In this article, the solar cell parameters (within the one-diode solar cell model) are obtained with less than 10% error, integrating the Co-Content function using up to order 6 Simpson integration method, and as a function of the number of measured points per volt and a percentage noise of the maximum current. It is shown, that less than 10% error (in some cases around 1%) can be obtained, in case the percentage noise is as larger as 0.1%, using higher order Simpson integration than 1, the usually used trapezoidal integration method.

Several weather catastrophes, such as drought, fires, hurricanes, extreme temperatures (either cold or hot), volcano eruptions, and floods, are striking nowadays all around the world, due to global warming that is taking place, due to the human-emitted CO2 gases to the atmosphere.1–4 Humanity is in a desperate situation, if it is willing to survive, to reduce as fast as possible, all the emission of green-house gases to the atmosphere, to avoid the climate catastrophe is getting into.1–4 Coincidentally, it is predicted that world energy demand will increase to 30 TW by 2050, from its today's value of 10 TW.5 Photovoltaic energy, obtained via solar panels and solar cells, is a suitable candidate to achieve both targets, as it is friendly energy with nature.6 

The one diode model is the simplest model to explain a solar cell7 and consists of five parameters, which are the light current (Ilig), also known as photocurrent, the saturation current (Isat), the ideality factor (n), the series resistance (Rs), and the shunt resistance (Rsh). The current–voltage (IV) and the electric circuit of the one diode model are shown in Sec. II. It is important for researchers to accurately obtain Isat, Ilig, n, Rsh, and Rs, as they yield valuable information of the physical phenomena that are happening inside the solar cell. For example, Isat is related to any Auger, Shockley–Read–Hall, contact, and/or surface recombination mechanisms,8 while Ilig yields information on the donor and acceptor densities, as the charge carriers' lifetime.9 In addition, n reveals transport mechanisms: in case it is 1, there is minority carrier diffusion, while there is recombination and/or generation of charge carriers inside the depletion region when its value is 2.10 Regarding the resistances Rs and Rsh, it is expected that the former one is as small as possible, to hinder a voltage drop, as it is related to the Ohmic contact quality,11,12 while on the contrary, the latter one is related to the crystal quality and it is expected to be as large as possible. A small value of Rsh reveals bad crystal quality, providing alternative charge carriers' current paths, which cause a current lost.13 The impact of Isat, Ilig, n, Rsh, and Rs on the solar cell performance is described in Refs. 8–13.

It is not a straight procedure to deduce the solar cell parameters, and thus, a variety of methods have been proposed to achieve this. However, it is necessary to either suppose some valid estimations on one or more of the solar cell parameters,14–21 or it is necessary to do a number of different luminescence and/or maximum power point measurements under different irradiation conditions.14,15,19,22–27

Only a little number of methods can be applied in a single current–voltage (IV), with no assumptions or/and independent of light conditions.7,28–31 Briefly, in Refs. 7 and 29, two procedures were proposed to obtain more accurately the solar cell parameters. Afterward, these procedures were incorporate into iterative cycles in Refs. 28 and 30, where it was shown that the Cheung method, which was originally proposed for Schottky contacts,32–37 can be also applied to the equation of one diode model solar cell. Finally, Ortiz-Conde et al.31 showed in 2006 that the IV curve can be expressed using the Lambert function, then proposing Co-Content function C C V , I = 0 V I I s c d V, where I s c = I ( V = 0 ) is the short-circuit current, and finally, fitting C C V , I to C V 1 V + C I 1 I I S C + C V 2 V 2 + C I 2 I I S C 2 +  C I 1 V 1 V I I S C, they showed that Isat, n, Ilig, Rs, and Rsh can be deduced from the fitting constants.

Outstanding solar cell parameter extraction was obtained, when the Ortiz-Conde et al.31 method was applied to noiseless IV curves (a percentage noise p n = 0 %), and a density of measured points per voltage P V = 1000 measured points V (see Sec. 3 in Ref. 7). Nonetheless, unfeasible extraction occurred when the Ortiz-Conde et al.31 technique was implemented in measured IV curves, due to the deleterious presence of noise (see Sec. 4 in Ref. 29). This shows that the Ortiz-Conde et al.31 technique depends strongly on P V and p n. At the same time, it has been exposed that the harmful effect of the noise can be minimized in some cases, increasing the value of P V.38–41 In those studies,38–41 the trapezoidal integration technique was used to obtain C C V , I, and the increase in P V allowed to diminish the deleterious effect of the noise in some cases, yielding a more accurate C C V , I, and hence more accurate solar cell parameter extraction. The drawback of this procedure is that p n should be increased, in some cases to as high as 50 001  measured points V. This causes time consumption and a larger set of data that need to be processed. Thus, it would be very convenient for researchers, if the C C V , I could be obtained more accurately, with just 101  measured points V or less, and still deduce accurate solar cell parameters, using the Ortiz-Conde et al. method.31 This explains the intention of this article: to explore 2-, 3-, 4-, 5-, and 6-order Simpson integration method of the C C V , I function, and its effect on the solar cell parameter extraction, within the one-diode solar cell model, and using 101  measured points V or less. Comparison with the 1-order Simpson integration, also known as trapezoidal integration, is also done and given.

This article has the following organization. This brief introduction is followed by Sec. II, where the current–voltage (IV) curves simulation is explained, as a function of p n and P V. Also, in Sec. II, a brief summary of the Ortiz-Conde et al. method31 is given, as the implementation of the 1-, 2-, 3-, 4-, 5-, and 6-order Simpson integration methods to calculate C C V , I more accurately. In Sec. III, an example of the application of the Ortiz-Conde et al. method31 is given: C C V , I are calculated using 1-, 2-, 3-, 4-, 5-, and 6-order Simpson integration method. Once it is done, R s ± Δ R s, R s h ± Δ R s h , n ± Δ n , I lig ± Δ I lig , I sat are deduced, and finally, the percentage errors of Rs, Rsh, n, Ilig, and Isat, as a function of the integration order, are computed and reported. Afterward, this is extended to the calculation of all the percentage errors of Rs, Rsh, n, Ilig, and Isat, as a function of all p n, P V values, and order 1 to 6 Simpson integration. The results and their analysis are also given. Finally, in Sec. IV, conclusions are reported.

