1. Introduction
Perovskite absorbers have been widely used in solar cells due to their suitable optoelectronic properties [
1], such as high carrier mobility [
2], tunable bandgap and long carrier lifetime [
3]. Perovskite solar cells (PSCs) have demonstrated great power conversion efficiency (PCE) improvement in a very fast pace [
4,
5], which nowadays exceeds 25%, overcoming other third-generation technologies [
6]. The planar structure is characterized by ease and low-cost fabrication and consists of a compact electron transport layer (ETL), the absorber, a hole transport layer (HTL) and the metal contacts. Hysteresis phenomenon and stability issues are the most important drawbacks of planar PSCs. The functionality of the ETL, as well as its interface with the perovskite, are crucial for the overall PSCs performance and stability [
7,
8]. Several interface engineering approaches [
9,
10,
11,
12] have been implemented to confront the abovementioned issues. Recently, manganese-based porphyrin was incorporated in planar PSCs [
13] as an electron transport mediator which facilitated the electron transfer, favored the growth of more homogenous perovskite films with larger and better crystallized grains and, thus, resulted in enhanced PCE and improved stability.
There are three common models that have been developed to describe the solar cell: single-diode, double-diode, and triple-diode models. Such models depend on the diodes as their main components, along with some resistors to characterize the properties of the PV solar cell. However, the main issue that needs to be faced is finding the optimal values for individual parameters, such as impedance, diode ideal factors, and saturation currents, in order to obtain successful modeling. The single-diode model is considered the basic one and has a single diode. The development of a more detailed model would be more appropriate to represent a wide range of PV systems considering different conditions [
14]. To increase the accuracy of conventional model, some modified models are developed. Dalia et al. [
14] suggested an improved double-diode model. They added an extra resistor in series with the second diode to consider the influence of grain boundary regions. An improved triple-diode model was therein suggested and compared with the improved double-diode model. The results proved that the accuracy of an improved triple-diode model is better than an improved double-diode model [
15].
In this paper, the modified TDM is examined to model the perovskite solar cells using the bald eagle search (BES) algorithm. Two perovskite-based devices, namely, control and modified, were experimentally fabricated, characterized and tested in the lab. Characteristic I–V parameters of the fabricated devices were recorded at standard condition. A comparison study is presented with other optimization algorithms of grey wolf optimizer (GWO), particle swarm optimizer (PSO), moth flame optimizer (MFO), (SCA) and (SSA), where the superiority of the proposed BES algorithm is proved.
The main contributions of this paper are summarized as follows:
a new application of the bald eagle search algorithm to identify the parameters of the modified TDM for perovskite solar cells;
the obtained results by the bald eagle search algorithm are compared with other methods;
the accuracy and superiority of the bald eagle search algorithm in determining parameters of the modified TDM for perovskite solar cells are proved.
The rest of the paper is arranged as follows:
Section 2 presents the details of the experimental work. The mathematical representation of the modified TDM and problem formulation is presented in
Section 3. A brief overview of the bald eagle search algorithm is explained in
Section 4. The obtained results are presented and discussed in
Section 5. Finally,
Section 6 outlines the main findings of the current research work.
3. Triple-Diode Model and Problem Formulation
Figure 3 shows the TDM equivalent circuit, while
Figure 4 shows the modified TDM equivalent circuit of PV. The mathematical model of both TDM and modified TDM are presented in this section.
From the equivalent circuit shown in
Figure 3, the solar cell output current is represented by the following equation [
14].
where
Ipv denotes the photo-generated current.
ID1,
ID2,
ID3 denote the current in the first, second and third diodes, respectively.
Rs is the series resistance.
Rp denotes the shunt resistance.
The diodes currents can be estimated as shown in Equation (2), based on the Shockley formula as follows:
where
n1,
n2, and
n3 denote the ideality factor of
D1,
D2, and
D3, respectively;
and
are the saturation currents of diodes
D1,
D2, and
D3, respectively.
