Photovoltaic module model determination by using the Tellegen’s theorem

Renewable Energy 152 (2020) 409e420

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Photovoltaic module model determination by using the Tellegen’s theorem
rez Abril b, * Rodolfo Manuel Arias García a, Ignacio Pe a b


n Villa Clara, Santa Clara, Cuba Empresa COPEXTEL.SA, Divisio Universidad Central “Marta Abreu” de Las Villas. Centro de Estudios Electroenerg
eticos, Carretera de Camajuaní km 5, Santa Clara, Villa Clara, Cuba

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 July 2019 Received in revised form 30 November 2019 Accepted 11 January 2020 Available online 13 January 2020

This paper proposes a generalized methodology to calculate the parameters of the single-diode or double-diode models of a photovoltaic module. The presented method relies in the solution by a simple iterative process of the set of equations obtained from the photovoltaic-module’s equivalent circuit. The use of the generalized form of Tellegen’s theorem allows the solving of the equivalent circuit’s equations in an exact way, without making considerations or approximations commonly used in previous works. The parameters of the equivalent circuit are obtained from the standard conditions of: open-circuit, short-circuit and maximum power, provided by the manufacturer in the data sheet of the photovoltaic module. The presented method is applied to obtain the models of several commercial modules. The accuracy of the obtained parameters is greater than that of the results previously determined by other authors. The curves calculated with the obtained models matches the experimental curves supplied in the manufacturer’s data. © 2020 Elsevier Ltd. All rights reserved.

Keywords: PV-Module Double-diode model Single-diode model

1. Introduction As a result of the high investment cost of PV-generating installations and to make optimal use of solar energy in photovoltaic systems, is required estimate the behavior of the PV-system from the design stage. The accuracy of the model of the whole PV-generating system is dominated by the precision of the model that represents the PVmodule [1,2]. Normally, two equivalent circuits have been used by researchers to represent the behavior of the PV-module: the singlediode model with five parameters and the double-diode model with seven parameters [3,4]. The single-diode model considers only the diffusion process of carriers in the semiconductor cell. Instead, the double-diode model incorporates the effect of the losses by recombination in the deflection region, which increases the accuracy of the model in the vicinity of the open circuit voltage of the module [2,3]. In both cases, the parameters of the equivalent circuit are obtained from the standard conditions of: open-circuit, short-circuit and maximum power, provided by the manufacturer in the data

* Corresponding author. E-mail addresses: [email protected] (R.M. Arias García), iperez@
rez Abril). uclv.edu.cu (I. Pe https://doi.org/10.1016/j.renene.2020.01.048 0960-1481/© 2020 Elsevier Ltd. All rights reserved.

sheet of the PV-module. Several methods have been proposed to obtain the parameters of the equivalent circuit: analytical and numerical methods [1,2,4e6], evolutionary algorithms [7], differential evolution [8], artificial immune systems [9], hybrid flower pollination with clonal selection algorithm [10], fireworks algorithm [11], etc. A very complete review of the existing research methods for the PV cell model parameter estimation is presented by Ref. [12]. However, in order to simplify the obtaining of the parameters of the model, they are commonly considered many assumptions and simplifications in the formulation of the equations of the equivalent circuit to be determined. As result of these simplifications, the results obtained with some of the published methods are not as accurate as might be. In the determination of the single-diode model, the most frequent considerations are: 1) the ideality factor of the diode n1 is estimated [4,9,13,14]; 2) the photocurrent Iph is considered equal to the PV-module’s short-circuit current [4,13,15e22]; 3) the inverse saturation current I01 is neglected for the calculation of the photocurrent [4,16,23]; and 4) the resistance in parallel Rp is neglected to calculate the current of reverse saturation [4,9,17,19]. Besides, in the determination of the double-diode model, other simplifications are added: 5) the values of the reverse saturation currents of both diodes I01 and I02 are considered equal [1,2,5,24]

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(this consideration is in contradiction to the well-known fact that the inverse saturation current of the second diode is three to seven times greater than that of the diode that considers the diffusion process [1,5,25]); and 6) the ideality factors of both diodes are assumed by different criteria. Several researchers [6,26] assume the ideality factors n1 ¼ 1 and n2 ¼ 2 based on the approximations of the Shockley-Read-Hall’s recombination in the deflection region of the photodiode. This assumption is widely used; but it is not always true as is remarked in Refs. [1,2,5]. Instead, the ideality factors are calculated in Refs. [1,2,5,9,24,28] by considering n1 ¼ 1, n2  1.2 (in Refs. [1,2,5,24] both ideality factors are related by assuming that the saturation currents of both diodes I01 and I02 are equal). Besides, in Ref. [8] are used the intervals 1  n1  2, 2  n2  4. The ideality factor of the diodes are calculated in Ref. [6] under the consideration of that n1þn2 ¼ 3 for multi-crystalline cells and thin-film cells, or that n1þn2 ¼ 4 for amorphous silicon cells. However, the reference [8] explains that these relationships have no physical basis and are not always reliable. Finally, the reference [27] employs the intervals 1  n1  2, n2 ¼ 2. In this work, the set of equations obtained from the PV-module’s equivalent circuit of the single-diode or double-diode models is solved by using a simple iterative process. The use of the generalized form of Tellegen’s theorem allows the solving of the equivalent circuit’s equations in an exact way, without making considerations or approximations commonly used in previous works. The parameters of the equivalent circuit are obtained from the standard conditions of: open-circuit, short-circuit and maximum power, provided by the manufacturer in the data sheet of the photovoltaic module. The presented method is applied to obtain the model of several commercial modules. The accuracy of the obtained parameters is greater compared with that of the results previously determined by other authors. The curves calculated by the obtained models matches the experimental curves supplied in the manufacturer’s data. 2. Determination of the parameters of the PV-module’s model The equivalent circuit of the double-diode model of the PVmodule is represented in Fig. 1, where, if the branch of the diode D2 is eliminated, the single-diode model is obtained. The current source Iph represents the photocurrent generated by the PV-module, the current ID1 that circulates through the diode D1

represents the diffusion process, the current ID2 that circulates through the diode D2 represents the recombination process, the current Ip that circulates through the resistance in parallel Rp represents the internal dispersion losses, and Im is the output current of the PV-module. The output voltage of the PV-module Vm is obtained by subtracting from VD the voltage drop in the series resistance Rs of the circuit. The mathematical model of the Schockley’s diode [20] represents the current in each diode by:


   VDi  1 ¼ I0i bi ðVDi Þ IDi ðVDi Þ ¼ I0i exp Nc ni Vti

(1)