For completeness purpose, first, the one-diode solar cell model circuit and the IV equation of the one-diode solar cell model are shown in Fig. 1 and in Eq. (1).7 Afterward, an abridgment of the Ortiz-Conde et al. method31 is given,
I = I sat exp V I R s nkT 1 + V I R s R s h I lig .
(1)
FIG. 1.

One-diode solar cell model electric circuit.

FIG. 1.

One-diode solar cell model electric circuit.

Close modal
In 2006, Ortiz-Conde et al. proposed the Co-Content C C V , I function as31 
C C V , I = 0 V I I s c d V ,
(2)
where I s c = I V = 0 is the short-circuit current.
Ortiz-Conde et al. proved31 that after fitting Eq. (2) to a second-degree polynomial in two variables, namely, I I s c and V,
C V 1 V + C I 1 I I s c + C V 2 V 2 + C I 2 I I s c 2 + C I 1 V 1 V I I s c ,
(3)
Rs, Rsh, n, and Ilig could be calculated from the fitting constants C V 1, C I 1, C V 2, and C I 2 using31 
R s h = 1 2 C V 2 ,
(4)
R s = A 1 4 C V 2 ,
(5)
n = C V 1 A 1 + 4 C I 1 C V 2 4 V t h C V 2 ,
(6)
and
I lig = 1 + A C V 1 + I s c 2 2 C I 1 C V 2 ,
(7)
where V t h = k T is the thermal velocity, with T the absolute temperature and k the Boltzmann constant, and A = 1 + 16 C I 2 C V 2.
Once Rs, Rsh, n, and Ilig have been calculated, Isat is obtained evaluating
I sat = B I + I lig V I R s R s h ,
(8)
at the largest possible V, as it is explained in Sec. 4 in Ref. 7.

The function B is B = 1 exp V I R s nkT 1.

The first errors of Rs, Rsh, n, and Ilig, i.e., Δ R s, Δ R s h, Δ n, and Δ I lig, were deduced in Sec. 3.2 in Ref. 29, while the first error of Isat, namely, Δ I sat, was deduced in Sec. 2 in Ref. 38, and their expressions are
Δ R s h = Δ C V 2 2 C V 2 2 ,
(9)
Δ R s = 1 2 A 1 + 8 C V 2 C I 2 A C V 2 2 Δ C V 2 + 8 Δ C I 2 A ,
(10)
Δ n = A 1 4 V t h C V 2 Δ C V 1 + C V 1 A 1 + 8 C V 2 C I 2 4 V t h A C V 2 Δ C V 2 + 1 V t h Δ C I 1 + 2 C V 1 A V t h Δ C I 2 ,
(11)
Δ I lig = 1 + A 2 Δ C V 1 4 C I 2 C V 1 + I s c A + 2 C I 1 Δ C V 2 2 C V 2 Δ C I 1 4 C V 1 + I s c C V 2 A Δ C I 2 ,
(12)
Δ I sat = B 1 + R s R s h R s nkT B + 1 I + I lig V I R s R s h Δ I + B Δ I lig + B I 1 R s B + 1 nkT I + I lig V I R s R s h Δ R s + B V I R s R s h 2 Δ R s h B B + 1 I R s n 2 k T I + I lig V I R s R s h Δ n ,
(13)
where Δ I is the precision of the amperemeter.

The Ortiz-Conde et al. method31 application is given in Sec. III.

The C program reported in Ref. 7 was applied to simulate the IV curves reported in this study. For comparison purpose, the solar cell parameters used were n = 2.5, Rs = 1 Ω, Rsh = 1 kΩ, Isat = 1 μA, and Ilig = 1 mA, in the voltage range [0, 1 V], which is the same as reported in the previous reported studies.7,8,18,20,21 These are common parameters that second generation laboratory-made solar cells usually show,42–61 this means Rs and Isat are three orders of magnitude smaller than Rsh and Ilig, respectively.42–61 Ten IV curves were computed noiseless, i.e., p n = 0 %, using P V = 11, 21, 31, 41, 51, 61, 71, 81, 91, and 101  number o f points V, equally spaced voltage points in the voltage range, for each simulated curve. Afterward, the IV curves were simulated again, but with the inclusion of a random percentage noise of p n = 0% (noiseless), 0.001%, 0.005%, 0.01%, 0.05%, 0.1%, 0.5%, and 1% of the maximum current of I, Imax = I(V = 1), to do them comparable to the previous reported studies.7,8,18,20,21 The random noise was incorporated as a percentage of Imax, because one of the principal sources of noise is the precision of the amperemeter at Imax, causing that noise affects more at lower voltages, i.e., lower currents, which is what is usually seen in IV measurements. The case of a constant percentage noise and a percentage noise of the short-circuit current, in the zero volt to open-circuit voltage, is currently being investigated and will be reported elsewhere.

Briefly, in those preliminary studies,7,8,18,20,21 it was discussed the importance to accurately determine the Co-Content function to achieve accurate solar cell parameters deduction. In those previous studies, C C V , I was refined increasing the value of P V, up to 50 001  measured points V and the trapezoidal integration method, which is an order 1 integration Simpson, was used.7,8,18,20,21

An alternative procedure to integrate Eq. (1) with a higher precision is to use larger order integration Simpson methods.62 Briefly, the Simpson integration method consists of the following.62 When a function f x is integrated in an interval [a, b], then this interval is divided, respectively, into two, three, four, five, six, and seven equally spaced points, being x 0 = a, and x m = b, where m = 1, 2, 3, 4, 5, 6 is the integration order, respectively, and the integral is approximated by (inside the parenthesis an alternative name to the order integration is given)62 

  • Order m = 1 (trapezoidal):
    x 0 = a x 1 = b f x Δ x f x 0 + f x 1 2 ,
    (14)
  • Order m = 2 (parabolic, Newton–Cotes quadrature formula or 1/3 rule):
    x 0 = a x 2 = b f x Δ x 3 f x 0 + 4 f x 1 + f x 2 ,
    (15)
  • Order m = 3 (cubic or 3/8 rule):
    x 0 = a x 3 = b f x 3 Δ x 8 f x 0 + 3 f x 1 + 3 f x 2 + f x 3 ,
    (16)
  • Order m = 4 (Boole's rule):
    x 0 = a x 4 = b f x 2 Δ x 45 7 f x 0 + 32 f x 1 + 12 f x 2 + 32 f x 3 + 7 f x 4 ,
    (17)
  • Order m = 5:
    x 0 = a x 5 = b f x 5 Δ x 288 19 f x 0 + 75 f x 1 + 50 f x 2 +  50 f x 3 + 75 f x 4 + 19 f x 5 ,
    (18)
  • Order m = 6:
    x 0 = a x 6 = b f x Δ x 140 41 f x 0 + 216 f x 1 + 27 f x 2 + 272 f x 3 +  27 f x 4 + 216 f x 5 + 41 f x 6 ,
    (19)

    where Δ x = b a m.