Vt denotes the thermal voltage. It can be calculated using the following relation.
where
k denotes the Boltzmann constant,
T denotes the temperature of the PV panel,
Ns is the number of series solar cells, and
q denotes the electron charge.
Based on equivalent circuit shown in
Figure 3, the solar cell output current using the TDM can be obtained from the following relation [
14].
To increase the accuracy of the equivalent model, the modified TDM is proposed, where the losses in the defect region are included in the model and expressed via adding the modified series resistance;
Rsm in the third diode branch as shown in
Figure 4, and Equation (4) updated to be as shown in Equation (5):
Coidering Equation (5), the modified TDM model contains ten unknown parameters (Ipv, , , Rs, Rp, Rsm).
The ten unknown parameters of the modified TDM should be estimated accurately. An optimization process is used for obtaining the module parameters, where the root mean square error (
RMSE) between the estimated and experimental data is used as an objective function. The
RMSE is estimated using the following relation and should be minimum.
where
N denotes the number of datasets,
Im denotes the measured current, and
Ie denotes the estimated current.
5. Results and Discussion
The absorption spectra (not shown) of the perovskite deposited on pristine and modified TiO2 confirmed that the ETL modification did not significantly affect the optical properties of the perovskite absorber [
13].
The photovoltaic parameters of the PSCs including the short circuit current density (Jsc), the open circuit voltage (Voc), the fill factor (FF) and the power conversion efficiency (PCE) were determined for both control and modified devices under 1 sun illumination conditions [
13]. The modified PSCs demonstrated significantly higher photovoltaic performance reaching power conversion efficiency (PCE) of 18.7%, clearly outperforming that of the control device (16.9%). This was attributed to better crystalized and more homogenous perovskite films [
17,
18], associated with a smoother and more functional ETL/absorber interface [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34]. The PCE enhancement for the modified devices was associated with the increase of J
sc and FF parameters, while the V
oc value was almost the same (~1.04 V). Specifically, the best-performing modified PSC provided a short circuit current density of 23.52 mA cm
−2, which was 4% larger than that of the reference one. The observed increase was assigned to the presence of the porphyrin’s negative dipole moment at the interface, which facilitated the electrons transfer to the photoanode. An additional improvement was registered in the case of fill factor, where the modified device presented a higher FF value (0.75) compared to the reference one (0.71).
A techno-economic study for the third-generation PSCs in real water pumping system application for irrigation purpose was carried out in [
34], proving the superiority of the PSCs in comparison with the silicon-based technology solar cells where, for the same application, for the PSCsbased case there is a cost-saving and it needs only 3.42 m
2, as the installation area, which is very small in comparison with the corresponding area of conventional silicon-based solar cells that exceeds 36 m
2.
In the current study, the PSCs’ parameters estimation based on TDM and MTDM was made via BES algorithm to prove the enhanced efficiency and precision of the proposed algorithm.
Two PSC devices, namely, control and modified devices experimental data, have been used in this work. The devices were fabricated as explained in the experimental section, while the photovoltaic performance was tested at standard condition (25 °C, 1000 W/m2).
The population size and number of iterations are the same for all optimization algorithms used in this study for a fair comparison. The least
RMSE between the experimental data and the estimated current density of PSCs is the main purpose of the optimization procedure in this study. The parameters of both TDM and MTDM are the decision variables in the used optimization procedure.
Table 1 and
Table 2 summarize the best parameters of the PSC devices based on TDM and MTDM, estimated by different optimization algorithms besides the lower and upper limits of the PSC device parameters. For better evaluation of the proposed BES algorithm performance, the studied optimizers are implemented 30 times. Statistical analysis for the optimization algorithms used in this study is shown in
Table 3 and
Table 4. The details of 30 Runs for the control and modified PSCs recorded with TDM and MTDM are presented in
Table 5,
Table 6,
Table 7 and
Table 8. From
Table 1 and
Table 2, the estimated parameters agree with experimental ones with minor error values, while the best results are obtained by using both the proposed MTDM and the BES algorithm. For PSCs performance better understanding, parameters like diode ideality factors and diodes saturation currents values are very important but difficult to be determined experimentally. An important advantage of the simulation models (TDM, MTDM) is the ability to easily determine these parameters. Thus, the analysis and performance evaluation of the PSCs is further supported.