Where, I0i is the inverse saturation current, ni is the ideality factor and Vti is the thermal voltage of diode Di. Besides, Nc is the number of cells in the PV-module. The ideality factors of the diodes have a marked impact on the form adopted by the voltage-current characteristic of the PVmodule [4]. In the proposed methodology, instead of assuming these ideality factors, they are calculated in conjunction with the rest of the parameters of the model in order to guaranteeing the best fit of the voltage-current characteristic of the PV-module. In the case of the single-diode model, n1 is calculated by exploring in the interval (1  n1  2). In the case of the doublediode model, n1 is calculated by searching in the interval (1  n1  1.2) and n2 by searching in the interval (1.2  n2  2). The selected intervals for n1 and n2 are based not only in the physics of solar cells [29] but also contain the values and intervals assumed in mostly of the analyzed references. Besides, the selected intervals have produced very good results in the examples solved in this paper. The thermal voltage of each diode depends on the temperature of the cell T, the Boltzmann constant (K ¼ 1.3806503  10-23 J/K), and the electron charge (q ¼ 1.60217646  10-19 C). The dependence of Vt on these parameters, for each case, is expressed as Vt ¼ KT/q. Applying the laws of Kirchoff to the PV-module’s circuit (Fig. 1), the output current Im and the output voltage Vm of the module are related by:

. Im ¼ Iph  I01 b1 ðVD Þ  I02 b2 ðVD Þ  VD Rp

(2)

Vm ¼ VD  Im Rs

(3)

In the case of the single-diode model, the equation (1) is reduced to:

. Im ¼ Iph  I01 b1 ðVD Þ  VD Rp

Fig. 1. Equivalent circuit of double-diode model.

(4)

For certain conditions of irradiance, temperature of the cells and load of the PV-module, the current and voltage of output calculated by the model must match the current and voltage (Im, Vm) that are provided by the PV-module’s manufacturer. Three points of the voltage-current characteristic of the PVmodule: open-circuit (oc), short-circuit (cc) and maximum power (max) are sufficient to obtain the seven parameters of the doublediode model: the nominal currents Iphn, I01n, I02n, the ideality factors n1, n2, the resistance in parallel Rp and the series resistance Rs. The data of these three points correspond to standard conditions of irradiance of 1000 W/m2, temperature of the cells of 25  C, and an air mass of 1.5 (AM ¼ 1.5) [4].

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2.1. Determination of the nominal currents from standard conditions data

. Iphn VDx2 þ I01n b1 ðVDx1 ÞVDx2 þ I02n b2 ðVDx1 ÞVDx2 þ VDx1 VDx2 Rp

Using the three referred points of the voltage-current characteristic of the PV-module, the nominal currents Iphn, I01n and I02n for standard conditions can be obtained by solving a system of linear equations. In order to establish this system of equations is applied the generalized form of Tellegen’s theorem [30] to the circuit of the PV-module (Fig. 1). Then the variable Rp is eliminated by an algebraic procedure to obtain a linear equations system for the nominal currents Iphn, I01n and I02n which coefficients depend only of the ideality factors n1 and n2 of both diodes as well as the series resistance Rs. The generalized form of Tellegen’s theorem, constitutes a theorem of exceptional value and versatility, is simple and general. For its application is only required that two circuits with the same number of branches have the same incidence matrix, although the elements of each branch will be different. If fulfilling this condition, Ii’ and Vi’ are the branch currents and voltages of a network N0 and Ii” and Vi” are the branch currents and voltages of a network N00 , both networks with n branches, the following condition is complied. n X

Ii ’Vi } ¼

i¼1

n X

Ii }Vi ’ ¼ 0

(5)

i¼1

The reference [31] can be examined for more background on Tellegen’s theorem and its applicability. The form that adopt the PV-module’s circuit in each condition of: open circuit, short circuit or maximum power, is different and thus can be considered as a different circuit. These three circuits have the same incidence matrix, thus, the Tellegen’s theorem can be applied to any combination of two of these three circuits.

2

VDmax b1 ðVDcc Þ  VDcc b1 ðVDmax Þ V ¼ 4 VDcc b1 ðVDoc Þ  VDoc b1 ðVDcc Þ VDmax b1 ðVDoc Þ  VDoc b1 ðVDmax Þ

411

þ Imx1 Vmx2 ¼0 (6) . Iphn VDx1 þ I01n b1 ðVDx2 ÞVDx1 þ I02n b2 ðVDx2 ÞVDx1 þ VDx2 VDx1 Rp þ Imx2 Vmx1 ¼0 (7) Then, a new equation (8) is obtained by adding the equation (6) minus the equation (7) and the term of the resistance in parallel Rp is eliminated:

Iphn ½VDx2  VDx1  þ I01n ½b1 ðVDx1 ÞVDx2  b1 ðVDx2 ÞVDx1  þ … I02n ½b2 ðVDx1 ÞVDx2  b2 ðVDx2 ÞVDx1  ¼ ½Imx2 Vmx1  Imx1 Vmx2  (8) The three equations obtained by applying the Tellegen’s theorem to the combinations of the three standard conditions form the system of linear equation (9) which solving determines the currents Iphn, I01n and I02n.

VI ¼ P

Where the elements of the matrices: V, I and P are the following:

VDmax b2 ðVDcc Þ  VDcc b2 ðVDmax Þ VDcc b2 ðVDoc Þ  VDoc b2 ðVDcc Þ VDmax b2 ðVDoc Þ  VDoc b2 ðVDmax Þ

The values of voltages and currents of the branches of the PVmodule’s circuit for each one of the selected conditions or for each one of the three referred circuits are presented in Table 1. Applying the generalized form of Tellegen’s theorem to the possible conditions of the circuit (Fig. 1) and using the values of the electric quantities in standard conditions (Table 1), for each pair of conditions or (x1, x2) that can be: 1) x1 ¼ open circuit, x2 ¼ short circuit, 2) x1 ¼ maximum power, x2 ¼ open circuit and 3) x1 ¼ short circuit, x2 ¼ maximum power, two equations are obtained as follows:

(9)

3 VDcc  VDmax VDoc  VDcc 5 VDoc  VDmax

(10)

2

3 Icc Vmax P ¼ 4 Icc Voc 5 Imax Voc

(11)

3 I01n I ¼ 4 I02n 5 Iphn

(12)

2

The single-diode model is a particular case of the double-diode model where I02n is zero. By eliminating the second column of the matrix V and any of the three rows of the matrices V and P, the

Table 1 Voltages and currents of the branches of the PV-module’s circuit. Parameter

Open-circuit condition

Short-circuit condition

Maximum power condition

VD Iphn ID1 ID2 Ip Vm Im

VDoc ¼ Voc Iphn I01n b1 ðVDoc Þ I02n b2 ðVDoc Þ VDoc =Rp Voc 0

VDcc ¼ Icc Rs Iphn I01n b1 ðVDcc Þ I02n b2 ðVDcc Þ VDcc =Rp 0 Icc

VDmax ¼ Vmax þ Imax Rs Iphn I01n b1 ðVDmax Þ I02n b2 ðVDmax Þ VDmax Þ=Rp Vmax Imax

R.M. Arias García, I. P
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412

resulting two equations system allows the determination of I01n and Iphn for the single-diode model. 2.2. Determination of the resistance in parallel from standard conditions data Once the nominal currents Iphn, I01n and I02n have been obtained by solving the linear equations system (9), the value of the resistance in parallel Rp can be determined by using (2) to evaluate the point of maximum power.