In this study, x 0 = a = 0 V, x P V 1 = b = 1 V, and f V = I I S C. The Simpson integration methods were implemented in the following form. For example, if the order m = 5 was used, then the first five integrals x 0 x 1 f V, x 0 x 2 f V, x 0 x 3 f V, x 0 x 4 f V, and x 0 x 5 f V were calculated using Eqs. (3)–(7), respectively. Afterward, any subsequent integral, with p 6, was calculated using
x 0 x p f V = x 0 x p 5 f V + x p 5 x p f V ,
(20)
where x p 5 x p f V is calculated using Eq. (7), and x 0 x p 5 f V has been already calculated.
The same idea was used for any order of integration, i.e., in case the order is m, the first m integrals are calculated using Eq. (3) to [Eq. (3) + m − 1]. The subsequent integrals (with p m + 1) are calculated using
x 0 x p f V = x 0 x p m f V + x p m x p f V ,
(21)
where x 0 x p m f V has been already calculated, and x p m x p f V is calculated using [Eq. (3) + m − 1]. Of course, x 0 x 0 f V = 0.

For clarity purpose for the reader, a descriptive diagram is shown in Fig. 2, and a numerical example of the implementation of this procedure is given in Table I, for the case of m = 4.

FIG. 2.

Schematic diagram, on the steps done to obtain R s ± Δ R s, R s h ± Δ R s h , n ± Δ n , I lig ± Δ I lig , I sat.

FIG. 2.

Schematic diagram, on the steps done to obtain R s ± Δ R s, R s h ± Δ R s h , n ± Δ n , I lig ± Δ I lig , I sat.

Close modal
TABLE I.

Example on the implementation of the integration methodology, as explained in Fig. 2, using Simpson integration method order 4. First, in columns 1 and 2, namely, Col1 and Col2, the IV calculated using the C program reported in Ref. 7, is given. Afterward, in Col3, I–Isc is reported as the value of column 2 minus the value of (line 1, column 2), i.e., Col2 – (L1, Col2). Next, in Col4, an explanation of each integral used for each line is given. L1 is zero, while in L2 to L5, the trapezoidal, parabolic, cubic, and Boole's rule are used. From L6 downward, Eq. (10) is used, with m = 4. For clarity purposes for the reader, the numerical calculations are given. Finally, in Col5, the final value of CC(V, I) is reported.

Line/column Col1 Col2 Col3 Col4 Col5
  Voltage V (V)  Current I (mA)  f V = I I s c (mA) Col2 – (L1, Col2)  Explanation of which integration was used to calculate CC(V, I)  CC(V, I) (mW) 
L1  −0.999  0 V 0 V f V = 0 mW  0 mW 
L2  0.1  −0.895  0.104  0 V 0.1 V f V Δ x f 0 V + f 0. 1 V 2 = 0.1 V 0 + 0.104 mA 2 = 0.0052 mW  0.0052 mW 
L3  0.2  −0.778  0.221 
0 V 0.2 V f V Δ x f 0 V + 4 f 0.1 V + f 0.2 V = 0.1 V 3 0 mA + 4 0.104 mA + 0.221 mA = 0.021 233 mW
 
0.021 23 mW 
L4  0.3  −0.595  0.404 
0 V 0.3 V f V 3 Δ x 8 f 0 V + 3 f 0.1 V + 3 f 0.2 V + f 0.3 V = 3 0.1 V 8 0 mA + 3 0.104 mA + 3 ( 0.221 mA + 0.404 mA ) = 0.051 71 mW
 
0.051 71 mW 
L5  0.4  −0.107  0.892 
0 V 0.4 V f V 2 Δ x 45 7 f 0 V + 32 f 0.1 V + 12 f 0.2 V + 32 f 0.3 V + 7 f 0.4 V = 2 0.1 V 45 7 ( 0 mA ) + 32 0.104 mA + 12 ( 0.221 mA + 32 0.404 mA + 7 ( 0.892 mA ) ) = 0.111 78 mW
 
0.111 78 mW 
L6  0.5  1.761  2.760 
0 V 0.5 V f V = 0 V 0.1 V f V + 0.1 V 0.5 V f V = ( L 2 , Col 5 ) + 2 Δ x 45 7 f 0.1 V +  32 f 0.2 V + 12 f 0.3 V + 32 f 0.4 V + 7 f 0.5 V = L 2 , Col 5 + 2 0.1 V 45 7 ( 0.104 mA ) + 32 0.221 mA +  12 ( 0.404 mA + 32 0.892 mA + 7 ( 2.76 mA ) ) = 0.0052 mW + 0.268 631 111 mW = 0.273 83 mW
 
0.273 83 mW 
L7  0.6  9.109  10.108 
0 V 0.6 V f V = 0 V 0.2 V f V + 0.2 V 0.6 V f V = ( L 3 , Col 5 ) + 2 Δ x 45 7 f 0.2 V +  32 f 0.3 V + 12 f 0.4 V + 32 f 0.5 V + 7 f 0.6 V = L 3 , Col 5 + 2 0.1 V 45 7 ( 0.221 mA ) + 32 0.404 mA + 12 ( 0.892 mA + 32 2.75 mA + 7 ( 10.108 mA ) ) = 0.021 23 mW + 0.817 488 88 mW = 0.838 71 mW
 
