Automatic Algorithm Based on Simpson Seventh-Order Integration of Current Minus Short-Circuit Current: Extracting Photovoltaic Device Parameters Within One-Diode Model

Abstract

Simpson’s seventh-order integration has been implemented in an automatically executable program to integrate the current minus the short-circuit current. Then, a regression of this integral to a second-degree polynomial in two variables, namely the voltage and the short-circuit current, is performed, obtaining six regression constants. The series ($R_s$) and shunt resistance ($R_{sh}$), the ideality factor (n), and the saturation (I_sat) and light current (I_lig) are extracted from these regression constants. The standard errors of these five photovoltaic device parameters are also calculated. $R_s$, $R_{sh}$, n, I_lig, and I_sat can be extracted with less than 1% error when the percentage noise is $p_n<$ 0.05%, with just $N\ge 26$ ($N\ge 101$ for I_sat), in contrast with a value of $N\ge 751$, in the case of the trapezoidal integration method being used. The program calculates the photovoltaic device parameters in less than a second for 1001 data points, four seconds for 10,001 data points, and nineteen seconds for 20,001 data points, which is in striking contrast with the tenths of minutes when using the trapezoidal integration provided by the software Origin, as it has to be performed manually. It is worth mentioning that for the case of $p_n\ge$ 0.1%, both trapezoidal and Simpson seventh-order integration practically yield the same accuracy; nevertheless, the program outstands the trapezoidal integration, as it achieves the extraction in nineteen seconds or less. The results reported in this article are valid for the one-diode solar cell model, and might not be valid for other models.

1. Introduction

The climate change our planet is facing has become a global concern, and the situation is so dramatic that it has been referred to as climate catastrophe [1,2,3,4,5,6,7]. At the same time, the power demand forecast is that it will rise to 30 TW by year 2050 [8,9]. Solar energy, harvested through photovoltaic devices, is a suitable solution to tackle both problems, as it is a large source of nature-friendly free energy [8,9,10,11].

The one-diode photovoltaic model is the one most frequently used, due to its simplicity, as it is the model with the smallest number of parameters, namely just five parameters [12]. These five parameters are as follows: the light current (I_lig), the saturation current (I_sat), the shunt resistance (R_sh), the series resistance (R_s), and the ideality factor (n). The current–voltage (IV) equation and the electric circuit of the one-diode photovoltaic device model are given and commented in the next section. These five parameters provide important information during research, manufacture and production about the photovoltaic devices. I_sat is related to recombination mechanisms, like Shockley–Read–Hall, Auger, or surface recombination [13], whilst I_lig provides information regarding acceptor and donor lifetime and densities [14]. In the case of n = 1, the transport mechanism is minority carrier diffusion; however, in the case of n = 2, the governing transport mechanism is the generation and/or recombination of charge carriers within the depletion region [15]. R_sh is related to the crystal quality, and it is expected to be as large as possible to avoid current loss, whilst R_s is connected to the ohmic contact quality, and it is expected to be as little as possible, as a large value causes a large voltage loss [16,17].

Further details can be found in the literature [13,14,15,16,17].

An examination of the IV equation (see Equation (1)) reveals it cannot be solved for either I or V, complicating the extraction of the five photovoltaic device parameters. Several techniques have been brought forward to obtain them. Some suppositions on one or more of the five photovoltaic device parameters are necessary in some cases [18,19,20,21,22,23,24,25], whilst in other cases, different luminescence and/or maximum power point measurements are needed [18,19,23,26,27,28,29,30,31,32].