Rp ¼

VDmax Iphn  I01n b1 ðVDmax Þ  I02n b2 ðVDmax Þ  Imax

(13)

With I02n equals zero, this expression is valid to obtain Rp in the case of the single-diode model. 2.3. Determination of the slope of the voltage-current characteristic Under standard conditions, the power curve of the photovoltaic module only contains a local maximum, which coincides with the global maximum. The point of maximum power is found when the slope of the characteristic voltage-current equals the negative of the conductance of the load [18,23].

dIm Imax ðVmax ; Imax Þ ¼  dVm Vmax

(14)

This derivative, evaluated in the point of maximum power is obtained by:

In the first phase, the single-diode model’s parameters are determined by varying the ideality factor n within the range (1  n  2). Taken into account that the slope of the I/V-characteristic is reduced when the series resistance is increased; for each n, a loop increases the value of Rs from a minimum value Rsmin ¼ 0 up to a maximum value Rsmax to obtain the rest of the parameters of the single-diode model: Iph, I0 and Rp. Once a negative value of the slope’s error ε is found (means that the search overpassed the slope of the characteristic at the maximum power condition) the loop ends. The set of parameters that produce the minimum error of slope ‫׀‬ε‫ ׀‬are chosen as the resulting model’s parameters. The value of Rs is normally low in order to increase cell’s efficiency [29]. Besides, the influence of the series resistance is stronger when the device operates in the voltage source region of the I/V-characteristic (Vmax Vm  Voc) [4]. Thus, a maximum value of Rs would be [26] Rsmax ¼ (Voc e Vmax)/Imax, however a conservative value of Rsmax ¼ 3 U have been used for the calculation of the examples considered. This value has not been reached in none of the solved examples. In the second phase, the double-diode model’s parameters are determined by varying the ideality factors (n1 and n2) in the ranges (1  n1 1.2) and (1.2 < n2  2) respectively. For each pair of values n1 and n2, the resistance Rs is increased from 0.5 to 1.5 times the value of the series resistance obtained in the first phase of the algorithm (this interval considers that the value of Rs in the two-diode’s model is near the value of Rs in the single-diode model, but allows a range of variation to adjust the model). For each value of Rs, the rest of the model’s parameters: Iph, I01, I02 and Rp are calculated, as well as the slope in the maximum power point. Once a negative value of the

1 þ Rp ½ðI01n =k1n ÞexpðVDmax =k1n Þ þ ðI02n =k2n ÞexpðVDmax =k2n Þ dIm ðVmax ; Imax Þ ¼  dVm Rs þ Rp þ Rs Rp ½ðI01n =k1n ÞexpðVDmax =k1n Þ þ ðI02n =k2n ÞexpðVDmax =k2n Þ

Where k1n and k2n are constants evaluated as:

kin ¼ Nc ni Vtin

(16)

Again, by doing I02n equal zero, the expression (15) is valid to obtain the derivate dIm/dVm in the case of the single-diode model. When the parameters of the model have been correctly determined, the slope’s error ε, must be zero.

ε¼

dIm Imax ðVmax ; Imax Þ þ dVm Vmax

(17)

2.4. Determination of the model’s parameters from standard conditions If the ideality factors of both diodes (n1 and n2) as well as the series resistance Rs are known, the three currents Iphn, I01n and I02n, the resistance in parallel Rp, and the slope of the voltage-current characteristic can be determined solving (9), (13), and (15) in this order. The single-model diode model is a particular case of the doublediode model in which n ¼ n1 ¼ n2 and I0n ¼ I01n þ I02n. On the other hands, is reasonable to consider that the parameters of the singlediode model must be related with the parameters of the doublediode model. Based in this assumption, a two-phase algorithm that determines consecutively the single-diode model and the doublediode model parameters has proven to be very effective to solve the presented problem.

(15)

slope’s error ε is found, the loop that increments Rs ends. Again, the set of parameters that produce the minimum error of slope ‫׀‬ε‫ ׀‬are chosen as the resulting model’s parameters. This algorithm is described by the following pseudo-code:

R.M. Arias García, I. P
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Vocx ¼ Vocn þ Kv DT

413

(19)

Where, DT ¼ Tx-Tn (K) is the difference of temperature of the cell from the real conditions Tx with respect to the standard conditions Tn, Ki (A/K) is the variation factor of the photocurrent with temperature, Kv (V/K) is the variation factor of the open circuit voltage with temperature, and Gx and Gn (W/m2) are the corresponding irradiances in real and standard conditions. The equations (20) and (21) are used [16,17,19,22] to determine the reverse saturation currents I01x and I02x for real operating conditions.

 I01x ¼ I01n  I02x ¼ I02n

Tx Tn Tx Tn

3

3

 
qEg 1 1 exp  n1 K Tn Tx

(20)

 
qEg 1 1 exp  n2 K Tn Tx

(21)

Where Eg is the activation energy of the semiconductor band gap, for the crystalline silicon Eg ¼ 1.12 eV and for the amorphous silicon Eg ¼ 1.7 eV [32]. However, these expressions appear with some changes in the different references. In the reference [33], the exponent of the temperature relation is divided by the corresponding ideality factor of the diode (Tn/Tx)3/n. Besides, in some references the numerator and the denominator of the temperature ratio appear inverted: (Tn/ Tx)3 [4e6,28,33] and (Tx/Tn)3 [16,17,19,22,34]. In this work, instead of using directly the expression (20), the calculation of I01x is made by applying the Tellegen’s theorem to the equivalent circuit of the PV-module. Mixing the equations of the two open circuit conditions: the standard temperature Tn and irradiance Gn condition (represented by the photocurrent Iphn and the open circuit voltage Vocn) and the real temperature Tx and irradiance Gx condition (represented by the calculated photocurrent Iphx (18) and open circuit voltage Vocx (19)), is obtained the resulting expression for I01x as:

Fig. 2. Simplified flowchart of algorithm.

For easy understanding of the algorithm a simplified flowchart is given in Fig. 2. 2.5. Determination of currents and voltages under real operating conditions The work of the photovoltaic module under standard conditions


  
   Iphx Vocn  Iphn Vocx þ Vocx I01n exp Vkocn  1 þ Vocx Io2n exp Vkocn  1 1n 2n 
  
  
   I01x ¼ Vocx ’ 1  1 exp K exp  1 Vocn exp Vkocx  1 þ II02n n n k 2 1 01n 1x

2x

is unlikely; thus, the dependence of their electrical magnitudes respect to the variation of radiation and temperature must be established. They have been observed that the photocurrent (Iph) has a high dependence on irradiance and, to a lesser extent, on temperature. Besides, the reverse saturation currents I01 and I02 of diodes D1 and D2, as well as the open circuit voltage Voc, are all greatly dependent on temperature variations. The photocurrent Iphx and the open circuit voltage Vocx of the PVmodule for real operating conditions of irradiance and temperature are calculated from their standard values by means of:

Iphx ¼

 Gx  Iphn þ Ki DT Gn

(22)

(18)

Where K0 , k1x and k2x are constants evaluated as:

K’ ¼

qEg K



1 1  Tn Tx



kix ¼ Nc ni Vtix

(23)

(24)

Again, by doing I02n equal zero, the expression (22) is valid in the case for the single-diode model. Finally, instead of using the expression (20) the current I02x is calculated by using the relation:

I02x ¼ I01x


   I02n 1 1 exp K ’  n2 n1 I01n

(25)

R.M. Arias García, I. P
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414 Table 2 Data of the PV-modules in standard conditions. Data

Multi-crystalline KC-200GTa

Pmax (W) Vmax (V) Imax (A) Voc (V) Icc (A) Nc a

200.00 26.30 7.61 32.90 8.21 54

142.22 23.20 6.13 29.90 6.62 54

Mono-crystalline MSX-60

SP-70

SQ-150-PC

60.00 17.10 3.50 21.10 3.80 36

70.00 16.50 4.25 21.40 4.70 36

150.00 34.00 4.40 43.40 4.80 72

The KC-200GT includes the NOCT (800 W/m2 & 47  C).