0.838 71 mW 
L8  0.7  31.417  32.416 
0 V 0.7 V f V = 0 V 0.3 V f V + 0.3 V 0.7 V f V = L 4 , Col 5 + 2 Δ x 45 7 f 0.3 V + 32 f 0.4 V +  12 f 0.5 V + 32 f 0.6 V + 7 f 0.7 V = L 4 , Col 5 + 2 0.1 V 45 7 ( 0.404 mA ) + 32 0.892 mA + 12 ( 2.75 mA +  32 10.108 mA + 7 ( 32.416 mA ) ) = 0.051 71 mW + 2.732 17 mW = 2.783 88 mW
 
2.783 88 mW 
L9  0.8  75.396  76.395 
0 V 0.8 V f V = 0 V 0.4 V f V + 0.4 V 0.8 V f V = ( L 5 , Col 5 ) + 2 Δ x 45 7 f 0.4 V + 32 f 0.5 V + 12 f 0.6 V + 32 f 0.7 V + 7 f 0.8 V = L 5 , Col 5 + 2 0.1 V 45 7 ( 0.892 mA ) + 32 2.75 mA + 12 ( 10.108 mA + 32 32.416 mA + 7 ( 76.395 mA ) ) = 0.111 78 mW + 7.855 11 mW = 7.966 89 mW
 
7.966 89 mW 
L10  0.9  136.999  137.998 
0 V 0.9 V f V = 0 V 0.5 V f V + 0.5 V 0.9 V f V = ( L 6 , Col 5 ) + 2 Δ x 45 7 f 0.5 V + 32 f 0.6 V + 12 f 0.7 V + 32 f 0.8 V + 7 f 0.9 V = L 6 , Col 5 + 2 0.1 V 45 7 ( 2.75 mA ) + 32 10.108 mA + 12 ( 32.416 mA + 32 79.395 mA + 7 ( 137.998 mA ) ) = 0.273 83 mW + 18.8369 mW = 19.110 82 mW
 
19.110 82 mW 
L11  1.0  209.615  210.614 
0 V 1 V f V = 0 V 0.6 V f V + 0.6 V 1 V f V = ( L 7 , Col 5 ) + 2 Δ x 45 7 f 0.6 V + 32 f 0.7 V + 12 f 0.8 V + 32 f 0.9 V + 7 f 1 V = L 7 , Col 5 + 2 0.1 V 45 7 ( 10.108 mA ) + 32 32.416 mA + 12 ( 79.395 mA + 32 137.998 mA + 7 ( 210.614 mA ) ) = 0.838 71 mW + 35.337 96 mW = 36.176 67 mW
 
36.176 67 mW 
Line/column Col1 Col2 Col3 Col4 Col5
  Voltage V (V)  Current I (mA)  f V = I I s c (mA) Col2 – (L1, Col2)  Explanation of which integration was used to calculate CC(V, I)  CC(V, I) (mW) 
L1  −0.999  0 V 0 V f V = 0 mW  0 mW 
L2  0.1  −0.895  0.104  0 V 0.1 V f V Δ x f 0 V + f 0. 1 V 2 = 0.1 V 0 + 0.104 mA 2 = 0.0052 mW  0.0052 mW 
L3  0.2  −0.778  0.221 
0 V 0.2 V f V Δ x f 0 V + 4 f 0.1 V + f 0.2 V = 0.1 V 3 0 mA + 4 0.104 mA + 0.221 mA = 0.021 233 mW
 
0.021 23 mW 
L4  0.3  −0.595  0.404 
0 V 0.3 V f V 3 Δ x 8 f 0 V + 3 f 0.1 V + 3 f 0.2 V + f 0.3 V = 3 0.1 V 8 0 mA + 3 0.104 mA + 3 ( 0.221 mA + 0.404 mA ) = 0.051 71 mW
 
0.051 71 mW 
L5  0.4  −0.107  0.892 
0 V 0.4 V f V 2 Δ x 45 7 f 0 V + 32 f 0.1 V + 12 f 0.2 V + 32 f 0.3 V + 7 f 0.4 V = 2 0.1 V 45 7 ( 0 mA ) + 32 0.104 mA + 12 ( 0.221 mA + 32 0.404 mA + 7 ( 0.892 mA ) ) = 0.111 78 mW
 
0.111 78 mW 
L6  0.5  1.761  2.760 
0 V 0.5 V f V = 0 V 0.1 V f V + 0.1 V 0.5 V f V = ( L 2 , Col 5 ) + 2 Δ x 45 7 f 0.1 V +  32 f 0.2 V + 12 f 0.3 V + 32 f 0.4 V + 7 f 0.5 V = L 2 , Col 5 + 2 0.1 V 45 7 ( 0.104 mA ) + 32 0.221 mA +  12 ( 0.404 mA + 32 0.892 mA + 7 ( 2.76 mA ) ) = 0.0052 mW + 0.268 631 111 mW = 0.273 83 mW
 
0.273 83 mW 
L7  0.6  9.109  10.108 
0 V 0.6 V f V = 0 V 0.2 V f V + 0.2 V 0.6 V f V = ( L 3 , Col 5 ) + 2 Δ x 45 7 f 0.2 V +  32 f 0.3 V + 12 f 0.4 V + 32 f 0.5 V + 7 f 0.6 V = L 3 , Col 5 + 2 0.1 V 45 7 ( 0.221 mA ) + 32 0.404 mA + 12 ( 0.892 mA + 32 2.75 mA + 7 ( 10.108 mA ) ) = 0.021 23 mW + 0.817 488 88 mW = 0.838 71 mW
 
0.838 71 mW 
L8  0.7  31.417  32.416 
0 V 0.7 V f V = 0 V 0.3 V f V + 0.3 V 0.7 V f V = L 4 , Col 5 + 2 Δ x 45 7 f 0.3 V + 32 f 0.4 V +  12 f 0.5 V + 32 f 0.6 V + 7 f 0.7 V = L 4 , Col 5 + 2 0.1 V 45 7 ( 0.404 mA ) + 32 0.892 mA + 12 ( 2.75 mA +  32 10.108 mA + 7 ( 32.416 mA ) ) = 0.051 71 mW + 2.732 17 mW = 2.783 88 mW
 
