114 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

Analytical Methods for Mathematical Modeling of Dye-Sensitized Solar 
Cells (DSSCs) Performance for Different Local Natural Dye 

Photosensitizers  

 

 J. B. Yerima1,a, S. C Ezike1, Dunama William2, and Alkali Babangida3  

1Department of Physics, Modibbo Adama University Yola, Nigeria 

2Department of Mathematics, Modibbo Adama University Yola, Nigeria 

3Department of Science Education, COE Azare, Bauchi State, Nigeria 

aEmail: bjyerima@gmail.com 
 
Abstract. In this paper, a new approach to generate the modified ideal diode factor of solar cells was 
developed which overcomes the problem of assuming its value a constant. Five models were employed to 
calculate the five-model parameters of one standard solar cell and fourteen DSSCs with varying 
photosensitizers. The results exhibit the conversion efficiencies of the solar cells studied lies in the range 
2.57% ≤ η ≤ 0.03%.  In particular, the standard cell has the highest efficiency 3.02% followed by DSSCs 
with photosensitizers: bitter gourd (2.57%), mango (1%), and bougainvillea (0.83%). Also, the five model 
parameters calculated are all positive for El Tayyan model and the rest of the models show discrepancies 
of varying degrees. Furthermore, despite the existence of these discrepancies, the results reveal good fit 
between the model data and experimental data I-V curves.  This suggests the tendency or possibility that 
irregular parameters may be desirable for some applications. Thus, the discrepancies found in the 
estimated parameters can serve as a vital assessment criterion and tool for researchers and engineers in 
selecting the appropriate parameter estimation method for their applications. 
 

Keywords: Analytical methods, Mathematical modeling, DSSCs, Photosensitizers, Irregular 
parameters, Conversion efficiency. 

1. Introduction 

The application of renewable energy is becoming more popular in modern societies and among 
the various sources, photoelectric energy is one of high demand in terms of increase of installed 
power. In addition, many specific applications such as satellites and spacecraft have motivated 
researchers to study the characteristics of photovoltaic cells, how to improve their power 
generation in the last three decades, and describe mechanisms that control the conversion of 
solar radiation into electric power [1-6]. Also, tremendous efforts have been made to obtain 
equivalent electrical/mathematical models to explain the behavior of solar cells under various 
conditions such as different radiation levels, photosensitizers, and cell temperatures. 

An electrical model consists in a simple electric circuit whose behavior matches the real behavior 
of solar cell [29]. The application of circuit models together with relevant electric parameters is 
very crucial to optimize the power derived from the cell working under real conditions.   
Furthermore, the application of equivalent circuit models makes the simulation of more 
complicated power systems that include solar cell panels possible. In practice, in space 
applications these power complicated systems include batteries and programmed power 
consumption, with important temperature gradients and different radiation levels affecting the 



  
 
 
 
 

 

115 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

output voltage of the solar cell/panel and must be maximized to ensure the survival of the 
spacecraft. 
 
The photovoltaic effect leads to the conversion of sunlight radiation falling on solar cells into 
usable electric energy. In general, the simplest way to characterize a solar cell is by considering 
a current source connected in parallel to an ideal diode. The current-voltage equation that 
explains the theory behind the behavior of the solar cell is called Shockley’s ideal diode equation 
given by  

     𝐼 = 𝐼𝑝ℎ − 𝐼𝑜 (𝑒
𝑉

𝑛𝑉𝑇 − 1)     (1) 

 
where the first term Iph, is the photocurrent delivered by the constant current source, the second 
term is the ideal recombination current from the diffusion and recombination of electrons and 
holes in p-n junction sides of the cell, Io is the reverse saturation current corresponding to it, VT is 
the thermal voltage, and n is the diode ideality factor. The thermal voltage VT is defined by 
 

      𝑉𝑇 =
𝑘𝑇

𝑞
      (2)  

where T is the absolute temperature, q is the charge of the electron, and k is the Boltzmann’s 
constant. 
 