3. Examples of application The presented methodology has been verified with the experimental data of several photovoltaic modules of different manufacturing technologies. The manufacturer data sheets of the used PV-modules in standard conditions [35e38] are shown in Table 2. Besides, are shown the data for the PV-module KC-200GT in NOCT conditions. For each one of the four presented PV-modules, two proposed models of single-diode and of double-diode were obtained by using the presented method. The Table 3 presents, for each PV-module’s case, a comparison between the models’ parameters obtained in different references including the models proposed in this work (in bold characters). All the parameters of the models in this work have been rounded to only four decimals. As can be seen (Table 3), there is great dispersion between the results obtained by different authors for the double-diode model of each PV-module, this occur mainly with the parameters I01n, I02n, Rp, n1 and n2. Instead, the parameters Iphn and Rs are more similar in all the double-diode models of different authors and even in the proposed single-diode model. The parameters of the model are calculated from the experimental data of the PV-module under standard conditions. Thus, the photocurrent generated by the PV-module has a constant value for any condition imposed by the load: open circuit, short circuit or maximum power. If the parameters of the model have been correctly determined, the photocurrent Iph calculated for any of these load conditions by applying the equations (2) and (3) with the data corresponding to that condition must be equal to the current Iphn obtained as parameter of the model.

On the other hand, if the parameters of the model are well determined, the output current Im and voltage Vm calculated by the model must equal the experimental data of the PV-module for any condition. In order to evaluate the correctness of the proposed methodology, the Table 4 shows a comparison of the errors in percent of the nominal photocurrent and of the output current and voltage of the four commercial PV-modules in standard conditions for all the models presented in Table 3. The errors in the calculation of the nominal photocurrent (ε-Iphn) are the relative difference between the nominal photocurrent parameter and the photocurrent obtained for each characteristic condition: open circuit (Voc, 0), short circuit (0, Icc) and maximum power (Vmax, Imax). On the other hand, the errors in the calculation of the PVmodule’s output currents and voltages (ε-Im and ε-Vm) are the relative difference between the experimental data of currents and voltages of Table 2 with respect to the currents and voltages calculated by the compared models in the characteristic conditions: open circuit (Im ¼ 0), short circuit (Vm ¼ 0) and maximum power (load resistance Rload ¼ Vmax/Imax). The comparison between the different models (Table 4) shows that the double-diode and the single-diode models proposed in this work have the minimum relative errors in all cases. Besides, these little errors are mainly due to the rounding of the proposed models’ parameters to only four decimals. The high precision of the results obtained by applying the proposed methodology relies in the use of a set of exact equations obtained from the equivalent circuit, which guarantee a correct balance of power in the circuit. The achieved precision is also a result of the correct selection of the intervals for n1, n2 and Rs. The worst results in Table 4 are the presented by Refs. [1,2,5,6] for the open-circuit condition. These errors are due to these methods calculate the inverse saturation current of both diodes ignoring the resistance in parallel. In Refs. [9,27] there are conditions in which the relative error is positive, which means that the calculated photocurrent is less than the short-circuit current. This lacks of physical sense and significantly altering the power balance of the circuit. The precision of the double-diode model obtained for the PVmodule KC 200 GT [35] is shown graphically by reproducing their voltage-current characteristics for different conditions of irradiance and temperature (Fig. 3 and Fig. 4). The markers represent

Table 3 Parameters of the double-diode models presented in different references for four commercial PV-modules and the proposed double-diode (2D) and single-diode (1D) models. PV-module

Reference

Iphn (A)

I01n (A)

I02n (A)

Rs (U)

Rp (U)

n1

n2

KC 200 GT Multi-crystalline

[9] [1,2,5] [27] [6] 2D-model 1D-model

8.2100 8.2100 8.2100 8.2222 8.2162 8.2132

1.1100  1008 4.2180  1010 3.9878  1010 2.2446  1009 1.5724x1009 9.6735x1008

1.8700  1010 4.2180  1010 1.0329  1009 4.2008  1009 3.9345x1008

0.3030 0.3200 0.3400 0.3045 0.2619 0.2310

343.1000 160.5000 179.4992 196.8896 345.9071 594.3089

1.2000 1.0000 1.0000 1.0800 1.1100 1.3000

1.0000 1.2000 2.0000 1.3200 1.2700

MSX-60 Multi-crystalline

[27] [1,2,5] [6] 2D-model 1D-model

3.7984 3.8000 3.8084 3.8049 3.8049

5.7600  1008 4.7040  1010 4.8723  1010 3.2656x1009 2.4553x1009

1.4900  1007 4.7040  1010 6.1528  1010 9.4800x1009

0.2510 0.3500 0.3692 0.2982 0.3300

599.4010 176.4000 169.0471 232.2989 197.9614

1.2700 1.0000 1.0003 1.1100 1.0800

1.7129 1.2000 1.9997 1.2400

SP-70 Mono-crystalline

[2,5] [9] 2D-model 1D-model

4.7000 4.7000 4.7179 4.7057

4.2060  1010 1.8700  1010 7.8528x1009 1.1259x1006

4.2060  1010 1.0100  1008 1.3297x1008

0.5100 0.5020 0.4580 0.3360

91.0000 264.9000 119.9314 277.9799

1.0000 1.0000 1.1700 1.5200

1.2000 1.2000 1.2500

SQ 150-PC Mono-crystalline

[5] 2D-model 1D-model

4.9000 4.8148 4.8032

3.1060  1010 6.5058x1010 2.8014x1007

3.1060  1010 3.7620x1009

0.9000 0.8801 0.6130

275.0000 285.1529 928.4235

1.0000 1.0400 1.4100

1.2000 1.2500

R.M. Arias García, I. P
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415

Table 4 Errors in percent of the photocurrent ε-Iph, of the output current ε-Im and of the output voltage ε-Vm calculated by the models shown in Table 3 for the three standard conditions. PV-module

Reference

PV-module condition Open circuit

Short circuit

Maximum power

ε-Iphn

ε-Vm

ε-Iphn

ε-Im

ε-Iphn

ε-Im

(Voc,0)

(Im ¼ 0)

(0,Icc)

(Vm ¼ 0)

(Vmax, Imax)

(Rload ¼ Vmax/Imax)