2.783 88 mW 
L9  0.8  75.396  76.395 
0 V 0.8 V f V = 0 V 0.4 V f V + 0.4 V 0.8 V f V = ( L 5 , Col 5 ) + 2 Δ x 45 7 f 0.4 V + 32 f 0.5 V + 12 f 0.6 V + 32 f 0.7 V + 7 f 0.8 V = L 5 , Col 5 + 2 0.1 V 45 7 ( 0.892 mA ) + 32 2.75 mA + 12 ( 10.108 mA + 32 32.416 mA + 7 ( 76.395 mA ) ) = 0.111 78 mW + 7.855 11 mW = 7.966 89 mW
 
7.966 89 mW 
L10  0.9  136.999  137.998 
0 V 0.9 V f V = 0 V 0.5 V f V + 0.5 V 0.9 V f V = ( L 6 , Col 5 ) + 2 Δ x 45 7 f 0.5 V + 32 f 0.6 V + 12 f 0.7 V + 32 f 0.8 V + 7 f 0.9 V = L 6 , Col 5 + 2 0.1 V 45 7 ( 2.75 mA ) + 32 10.108 mA + 12 ( 32.416 mA + 32 79.395 mA + 7 ( 137.998 mA ) ) = 0.273 83 mW + 18.8369 mW = 19.110 82 mW
 
19.110 82 mW 
L11  1.0  209.615  210.614 
0 V 1 V f V = 0 V 0.6 V f V + 0.6 V 1 V f V = ( L 7 , Col 5 ) + 2 Δ x 45 7 f 0.6 V + 32 f 0.7 V + 12 f 0.8 V + 32 f 0.9 V + 7 f 1 V = L 7 , Col 5 + 2 0.1 V 45 7 ( 10.108 mA ) + 32 32.416 mA + 12 ( 79.395 mA + 32 137.998 mA + 7 ( 210.614 mA ) ) = 0.838 71 mW + 35.337 96 mW = 36.176 67 mW
 
36.176 67 mW 

This integration procedure was implemented on all the combinations of p n and P V and using integrations orders of m = 1, 2, 3, 4, 5, and 6, to calculate each C C V , I. The Ortiz-Conde et al. method31 was applied to each one, i.e., each C C V , I was fitted to Eq. (2), Rs, Rsh, n, and Ilig were calculated according to Eqs. (3)–(6) in Ref. 31, and Isat was calculated using Eq. (11) in Ref. 7, evaluated at the largest voltage, i.e., V = 1, as it is explained in Sec. 4 in Ref. 7. Once the Rs, Rsh, n, Ilig, and Isat were obtained, the percentage errors relative to the original Rs = 1 Ω, Rsh = 1 kΩ, n = 2.5, Ilig = 1 mA, and Isat = 1 μA were calculated. The results are shown in Figs. 5–9 in Sec. III.

For completeness purpose, an example of the IV curves simulated with p n = 1 %, and all the studied P V values, is shown in Fig. 3. To show the IV curves clearer for the reader, the scale is shown for the 101  number o f points V (black line) case, and each subsequent IV curve upward is multiplied by 1.2, 1.5, 1.9, 2.7, 3.7, 5.2, 7.2, 10, and 15, respectively. In the inset, the same data are given, in linear scale. The scale is shown for the 101  number o f points V (black line), and each subsequent IV curve upward is shifted +800 μA relative to the previous one.

FIG. 3.

Absolute I vs V in y-logarithmic scale for the 1% noise case: 101  number o f points V (black line), 91  number o f points V (red line), 81  number o f points V (light green line), 71  number o f points V (dark blue line), 61  number o f points V (magenta), 51  number o f points V (light blue line), 41  number o f points V (purple line), 31  number o f points V (wine line), 21  number o f points V (dark green line), and 11  number o f points V (orange line). The scale is shown for the 101  number o f points V (black line) case, and each subsequent IV curve upward is multiplied by 1.2, 1.5, 1.9, 2.7, 3.7, 5.2, 7.2, 10, and 15, respectively, to separate them and render them clearer for the reader. For clarity purposes, in the insets, the same data are shown, but in linear scale. The scale is shown for the 101  number o f points V (black line), and each subsequent IV curve upward is shifted +800 μA relative to the previous one.

FIG. 3.

Absolute I vs V in y-logarithmic scale for the 1% noise case: 101  number o f points V (black line), 91  number o f points V (red line), 81  number o f points V (light green line), 71  number o f points V (dark blue line), 61  number o f points V (magenta), 51  number o f points V (light blue line), 41  number o f points V (purple line), 31  number o f points V (wine line), 21  number o f points V (dark green line), and 11  number o f points V (orange line). The scale is shown for the 101  number o f points V (black line) case, and each subsequent IV curve upward is multiplied by 1.2, 1.5, 1.9, 2.7, 3.7, 5.2, 7.2, 10, and 15, respectively, to separate them and render them clearer for the reader. For clarity purposes, in the insets, the same data are shown, but in linear scale. The scale is shown for the 101  number o f points V (black line), and each subsequent IV curve upward is shifted +800 μA relative to the previous one.

Close modal

The effect of p n = 1 % noise of I max can be seen in the IV curves, especially for low voltages, when I is smaller, becoming more noticeable.

An example of the application of the Ortiz-Conde et al. method,31 for order 1 to order 6 Simpson integration, is shown in Fig. 4, for the P V = 41 measured points V and p n = 0.01 % case.

FIG. 4.

Calculation of C C V , I using Eq. (2) and fitting to a second order polynomial order on V and I I s c, namely, a V + b I I S C + c V 2 + d I I S C 2 + f V I I S C [Eq. (3)] for IV curves simulated with P V = 41 measured points V and p n = 0.01 %, and Simpson integration using order (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6. The parameters C V 1 , C I 1 , C V 2 , and C I 2 are C V 1 = a , C I 1 = b , C V 2 = c , and C I 2 = d, respectively.

FIG. 4.

Calculation of C C V , I using Eq. (2) and fitting to a second order polynomial order on V and I I s c, namely, a V + b I I S C + c V 2 + d I I S C 2 + f V I I S C [Eq. (3)] for IV curves simulated with P V = 41 measured points V and p n = 0.01 %, and Simpson integration using order (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6. The parameters C V 1 , C I 1 , C V 2 , and C I 2 are C V 1 = a , C I 1 = b , C V 2 = c , and C I 2 = d, respectively.

