1. Introduction
In recent studies [
1,
2,
3,
4], a large number of optical and electrical tests are required for the production and application of solar cells in order to analyze their performance and reliability, which could be represented by
I-V characteristics. The
I-V curves can be re-expressed by a single-diode lumped-parameter equivalent circuit model [
5,
6] to assist evaluations and modeling for solar cells. In the modelling of solar cells, an accurate extraction of the parameters in a single-diode model helps researchers to define the directions of solar cells’ process optimizations and achieve the implementation of single-diode model in optoelectronic device modeling and circuit design. Therefore, a general parameter estimation procedure is extremely significant for the single-diode model of solar cells to both accurately and validly simulate its
I-V characteristics.
There are various methods [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19] for extracting the parameters of single-diode model. These methods are usually divided into two categories. The procedure of the first category [
7,
8,
9,
10,
11] is to obtain several special points from the
I-V characteristics, such as the open-circuit voltage point, the short-circuit current point, and the voltage and current at the maximum power point, and then perform calculations by directly using these key points. In these schemes, computation efficiency is guaranteed, but nonlinear and implicit equations lead to convergence problems when solving exponential equations. The procedure of the second category [
12,
13,
14,
15,
16,
17,
18,
19] is to use data modeling or curve fitting to extract parameters. These methods require as many points on the curve as possible, and they are time- and labor-consuming due to a number of tests required. Among these approaches, many intelligent algorithms have also been adopted. The intelligent algorithm method [
17,
18,
19] is to obtain the
I-V and
P-V characteristic curves by optimizing and fitting, which can minimize errors. However, the computational efficiency of this method is low. Intelligent algorithms can achieve a comparable accuracy depending on the fitting standards and algorithms. These approach require a good trade-off between accuracy and efficiency. Therefore, it is necessary for single-diode models to accurately and efficiently perform the
I-V curve’s simulation with the parameter extraction procedure.
In this paper, we propose an analytical algorithm for acquiring the parameters of a single-diode model. Firstly, we deduce the expressions of terminal I-V equations, and then substitute the relevant factors obtained from the datasheet into the I-V equations to set up a transcendental equations’ set. Secondly, we derive the analytical solution of model parameters from the set of transcendental equations, including as few simplifications as possible. Thirdly, we utilize the simulation results and experimental data measured from perovskite solar cells and PV panels to verify the accuracy, efficiency, and universality of the parameter extraction algorithm we propose. The results prove that such an algorithm can have an important effect on extracting model parameters in the single-diode lumped-parameter equivalent circuit of solar cells.
2. Algorithm of Extracting Model Parameters in the Single-Diode Lumped-Parameter Equivalent Circuit
A single-diode lumped-parameter equivalent circuit is shown in
Figure 1 [
20]. Here, there are five parameters that need to be defined to simulate the
I-V characteristics of solar cells.
According to Kirchhoff’s current law and Schottky’s diode ideal current equation, the relationship between the terminal current and voltage (
I-V) can be written by [
20]:
Equation (1) is a transcendental function, where I is the output current and V is the output voltage. Iph, Is, a, Rs and Rsh are five parameters that need to be extracted, where Iph is the photovoltaic current source, Is is the reverse saturation current of the diode, a is the ideality factor of the diode, Rs is the series resistance, and Rsh is the parallel resistance.
In order to calculate the {
Iph, Is, a, Rs, Rsh} values, we require at least five equations. On the one hand, we can obtain three equations at three special points in the
I-V characteristic, including the open-circuit voltage point, short-circuit current point and the maximum power point. It is noted that these three points can be found out in datasheet. On the other hand, the slopes of the
I-V curve at these three points can be acquired from the reconstructed experimental results or PV manufacturers’ data [
21,
22]. Notably, these three special points are directly substituted into (1), yielding the following nonlinear equations. At the short-circuit current point (
V = 0,
I =
Isc), Equation (1) becomes:
At the open-circuit voltage point (
V =
Voc,
I = 0), Equation (1) is reformulated as:
At the maximum power point (
V =
Vm,
I =
Im), Equation (1) is rewritten as:
Equations (2)–(4) are three independent equations with five unknown variables:
Iph,
Is,
a,
Rs and
Rsh. Additionally, another three equations could be established according to the slopes of
I-V characteristics at the three simultaneous special points. By differentiating
V with respect to
I in Equation (1), we obtain:
The slopes of the
I-V curve at the short-circuit current and the open-circuit voltage points can be expressed by
Rsc and
Roc, respectively:
By substituting Equations (6) and (7) into (5), the following equations are acquired as:
In addition, the derivative of power to voltage is zero at the maximum power point, so the slope of this point can be obtained as:
Then, by substituting Equation (11) into (5), we obtain:
Six equations—(2), (3), (4), (8), (9), and (12)—form a transcendental equation set, where five model parameters can be solved analytically.
The results are as follows:
where:
and:
where:
and:
3. Verification and Discussion
In this section, numerical simulations and reconstructed experimental data are necessary to validate the parameter estimation algorithm of the solar cell model. In order to verify the accuracy and efficiency of our parameter estimation method, the original I-V curve based on numerical simulations with the settled parameters is contrasted, with the curves simulated by using the estimated parameters. Furthermore, the experimental data from solar cells of the I-V curves are used to validate the universality of our parameter extraction algorithm.
