp I mp  N j 1  are used. In Eqs. (49) and (50) Viss,j and Iiss,j are the voltage and current of the j-th point extracted from the I-V characteristics issued by manufacturers, Icalc,j is the value of the current calculated in correspondence with Viss,j and N is the number of extracted points. In Tables 7 and 8 the values of the ADI, referred to the I-V curves at T=Tref and G=Gref, are listed. In order to facilitate the interpretation of the data listed in the tables, the maximum and the minimum values for each curve are written in underscored or bold type, respectively. Table 7. Comparison between the values of the ADI at temperature T=25 °C PV PANEL Shell SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

369 370

IRRADIANCE [W/m2]

TWO-DIODE MODEL

200

400

600

800

1000

Hejri et al. model

0.638

0.957

0.752

0.593

0.957

Elbaset et al. model

1.937

1.800

1.413

1.003

1.071

New procedure

0.752

0.729

0.934

1.162

1.048

Hejri et al. model

1.301

1.927

1.590

1.253

0.915

Elbaset et al. model

0.783

0.988

0.783

0.783

0.494

New procedure

0.566

0.867

1.156

0.795

0.494

Hejri et al. model

1.028

2.465

3.620

3.902

3.930

Elbaset et al. model

3.282

4.395

4.353

2.550

0.324

New procedure

0.183

0.507

0.563

0.845

0.324

Hejri et al. model

0.833

0.714

1.072

1.548

1.191

Elbaset et al. model

4.167

3.096

2.143

0.952

0.833

New procedure

0.476

0.833

1.191

1.548

0.833

Hejri et al. model

0.998

1.165

0.832

0.388

1.165

Elbaset et al. model

1.830

2.662

2.662

1.609

0.555

New procedure

1.054

0.998

0.832

0.388

0.666

Hejri et al. model

3.090

3.972

3.468

2.711

1.324

Elbaset et al. model

1.387

2.018

2.207

2.333

0.694

New model

0.126

0.315

0.694

0.441

0.504

Hejri et al. model

2.988

4.398

4.315

3.237

1.909

Elbaset et al. model

9.876

10.788

8.133

4.398

0.332

New procedure

0.581

0.581

0.747

0.830

0.581

Table 8. Comparison between the values of the ADI at irradiance G =1000 W/m2 PV PANEL Shell

TWO-DIODE MODEL

TEMPERATURE [°C] 25

50

55

60

70

75

Hejri et al. model

0.957

0.798

-

1.276

-

-

Elbaset et al. model

1.071

1.231

-

0.798

-

-

18

ACCEPTED MANUSCRIPT SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388

New procedure

1.048

1.413

-

1.276

-

-

Hejri et al. model

0.915

3.361

-

-

-

7.805

Elbaset et al. model

0.494

3.168

-

-

-

6.854

New procedure

0.494

0.879

-

-

-

1.337

Hejri et al. model

3.930

5.099

-

-

-

6.733

Elbaset et al. model

0.324

1.352

-

-

-

2.986

New procedure

0.324

0.535

-

-

-

1.071

Hejri et al. model

1.191

-

7.620

-

11.311

-

Elbaset et al. model

0.833

-

1.429

-

1.429

-

New procedure

0.833

-

2.143

-

2.977

-

Hejri et al. model

1.165

1.331

-

-

-

2.773

Elbaset et al. model

0.555

1.498

-

-

-

3.883

New procedure

0.666

1.054

-

-

-

1.442

Hejri et al. model

1.324

2.963

-

-

-

2.648

Elbaset et al. model

0.694

2.207

-

-

-

2.837

New procedure

0.504

1.513

-

-

-

0.631

Hejri et al. model

1.909

-

1.162

-

1.660

-

Elbaset et al. model

0.332

-

2.241

-

2.158

-

New procedure

0.581

-

1.411

-

0.747

-

From the data in Table 7 it can be observed that, among the compared model of Hejri et al. and Elbaset et al., the best accuracy in representing the I-V curves at constant temperature is achieved by the Elbaset et al. model for the Kyocera PV panel, with values of the ADI varying between 0.494% and 0.988%. The same model also presents the lowest accuracy for the Pramac PV device, with values of the ADI ranging from 0.332% to 10.788%. The I-V characteristics evaluated with the new procedure yield values of the ADI that vary from 0.126% to 0.694% for the Soltecture PV module, in the best case. The smallest precision is reached for the Exiom PV device with values of the ADI contained from 0.476% to 1.548%. Considering the cell temperature variation, the data of the ADI listed in Table 8 show that, among the compared models, the greater accuracy is given by the Hejri et al. model, which shows values of the ADI that range from 0.798% to 1.276% for the Shell PV module. The smallest precision is evaluated with the same model that provides values of the ADI varying between 1.191% and 11.311% for the Exiom PV panel. By means of the new procedure it is possible to obtain the greatest precision, ranging from 0.324% to 1.071%, for the Sanyo PV module, and the smallest accuracy, which varies from 0.833% to 2.977%, for the Exiom PV device. Table 9. Comparison between the values of the ADP at temperature T=25 °C PV PANEL Shell SQ 150-PC Kyocera KD245GH-4FB2