 

Figure 1. Electrical equivalent circuit of the single-diode solar cell 

 
In order to modify equation (1) to better fit the solar cell behavior, shunt resistor Rsh and series 
resistor Rs are usually added to the circuit (Fig.1). The shunt resistor is connected in parallel with 
the source and diode and it represents the current leakage through the high conductivity shunts 
across the p-n junction while the series resistor is added in series and it represents the losses in 
cell solder bonds, interconnection, junction box, and so on [4, 7]. Also, a dimensionless constant, 
n, called ideality/quality factor or emission coefficient is added to the term of the recombination 
current in the p- and n-sides and it takes into account the deviation of the diodes from the Shockley 
diffusion theory. The value of this factor depends on the ratio between current I and voltage V of 
the cell (10). The one-diode and two-resistors circuit model is then defined by equation (3) 



  
 
 
 
 

 

116 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

 𝐼 = 𝐼𝑝ℎ − 𝐼𝑜 [𝑒𝑥𝑝(
𝑉+𝐼𝑅𝑠

𝑛𝑉𝑇
) − 1] −

𝑉+𝐼𝑅𝑠

𝑅𝑠ℎ
       (3) 

 
The modified form a of the ideal factor n of the diode is defined by equation (4) 
 

 𝑎 = 𝑛𝑁𝑠𝑉𝑇 = 𝑛𝑁𝑠
𝑘𝑇

𝑞
         (4) 

 
where Ns is the number of solar cells connected in series i.e. Ns=1 for single-diode solar cell.  
 
In another vein, it has been reported by some authors [8, 9] that the easiest and more commonly 
used 1-diode/2-resistors model exactly reveals the behavior of the solar cell around the maximum 
power point, that is, at high voltage levels. This model simplifies the study of the solar cell behavior 
as a function of the various circuit variables [10-13], and has been employed to examine the effect 
of the irradiance and the temperature on the cell behavior.  
 
Once the circuit model has been selected to investigate a specific solar cell, it must be modified, 
that is, the value of the circuit parameters must be calculated as accurately as possible. These 
calculations can be based on calibration results of the cell such that once the I-V curve are 
obtained under definite irradiance and temperature conditions in a laboratory, the parameters of 
the model can be adjusted to give the best possible fit to this curve [14, 18-23, 46]. Nevertheless, 
sometimes the only data available to adjust the chosen circuit model comes from the 
manufacturer and it is restricted to only certain points on the I-V graph (short-circuit, open circuit, 
and maximum power points) [10, 16, 23-28]. In a nut shell, with respect to the existing methods 
to adjust the parameters of the chosen circuit model, some of them are numerical [11, 12, 15, 
29,30, 46] while some others are analytical [2, 31-33, 45]. 
 
Analytical methods are preferred because they are simple and fast. Obviously, these methods 
usually rely on experimental behavior of the I-V curve, that is, they demand wide range of testing 
results [34]. Alternatively, some researchers have established numerical methods to adjust the 
electric circuit parameters to the listed characteristic point of the curve [16, 17, 22, 28]. This 
technique is quite motivating, as it requires only a few data to allow final users to explain the 
performance of photoelectric devices.  
 
In this paper, analytical methods for photovoltaic equivalent electric circuit parameters extraction 
from experimental observations are reported. These approaches are based only on the points on 
the I-V curve (short-circuit, open circuit, and maximum power points) for modeling DSSCs 
performance for different local natural dye photosensitizers. The approach to the experimental 
parameter extraction problem for DSSC systems studied does not seem to have been studied as 
yet.  

2. Modelings 

 
2.1 El Tayyan Model 

The El Tayyan model [35] for generating the I-V characteristics of a solar cell or PV module is 
given by  



  
 
 
 
 

 

117 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

 𝐼 = 𝐼𝑠𝑐 − 𝐶1⁡𝑒
−
𝑉𝑜𝑐
𝐶2 (𝑒

𝑉

𝐶2 − 1)        (5) 

 
where C1 and C2 are coefficients of the model equation with units of current and voltage 
respectively. Using the short-circuit (SC), open-circuit (OC) and maximum power point (MPP) 

conditions and assuming 
𝑉𝑜𝑐

𝐶2
≫ 1⁡𝑎𝑛𝑑⁡

𝑉𝑚𝑝

𝐶2
≫ 1 yields  

 

 𝐶1 =
𝐼𝑠𝑐−𝐼𝑚𝑝

𝑒

𝑉𝑚𝑝−𝑉𝑜𝑐
𝐶2

          (6) 

 

 𝐶2 =
𝑉𝑚𝑝−𝑉𝑜𝑐

𝑊−1((1−
𝑉𝑜𝑐
𝑉𝑚𝑝

)(
𝐼𝑚𝑝

𝐼𝑠𝑐
))

         (7) 

 
where W-1 is the lower branch of the Lambert W function. Substituting the values of the currents 
and voltages at the characteristic points SC(Isc, 0), OC(0, Voc) and MPP(Imp, Vmp) on the I-C curve 
in equations (6) and (7) the values of C1 and C2 can be determined. Also, substituting the values 
of C1 and C2 in equation (5) yields the desired El Tayyan empirical model for a single-diode solar 
model. Subsequently, for any given value of voltage V, the output current I in equation (5) can be 
calculated. 
 