ε-Vm

KC 200 GT Multi-crystalline

[9] [1,2,5] [27] [6] 2D-model 1D-model

0.8121 7.8864 0.0023 0.0750 0.0002 0.0005

0.0381 0.3285 0.0001 0.0035 0.0000 0.0000

0.0883 0.1994 0.1894 0.0060 0.0002 0.0001

0.0882 0.1990 0.1891 0.0060 0.0002 0.0001

0.2692 0.1658 0.0661 0.0663 0.0002 0.0001

0.1308 0.0816 0.0319 0.0327 0.0001 0.0001

0.1308 0.0816 0.0319 0.0327 0.0001 0.0001

MSX-60 Multi-crystalline

[27] [1,2,5] [6] 2D-model 1D-model

0.1082 6.5570 7.1091 0.0003 0.0360

0.0061 0.2876 0.3103 0.0000 0.0018

0.0840 0.1984 0.0026 0.0006 0.0377

0.0840 0.1980 0.0026 0.0006 0.0377

0.0466 0.0513 0.3137 0.0006 0.0376

0.0233 0.0260 0.1541 0.0003 0.0191

0.0233 0.0260 0.1541 0.0003 0.0191

SP-70 Mono-crystalline

[2,5] [9] 2D-model 1D-model

8.2991 2.0281 0.0012 0.0027

0.3622 0.0988 0.0001 0.0002

0.5604 0.1895 0.0010 0.0004

0.5573 0.1891 0.0010 0.0004

0.2363 1.5865 0.0010 0.0001

0.1165 0.7324 0.0005 0.0001

0.1165 0.7324 0.0005 0.0001

SQ 150-PC Mono-crystalline

[5] 2D-model 1D-model

4.3265 0.0007 0.0001

0.1866 0.0000 0.0000

1.7202 0.0003 0.0006

1.7503 0.0003 0.0006

1.9877 0.0003 0.0006

0.9287 0.0002 0.0003

0.9287 0.0002 0.0003

Fig. 3. Voltage-current characteristics of the KC200GT module under standard and NOCT conditions calculated by the double-diode model.

experimental points extracted from the curves presented in the manufacturer’s data sheet. The Fig. 3 shows the voltage-current characteristics of the KC200GT PV-module in standard and NOCT conditions calculated by using the double-diode model proposed in Table 3 for this PVmodule. As can be seen (Fig. 3), the experimental points extracted from datasheet of the PV-module (Table 2) seem to match exactly with the characteristic points of the presented curves. As is shown in Table 4, the relative errors for the standard condition are: ε-Voc ¼ 0, ε-Icc ¼ 0.0002% and ε-Imax ¼ 0.0001%. Besides, the relative errors of characteristic points for the NOCT condition are calculated as: ε-Voc ¼ 0, ε-Icc ¼ 0.0016% and ε-Imax ¼ 0.0589%. The Fig. 4a and b shows the voltage-current characteristics of the PV-module with different conditions of temperature and irradiance. These characteristics have been calculated by using the double-

Fig. 4a. Voltage-current characteristics of the KC200GT module calculated by the double-diode model for temperature 25  C.

diode model proposed for this PV-module (Table 3). As can be seen, always the experimental points extracted from datasheet of the PVmodule are seem to be contained in the presented curves. Finally, the Fig. 5 compares the voltage-current characteristics of the double-diode model (continuous line) with the single-diode model (dash line), both obtained with the proposed methodology for different irradiance conditions and 25  C of temperature. As can be seen (Fig. 5), the characteristics obtained by both models are very much alike. Mostly of the experimental points extracted from the curves in manufacturer data sheet are seem to be contained in both of the obtained characteristics. However, in the zone near to the open circuit condition, and above all for low values of irradiance, the experimental points are outside the characteristic obtained by the single-diode model. The Table 5 presents a summary of the absolute errors of the characteristics voltage-current of the proposed double-diode

R.M. Arias García, I. P
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416

Fig. 4b. Voltage-current characteristics of the KC200GT module calculated by the double-diode model for irradiance 1000 W/m2.

model with respect to a set of points taken from the experimental curves. With the purpose of comparing the single-diode model with the double-diode model results, the single-diode model errors are presented for the module KC-200GT. In order to evaluate the correctness of the I/V characteristics obtained by the proposed models for different conditions of irradiance and temperature, the root mean square error (εrms) and the maximum error (εmax) are calculated from data in the Table A.1, Table A.2, Table A.3 and Table A.4 in the appendix. As is shown in Table 5, for the module KC-200GT the greater errors in the characteristics for 25  C of temperature (Fig. 4a) are of only 0.9 mA (rms) and 2.2 mA (max) (characteristic at 600 W/m2 of irradiance). Besides, for an irradiance of 1000 W/m2 (Fig. 4b), the greater errors are of only 6.8 mA (rms) and 26.5 mA (max) (characteristic at 75  C of temperature). These maximum errors are only a 0.09% and a 0.35% of the current of this PV-module in the point of maximum power Imax ¼ 7.61 A. The maximum errors for the module MSX-60 are of 0.3 mA (rms) and 0.7 mA (max) (characteristic at 1000 W/m2 and 25  C), that is, the 0.009% and the 0.02% of the current at maximum power Imax ¼ 3.5 A. The maximum errors for the module SP-70 are of 2.6 mA (rms) and 6.8 mA (max) (characteristic at 1000 W/m2 and 30  C), that is, the 0.06% and the 0.16% of the current at maximum power Imax ¼ 4.25 A. Finally, the maximum errors for the module SQ-150PC are of 0.5 mA (rms) and 1.2 mA (max) (characteristics at 1000 W/m2), that is, the 0.011% and the 0.027% of the current at maximum power Imax ¼ 4.4 A. This results proof the validity of the proposed methodology to obtain the parameters of the equivalent circuit, as well as the accuracy of the equations obtained to model the PV-module. The errors shown by the Table 5 for the module KC-200GT confirms the advantage in precision of the double-diode model over the single-diode model. The root mean square error (εrms) and the maximum error (εmax) of the characteristics obtained with the single-diode model are several times greater that the obtained with the double-diode model. 4. Conclusion

Fig. 5. Voltage-current characteristics of the KC200GT module for temperature of 25  C calculated by the single-diode and the double-diode models.

A generalized methodology is developed and validated, which obtain very precise parameters for the equivalent circuit of the single and the double diode models by employing the experimental data provided by the manufacturers in the datasheet of the PVmodules. In the development of this method they have not been used any approximations or considerations.