TWO-DIODE MODEL

IRRADIANCE [W/m2] 200

400

600

800

1000

Hejri et al. model

0.673

1.048

0.824

0.572

1.046

Elbaset et al. model

1.896

1.862

1.416

0.969

1.135

New procedure

0.648

0.709

0.902

1.158

1.113

Hejri et al. model

1.337

2.020

1.648

1.262

0.945

Elbaset et al. model

0.687

0.986

0.814

0.833

0.525

New procedure

0.548

0.924

1.291

0.900

0.536

19

ACCEPTED MANUSCRIPT Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

389 390

1.063

2.701

4.039

4.380

4.412

Elbaset et al. model

3.556

4.868

4.856

2.828

0.352

New procedure

0.168

0.541

0.627

0.972

0.373

Hejri et al. model

0.782

0.604

1.022

1.635

1.152

Elbaset et al. model

3.990

3.107

2.132

1.093

0.770

New procedure

0.425

0.744

1.158

1.477

0.761

Hejri et al. model

1.004

1.315

0.981

0.467

1.391

Elbaset et al. model

1.730

2.839

3.116

1.945

0.532

New procedure

0.874

0.933

0.901

0.428

0.674

Hejri et al. model

3.100

4.212

3.738

2.931

1.279

Elbaset et al. model

1.261

2.068

2.344

2.484

0.632

New procedure

0.083

0.309

0.704

0.370

0.478

Hejri et al. model

2.790

4.514

4.724

3.622

2.198

Elbaset et al. model

7.806

10.078

8.326

4.726

0.338

New procedure

0.504

0.482

0.703

0.761

0.557

Table 10. Comparison between the values of the ADP at irradiance G =1000 W/m2 PV PANEL Shell SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

391 392 393 394 395 396

Hejri et al. model

TWO-DIODE MODEL

TEMPERATURE [°C] 25

50

55

60

70

75

Hejri et al. model

1.046

1.045

-

0.711

-

-

Elbaset et al. model

1.135

1.146

-

0.660

-

-

New procedure

1.113

1.349

-

1.151

-

-

Hejri et al. model

0.945

3.276

-

-

-

7.323

Elbaset et al. model

0.525

3.155

-

-

-

6.349

New procedure

0.536

0.855

-

-

-

1.188

Hejri et al. model

4.412

5.415

-

-

-

6.812

Elbaset et al. model

0.352

1.446

-

-

-

3.054

New procedure

0.373

0.587

-

-

-

1.081

Hejri et al. model

1.152

-

8.236

-

11.723

-

Elbaset et al. model

0.770

-

1.233

-

1.097

-

New procedure

0.761

-

2.108

-

2.666

-

Hejri et al. model

1.391

1.109

-

-

-

1.970

Elbaset et al. model

0.532

1.358

-

-

-

3.223

New procedure

0.674

0.953

-

-

-

1.145

Hejri et al. model

1.279

2.717

-

-

-

2.130

Elbaset et al. model

0.632

1.997

-

-

-

2.216

New procedure

0.478

1.400

-

-

-

0.509

Hejri et al. model

2.198

-

1.198

-

1.540

-

Elbaset et al. model

0.338

-

2.452

-

2.188

-

New procedure

0.557

-

1.397

-

0.629

-

The values of the ADP in Tables 9 and 10 confirm the data of the ADI listed in Tables 7 and 8. Among the compared models, the Elbaset et al. model presents both the greatest and the smallest accuracy for the I-V curves at constant temperature. The model provides values of the ADP respectively ranging from 0.525% to 0.986%, for the Kyocera PV panel, and from 0.338% to

20

ACCEPTED MANUSCRIPT 397 398 399 400 401 402 403 404 405 406 407 408 409

10.078%, for the Pramac PV device. With the new procedure the best and worst performances are again obtained for the Soltecture and the Exiom PV panels, respectively. The ADP varies between 0.083% and 0.704%, in the first case, and between 0.425% and 1.477%, in the second case. Considering the I-V curves at constant solar irradiance, the Hejri et al. model gives both the greatest and the smallest precisions for the Shell and the Exiom PV modules, respectively. The smallest ADP ranges from 0.711% to 1.046%; the greatest ADP varies between 1.152% and 11.723%. The greatest and smallest accuracies are again calculated with the new procedure for the Sanyo and the Exiom PV devices. Values of the ADP contained in the range from 0.373% to 1.081% and from 0.761% to 2.666% are respectively provided. In order to assess the range of dispersion of the results, Tables 11-14 list the percentage ratio of the maximum difference of current to the rated current at the MPP, and of the maximum difference of power to the rated power at the MPP, evaluated using the following relations: 100 (51) MDI  MAX  I calc , j  I iss , j  I mp 100 MAX Viss , j I calc , j  Viss , j I iss , j  Vmp I mp

(52)

410

MDP 

411 412 413 414

In the tables, the upper and lower bounds of the MDI and MDP are indicated with underscored or bold type, respectively. Table 11. Comparison between the values of the MDI at temperature T=25 °C PV PANEL Shell SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