Only few methodologies directly process the IV curves independently of any previous assumption or illumination conditions [7,33,34,35]. For example, in [7,34], two methodologies were proposed to extract the five solar cell parameters. These proposals were later incorporated inside two iterative cycles in [34], where the Cheung method, originally suggested to extract R_s and n from a Schottky contact [36], was proved to be appliable to photovoltaic device IV curves [34]. Finally, in 2006, Ortiz–Conde et al. [35] proposed an IV equation for photovoltaic devices, based on the Lambert function. Afterwards, they defined the Co-Content function as $CC\left(V,I\right)= {\int }_0^V\left(I-I_{sc}\right)dV$, ($I_{sc}=I(V=0)$, which refers to the short-circuit current [35]. In their pioneering article, they proved that after performing the regression of $CC\left(V,I\right)$ to ${C_{V0}+C}_{V1}V+C_{V2}{V}^2+C_{I1}\left(I-I_{sc}\right)+C_{I2}{\left(I-I_{sc}\right)}^2+C_{I1V1}V\left(I-I_{sc}\right)$, the elements I_sat, n, I_lig, R_s, and R_sh can be deduced from $C_{V0}$, $C_{V1}$, $C_{V2}$, $C_{I1}$, $C_{I2},$ and $C_{I1V1}$ [35].

When the Ortiz–Conde et al. [35] method was applied to the ideal, i.e., noiseless IV curves, with a percentage noise $p_n=0\%$, and a density of measured points per voltage $P_V=1000 {\displaystyle \frac{measured points}{V}}$ (see Section 3 in [7]), outstanding photovoltaic device parameter extraction was achieved [7]. However, when it was implemented in CIGS- and CdTe-based photovoltaic-device-measured IV curves, unrealistic parameter extraction occurred due to the limiting presence of noise (see Section 4 in [34]). This reveals that the correct calculation of $CC\left(V,I\right)$ is a pivotal step in this methodology, as the integration strongly depends on $P_V$ and $p_n$. Further research revealed that the adverse effect of the noise can be reduced in some cases, increasing the value of $P_V$ [37]. In those articles [37], $CC\left(V,I\right)$ was calculated by applying the trapezoidal integration technique and escalating $P_V$ minimized the adverse effect of the noise, providing a more faultless $CC\left(V,I\right)$ in some cases; hence, the photovoltaic device parameter obtention was more accurate. A large weakness of escalating $P_V$, is that values as big as 50,001 ${\displaystyle \frac{measured points}{V}}$ are necessary, causing computation and time analysis to vastly increase. Alternative solutions to deal with this issue were explored: the application of the Newton–Cotes quadrature formula, the 3/8 rule, and Simpson’s integration methods were explored to reduce the adverse effect of noise [38], while in other reports, the $CC\left(V,I\right)$ was computed by first performing a polynomial fitting of $I-I_{sc}$, and then integrating it [39]. Enhancement was obtained using these suggestions; nevertheless, software like Labview and Origin are needed to implement them, leading to researchers needing to spending money on these software licenses and to dedicate time to implement them. Thus, it would be very convenient for the photovoltaic device research community to be supplied with an executable program that automatically, quickly, and accurately deduced the five photovoltaic parameters and their standard deviations, together with any matrixes and vectors involved in the computation, from the photovoltaic device IV curves.

This short discussion explains the reason of this research: to supply the photovoltaic device research community with an automatically executable program that extracts from the IV curves the five photovoltaic parameters and any other valuable information, based on the Ortiz–Conde et al. method [35], and using the Simpson seventh-order integration method.

It is worth mentioning here that the Ortiz–Conde et al. method was designed for the one-diode solar cell model [35]. This is the model used in second-generation solar cells. However, in third-generation solar cells, like perovskites- and kesterites-based solar cells, a drift–diffusion model is used [40,41,42,43,44,45,46,47]. Further information about this model can be found in the literature [40,41,42,43,44,45,46,47].

The structure of this publication is as follows. Section 2 briefly discusses the one-diode photovoltaic device model and the Ortiz–Conde et al. method [35]. In Section 3, the theory of the polynomial fitting of $CC\left(V,I\right)$ to ${C_{V0}+C}_{V1}V+C_{V2}{V}^2 {+C}_{I1}\left(I-I_{SC}\right)+C_{I2}{\left(I-I_{SC}\right)}^2+C_{I1V1}V\left(I-I_{SC}\right)$ is discussed. Results are then revealed in Section 4, and finally the conclusions are summarized in Section 5.