Table 5 Absolute errors of the characteristics I/V of the proposed models with respect to points taken from the experimental curves. Conditions

KC-200GT 1D-model

2D-model

MSX-60

SP-70

SQ-150PC

2D-model

2D-model

2D-model

Irradiance (W/m2) Temp (0C) ε-rms (mA) ε-max (mA) ε-rms (mA) ε-max (mA) ε-rms (mA) ε-max (mA) ε-rms (mA) ε-max (mA) ε-rms (mA) ε-max (mA) 1000 1000 1000 1000 1000 800 800 600 400 200

75 60 50 30 25 47 25 25 25 25

22.4 e 28.2 e 28.6 11.9 52.6 44.4 34.2 21.9

41.5 e 58.8 e 59.9 24.3 196.0 167.2 124.4 70.9

6.8 e 5.2 e 0.2 1.6 0.4 0.9 0.3 0.4

26.5 e 9.6 e 0.5 3.9 1.0 2.2 0.6 0.9

0.2 e 0.2 e 0.3 e 0.2 0.3 0.3 0.3

0.6 e 0.3 e 0.7 e 0.4 0.5 0.5 0.4

e 2.1 e 2.6 1.8 e 2.1 2.0 2.1 2.3

e 5.4 e 6.8 3.2 e 3.3 3.3 3.4 3.6

e 0.5 e 0.5 0.5 e 0.3 0.2 0.2 0.3

e 1.2 e 1.2 1.2 e 0.8 0.4 0.4 0.4

R.M. Arias García, I. P
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The generalized form of the Tellegen’s theorem is used to determining the linear system of equation (9) of the PV-module in standard conditions. This procedure allows the decoupled manner in which the three currents Iphn, I01n and I02n, the parallel resistance Rp, and the slope of the voltage-current characteristic of the PVmodule are determined. Besides, this theorem is used also to determine the equation (22) for calculating the inverse saturation current I01x under real operating conditions. This equation (22) has the advantage of achieving an exact power balance in the equivalent circuit under any operating condition of the photovoltaic module, which significantly improves the accuracy of the results. The set of equations as well as the solution method presented in this paper are the basis for the development of the dynamic model for the simulation of the PV-modules or PV-arrays in real conditions.

417

Author contributions section Rodolfo Manuel Arias García: Conceptualization, Methodology,
rez Abril: Software, WritingSoftware, Data curation, Ignacio Pe Original draft preparation, Reviewing and Editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix

Table A.1 Experimental points Vm, Im of the module KC-200GT’s characteristics and the currents I1D, I2D calculated by the single-diode model and the double-diode model respectively. 1000 W/m2 & 75  C

1000 W/m2 & 50  C

1000 W/m2 & 25  C

Vm(V)

Im(A)

I1D(A)

I2D(A)

Vm(V)

Im(A)

I1D(A)

I2D(A)

Vm(V)

Im(A)

I1D(A)

I2D(A)

0.0500 3.3500 6.6500 10.1500 13.6500 17.1500 20.0500 21.8500 23.0500 23.9500 24.6500 25.2500 25.7500 26.2500 26.7500

8.3690 8.3590 8.3490 8.3340 8.2970 8.1330 7.5630 6.6250 5.6390 4.6210 3.6670 2.7390 1.8900 0.9758 0.0000

8.3688 8.3625 8.3553 8.3428 8.3040 8.1284 7.5471 6.6488 5.6569 4.6543 3.7085 2.7801 1.9246 0.9970 0.0000

8.3687 8.3590 8.3487 8.3339 8.2967 8.1330 7.5626 6.6515 5.6390 4.6206 3.6672 2.7390 1.8900 0.9758 0.0000

0.0250 2.8250 6.1250 9.5250 12.8200 16.2200 19.9200 22.9200 24.8200 26.2200 27.2200 28.0200 28.7200 29.3200 29.8200

8.2980 8.2810 8.2720 8.2620 8.2500 8.2270 8.1150 7.6640 6.8170 5.6570 4.4740 3.3050 2.1250 1.0050 0.0000

8.2894 8.2843 8.2781 8.2714 8.2625 8.2397 8.1201 7.6573 6.8226 5.6898 4.5264 3.3638 2.1774 1.0395 0.0106

8.2893 8.2813 8.2717 8.2616 8.2499 8.2266 8.1155 7.6650 6.8198 5.6621 4.4809 3.3131 2.1342 1.0146 0.0000

0.0000 4.2000 8.3000 12.5000 16.5000 20.2000 23.5000 26.3000 27.9000 29.3000 30.4000 31.2000 31.9000 32.4000 32.9000

8.2100 8.1980 8.1860 8.1740 8.1610 8.1360 8.0350 7.6100 6.9150 5.7850 4.4580 3.2390 2.0060 1.0360 0.0000

8.2100 8.2023 8.1947 8.1868 8.1772 8.1534 8.0448 7.6100 6.9251 5.8212 4.5148 3.2989 2.0535 1.0653 0.0000

8.2100 8.1979 8.1861 8.1739 8.1607 8.1360 8.0348 7.6100 6.9152 5.7849 4.4580 3.2390 2.0056 1.0365 0.0000

800 W/m2 & 47  C Vm(V) Im(A)

I1D(A)

I2D(A)

800 W/m2 & 25  C Vm(V) Im(A)

I1D(A)

I2D(A)

600 W/m2 & 25  C Vm(V) Im(A)

I1D(A)

I2D(A)

0.0000 1.8620 4.3620 7.3620 10.5600 13.8600 16.9600 20.4600 23.2000 25.3600 26.8600 27.9600 28.7600 29.3600 29.9000

6.6239 6.6205 6.6159 6.6103 6.6039 6.5950 6.5751 6.4780 6.1308 5.3642 4.2597 3.0371 1.9023 0.9141 0.0000

6.6239 6.6186 6.6113 6.6027 6.5931 6.5814 6.5610 6.4718 6.1301 5.3754 4.2544 3.0178 1.8810 0.9001 0.0000

0.0000 3.0910 6.7910 10.4900 13.4900 16.6900 19.6900 22.6900 25.6900 27.7900 29.2900 30.3900 31.1900 31.8900 32.5906

6.5680 6.5623 6.5555 6.5487 6.5429 6.5353 6.5206 6.4664 6.2108 5.5921 4.6286 3.5028 2.4243 1.2961 0.1960

6.5680 6.5591 6.5484 6.5378 6.5289 6.5184 6.5025 6.4523 6.2114 5.5983 4.6212 3.4800 2.3964 1.2748 0.0000

0.0000 2.2920 4.9920 7.9920 10.9700 13.3900 16.2900 19.1900 22.4900 25.1900 27.7900 29.5900 30.5900 31.3900 32.1916

4.9260 4.9218 4.9168 4.9113 4.9058 4.9012 4.8948 4.8843 4.8442 4.7000 4.1585 3.1536 2.2133 1.2204 0.1672

4.9260 4.9194 4.9116 4.9030 4.8944 4.8873 4.8782 4.8659 4.8282 4.6956 4.1707 3.1611 2.2112 1.2139 0.0000

400 W/m2 & 25  C Vm(V) Im(A)

I1D(A)

I2D(A)

200 W/m2 & 25  C Vm(V) Im(A)

I1D(A)

I2D(A)

0.0294 2.8290 6.0290 9.1290 12.1300 14.5300 17.1300 19.8300 22.1300 24.4300 26.8300 28.7300 29.9300 30.8300 31.6294

3.2839 3.2788 3.2729 3.2672 3.2616 3.2570 3.2511 3.2409 3.2191 3.1520 2.9118 2.3621 1.6753 0.9045 0.1244