TWO-DIODE MODEL

IRRADIANCE [W/m2] 200

400

600

800

1000

Hejri et al. model

2.849

3.122

2.917

1.687

-3.783

Elbaset et al. model

-5.515

-4.946

-3.624

-3.054

3.191

New procedure

-1.413

1.481

2.598

3.077

3.191

Hejri et al. model

3.096

4.553

3.589

2.722

-2.457

Elbaset et al. model

-1.506

2.505

-2.325

2.313

-1.542

New procedure

1.108

-2.445

-4.433

-3.216

1.542

Hejri et al. model

3.620

5.325

4.198

3.184

-2.874

Elbaset et al. model

9.339

12.593

11.903

6.916

-1.423

New procedure

-0.451

-1.141

-1.620

-3.113

-1.395

Hejri et al. model

-2.143

-2.024

-2.738

-3.929

-3.572

Elbaset et al. model

-8.334

-7.025

-6.191

-5.239

2.500

New procedure

-0.952

1.548

2.500

3.096

2.500

Hejri et al. model

-5.325

-5.491

-4.105

-1.720

4.382

Elbaset et al. model

-8.320

-11.426

-10.761

-6.101

1.553

New procedure

-1.830

1.997

2.108

1.276

-2.718

Hejri et al. model

9.836

13.367

10.971

9.584

-4.288

Elbaset et al. model

4.288

7.566

7.377

8.701

1.261

New procedure

0.252

-2.333

-3.342

-0.757

-1.450

Hejri et al. model

-10.622

-18.589

-18.672

-13.776

-7.552

Elbaset et al. model

-18.257

-22.407

-17.593

-9.046

0.913

New procedure

-1.328

-0.913

1.411

1.826

-1.411

415 416 21

ACCEPTED MANUSCRIPT 417

Table 12. Comparison between the values of the MDI at irradiance G =1000 W/m2

Shell SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

418 419 420 421 422 423 424 425 426 427 428 429 430 431

TEMPERATURE [°C]

TWO-DIODE MODEL

PV PANEL

25

50

55

60

70

75

Hejri et al. model

-3.783

-2.302

-

3.920

-

-

Elbaset et al. model

3.191

-5.333

-

2.188

-

-

New procedure

3.191

4.399

-

3.966

-

-

Hejri et al. model

-2.457

8.986

-

-

-

18.297

Elbaset et al. model

-1.542

7.420

-

-

-

14.743

New procedure

1.542

2.855

-

-

-

4.505

Hejri et al. model

-2.874

14.256

-

-

-

16.594

Elbaset et al. model

-1.423

4.564

-

-

-

10.114

New procedure

-1.395

-2.254

-

-

-

-2.747

Hejri et al. model

-3.572

-

-19.169

-

-25.598

-

Elbaset et al. model

2.500

-

3.453

-

3.691

-

New procedure

2.500

-

-4.762

-

-4.762

-

Hejri et al. model

4.382

-3.106

-

-

-

-5.824

Elbaset et al. model

1.553

-3.273

-

-

-

-6.601

New procedure

-2.718

-2.995

-

-

-

-3.605

Hejri et al. model

-4.288

-8.575

-

-

-

-6.242

Elbaset et al. model

1.261

-4.729

-

-

-

-5.170

New procedure

-1.450

-5.675

-

-

-

-1.702

Hejri et al. model

-7.552

-

-6.888

-

-9.295

-

Elbaset et al. model

0.913

-

8.299

-

8.382

-

New procedure

-1.411

-

3.983

-

1.992

-

The data listed in Table 11 show that, among the compared models of Hejri et al. and Elbaset et al., the greatest MDI is evaluated by the Elbaset et al. model for the Pramac PV module with a value of -22.407%. For the same PV device the Elbaset et al. model also provides the smallest MDI with a value equal to 0.913%. The new procedure permits to reach values of the MDI ranging from 0.252% to -4.433% for the Soltecture and the Kyocera PV devices, respectively. Considering the data listed in Table 12, which refers to the I-V curves at constant solar irradiance, the greatest accuracy among the compared models is given by Elbaset et al. model that reaches a value of the MDI equal to 0.913% for the Pramac PV module. The greatest MDI, which is -25.598%, is obtained by the Hejri et al. model for the Exiom PV panel. With the new procedure it is possible to evaluate the I-V characteristics with MDI ranging from -1.395% to -5.675% for the Sanyo and the Soltecture PV devices, respectively. Table 13. Comparison between the values of MDP at temperature T=25 °C PV PANEL Shell SQ 150-PC Kyocera