Close modal

A summary of the previous application of the Ortiz-Conde et al. method31 is given in Table II, with C V 1 , C I 1 , C V 2 , C I 2, Δ C V 1 , Δ C I 1 , Δ C V 2 , and Δ C I 2, obtained from Fig. 4, together with the calculated R s + Δ R s, R s h + Δ R s h , n + Δ n , I lig + Δ I lig , I sat, using Eqs. (4)–(12), and the percentage errors relative to the original Rs = 1 Ω, Rsh = 1 kΩ, n = 2.5, Ilig = 1 mA, and Isat = 1 μA.

TABLE II.

C V 1 , C I 1 , C V 2 , C I 2, Δ C V 1 , Δ C I 1 , Δ C V 2 , and Δ C I 2 parameters obtained from Fig. 4. Deduction of R s ± Δ R s, R s h ± Δ R s h , n ± Δ n , I lig ± Δ I lig , I sat, using Eqs. (4)–(12), and finally the percentage errors of R s, R s h , n , I lig , and I sat, relative to the original Rs = 1 Ω, Rsh = 1 kΩ, n = 2.5, Ilig = 1 mA, and Isat = 1 μA.

C V 1 , C I 1 , C V 2 , C I 2, Δ C V 1 , Δ C I 1 , Δ C V 2 , and Δ C I 2 obtained from Fig. 4.
Order C V 1 ± Δ C V 1 C V 2 ± Δ C V 2 C I 1 ± Δ C I 1 C I 2 ± Δ C I 2
(−5.6 ± 0.6) × 10−5  (4.7 ± 0.1) × 10−4  (6.92 ± 0.07) × 10−2  (5.07 ± 0.01) × 10−1 
(−5.8 ± 0.9) × 10−5  (4.7 ± 0.2) × 10−4  (6.86 ± 0.09) × 10−2  (5.06 ± 0.02) × 10−1 
(−7.2 ± 0.6) × 10−5  (5.1 ± 0.1) × 10−4  (6.77 ± 0.08) × 10−2  (5.04 ± 0.01) × 10−1 
(−7.7 ± 0.8) × 10−5  (4.9 ± 0.2) × 10−4  (6.75 ± 0.09) × 10−2  (5.04 ± 0.02) × 10−1 
(−5.5 ± 0.6) × 10−5  (4.9 ± 0.1) × 10−4  (6.83 ± 0.08) × 10−2  (5.05 ± 0.02) × 10−1 
(−6.1 ± 0.8) × 10−5  (4.5 ± 0.2) × 10−4  (8.02 ± 0.09) × 10−2  (5.23 ± 0.02) × 10−1 
C V 1 , C I 1 , C V 2 , C I 2, Δ C V 1 , Δ C I 1 , Δ C V 2 , and Δ C I 2 obtained from Fig. 4.
Order C V 1 ± Δ C V 1 C V 2 ± Δ C V 2 C I 1 ± Δ C I 1 C I 2 ± Δ C I 2
(−5.6 ± 0.6) × 10−5  (4.7 ± 0.1) × 10−4  (6.92 ± 0.07) × 10−2  (5.07 ± 0.01) × 10−1 
(−5.8 ± 0.9) × 10−5  (4.7 ± 0.2) × 10−4  (6.86 ± 0.09) × 10−2  (5.06 ± 0.02) × 10−1 
(−7.2 ± 0.6) × 10−5  (5.1 ± 0.1) × 10−4  (6.77 ± 0.08) × 10−2  (5.04 ± 0.01) × 10−1 
(−7.7 ± 0.8) × 10−5  (4.9 ± 0.2) × 10−4  (6.75 ± 0.09) × 10−2  (5.04 ± 0.02) × 10−1 
(−5.5 ± 0.6) × 10−5  (4.9 ± 0.1) × 10−4  (6.83 ± 0.08) × 10−2  (5.05 ± 0.02) × 10−1 
(−6.1 ± 0.8) × 10−5  (4.5 ± 0.2) × 10−4  (8.02 ± 0.09) × 10−2  (5.23 ± 0.02) × 10−1 
Solar cell parameters obtained using Eqs. (4)–(12), using the previous values of C V 1 , C I 1 , C V 2 , C I 2, Δ C V 1 , Δ C I 1 , Δ C V 2 , and Δ C I 2
Order R s ± Δ R s ( ) R s h ± Δ R s h ( ) n ± Δ n I lig ± Δ I lig ( mA ) I sat ( μ A )
1.014 ± 0.005  1055 ± 29  2.68 ± 0.03  0.99 ± 0.1  2.43 
1.001 ± 0.007  1036 ± 41  2.51 ± 0.04  0.995 ± 0.2  1.1 
1.003 ± 0.005  972 ± 25  2.52 ± 0.03  1.01 ± 0.1  1.16 
0.998 ± 0.006  997 ± 31  2.48 ± 0.03  1 ± 0.2  0.93 
0.999 ± 0.006  989 ± 27  2.49 ± 0.03  1 ± 0.1  0.99 
1 ± 0.01  994 ± 51  2.52 ± 0.06  1 ± 0.3  1.15 
Solar cell parameters obtained using Eqs. (4)–(12), using the previous values of C V 1 , C I 1 , C V 2 , C I 2, Δ C V 1 , Δ C I 1 , Δ C V 2 , and Δ C I 2
Order R s ± Δ R s ( ) R s h ± Δ R s h ( ) n ± Δ n I lig ± Δ I lig ( mA ) I sat ( μ A )
1.014 ± 0.005  1055 ± 29  2.68 ± 0.03  0.99 ± 0.1  2.43 
1.001 ± 0.007  1036 ± 41  2.51 ± 0.04  0.995 ± 0.2  1.1 
1.003 ± 0.005  972 ± 25  2.52 ± 0.03  1.01 ± 0.1  1.16 
0.998 ± 0.006  997 ± 31  2.48 ± 0.03  1 ± 0.2  0.93 
0.999 ± 0.006  989 ± 27  2.49 ± 0.03  1 ± 0.1  0.99 
1 ± 0.01  994 ± 51  2.52 ± 0.06  1 ± 0.3  1.15 
Percentage errors of Rs, Rsh, n, Ilig, and Isat, relative to the original values of Rs = 1 Ω, Rsh = 1 kΩ, n = 2.5, Ilig = 1 mA, and Isat = 1 μA
Order Percentage error of R s (%) Percentage error of R s h (%) Percentage error of n (%) Percentage error of I lig (%) Percentage error of I sat (%)
1.36  5.59  7.26  0.95  143.7 
0.11  3.64  0.59  0.53  10.02 
0.27  2.79  0.97  1.08  15.8 
0.2  0.2  0.67  0.04  6.65 
0.08  1.1  0.23  0.17  1.03 
0.29  0.55  0.95  0.28  15.6 
Percentage errors of Rs, Rsh, n, Ilig, and Isat, relative to the original values of Rs = 1 Ω, Rsh = 1 kΩ, n = 2.5, Ilig = 1 mA, and Isat = 1 μA
Order Percentage error of R s (%) Percentage error of R s h (%) Percentage error of n (%) Percentage error of I lig (%) Percentage error of I sat (%)
1.36  5.59  7.26  0.95  143.7 
0.11  3.64  0.59  0.53  10.02 
0.27  2.79  0.97  1.08  15.8 
0.2  0.2  0.67  0.04  6.65 
0.08  1.1  0.23  0.17  1.03 
0.29  0.55  0.95  0.28  15.6 