3.2839 3.2759 3.2666 3.2577 3.2490 3.2420 3.2336 3.2219 3.2010 3.1402 2.9156 2.3770 1.6878 0.9099 0.0000

0.0682 2.5680 5.1680 8.0680 10.8700 13.5700 16.3700 18.6700 20.8700 22.7700 24.6700 26.5700 28.2700 29.5700 30.6682

1.6419 1.6373 1.6325 1.6272 1.6220 1.6170 1.6113 1.6053 1.5955 1.5766 1.5290 1.4005 1.1102 0.6529 0.0709

1.6418 1.6346 1.6271 1.6188 1.6107 1.6029 1.5944 1.5865 1.5757 1.5576 1.5142 1.3951 1.1154 0.6604 0.0000

6.6200 6.6190 6.6110 6.6030 6.5930 6.5810 6.5610 6.4720 6.1300 5.3740 4.2530 3.0160 1.8780 0.8972 0.0000

3.2840 3.2760 3.2670 3.2580 3.2490 3.2420 3.2340 3.2220 3.2010 3.1400 2.9160 2.3770 1.6880 0.9105 0.0000

6.5680 6.5590 6.5480 6.5380 6.5290 6.5180 6.5030 6.4520 6.2110 5.5980 4.6210 3.4790 2.3960 1.2740 0.0000

1.6420 1.6350 1.6270 1.6190 1.6110 1.6030 1.5940 1.5870 1.5760 1.5580 1.5140 1.3950 1.1160 0.6613 0.0000

4.9260 4.9190 4.9120 4.9030 4.8950 4.8870 4.8780 4.8660 4.8280 4.6950 4.1700 3.1600 2.2090 1.2120 0.0000

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Table A.2 Experimental points Vm, Im of the module MSX-60’s characteristics and the current-I2D calculated by the double-diode model. 1000 W/m2 & 75  C

1000 W/m2 & 50  C

1000 W/m2 & 25  C

800 W/m2 & 25  C

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

0.0000 1.5000 2.7000 3.9000 5.1000 6.3000 7.5000 8.7000 9.9000 11.1000 12.3000 13.5000 14.7000 15.9000 17.1000

3.9498 3.9434 3.9382 3.9329 3.9274 3.9212 3.9134 3.9012 3.8780 3.8261 3.7030 3.4146 2.7992 1.6829 0.0000

3.9498 3.9433 3.9381 3.9328 3.9273 3.9211 3.9133 3.9012 3.8779 3.8261 3.7027 3.4140 2.7986 1.6829 0.0000

0.0000 2.2000 3.5000 4.8000 6.1000 7.4000 8.7000 10.0000 11.3000 12.6000 13.9000 15.2000 16.5000 17.8000 19.1000

3.8749 3.8655 3.8600 3.8544 3.8488 3.8429 3.8365 3.8284 3.8149 3.7855 3.7084 3.4968 2.9591 1.8349 0.0000

3.8749 3.8655 3.8599 3.8542 3.8486 3.8427 3.8363 3.8281 3.8148 3.7854 3.7083 3.4965 2.9588 1.8352 0.0000

0.0000 1.6000 3.1000 4.6000 6.1000 7.6000 9.1000 10.6000 12.1000 13.6000 15.1000 17.1000 18.1000 19.6000 21.1000

3.8000 3.7932 3.7868 3.7804 3.7740 3.7676 3.7610 3.7542 3.7461 3.7327 3.6987 3.5000 3.1890 2.0962 0.0000

3.8000 3.7931 3.7867 3.7802 3.7738 3.7673 3.7608 3.7539 3.7458 3.7325 3.6986 3.5000 3.1891 2.0969 0.0000

0.0000 2.6937 4.0937 5.4937 6.8937 8.2937 9.6937 11.0937 12.4937 13.8937 15.2937 16.6937 18.0937 19.4937 20.8937

3.0400 3.0285 3.0225 3.0166 3.0106 3.0046 2.9985 2.9920 2.9841 2.9712 2.9401 2.8440 2.5360 1.6929 0.0000

3.0400 3.0284 3.0224 3.0164 3.0104 3.0043 2.9982 2.9916 2.9838 2.9710 2.9400 2.8440 2.5358 1.6929 0.0000

600 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

400 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

200 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

0.0000 2.4278 3.8278 5.2278 6.6278 8.0278 9.4278 10.8278 12.2278 13.6278 15.0278 16.4278 17.8278 19.2278 20.6278

2.2800 2.2696 2.2636 2.2575 2.2515 2.2455 2.2394 2.2330 2.2259 2.2158 2.1947 2.1336 1.9355 1.3460 0.0000

0.0000 2.0529 3.4529 4.8529 6.2529 7.6529 9.0529 10.4529 11.8529 13.2529 14.6529 16.0529 17.4529 18.8529 20.2529

1.5200 1.5112 1.5052 1.4991 1.4931 1.4871 1.4810 1.4748 1.4682 1.4600 1.4460 1.4104 1.2983 0.9418 0.0000

0.0000 1.4121 2.8121 4.2121 5.6121 7.0121 8.4121 9.8121 11.2121 12.6121 14.0121 15.4121 16.8121 18.2121 19.6121

0.7600 0.7539 0.7479 0.7419 0.7359 0.7299 0.7238 0.7177 0.7115 0.7046 0.6955 0.6781 0.6302 0.4758 0.0000

2.2800 2.2696 2.2637 2.2577 2.2517 2.2457 2.2397 2.2334 2.2263 2.2161 2.1949 2.1337 1.9358 1.3465 0.0000

1.5200 1.5113 1.5053 1.4993 1.4933 1.4874 1.4813 1.4752 1.4686 1.4604 1.4463 1.4106 1.2985 0.9423 0.0000

0.7600 0.7540 0.7480 0.7420 0.7361 0.7301 0.7241 0.7181 0.7119 0.7050 0.6959 0.6783 0.6304 0.4760 0.0000

Table A.3 Experimental points Vm, Im of the module SP-70’s characteristics and the current-I2D calculated by the double-diode model. 1000 W/m2 & 60  C

1000 W/m2 & 30  C

1000 W/m2 & 25  C

800 W/m2 & 25  C

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

0.0000 1.8400 3.1400 4.4400 5.7400 7.0400 8.3400 9.6400 10.9400 12.2400 13.5400 14.8400 16.1400 17.4400 18.7400

4.7697 4.7550 4.7445 4.7339 4.7230 4.7111 4.6966 4.6748 4.6329 4.5382 4.3153 3.8327 2.9581 1.6580 0.0000

4.7697 4.7544 4.7435 4.7325 4.7213 4.7091 4.6945 4.6729 4.6319 4.5387 4.3170 3.8330 2.9543 1.6526 0.0000

0.0000 1.5200 3.0200 4.5200 6.0200 7.5200 9.0200 10.5200 12.0200 13.5200 15.0200 16.5200 18.0200 19.5200 21.0200

4.7100 4.6978 4.6858 4.6738 4.6618 4.6496 4.6370 4.6229 4.6031 4.5628 4.4517 4.1275 3.3402 1.9353 0.0000