TWO-DIODE MODEL

IRRADIANCE [W/m2] 200

400

600

800

1000

Hejri et al. model

3.312

3.626

3.348

1.909

-4.705

Elbaset et al. model

-6.193

-5.883

-4.409

-3.787

3.752

New model

-1.271

1.712

2.969

3.571

3.754

Hejri et al. model

3.418

5.030

3.961

2.993

-2.866

Elbaset et al. model

-1.286

2.760

-2.793

2.548

-1.899

22

ACCEPTED MANUSCRIPT KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

432 433

1.058

-2.846

-5.284

-3.940

-1.817

Hejri et al. model

3.506

9.028

13.626

14.537

14.615

Elbaset et al. model

10.448

14.143

13.568

7.886

-1.765

New model

-0.452

-1.315

-1.989

-3.818

-1.730

Hejri et al. model

-2.449

-2.579

-3.646

-5.165

-4.674

Elbaset et al. model

-9.397

-8.534

-8.050

-6.839

-2.512

New model

-0.937

1.787

2.913

3.596

-2.515

Hejri et al. model

-6.570

-7.151

-5.556

-2.373

5.534

Elbaset et al. model

-9.704

-14.484

-14.349

-8.349

1.975

New model

-1.915

2.558

2.645

1.711

-2.975

Hejri et al. model

10.243

15.113

12.910

11.698

-5.204

Elbaset et al. model

4.474

8.528

8.625

10.616

-1.485

New model

0.246

-2.628

-3.888

0.904

-1.792

Hejri et al. model

-11.374

-21.820

-22.876

-17.096

-9.415

Elbaset et al. model

-17.778

-25.749

-21.561

-11.561

1.113

New model

-1.482

0.997

1.764

2.295

-1.791

Table 14. Comparison between the values of the MDP at irradiance G =1000 W/m2 PV PANEL Shell SQ 150-PC Kyocera KD245GH-4FB2 Sanyo HIT-240 HDE4 Exiom EXTF-75 Solar Frontier SF130-L Soltecture Linion 90 Pramac MCPH P7 125

434 435 436 437

New model

TWO-DIODE MODEL

TEMPERATURE [°C] 25

50

55

60

70

75

Hejri et al. model

-4.705

-2.694

Elbaset et al. model

3.752

-6.172

-

4.381

-

-

-

-2.439

-

-

New procedure

3.754

4.598

-

3.965

-

-

Hejri et al. model Elbaset et al. model

-2.866

10.239

-

-

-

18.733

-1.899

8.450

-

-

-

14.620

New procedure

-1.817

2.854

-

-

-

4.045

Hejri et al. model

14.615

15.858

-

-

-

17.474

Elbaset et al. model

-1.765

5.440

-

-

-

11.252

New procedure

-1.730

-2.612

-

-

-

-2.937

Hejri et al. model

-4.674

-

-23.528

-

-29.474

-

Elbaset et al. model

-2.512

-

2.883

-

2.548

-

New procedure

-2.515

-

-5.608

-

-5.371

-

Hejri et al. model

5.534

-2.707

-

-

-

-4.835

Elbaset et al. model

1.975

-3.245

-

-

-

-7.280

New procedure

-2.975

-3.219

-

-

-

-3.613

Hejri et al. model

-5.204

-9.302

-

-

-

-5.840

Elbaset et al. model

-1.485

-5.055

-

-

-

-4.713

New procedure

-1.792

-6.440

-

-

Hejri et al. model

-9.415

-

-8.071

-

-10.236

-

Elbaset et al. model

1.113

-

9.488

-

9.196

-

New procedure

-1.791

-

4.467

-

2.198

-

-1.726

The data contained in Tables 13 and 14 confirm for the MDP the same results obtained for the MDI. The Elbaset et al. model is the most precise and the most imprecise at the same time. For the Pramac PV module such a model reaches values of the MDP contained in the range from 1.113% to

23

ACCEPTED MANUSCRIPT 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479

-25.749%. The MDP obtained with the new procedure varies from a minimum value of -0.452%, for the Sanyo PV panel, to a maximum value of -5.284, for the Kyocera PV module. From the data listed in Table 14 it is possible to observe that the greatest accuracy among the compared models is given by the Elbaset et al. model that yields a value of the MDP equal to 1.113% for the Pramac PV module. Adversely, the Hejri et al. model reaches the greatest MDP for the Exiom PV device, which is -29.474%. The new procedure provides I-V characteristics evaluated with MDP ranging between -1.726% and -6.440% for the Soltecture PV panel. The data contained in Tables 7-14 numerically confirm that the precision achieved by the Hejri et al. and Elbaset et al. models in representing the performance curves of the analysed PV panels is significantly influenced by the typology of the considered PV device. Actually, a given model can provide very accurate results for a PV module and be quite imprecise when a different PV device is considered. The following values of the variation range of the ADI, ADP, MDI and MDP permit to better appreciate the global performances given by the examined two-diode models: Variation range of the ADI:  Hejri et al. model : 0.388% ÷ 11.311%  Elbaset et al. model : 0.324% ÷ 10.788%  New procedure : 0.126% ÷ 2.977% Variation range of the ADP:  Hejri et al. model : 0.467% ÷ 11.723%  Elbaset et al. model : 0.338% ÷ 10.078%  New pocedure : 0.083% ÷ 2.666% Variation range of the MDI:  Hejri et al. model : -25.598% ÷ 18.297%  Elbaset et al. model : -22.407% ÷ 14.743%  New procedure : -5.675% ÷ 4.505% Variation range of the MDP:  Hejri et al. model : -29.474% ÷ 18.733%  Elbaset et al. model : -25.749% ÷ 14.620 %  New procedure : -6.440% ÷ 4.598% It is evident that, in general terms, the new procedure results more accurate than the Hejri et al. and Elbaset et al. models. The proposed procedure calculates the I-V characteristics with very small values of the ADI and ADP. Moreover, even in the worst case, the values of the MDI and MDP are comparable with the performance data tolerance declared by the manufacturer, which usually does not exceed +10/−5% for the maximum power at the SRC. 6. Conclusions A new analytical procedure to evaluate the parameters of the two-diode model has been implemented. The parameters are calculated as a solution of six equations are derived from the short circuit point, open circuit point, MPP and the derivative in correspondence with the same points and, by means of a double iterative procedure, parameters n1, I01,ref, I02,ref, IL,ref, Rs, Rsh are evaluated. The seventh condition, which is necessary to calculate parameter n2, is based on the assumption that only the positive values of the saturation currents at the SRC have to be considered. Moreover,