2. Overview of the Ortiz–Conde et al. Method

For completeness purposes, first, the one-diode photovoltaic model circuit and the IV equation of the one-diode photovoltaic model are given in Figure 1 and in Equation (1) [7]. An overview of the Ortiz–Conde et al. method [35] is given afterwards.

$$I=I_{sat}\left(exp\left({\displaystyle \frac{V-I{R}_s}{nkT}}\right)-1\right)+{\displaystyle \frac{V-I{R}_s}{R_{sh}}}-I_{lig}$$

(1)

Ortiz–Conde et al. proposed, in 2006, the Co-Content $CC\left(V,I\right)$ function as follows [35]:

$$CC\left(V,I\right)= {\int }_0^V\left(I-I_{sc}\right)dV,$$

(2)

where $I_{sc}=I\left(V=0\right)$ is the short-circuit current.

In their study, they proved that after performing polynomial fitting of Equation (2) to the next second-degree polynomial in two variables, namely $V$ and $I-I_{SC}$,

$$C_{V0}+C_{V1}V+C_{V2}{V}^2+C_{I1}\left(I-I_{SC}\right){+C}_{I2}{\left(I-I_{SC}\right)}^2+C_{I1V1}V\left(I-I_{SC}\right),$$

(3)

R_s, R_sh, n, and I_lig parameters could be extracted from the regression constants $C_{V1}$, $C_{I1}$, $C_{V2}$, and $C_{I2}$ using the following [35]:

$$R_{sh}={\displaystyle \frac{1}{2C_{V2}}},$$

(4)

$$R_s={\displaystyle \frac{A-1}{4C_{V2}}},$$

(5)

$$n={\displaystyle \frac{C_{V1}\left(A-1\right)+4C_{I1}{C}_{V2}}{4V_{th}{C}_{V2}}}, \mathrm{a}\mathrm{n}\mathrm{d}$$

(6)

$$I_{lig}=-{\displaystyle \frac{\left(1+A\right)\left(C_{V1}+I_{sc}\right)}{2}}-2C_{I1}{C}_{V2},$$

(7)

where $V_{th}=kT$ is the thermal velocity, whilst $T$ the absolute temperature, $k$ the Boltzmann constant, and $A=\sqrt{1+16C_{I2}{C}_{V2}}$.

I_sat is then computed by evaluating the following:

$$I_{sat}=B\left(I+I_{lig}-{\displaystyle \frac{V-I{R}_s}{R_{sh}}}\right),$$

(8)

at the largest possible $V$ (see Section 4 in [7] for explanation).

The function $B$ is given by $B={\displaystyle \frac{1}{exp\left(\right(V-I{R}_s)/nkT)-1}}$.

The expressions for the first-order errors of R_s, R_sh, n, and I_lig, i.e., $∆R_s$, $∆R_{sh}$, $∆n$, and $∆I_{lig}$, as function of the standard deviations of $C_{V0}$, $C_{V1}$, $C_{V2}$, $C_{I1}$, $C_{I2}$, $C_{V1I1}$, i.e., $∆C_{V0}$, ${∆C}_{V1}$, ${∆C}_{V2}$, ${∆C}_{I1}$, $∆C_{I2}$, and $∆C_{V1I1}$ were deduced in Section 3.2 in [34], and are given by the following:

$${∆R}_{sh}=-{\displaystyle \frac{∆C_{V2}}{2C_{V2}^2}},$$

(9)

$$∆R_s={\displaystyle \frac{1}{2}}\left\{\left({\displaystyle \frac{A-\left(1+8C_{V2}{C}_{I2}\right)}{A{C}_{V2}^2}}\right)∆C_{V2}+{\displaystyle \frac{8∆C_{I2}}{A}}\right\},$$