It is worth noting that since the El Tayyan empirical model has only two coefficients, C1 and C2, it 
means it is a two-parameter model. This fact can be established by comparing equations (1) and 
(5) which yields the two parameters Io and a of the five model parameters of a single-diode solar 
cell given by 
 

 𝐼𝑜 = 𝐶1𝑒
−
𝑉𝑜𝑐
𝐶2           (8) 

 
 𝐶2 = 𝑛𝑉𝑇 = 𝑎          (9) 
 
The validity of equations (8) and (9) can be justified. Thus, equation (8) is valid since Io has same 
units of current with C1 as the exponent is dimensionless whereas equation (9) is valid since it is 
equal to equation (4). The modified ideality factor a of the diode is not usually available on the 
manufacturer’s datasheet and is not easily deduced on the I-V curve which makes some authors 
[39, 40] to guess its value and others [41-44, 45] reduce their models to less than five parameters. 
The advantage of the El Tayyan model over other existing models is that it doesn’t require the 
knowledge of the internal PV system parameters and involve less computational efforts or extra 
measurements. It is against this background, El Tayyan model combined with two models [39, 
40] that assumed constant a produced two additional models, El Tayyan-Cubas model and El 
Tayyan-Senturk model. Therefore, in this work, five models are used to calculate the five model 
parameters (a, Iph, Io, Rs, Rsh) to generate model I-V curves for fifteen DSSCs for comparison 
purposes.  
 
 
 
 
 



  
 
 
 
 

 

118 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

2.2 Cubas model 

The Cubas model [39] assumed the ideal diode factor n=1.1 for the silicon cells studied and the 
modified diode factor a is given by equation (4). The following auxiliary parameters A, B, C, and 
D in equations (10-13) are used to calculate Rs in equation (14) 
 

 𝐴 =
𝑎

𝐼𝑚𝑝
           (10) 

 𝐵 =
𝑉𝑚𝑝(𝐼𝑠𝑐−2𝐼𝑚𝑝)

(𝑉𝑚𝑝𝐼𝑠𝑐+𝑉𝑜𝑐(𝐼𝑚𝑝−𝐼𝑠𝑐))
        (11) 

 𝐶 =
𝑉𝑜𝑐−2𝑉𝑚𝑝

𝑎
+

𝑉𝑚𝑝𝐼𝑠𝑐−𝑉𝑜𝑐𝐼𝑚𝑝

(𝑉𝑚𝑝𝐼𝑠𝑐+𝑉𝑜𝑐(𝐼𝑚𝑝−𝐼𝑠𝑐))
       (12) 

 𝐷 =
𝑉𝑚𝑝−𝑉𝑜𝑐

𝑎
          (13) 

 𝑅𝑠 = 𝐴[𝑊−1(𝐵𝑒
𝐶) − (𝐷 + 𝐶)]        (14) 

 
where W-1 is the lower branch of the Lambert W function. The remaining parameters Rsh, Io. and 
Iph are calculated by equations (15), (16), and (17) respectively. 
 

 𝑅𝑠ℎ =
(𝑉𝑚𝑝−𝐼𝑚𝑝𝑅𝑠)(𝑉𝑚𝑝−𝑅𝑠(𝐼𝑠𝑐−𝐼𝑚𝑝)−𝑎)

(𝑉𝑚𝑝−𝐼𝑚𝑝𝑅𝑠)(𝐼𝑠𝑐−𝐼𝑚𝑝)−𝑎𝐼𝑚𝑝
       (15) 

 𝐼𝑜 = [𝐼𝑠𝑐 (1 +
𝑅𝑠

𝑅𝑠ℎ
) −

𝑉𝑜𝑐

𝑅𝑠ℎ
]𝑒

−
𝑉𝑜𝑐
𝑎         (16) 

 𝐼𝑝ℎ = 𝐼𝑠𝑐 (1 +
𝑅𝑠

𝑅𝑠ℎ
)         (17) 

 
2.3 El Tayyan-Cubas model 

In this model, the modified ideal diode factor a in Cubas model is replaced by the second 
characteristic coefficient El Tayyan’s model i.e., C2=a and equations (10-17) remain the same.  
 