24

ACCEPTED MANUSCRIPT 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522

it was observed that the most precise results are obtained using the smallest value of n2, for which it is also n2>n1 and all the model parameters are positive at the SRC. The new model proved to be suitable to correctly represent the electrical behaviour of PV modules even for solar irradiance and cell temperature different from 1000 W/m² and of 25°C. The effectiveness of the new procedure was tested by comparing the calculated I-V curves with the issued characteristics of seven PV modules belonging to different construction typologies: Shell SQ 150-PC (monocrystalline), Kyocera KD245GH-4FB2 (polycrystalline), Sanyo HIT-240 HDE4 (HIT), Exiom EXTF-75 (amorphous silicon), Solar Frontier SF130-L (CIS), Soltecture Linion 90 (CIGS) and Pramac MCPH P7 125 (tandem). Additionally, a comparison with the I-V curves calculated using the two-diode models proposed by Hejri et al. and Elbaset et al. was performed. The results turned out that the approaches used by the alternative models are often less precise than the proposed model. Actually, although the Hejri et al. and Elbaset et al. models can reach some good performances when applied to a specific typology of PV module, they globally result less accurate because they are penalised by the initial assumptions that do not permit to define seven independent parameters. Conversely, the new procedure leads to a very precise representation of the electrical behaviour of the considered PV modules, regardless of the various construction typologies and for solar irradiance and temperature different from the SRC. The calculated variation ranges of the error parameters ADI (0.126% ÷ 2.977%), ADP (0.083% ÷ 2.666%), MDI (-5.675% ÷ 4.505%) and MDP (-6.440% ÷ 4.598%) reflect the higher reliability of the proposed procedure. Nomenclature a1, a2 diode shape factors G solar irradiance [W/m2] Gref solar irradiance at SRC (1000 W/m2) i normalised current generated by the panel I current generated by the panel [A] Icalc,j current of the calculated point on the I-V characteristic [A] IL photocurrent [A] IL,ref photocurrent at SRC [A] Imp current in the maximum power point [A] Imp,ref current in the maximum power point at SRC [A] Isc short circuit current of the panel [A] Isc,ref short circuit current of the panel at SRC [A] Iiss,j current of the j-th point extracted from the issued I-V characteristics [A] Ix current in correspondence with point x [A] I01, I02 diode saturation current [A] I01,200, I02,200 diode saturation current at G=200 W/m2 and T= Tref [A] k Boltzmann constant [J/K] n1, n2 diode quality factors [V/K] N number of points extracted from the I-V characteristics issued by manufacturers Ns number of cells connected in series P power generated by the PV panel [W] Pmp power in the maximum power point [W]

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ADI ADP MDI MDP q RIo1, RIo2 Rs Rso Rso,200 Rs,ref Rsh Rsho Rsho,200 Rsh,ref Rx T Tref v V Vcalc,j Viss,j Vmp Vmp,ref Voc Voc,ref Voc,200 V0.8Isc ΔRso,C ΔRso,V ΔRsho,C ΔRsho,V μI,sc μV,oc

percentage ratio of the of the mean absolute difference of current to the current at the maximum power point in the SRC percentage ratio of the of the mean absolute difference of power to the power at the maximum power point in the SRC percentage ratio of the of the maximum difference of current to the current at the maximum power point in the SRC percentage ratio of the of the maximum difference of power to the power at the maximum power point in the SRC electron charge [C] ratio of I01,200 to I01,ref and of I02,200 to I02,ref series resistance [Ω] negative reciprocal of slope of the I-V characteristic for V = Voc,ref and I = 0 [Ω] negative reciprocal of slope of the I-V characteristic for G=200 W/m2, V = Voc and I = 0 [Ω] series resistance at SRC [Ω] shunt resistance [Ω] negative reciprocal of the slope of the I-V characteristic for V = 0 and I = Isc,ref [Ω] negative reciprocal of the slope of the I-V characteristic for G=200 W/m2, V = 0 and I = Isc [Ω] shunt resistance at SRC [Ω] negative reciprocal of the slope of the I-V characteristic in correspondence with the point x [Ω] temperature of the PV cell [°K] temperature of the PV panel at SRC (25°C – 298.15°K) normalised voltage generated by the PV panel voltage generated by the PV panel [V] voltage of the calculated point on the I-V characteristic [V] voltage of the j-th point extracted from the issued I-V characteristics [V] voltage in the maximum power point [V] voltage in the maximum power point at SRC [V] open circuit voltage of the PV panel [V] open circuit voltage of the PV panel at SRC [V] open circuit voltage of the I-V characteristic at G = 200 W/m2 and T = Tref [V] voltage of the point on the SRC I-V curve in which it is I = 0.8 Isc,ref [V] relative difference between Rx and Rso,200 relative difference between 5Rso and Rso,200 relative difference between Rsho and Rsho,200 relative difference between 5Rsho and Rsho,200 short circuit current temperature coefficient [A/°C] open circuit current temperature coefficient [V/°C]