4.7099 4.6973 4.6848 4.6724 4.6599 4.6473 4.6343 4.6199 4.6001 4.5606 4.4511 4.1275 3.3363 1.9285 0.0000

0.0000 1.9000 3.4000 4.9000 6.4000 7.9000 9.4000 10.9000 12.4000 13.9000 15.4000 16.5000 18.4000 19.9000 21.4000

4.7000 4.6848 4.6728 4.6608 4.6488 4.6366 4.6241 4.6103 4.5912 4.5530 4.4472 4.2500 3.3472 1.9378 0.0000

4.7000 4.6842 4.6717 4.6592 4.6467 4.6342 4.6213 4.6071 4.5880 4.5505 4.4463 4.2500 3.3477 1.9374 0.0000

0.0000 1.6900 3.1900 4.6900 6.1900 7.6900 9.1900 10.6900 12.1900 13.6900 15.1900 16.6900 18.1900 19.6900 21.1900

3.7600 3.7465 3.7345 3.7225 3.7105 3.6984 3.6861 3.6731 3.6572 3.6308 3.5661 3.3730 2.8387 1.7204 0.0000

3.7600 3.7459 3.7335 3.7210 3.7085 3.6960 3.6833 3.6699 3.6539 3.6278 3.5645 3.3735 2.8387 1.7173 0.0000

600 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

400 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

200 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

0.0000 2.4000 3.8000 5.2000 6.6000 8.0000 9.4000 10.8000 12.2000 13.6000 15.0000 16.4000 17.8000 19.2000 20.6000

2.8200 2.8000 2.7884 2.7768 2.7651 2.7534 2.7415 2.7291 2.7145 2.6925 2.6450 2.5151 2.1573 1.3528 0.0000

0.0000 1.8600 3.2600 4.6600 6.0600 7.4600 8.8600 10.2600 11.6600 13.0600 14.4600 15.8600 17.2600 18.6600 20.0600

1.8800 1.8645 1.8529 1.8413 1.8296 1.8180 1.8062 1.7942 1.7813 1.7651 1.7374 1.6712 1.4843 0.9952 0.0000

0.0000 2.4300 3.7300 5.0300 6.3300 7.6300 8.9300 10.1300 11.4300 12.7300 14.0300 15.3300 16.6300 17.9300 19.2300

0.9400 0.9198 0.9090 0.8982 0.8874 0.8766 0.8657 0.8555 0.8441 0.8311 0.8134 0.7807 0.7021 0.4951 0.0000

2.8200 2.8008 2.7896 2.7784 2.7672 2.7559 2.7444 2.7323 2.7178 2.6956 2.6470 2.5149 2.1556 1.3527 0.0000

1.8800 1.8651 1.8539 1.8427 1.8315 1.8203 1.8090 1.7973 1.7847 1.7685 1.7404 1.6725 1.4832 0.9935 0.0000

0.9400 0.9206 0.9102 0.8998 0.8894 0.8789 0.8685 0.8586 0.8475 0.8347 0.8169 0.7835 0.7033 0.4946 0.0000

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Table A.4 Experimental points Vm, Im of the module SQ-150PC’s characteristics and the current-I2D calculated by the double-diode model 1000 W/m2 & 60  C

1000 W/m2 & 30  C

1000 W/m2 & 25  C

800 W/m2 & 25  C

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

Vm(V)

Im(A)

I2D(A)

0.0000 4.4000 7.0000 9.6000 12.2000 14.8000 17.4000 20.0000 22.6000 25.2000 27.8000 30.4000 33.0000 35.6000 38.2000

4.8488 4.8334 4.8242 4.8150 4.8057 4.7960 4.7850 4.7700 4.7424 4.6759 4.4977 4.0541 3.1642 1.7807 0.0000

4.8488 4.8334 4.8243 4.8152 4.8059 4.7962 4.7852 4.7702 4.7426 4.6760 4.4976 4.0533 3.1630 1.7797 0.0000

0.0000 4.8900 7.7900 10.6900 13.5900 16.4900 19.3900 22.2900 25.1900 28.0900 30.9900 33.8900 36.7900 39.6900 42.5900

4.8069 4.7898 4.7796 4.7694 4.7592 4.7490 4.7385 4.7271 4.7120 4.6815 4.5905 4.2905 3.4904 2.0180 0.0000

4.8070 4.7899 4.7797 4.7696 4.7594 4.7492 4.7388 4.7274 4.7123 4.6818 4.5907 4.2900 3.4892 2.0169 0.0000

0.0000 4.4000 7.4000 10.4000 13.4000 16.4000 19.4000 22.4000 25.4000 28.4000 31.4000 34.0000 37.4000 40.4000 43.4000

4.8000 4.7845 4.7740 4.7635 4.7529 4.7424 4.7317 4.7204 4.7065 4.6812 4.6068 4.4002 3.5801 2.0891 0.0000

4.8000 4.7846 4.7741 4.7636 4.7531 4.7426 4.7319 4.7207 4.7068 4.6815 4.6070 4.4000 3.5790 2.0879 0.0000

0.0000 5.2900 8.1900 11.0900 13.9900 16.8900 19.7900 22.6900 25.5900 28.4900 31.3900 34.2900 37.1900 40.0900 42.9800

3.8400 3.8214 3.8112 3.8010 3.7909 3.7807 3.7703 3.7596 3.7471 3.7270 3.6759 3.5050 2.9748 1.8018 0.0000

3.8400 3.8215 3.8114 3.8012 3.7911 3.7809 3.7706 3.7599 3.7474 3.7274 3.6762 3.5050 2.9742 1.8010 0.0000

600 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

400 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

200 W/m2 & 25  C Vm(V) Im(A)

I2D(A)

0.0000 4.7500 7.6500 10.5500 13.4500 16.3500 19.2500 22.1500 25.0500 27.9500 30.8500 33.7500 36.6500 39.5500 42.4500

2.8800 2.8634 2.8532 2.8431 2.8330 2.8228 2.8126 2.8022 2.7909 2.7759 2.7456 2.6531 2.3422 1.5119 0.0000

0.0000 4.8000 7.6000 10.4000 13.2000 16.0000 18.8000 21.6000 24.4000 27.2000 30.0000 32.8000 35.6000 38.4000 41.2000

1.9200 1.9032 1.8934 1.8836 1.8738 1.8640 1.8542 1.8442 1.8338 1.8213 1.8010 1.7496 1.5841 1.0882 0.0000

0.0000 4.7000 7.4000 10.1000 12.8000 15.5000 18.2000 20.9000 23.6000 26.3000 29.0000 31.7000 34.4000 37.1000 39.8000

0.9600 0.9436 0.9341 0.9247 0.9152 0.9058 0.8963 0.8868 0.8771 0.8666 0.8533 0.8290 0.7641 0.5613 0.0000

2.8800 2.8633 2.8531 2.8430 2.8328 2.8226 2.8123 2.8019 2.7906 2.7756 2.7452 2.6528 2.3423 1.5123 0.0000

1.9200 1.9031 1.8933 1.8835 1.8737 1.8638 1.8540 1.8440 1.8335 1.8210 1.8006 1.7492 1.5839 1.0881 0.0000

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