The
I- and
P-V characteristics, based on the numerical simulations with the settled parameters, are shown in
Figure 2 and
Figure 3, respectively. By using our above analytical algorithm of parameter extraction, we can define all five model parameters listed in
Table 1 and reproduce
I- and
P-V curves with the extracted parameters in
Figure 2 and
Figure 3. Here, we notice that
I- and
P-V curves have a good agreement by using the estimated and settled parameters. The absolute errors of the two curves in
Figure 2 are within 2 × 10
−5 A, and those in
Figure 3 are less than 2.5 × 10
−5 W. It is observed that the maximum errors of
P-V characteristics is located near the open circuit voltage point (
Voc, 0). The reason for this is that the coefficient
of
Im is ignored, and the coefficient
of
Vm and
Voc is replaced by
, in the processs of equality simplification. In
Table 1, we can observe that the minimum relative error between the settled and extracted parameters is below 5‰. In
Table 2, it also can be seen that the relevant factors obtained from the original curve and calculated from
I-V curve with the extracted parameters have a high consistency.
Figure 4 and
Figure 5 show the reconstructed experimental data [
20] and simulation results by using the parameters estimated through our method described above.
Figure 6 illustrates the absolute errors in
Figure 4. The estimated parameter values are summarized in
Table 3, and their relevant factors, calculated from our parameter estimation method, are shown in
Table 4, where these factors of the curve obtained by polynomial fitting and the curve solved by the extracted parameter values are also compared.
In reference [
23], prepared CuGaS
2 quantum dots (CGS QDs) were used as hole-transport materials (HTM) for perovskite solar cells (PSCs). Due to the inevitable flaws of inorganic nanoparticles, which could have an influence on the properties of PSCs, this study accurately controlled the Cu content and systematically studied its impact on the photovoltaic performance of solar cells.
First, good agreements are observed in
Figure 4 and
Figure 5. In addition, the absolute error of the two curves in
Figure 6 is within 4 × 10
−4 A. The error curve varies in wave shape. The voltage relationship of the error peak value is consistent with the relationship of the maximum power voltage value. The relationship of the maximum power voltage value from small to large is 0.6:1,1.2:1,0.8:1 (
Cu:Ga, molar ratio) in order, and the voltage relationship corresponding to the error peak value is also 0.6:1,1.2:1,0.8:1 (
Cu:Ga, molar ratio).
Second, the impact of parameters on the curve can also be shown from the curves. The parameter
Iph is very approaching to the short-circuit current
Isc, which represents the intercept with the ordinate axis of the
I-V curve. It can be seen that the order of intercept from large to small is 0.8:1,0.6:1,1.2:1 (
Cu:Ga, molar ratio) in
Figure 4, and the parameter value
Iph also conforms to the law of the image. The parallel resistance value
Rsh and the series resistance value
Rs are consistent with the slope of the curve at
0 V <
V <
Vm and
Vm <
V <
Voc. On the one hand, it can be observed that, when
0 V <
V <
Vm, as the ratio of Cu to Ga, decreases, the slope also decreases, and the value of
Rsh also decreases, as shown in
Table 3. On the other hand, for series resistance
Rs, when
Vm <
V <
Voc the order of slope from large to small is 1.2:1,0.6:1,0.8:1(
Cu:Ga, molar ratio). From
Table 3, it can be seen that the relationship of
Rs is 1.2:1 > 0.6:1 > 0.8:1(
Cu:Ga, molar ratio) similarly. Apart from this, both
Is and
a increase as the
Cu:Ga ratio decreases.
Third, the relevant factors (including Isc, Voc, Pm, Roc, Rsc and FF) of I- and P-V curves acquired by the calculation of extracted parameters are compared with those acquired by experimental curve fitting. It is worth noting that the symbols’ parameters have a good correlation with the lines’ parameters, and their maximum relative error is below 1%. The filling factor FF of the PSCs based on CuCaS2 increases from approximately 70.9209% to 71.2947%, while the Cu content is reduced from 1.2:1 to 0.8:1 (Cu:Ga, molar ratio). Nevertheless, as the Cu content further decreases to 0.6:1, the property of PSCs becomes worse.
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11 and
Figure 12 illustrate the modeled
I- and
P-V characteristics regarding the experimental data for the SM55 PV model [
24] under different irradiance and temperature levels. The results show that the simulation results of the proposed model match closely with the experimental measurements. The absolute errors between modeled and measured results are also shown. The variable is irradiance, which ranged from 200 W/m
2 up to 1000 W/m
2 at a certain temperature of 25 °C, as shown in
Figure 7 and
Figure 8. When the irradiance increases, the short-circuit current
Isc will increase, but the open-circuit voltage
Voc has little change. Furthermore, the accuracy of the proposed algorithm was experimentally evaluated.
Figure 9 shows the absolute errors in
Figure 7. The absolute error of the two curves was below 0.1 A, and the shape of the error change is the waveform. Under fixed-temperature conditions, the voltage values of the maximum power point are almost the same, as well as the positions of the maximum error. Similarly, the variable that is controlled in
Figure 10 and
Figure 11 is temperature, increasing from 20 °C to 60 °C, and the irradiance is kept at a constant value of 1000 W/m
2. The increasing temperature reduces
Voc, and
Isc does not change much.
Figure 12 shows the absolute errors in
Figure 10. The absolute error of the two curves is less than 0.1 A, and the error varies in different waves. Under different temperature conditions, the voltage values of the maximum power point and the voltage values of the maximum error decrease with the increase in temperature. The five parameters are also extracted by using the equations in the previous section, given in
Table 5 and
Table 6. Therefore, the proposed algorithm has a high accuracy and can simulate the
I- and
P-V curves for a PV module under any operating conditions. Under dim indoor light conditions,
RS and
Rsh play quite different roles from those under outdoor sunlight. Under the outdoor condition, the photovoltaic performance is likely to be governed by
RS [
25,
26].