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References [1] Virtuani, A, Pavanello D, Friesen, G. Overview of temperature coefficients of different thin film photovoltaic technologies. 25th European Photovoltaic Solar Energy Conference and Exhibition/5th World Conference on Photovoltaic Energy Conversion 2010: 6-10. [2] Tossa AK, Soro YM, Thiaw L, Azoumah Y, Sicot L, Yamegueu D, Lishou C, Coulibaly Y, Razongles G. Energy performance of different silicon photovoltaic technologies under hot and harsh climate. Energy 2013 ; 103: 261-270. [3] Meyer EL, van Dyk EE. Characterization of degradation in thin-film photovoltaic module performance parameters. Renewable Energy 2003; 28: 1455-1469. [4] Mendoza-Pérez R, Sastre-Hernández J, Contreras-Puente G, Vigil-Galán O. CdTe solar cell degradation studies with the use of CdS as the window material. Solar Energy Materials & Solar Cells 2009; 93: 79-84. [5] Muñoz-García M, Marin O, Alonso-García MC, Chenlo F. Characterization of thin film PV modules under standard test conditions: results of indoor and outdoor measurements and the effects of sunlight exposure. Solar Energy 2012; 86: 3049-3056. [6] Jordan DC, Kurtz SR. Photovoltaic degradation rates – an analytical review. Progress in Photovoltaics: Research and Applications 2013; 21: 12-29. [7] IEC 60904-3 (Ed. 2), Photovoltaic devices - Part 3: Measurement principles for terrestrial photovoltaic (PV) solar devices with reference spectral irradiance data. 2008. [8] Chan DSH, Phang JCH. Analytical methods for the extraction of solar-cell single- and doublediode model parameters from I-V characteristics. IEEE Transactions on Electron Devices 1987; 34: 286-293. [9] Enebish N, Agchbayar D, Dorjkhand S, Baatar D, Ylemj I. Numerical analysis of solar cell current-voltage characteristics. Solar Energy Materials & Solar Cells 1993; 29: 201-208. [10] Hovinen A. Fitting of the solar cell IV-curve to the two diode model. Physica Scripta 1994; T54: 175-176. [11] Gow JA, Manning CD. Development of a photovoltaic array model for use in powerelectronics simulation studies. IEE Proceedings - Electric Power Applications 1999; 146-2: 193200. [12] Haouari-Merbah M, Belhamel M, Tobías I, Ruiz JM. Extraction and analysis of solar cell parameters from the illuminated current-voltage curve. Solar Energy Materials & Solar Cells 2005; 87: 225-233. [13] Hejri M, Mokhtari H, Azizian MR, Ghandhari M, Söder L. On the parameter extraction of a five-parameter double-diode model of photovoltaic cells and modules. IEEE Journal of Photovoltaics 2014; 4: 915-923. [14] Masmoudi F, Ben Salem , Derbel N. Single and double diode models for conventional monocrystalline solar cell with extraction of internal parameters. 13th International Multi-Conference on Systems, Signals & Devices (SSD) 2016: 720-728. [15] Sallem S, Marrekchi A, Kammoun S, Kammoun MBA. Multi-PV technologies method for parameters estimation of two-diode model. The European Physical Journal Plus 2016; 131: 1-10. [16] Salam Z, Ishaque K, Taheri H. An improved two-diode photovoltaic (PV) model for PV system. Joint International Conference on Power Electronics, Drives and Energy Systems & 2010 Power India 2010: 1-5.