(10)

$$∆n=\left({\displaystyle \frac{A-1}{4V_{th}{C}_{V2}}}\right)∆C_{V1}+{\displaystyle \frac{C_{V1}\left(A-\left(1+8C_{V2}{C}_{I2}\right)\right)}{4V_{th}A{C}_{V2}}}∆C_{V2}+{\displaystyle \frac{1}{V_{th}}}∆C_{I1}+{\displaystyle \frac{2C_{V1}}{A{V}_{th}}}∆C_{I2},$$

(11)

$$∆I_{lig}=-\left({\displaystyle \frac{1+A}{2}}\right)∆C_{V1}-\left\{{\displaystyle \frac{4C_{I2}\left(C_{V1}+I_{sc}\right)}{A}}+2C_{I1}\right\}∆C_{V2}-2C_{V2}∆C_{I1}-{\displaystyle \frac{4\left(C_{V1}+I_{sc}\right)C_{V2}}{A}}∆C_{I2},$$

(12)

while the first error of I_sat, i.e., $∆I_{sat}$, was deduced in Section 2 in the following [38]:

$$\begin{gathered} ∆I_{sat}=B\left\{1+{\displaystyle \frac{R_s}{R_{sh}}}-\left({\displaystyle \frac{R_s}{nkT}}\right)\left\{B+1\right\}\left\{I+I_{lig}-{\displaystyle \frac{V-I{R}_s}{R_{sh}}}\right\}\right\}∆I+B∆I_{lig}+BI\left\{{\displaystyle \frac{1}{R_s}}- \\ \left({\displaystyle \frac{B+1}{nkT}}\right)\left(I+I_{lig}-{\displaystyle \frac{V-I{R}_s}{R_{sh}}}\right)\right\}∆R_s+B\left({\displaystyle \frac{V-I{R}_s}{{R_{sh}}^2}}\right)∆R_{sh}-B\left(B+1\right)\left({\displaystyle \frac{I{R}_s}{n^{2}kT}}\right)\left(I+I_{lig}-{\displaystyle \frac{V-I{R}_s}{R_{sh}}}\right)∆n, \end{gathered}$$

(13)

where $∆I$ is the amperemeter precision.

The application of the Ortiz–Conde et al. method [35] is given in the next section.

3. IV Computation and Their Mathematical Analysis

The IV simulated curves used in this article were built using the C program reported in [7]. The photovoltaic device parameters used in this article were $n=2.5$, R_sh = 1 kΩ, R_s = 1 Ω, I_lig = 1 mA, and I_sat = 1 µA, in the [0 V, 1 V] voltage range. R_sh, R_s, I_sat, and I_lig are powers of 10 to facilitate comparison with the extracted photovoltaic device parameters from the program; at the same time, in general, second-generation laboratory-made photovoltaic devices have this tendency in these parameters: R_sh and I_lig are three orders of magnitude larger than R_s and I_sat, respectively [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69]. As a first step, the IV curves were simulated noiseless, i.e., $p_n=0\%$, and then, a second group of IV curves was built, adding $p_n=0.001\%$, 0.005%, 0.01%, 0.05%, 0.1%, and 0.5% of $I_{max}=I\left(V=1V\right)=209.61 \mathrm{mA}$ to evaluate the accuracy of the program, extracting the photovoltaic device parameters IV curves. The expression for adding the noise is as follows:

$$I_{with noise}=I_{without noise}+r\times p_n\times I_{max}$$

(14)

where $r$ is a random number between −1 and 1. An example of some of the IV curves, with a $p_n=$ 0.1% of $I_{max}$ is shown in Figure 2.

Let us assume that we deal with an IV data set of N data pairs, i.e., $\{\left(V_i, I_i\right) with i=0, 1, \dots , N-1\}$. In this manuscript, the first data point is indexed as zero, meaning that the last data point is the N − 1 point.