The Ortiz-Conde et al. method31 was used in all the combinations of P V and p n, and C C V , I was integrated, applying Simpson integration orders of 1, 2, 3, 4, 5, and 6. Afterward, Rs, Rsh, n, Ilig, and Isat were calculated, using Eqs. (4)–(12). Finally, the percentage errors of these calculated Rs, Rsh, n, Ilig, and Isat were obtained, relative to the initial Rs = 1 Ω, Rsh = 1 kΩ, n = 2.5, Ilig = 1 mA, and Isat = 1 μA values, and they are shown in Figs. 5–9, as a function of P V , p n, and the integration order.

FIG. 5.

Percentage errors of R s obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

FIG. 5.

Percentage errors of R s obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

Close modal
FIG. 6.

Percentage errors of Rsh obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

FIG. 6.

Percentage errors of Rsh obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

Close modal
FIG. 7.

Percentage errors of n obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

FIG. 7.

Percentage errors of n obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

Close modal
FIG. 8.

Percentage errors of Ilig obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

FIG. 8.

Percentage errors of Ilig obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

Close modal
FIG. 9.

Percentage errors of Isat obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

FIG. 9.

Percentage errors of Isat obtained using Simpson integration of orders 1 to 6 for (a) 0% noise, (b) 0.001% noise, (c) 0.005% noise, (d) 0.01% noise, (e) 0.05% noise, (f) 0.1% noise, (g) 0.5% noise, and (h) 1% noise of Imax.

Close modal

As can be seen from Figs. 5(a), 6(a), 7(a), 8(a), and 9(a), when the curves are noiseless, the use of order 2 integration or larger drastically diminishes the percentage errors in all the solar cell parameters, compared with the order 1 case, revealing that using 2 to 6 order Simpson integration effectively yields a more accurate C C V , I. It is worth mentioning that the percentage errors are smaller than those obtained using the trapezoidal (order 1) integration reported in Ref. 32, where it was necessary to use P V = 5001 measured points V, to reach percentage errors of 0.05%, 0.001%, 0.01%, and 0.0001%, for Rs, Rsh, n, and Ilig, while, as shown here, same percentage errors can be obtained, using only order 2 integration and P V = 21 measured points V, or less. This shows that using order 2 or larger Simpson integration is a useful tool to obtain the same precision in the solar cell parameters, using a number of measured points two orders smaller, allowing the researchers to use less measurement time, and dealing with smaller amount of data.

The same effect and results can be seen in the case of p n = 0.001% [see Figs. 5(b), 6(b), 7(b), 8(b), and 9(b)], and the curves are practically undistinguishable from the previous case. This suggests that similar results will be obtained for any p n smaller than 0.001%. It is again possible to determine all the solar cell parameters with less than 10%, using a 2-order Simpson integration and just P V = 21 measured points V. Any larger value of p n or P V yields smaller percentage errors.

In the case of p n = 0.005% [Figs. 5(c), 6(c), 7(c), 8(c), and 9(c)], the curves start to appear slightly scattered due to the presence of noise. Nevertheless, it is again possible to determine all the solar cell parameters, except Isat, with less than 10% percentage error, using an order 2 Simpson integration and just P V = 21 measured points V. For the case of Isat, it is necessary to use a value of P V = 61 measured points V or larger, and order 2 or larger Simpson integration, to obtain it with less than 10% percentage error.

Something similar happens for the p n = 0.01% case [Figs. 5(d), 6(d), 7(d), 8(d), and 9(d)]; however, it is necessary to use P V = 101 measured points V, and order 2 or larger Simpson integration, to deduce Isat, with less than 10% percentage error.

Once p n = 0.05% [Figs. 5(e), 6(e), 7(e), 8(e), and 9(e)], a decrease in the percentage errors can be observed when changing from P V = 11 measured points V to P V = 31 measured points V, when using order 2 or larger. The percentage errors do not decrease further once P V = 41 measured points V or larger, and they stay around to 1%, 6%, 5%, 1%, and 50%, for Rs, Rsh, n, Ilig, and Isat, respectively. Then, in this case, it is no possible to obtain Isat, with less than 10% error, but, nevertheless, it is 50%. All the other parameters can be obtained with less than 10%, using order 2 or larger integration, and P V = 41 measured points V or larger. It is worth comparing with the results in Fig. 5(b) in Ref. 32, where it was necessary to use P V = 5001 measured points V to achieve the same results. This again reveals that the using order 2 or larger Simpson integration can drastically reduce the number of measured points by two orders, reducing measurement time, and using smaller amount of data.