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[17] Ishaque K, Salam Z, Taheri, H. Accurate MATLAB Simulink PV system simulator based on a two-diode model. Journal of Power Electronics 2011; 11-2:179-187. [18] Ishaque K, Salam Z, Taheri, H. Simple, fast and accurate two-diode model for photovoltaic modules. Solar Energy Materials & Solar Cells 2011; 95: 586-594. [19] Marrekchi A, Sallem S, Kammoun MBA, Tina GM. A new five parameters estimation method for two-diode model of PV module. IREC2015 The Sixth International Renewable Energy Congress 2015: 1-6. [20] Duong MQ, Nguyen HH, Leva S, Mussetta M, Sava GN, Costinas S. Performance analysis of a 310 Wp photovoltaic module based on single and double diode model. International Symposium on Fundamentals of Electrical Engineering (ISFEE) 2016: 1-6. [21] Nguyen HH, Duong MQ. High-performance coordination for accurate Matlab Simulink PV module simulator based on a two-diode model. IEEE International Conference on Sustainable Energy Technologies (ICSET) 2016: 379-383. [22] Gupta S. Tiwari H. Fozdar M. Chandna V. Development of a two diode model for photovoltaic modules suitable for use in simulation studies. Power and Energy Engineering Conference (APPEEC), Asia-Pacific 2012: 1-4. [23] Babu BC, Gurjar S. A novel simplified two-diode model of photovoltaic (PV) module. IEEE Journal of Photovoltaics 2014; 4-4: 1156-1161. [24] Sangeetha RS, Jayan MV, Pradish M. An improved technique for predicting characteristics of two-diode based PV model. Energy Procedia 2017; 117: 870-877. [25] Elbaset AA, Ali H, Abd-El Sattar M. Novel seven-parameter model for photovoltaic modules. Solar Energy Materials & Solar Cells 2014; 130: 442-455. [26] Shannan NM, Yahaya NZ, Singh B. Two diode model for parameters extraction of PV module. IEEE Conference on Energy Conversion (CENCON) 2014: 260-264. [27] Orioli A, Di Gangi A. A criterion for rating the usability and accuracy of the one-diode models for photovoltaic modules. Energies 2016; 9-6; 427: 1-48. [28] Franzitta V, Orioli A, Di Gangi A. A criterion for rating the usability and accuracy of the simplified one-diode models for photovoltaic modules. Energies 2016; 9-12; 1019: 1-41. [29] Franzitta V, Orioli A, Di Gangi A. A criterion for rating the usability and accuracy of the twodiode models for photovoltaic modules. Energies 2017, 10-4, 564: 1-32. [30] Lo Brano V, Orioli A, Ciulla G, Di Gangi A. An improved five-parameter model for photovoltaic modules. Solar Energy Materials & Solar Cells 2010; 94: 1358-1370. [31] Lo Brano V, Orioli A, Ciulla G. On the experimental validation of an improved five-parameter model for silicon photovoltaic modules. Solar Energy Materials & Solar Cells 2012; 105: 27-39. [32] http://www.physics.arizona.edu/~cronin/Solar/TEP%20module%20spec%20sheets/Shell%20 SQ150.pdf [33] http://www.australiansolar.com.au/images/KD245GH-4FB2.pdf [34] https://www.oeko-energie.de/downloads/sanyo_hit-240_235_hde4__de__web.pdf [35] http://www.exiomsolution.com/download/thinexiom2.pdf [36] http://www.vindogsol.dk/CIS%20SF%20130%20modul%20datablad.pdf [37] http://www.soltecture.com/uploads/media/Datasheet_LINION_L_EN_REV2.3_01.pdf [38] http://amysolar.com/amysolar/docs/PRAMAC_MCPH%20P7_FR_v3.1.pdf

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ACCEPTED MANUSCRIPT 649 650 651

[39] Orioli A, Di Gangi A. A procedure to calculate the five-parameter model of crystalline silicon photovoltaic modules on the basis of the tabular performance data. Applied Energy 2013; 102: 1160-1177.

652 653

APPENDIX A

654 655

Equations to calculate the model parameters at SRC. From Eqs. (3) and (7), the photocurrent and the shunt resistance are extracted:

656 657

I L ,ref  I 01,ref

Rsh ,ref 

 VnocT,ref   VnocT,ref  V 1 ref e  1  I 02,ref  e 2 ref  1  oc ,ref     Rsh ,ref     Rsho  Rs ,ref

1   Rs ,ref

 I  Rsho   01 e  n1Tref

I sc , ref Rs , ref n1Tref

I  02 e n2Tref

(A1) (A2) I sc , ref Rs , ref n2Tref

  

658 659

in order to be substituted in Eqs. (4) and (5), which respectively become:

m11 I 01,ref  m12 I 02,ref  A

(A3)

660

m21 I 01,ref  m22 I 02,ref  B

(A4)

661

where: Voc , ref

662

n1Tref

m11  e

Voc , ref

663

664

n2Tref

m12  e

 nsc1T,refref  e 

 V  I sc ,ref Rs ,ref  1  oc ,ref  n2Tref 

 nsc2T,refref  e 

Voc , ref

Vmp , ref  I mp , ref Rs , ref

n1Tref

n1Tref

m21  e

I

 V  I sc ,ref Rs ,ref  1  oc ,ref  n1Tref 

e

Voc , ref

Vmp , ref  I mp , ref Rs , ref

n2Tref

n2Tref



(A5)

I

(A6)

Vmp ,ref  I mp ,ref Rs ,ref  Voc ,ref n1Tref Vmp ,ref  I mp ,ref Rs ,ref  Voc ,ref

I sc , ref

e

n1Tref

e

n2Tref

(A7)

I sc , ref

(A8)

665

m22  e

666

A  I sc ,ref 

667

B  I mp ,ref 

668 669

Currents I01,ref and I02,ref can be obtained by solving Eqs. (11) and (12):

670

I 01,ref 

(A11)

671

I 02,ref

(A12)

e



n2Tref

I sc ,ref Rs ,ref  Voc ,ref

(A9)

Rsho  Rs ,ref I mp ,ref Rs ,ref  Vmp ,ref  Voc ,ref

(A10)