From both
Table 3 and
Table 4, the superiority of the proposed BES algorithm is evident for all studied cases. It is clear that the proposed BES algorithm has the best performance compared with other optimizers used in this work. From
Table 3, for the control device under the case of TDM, the mean
RMSE values have fluctuated between 1.660 × 10
−5 and 2.039 × 10
−4. The minimum mean
RMSE of 1.660 × 10
−5 is accomplished by the BES algorithm flowed by 3.176 × 10
−5 applying the MFO algorithm. The minimum cost function of 1.379 × 10
−5 is obtained by the BES algorithm followed by MFO and GWO algorithms, respectively. On the other hand, the mean
RMSE values for the modified device have fluctuated between 1.689 × 10
−5 and 2.438 × 10
−4. The minimum mean
RMSE of 1.689 × 10
−5 is accomplished by the BES algorithm flowed by 4.117 × 10
−5 applying the MFO algorithm. The minimum cost function of 1.632 × 10
−5 was obtained by the BES algorithm followed by the MFO algorithm. From
Table 4, for the control device under the case of MTDM, the mean
RMSE values have fluctuated between 1.570 × 10
−5 and 2.557 × 10
−4. The minimum mean
RMSE of 1.570 × 10
−5 is accomplished by the BES algorithm flowed by 2.470 × 10
−5 applying the MFO algorithm. The minimum cost function of 1.379 × 10
−5 is obtained by the BES algorithm followed by MFO and GWO algorithms, respectively. In the other side, for the modified device, the mean
RMSE values have fluctuated between 1.536 × 10
−5 and 2.048 × 10
−4. The minimum mean
RMSE of 1.536 × 10
−5 is accomplished by the BES algorithm flowed by 6.024 × 10
−5 applying the MFO algorithm. The minimum cost function of 1.499 × 10
−5 is obtained by the BES algorithm followed by the MFO algorithm.
The comparison between experimentally obtained and estimated I–V and P–V curves for both control and modified devices used in this study are shown in
Figure 8,
Figure 9 and
Figure 10. The comparison revealed that the experimental obtained and estimated I–V and P–V curves are almost identical, which is an indication of the efficiency of both the suggested model and the optimization algorithm in simulating the performance of the PSCs. The estimated parameters of the PSCs are matched very well with the experimental ones. From previous discussion for
Table 1,
Table 2,
Table 3 and
Table 4, applying BES algorithm in this study for PSCs parameters estimation via both TDM and MTDM provides the best performance results at minimum errors.
The convergence curves through the parameter identification process of both control and modified devices based on both TDM and MTDM per applying different optimizers are summarized, respectively, in
Figure 11 and
Figure 12. From these figures, it is obvious that the proposed BES algorithm performance is the best among the other used algorithms. The fast convergence and the lowest cost function resulted from the BES algorithm in all studied cases, which is proof also of the superiority of the BES in comparison with the other optimizers used in this study.
Further investigation of absolute current density error for both control and modified PSCs devices via using different optimization algorithms applying both TDM and MTDM, as explained in
Figure 13 and
Figure 14, respectively. It is very easy and clear to notice that the proposed BES in both devices, and under both TDM and MTDM, has the highest coefficient of determination and the lowest mean absolute error values. This also proves the accuracy, superiority, and efficiency of the suggested BES algorithm in estimating the best parameters of both control and modified PSCs devices in both TDM and MTDM.
To prove the consistency of the proposed strategy, ANOVA test has been performed. ANOVA test results are provided in
Table 9, and its related graphical ranking is illustrated in
Figure 15. According to these results, the
p-value is much lower than the F value that confirms the difference between the provided performances. From
Figure 15, the BES can provide the optimal performance in terms of mean fitness. Moreover, its variations range is the lowest comparing with the other algorithms, which demonstrates its robustness.