The first IV data point is $\left(V_0=0 V, I_0=I_{sc}\right)$, and then the program calculates $I_{sc}$ as this first I data point, and, at the same time, the program computes the voltage step, i.e., $∆V$ as $∆V=$ $V_1- V_0$. This explains why one of the conditions of the IV data text file is that all the voltage points must be equally spaced (see text below for further details), as the program calculates $∆V$ using this expression, and presumes it to be of constant value through the whole computation. In following step, the program calculates $I-I_{sc}$ as ${\left( I-I_{sc}\right)}_i=I_i-I_{sc}$.

The program then computes $CC\left(V,I\right)$. For ${CC}_0$, its value is assigned to zero, as it is the integral form 0 V to 0 V (see Equation (2)):

$${CC}_0=0.$$

(15)

The next value, ${CC}_1$, is calculated using the trapezoidal integration formula [70]:

$${CC}_1={CC}_0+\left({\displaystyle \frac{{\left( I-I_{sc}\right)}_1 + {\left( I-I_{sc}\right)}_0 }{2}}\right)∆V,$$

(16)

while the Newton–Cotes quadrature formula [70] is used to calculate ${CC}_2$:

$${CC}_2={CC}_0+ {\displaystyle \frac{∆V}{3}}\left({\left( I-I_{sc}\right)}_2+4{\left( I-I_{sc}\right)}_1+{\left( I-I_{sc}\right)}_0\right),$$

(17)

It continues, calculating ${CC}_3$ using the 3/8 integration rule [70]:

$${CC}_3={CC}_0+ {\displaystyle \frac{3∆V}{8}}\left({\left( I-I_{sc}\right)}_3+3{\left( I-I_{sc}\right)}_2+3{\left( I-I_{sc}\right)}_1+{\left( I-I_{sc}\right)}_0\right),$$

(18)

The Boole’s formula [70] is used to compute ${CC}_4$:

$$\begin{gathered} {CC}_4={CC}_0+ {\displaystyle \frac{2∆V}{45}}\left(7{\left( I-I_{sc}\right)}_4+32{\left( I-I_{sc}\right)}_3+12{\left( I-I_{sc}\right)}_2+{32\left( I-I_{sc}\right)}_1+ \\ 7{\left( I-I_{sc}\right)}_0\right), \end{gathered}$$

(19)

and for the ${CC}_5$ and ${CC}_6$ cases, the Simpson fifth- and sixth-order integration methods [70] are used, respectively, as follows:

$$\begin{gathered} {CC}_5={CC}_0+ {\displaystyle \frac{5∆V}{288}}\left(19{\left( I-I_{sc}\right)}_5+75{\left( I-I_{sc}\right)}_4+50{\left( I-I_{sc}\right)}_3+{50\left( I-I_{sc}\right)}_2+ \\ 75{\left( I-I_{sc}\right)}_1+19{\left( I-I_{sc}\right)}_0\right),\mathrm{a}\mathrm{n}\mathrm{d} \end{gathered}$$

(20)

$$\begin{gathered} {CC}_6={CC}_0+{\displaystyle \frac{∆V}{140}}\left\{41{\left( I-I_{sc}\right)}_6+216{\left( I-I_{sc}\right)}_5+27{\left( I-I_{sc}\right)}_4+ \\ 272{\left( I-I_{sc}\right)}_3+27{\left( I-I_{sc}\right)}_2+216{\left( I-I_{sc}\right)}_1+41{\left( I-I_{sc}\right)}_0\right\}, \end{gathered}$$

(21)

Finally, the subsequent ones are calculated using the Simpson seventh-order integration [70]

$$\begin{gathered} {CC}_i={CC}_0+{\displaystyle \frac{7∆V}{17280}}\left\{751{\left( I-I_{sc}\right)}_i+3577{\left( I-I_{sc}\right)}_{i-1}+1323{\left( I-I_{sc}\right)}_{i-2}+ \\ 2989{\left( I-I_{sc}\right)}_{i-3}+2989{\left( I-I_{sc}\right)}_{i-4}+1323{\left( I-I_{sc}\right)}_{i-5}+ \\ 3577{\left( I-I_{sc}\right)}_{i-6}+751{\left( I-I_{sc}\right)}_{i-7},\right\} \end{gathered}$$

(22)

for values $i=7, \dots , N-1$.