Rsho  Rs ,ref

m22 A  m12 B m11m22  m12 m21 m11 B  m21 A  m11m22  m12 m21

29

ACCEPTED MANUSCRIPT 672 673

APPENDIX B

674 675 676 677 678

Equations to calculate the model parameters at conditions different from the SRC. To make the best choice in selecting constant or variable values of the model resistances it is suitable to consider the derivative of the I-V curve at the open circuit, in correspondence with the lowest value of the solar irradiance for which a I-V curve is provided by the manufactures, which usually is 200 W/m2: dI 1 (B1)  V Voc dV I 0 Rso ,200

679

G  200

680 681 682

Resistance Rso,200 can be obtained from the issued I-V characteristics following the rules of the approximate procedure described in [39]. The value of Rso,200 is then compared with resistance Rx related to the negative reciprocal derivative of the I-V characteristic at the SRC:

683

dI dV

684 685 686

687 688 689 690

I I x G 1000



1 Rx

(B2)

evaluated in correspondence with a specific point x on the SRC I-V curve. Point x is the point whose current is equal to the value calculated with the following expression:   1   1  (B3) I x  0.8 I sc ,ref  V0.8 Isc tan atn      atn    Rsho   Rsho ,200   where V0.8Isc is the voltage of the point on the SRC I-V curve in which it is I = 0.8 Isc,ref . Resistance Rsho,200 is the negative reciprocal derivative of the I-V characteristic in the open circuit point at G = 200 W/m2 and T = Tref: dI 1 (B4)  V 0 dV I  Isc ,200 Rsho ,200 G  200

691 692 693 694

Isc,200 is equal to the value of the short circuit current at G = 200 W/m2 and T = Tref. The decision to use constant or variable values for the series and shunt resistances is made on the basis of the following relative differences: 5 R  Rso ,200 R  Rso ,200 (B5) Rso ,V  so Rso ,C  x Rso ,200 Rso ,200

5 Rsho  Rsho ,200

Rsho  Rsho ,200

(B6)

695

Rsho ,V 

696 697 698 699

If ΔRso,V is smaller than ΔRso,C, resistance Rs(G) is better described using Eq. (11); if ΔRso,C is smaller than ΔRso,V, a the constant value of Rs,ref should be assumed for Rs(G). The analogous comparison between ΔRsho,V and ΔRsho,C permits to select the best way to represent shunt resistance Rsh(G) using Eq. (11) or the constant value Rsh,ref .

Rsho ,200

Rsho ,C 

Rsho ,200

30

ACCEPTED MANUSCRIPT 700 701 702

To define I01(G,Tref) and I02(G,Tref), Eqs. (3) and (6) are rewritten in correspondence with the lowest value of G for which a I-V curve for T=Tref is provided by manufactures, which usually is 200 W/m2:

703

 VnocT,200   VnocT,200  V 1 ref 0  I L (200, Tref )  I 01,200  e  1  I 02,200  e 2 ref  1  oc ,200     Rsh (200)     Voc , ref

Voc ,200

704

705 706 707 708 709 710

dI dV

V Voc ,200 I 0



(B7)

I 01,200 n1Tref I 02,200 n2Tref 1 e  e  n1Tref n2Tref Rsh (200) I  I 1 nT nT  1  Rs  200   01,200 e 1 ref  02,200 e 2 ref  n2Tref Rsh (200)   n1Tref  Voc ,200

Voc ,200



1

(B8)

Rso ,200

where I01,200 = I01(200,Tref) and I02,200 = I02(200,Tref), Voc,200 is the open voltage at G =200 W/m2 and T=Tref. Resistances Rs(200) and Rsh(200) are set constant, or are calculated with Eq. (11), on the basis of the comparison between the values of ΔRso,V, ΔRso,C, ΔRso,V, and ΔRso,C. Currents I01,200 and I02,200 can be calculated with the following expressions obtained by solving the Eq. (B7) and (B8) system: r C  r12 D (B9) I 01,200  22 r11r22  r12 r21

711

I 02,200 

712

where:

r11 D  r21C r11r22  r12 r21

(B10)

Voc ,200

713

r11  e

n1Tref

1

(B11)

1

(B12)

Voc ,200

714 715

r12  e r21 

n2Tref

Rso ,200  Rs (200) n1Tref Rso ,200  Rs (200)

Voc ,200

e

(B13)

n1Tref

Voc ,200

(B14)

716

r22 

717

C  I L (200, Tref ) 

718

D  1

719 720

Finally, currents I01(G,Tref) and I02(G,Tref) are calculate by linear interpolation between the respective values at G =1000 W/m2 and G = 200 W/m2: G  200   (B17) I 01 G , Tref  I 01,ref 1  RIo1   RIo1  800  

721 722



n2Tref

e

n2Tref

Voc ,200

(B15)

Rsh (200)

Rso ,200  Rs (200)

(B16)

Rsh (200)



G  200   I 02  G , Tref   I 02,ref 1  RIo2   RIo2  800  

(B18)

31

ACCEPTED MANUSCRIPT 723

where:

724

RIo1 

I 01,200 I 01,ref

RIo2 

I 02,200

(B19)

I 02,ref

725 726 727 728 729 730 731

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ACCEPTED MANUSCRIPT    

A procedure to get an improved two-diode model for PV devices is presented The procedure calculates seven truly independent model parameters The procedure is used to depict the I-V curves of seven different PV device typologies The procedure is checked by comparison with other recognised two-diode models