Something similar happens in the p n = 0.1% case [Figs. 5(f), 6(f), 7(f), 8(f), and 9(f)]: some decrease in the percentage errors can be seen for when increasing P V from P V = 11 measured points V to P V = 41 measured points V, and using 2 to 6 order integration. For any larger values, the percentage errors converge to approximately 2%, 8%, 8%, 2%, and 70%, for Rs, Rsh, n, Ilig, and Isat, respectively, using order 2 to 6 integration, and P V = 41 measured points V or larger. This is the limit case to obtain Rs, Rsh, n, and Ilig with 10% error or less. Comparing with the same p n case investigated using the trapezoidal integration method in Fig. 5(c) in Ref. 32, it is worth mentioning that in Ref. 32, the percentage errors for Rs, Rsh, n, and Ilig were approximately constant to 5%, 30%, 20%, and 1%, slightly larger compared to what was obtained here (except for Ilig). However, in Ref. 32, they were always around these values, with P V as large as P V = 50 001 measured points V, while in this study, with just P V = 41 measured points V or larger, and order 2 of integration or larger, the same results were obtained, showing again the usefulness to use order 2 or larger integration, with three orders of magnitude less measurement points.

Once p n = 0.5% or p n = 1%, no decrease in the percentage errors is observed, either increasing the integration order from 2 to 6 or increasing P V up to P V = 101 measured points V. The percentage errors stay around 20%, 60%, 60%, 30%, and 2 × 107%, for Rs, Rsh, n, Ilig, and Isat, respectively, in case p n = 0.5%. If p n = 1%, the percentage errors for Rs, Rsh, n, Ilig, and Isat are around 60%, 80%, 250%, 200%, and 2 × 107%, respectively. It is currently being investigated if larger order of Simpson integration, namely, 7, 8, 9 and 10, could decrease this percentage error. This will be reported elsewhere.

In Table III, a summary of the limitations of this methodology presented here, on the precision obtained on each solar cell parameter, is given. The minimum number of measured points and the minimum integration order m needed to obtain the solar cell parameters, with less than a10% percentage error or b1% percentage error, for any integration order equal or larger than m, as a function of the percentage noise, are given.

TABLE III.

Minimum value of PV and its respective minimum value integration order m to obtain the solar cell parameters in each column, with a percentage error of a10% or less, or b1% or less, for any integration order equal or larger than m.

Noise percentage Rs Rsh n Ilig Isat
Ideal (noiseless)  a21, m 2  a21, m 1  a21, m 2  a11, m 1  a21, m 2 
  b21, m 2  b31, m 2  b21, m 2  b21, m 1  bnot possible 
0.001%  a21, m 2  a21, m 1  a21, m 2  a11, m 1  a21, m 2 
  b21, m 2  b31, m 2  b21, m 2  b21, m 1  bnot possible 
0.005%  a21, m 2  a21, m 1  a21, m 2  a11, m 1  a71, m 2 
  b21, m 2  b41, m 2  b51, m 2  b21, m 1  bnot possible 
0.01%  a21, m 2  a21, m 2  a21, m 2  a11, m 1  a101, m 2 
  b21, m 2  b81, m 2  b81, m 2  b21, m 1  bnot possible 
0.05%  a21, m 2  a61, m 2  a41, m 2  a11, m 1  a,bnot possible 
  b81, m 2  bnot possible  bnot possible  b101. m 2   
0.1%  a31, m 2  a101, m 2  a101, m 2  a 11, m 1  a,bnot possible 
  bnot possible  bnot possible  bnot possible  bnot possible   
0.5%  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible 
1%  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible 
Noise percentage Rs Rsh n Ilig Isat
Ideal (noiseless)  a21, m 2  a21, m 1  a21, m 2  a11, m 1  a21, m 2 
  b21, m 2  b31, m 2  b21, m 2  b21, m 1  bnot possible 
0.001%  a21, m 2  a21, m 1  a21, m 2  a11, m 1  a21, m 2 
  b21, m 2  b31, m 2  b21, m 2  b21, m 1  bnot possible 
0.005%  a21, m 2  a21, m 1  a21, m 2  a11, m 1  a71, m 2 
  b21, m 2  b41, m 2  b51, m 2  b21, m 1  bnot possible 
0.01%  a21, m 2  a21, m 2  a21, m 2  a11, m 1  a101, m 2 
  b21, m 2  b81, m 2  b81, m 2  b21, m 1  bnot possible 
0.05%  a21, m 2  a61, m 2  a41, m 2  a11, m 1  a,bnot possible 
  b81, m 2  bnot possible  bnot possible  b101. m 2   
0.1%  a31, m 2  a101, m 2  a101, m 2  a 11, m 1  a,bnot possible 
  bnot possible  bnot possible  bnot possible  bnot possible   
0.5%  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible 
1%  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible  a,bnot possible 

Currently, Simpson integration of orders 7, 8, 9, and 10 are being investigated, together with alternative integration methods, such as Monte Carlo integration and ab initio calculations to enhance the results presented in this article. They will be published elsewhere.

The reader should be aware that these results are valid for the one-diode solar cell model. They are not necessarily valid, when other models are used to model the solar cells, such as two- or multi-diode solar cell models.

In this study, it has been shown that the use of order 2 integration or larger yields percentage errors below 10% for Rs, Rsh, n, and Ilig, in case p n = 0.1%, or smaller, and P V = 41 measured points V, or larger. To determine Isat with less than 10%, it is necessary that p n = 0.01% or smaller, and it is necessary to use P V = 101 measured points V , and order 2 or larger Simpson integration.

Once p n = 0.5% or p n = 1%, all the percentage errors are larger than 20%, for the former case, while they are larger than 60%, for the latter case, independently of order integration from 2 to 6, or P V as large as P V = 101 measured points V. It is presently being studied if the order of integrations of 7, 8, 9, and 10, or Monte Carlo integration and ab initio calculations, could decrease these percentage errors.

It is expected these results will provide solar cell and solar panel researchers a useful tool to obtain accurately and fast the five solar cell parameters, to help them in their research, to produce more efficient photovoltaics devices.

The scientific advice and support by Dr. Daniel Muenstermann, Professor Robert James Young, the Newton Funds (Grant No. ST/P003052/1), and Lancaster University is gratefully acknowledged.

The author has no conflicts to disclose.

Victor Tapio Rangel-Kuoppa: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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