A numerical example on the implementation of this integration procedure can be found in the Supplementary Material, where the polynomial fitting of $CC\left(V,I\right)$ to $C_{V0}+C_{V1}V+C_{V2}{V}^2+C_{I1}\left(I-I_{sc}\right)+C_{I2}{\left(I-I_{sc}\right)}^2+C_{I1V1}V\left(I-I_{sc}\right)$, to obtain $C_{V0}$, $C_{V1}$, $C_{V2}$, $C_{I1}$, $C_{I2},$ and $C_{I1V1}$, is explained. At the same time, a diagram flow of the program, as the numerical computational steps, can be found in the Supplementary Material, together with a summary of the conditions the IV data text file should have to be properly read by the program.

The program was run in IV data files containing 8, 9, 10, 11, 19, 26, 38, 51, 64, 76, 101, 251, 501, 751, 1,001, 2,501, 5,001, 7,501, 10,001, 12,501, 15,001, 17,501, and 20,001 data points for the no noise case, and $p_n=0.001\%,0.005\%,0.01\%$, 0.05%, and 0.1% of $I_{max}$ added noise, provided in the Supplementary Material. The results were compared to those obtained using the trapezoidal integration method in the following section to study the accuracy of the five deduced photovoltaic device parameters.

The computational time the program requires for each of the previous number of data points is worth mentioning here. The program delivers the total computational time in the Results files (see Supplementary Material for further details), and when plotting it as a function of $N$, a parabolic relation expression, namely $computational time=0.04+\left(-1.2\times {10}^{-4}\right)\times N+\left(5.4\times {10}^{-8}\right)\times {\left(N\right)}^2,$ is obtained, in case the user does not request the vector and matrixes to be saved. This means that the computational time is around four and nineteen seconds, when $N=10001$ and $N=20001$ data points, respectively. However, in case the user decides to save the vectors and matrices related to the computational steps (see Supplementary Material for further details), the total computational time also shows a parabolic relation as function of $N$, and in this case, $computational time=3.4+\left(-6\times {10}^{-3}\right)\times N+(7.2\times {10}^{-6})\times {\left(N\right)}^2$, which means that for the cases $N=10001$ and $N=20001$, the total computational time is 664 s = 11 min and 2764 s = 46 min, increasing abruptly compared with the former case. The user should be aware of the large increase in computational time when using the program, in case she/he decides to save the computational matrixes and vectors used by the program. Nevertheless, the duration of four and nineteen seconds, in case the user does not save the vectors and matrixes, is in striking contrast with the tenths of minutes required when implementing the trapezoidal integration method, as it has to be performed manually using software like Origin or Excell.

4. Results and Discussion

A compendium of the implementation of the program CCSimpsOrd7.exe is revealed in Table S1, available in the Supplementary Material, showing the $C_{V0} \pm {∆C}_{V0},$ $C_{V1}\pm {∆C}_{V1}, {C_{V2}\pm {∆C}_{V2},C}_{I1}\pm {∆C}_{I1}, C_{I2}\pm ∆C_{I2}$ and $C_{V1I1}\pm {∆C}_{V1I1}$ together with the calculated $R_s\pm ∆R_s$, $R_{sh}\pm ∆R_{sh}, n\pm ∆n, I_{lig}\pm ∆I_{lig}, I_{sat}\pm ∆I_{sat}$ in the case of noiseless IV curves.

The percentage errors, relative to the original values of $n=2.5$, R_s = 1 Ω, R_sh = 1 kΩ, I_sat = 1 µA, and $I_{ph}$ = 1 mA of the results reported in Table S1, are plotted in Figure 3 in green colour. Also, for comparison purposes, the percentage errors using the trapezoidal integration method provided by the software Origin are shown in blue colour. In Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, the percentage errors for the cases of $p_n=0.001\%,0.005\%,0.01\%$, 0.05%, 0.1%, and 0.5% of $I_{max}$ are exposed, respectively.

As seen from Table S1 and Figure 3, reasonable parameter extraction was achieved using the seventh-order Simpson integration method in the case of noise case, obtaining the five solar cell parameters with less than 1% error, and with just $N\ge 26$; in the case where the trapezoidal integration method was used, at least $N\ge 251$ was needed to extract all the five photovoltaic device parameters. Something similar happens in the case of $p_n=$0.001%, 0.005%, 0.01%, and 0.05 of $I_{max}$, for $R_s$, $R_{sh}$, n, and I_lig (see Figure 4, Figure 5, Figure 6 and Figure 7): they can be extracted with 1% error or less, with just $N\ge 26$; however, when it comes to I_sat, it is necessary to have at least $N\ge 101.$ For these $p_n$ cases, in case the trapezoidal integration method is used, it is necessary that $N\ge 751$ to deduce all the five solar cell parameters with less than 1% error. Clearly, for these values of $p_n$, the seventh-order integration method clearly yields more accurate values, with smaller values of N.

For the case of $p_n=$0.1% and 0.5% of $I_{max}$ (see Figure 8 and Figure 9), the seventh-order Simpson integration method does not give any benefit compared to using the trapezoidal integration method, and the percentage errors of $R_s$, $R_{sh}$, n, I_lig, and I_sat converge to 3%, 7%, 20%, 7%, and 100% in the case of $p_n=$0.1%, and to 12%, 70%, 180%, 30%, and 100% in the case of $p_n=$0.5%, for values of $N\ge 251$. In general, the trapezoidal integration method and the seventh-order Simpson integration yield the same results for $N\ge 501$, which suggest that 501 is the optimum number of points.

The reader should notice that results reported in this article are valid for the one-diode photovoltaic device model. They are not necessarily valid for other models.

5. Conclusions

The seventh-order Simpson integration method has been implemented in an executable program to more accurately compute the Co-Content function, and to extract with more precision the five photovoltaic device parameters, together with their standard deviations. It has been tested in ideal (noiseless) IV curves and IV curves with $p_n=$0.001%, 0.005%, 0.01%, 0.05%, 0.1%, and 0.5% noise of the maximum current. Excellent parameter extraction is obtained for $R_s$, $R_{sh}$, n, and I_lig, for the cases of $p_n<$0.05%, with less than 1% error, with just $N\ge 26$ for $R_s$, $R_{sh}$, n, and I_lig, and $N\ge 101$ for I_sat, respectively, in contrast with a value of $N\ge 751$, in case the trapezoidal integration method is used. For the case of $p_n=$ 0.1% and 0.5%, it is not possible to extract the five photovoltaic device parameters with less than 1% error: when $p_n\ge$0.1%, both trapezoidal and Simpson seventh-order integration practically yield the same accuracy; nevertheless, the program still outstands the trapezoidal integration regarding computational time, as it achieves the extraction in nineteen seconds or less, while the trapezoidal integration method has to be implemented manually in software like Origin, requiring at least tenths of minutes.

In future studies, other integration methods, such as Gauss quadrature, Monte Carlo integration, or higher-order Simpson integration, should be investigated. It is expected that higher-order Simpson integration, such as eighth-order, should yield a more accurate calculation of the Co-Content function and more accurate photovoltaic device parameter extraction.

The reader should be aware that the results reported in this article are valid for the one-diode solar cell model and might not be valid for other models.

It is expected that the program provided in this article will provide the photovoltaic solar energy research community with a useful tool to obtain the five photovoltaic device parameters in a fast and precise way, which will help them to research and obtain more efficient photovoltaics devices.