2.4 Senturk model  

This model is applicable to standard test conditions (STC) [37] and other conditions as well. This 
model assumes the diode factor n=1.2 and the modified ideal factor a is calculated using equation 
(4). Then, the experimental resistances Rsho and Rso are approximated by equations (18) and (19) 
respectively 

 𝑅𝑠ℎ𝑜 =
𝑉𝑚𝑝

𝐼𝑠𝑐−𝐼𝑚𝑝
          (18) 

 𝑅𝑠𝑜 =
𝑉𝑜𝑐−𝑉𝑚𝑝

2𝐼𝑚𝑝
          (19) 

 
Thereafter, Iph and Io are calculated by equations (20) and (21) respectively 
 

 𝐼𝑝ℎ =
(𝑅𝑠𝑜+𝑅𝑠ℎ𝑜)𝐼𝑠𝑐

𝑅𝑠ℎ𝑜
         (20) 

 𝐼𝑜 =
𝐼𝑝ℎ−

𝑉𝑜𝑐
𝑅𝑠ℎ𝑜

𝑒
𝑉𝑜𝑐
𝑎 −1

          (21) 

 
and finally, Rs via Phang’s model [32] and Rsh by equations (22) and (23) respectively. 



  
 
 
 
 

 

119 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

 𝑅𝑠 = 𝑅𝑠𝑜 −
𝑎

𝐼𝑜
𝑒
−𝑉𝑜𝑐
𝑎          (22) 

 𝑅𝑠ℎ =
𝑉𝑚𝑝+𝐼𝑚𝑝𝑅𝑠

𝐼𝑝ℎ−𝐼𝑚𝑝−𝐼𝑜(𝑒
𝑉𝑚𝑝+𝐼𝑚𝑝𝑅𝑠

𝑎 −1)

        (23) 

 
2.5 El Tayyan-Senturk model 

Similarly, in this model, a in Senturk model is set equal to C2 i.e., a=C2 and equations (18-23) 
remain the same. 
In practice, it is sometimes possible to differentiate the explicit model that produces the worst 
approximation to the data, but it is not possible to choose the best option with a proper criterion 
beyond a visual impression. For this reason, the results are usually analyzed using the normalized 
root mean-squared error (RMSE) denoted by in equation (24) 
 

 𝜀 =
1

𝐼𝑠𝑐
√
1

𝑁
∑ (𝐼𝑐𝑎𝑙,𝑗 − 𝐼𝑗)

2𝑁
𝑗=1         (24) 

 
Besides, the difference between the output current calculated with models and the one from the 
experimental data, related to the short-circuit current is given by equation (25) 

 𝜉 =
𝐼−𝐼𝑒𝑥𝑝

𝐼𝑠𝑐
          (25) 

 

3. Materials and Methods 

3.1 Materials 

The materials used in the work were chemicals, reagents and equipment viz: Ethanol, Titanium 
dioxide nano powder, Hydrochloric acid, Acetic acid, Mortar and pestle, Stainless steel mesh, 
Filter paper, Doctor tape C, Scanning Electron microscope, Solar energy simulator, 
Spectrophotometer, Magnetic stirrer, Glass bottle, Aluminium Foil, Glue, Adhesive tape, Acetone, 
Anhydrous alcohol, Strips of glass insulation spacers, Iodide electrolyte solution, N719 dye, plant 
dyes, Centrifuge machine, Sonicator, Fourier transform infrared spectrometer, Fluorine doped Tin 
oxide (FTO), Deionized water.  

3.2 Methods 

3.2.1 Extraction of plants dyes 

Fourteen plants parts were selected and allowed to dry on shade for two weeks. The dried fruits 
and flowers were grinded into fine powder using a blinder (Walmat BLSTVB-RVO-000). 0.5 g, 1 
g, and 2 g each of the selected samples were weighed on weighing balance and 5 % of the 
extracts were obtained respectively. The weighed powdered samples were collected in sterile 50 
ml falcon tubes. 20 ml of ethanol was then added and were vortexed. The solutions were 
sonicated using a sonicator (Branson SFX250) for 1 h at 40 C. The samples were then centrifuged 
at 1500 rpm in 40 C for 10 min. The solid residues were filtered out while the supernatants of the 
clear dry solutions were collected, and stored at 40 C before use. The containers were covered 
with aluminium foil to prevent damage from light exposure [47,48].  



  
 
 
 
 

 

120 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

3.2.2 Assemblage of solar cells 

Fluorine doped tin oxide (FTO) of resistance 14 Ω/cm2 (TCO 2215 salonix) substrate was cleaned 
in surfactant, deionized water, acetone and ethanol. Paste of TiO2 was prepared with 0.25 g of 
TiO2, 0.5 ml of acetic acid and mixture of deionized water and ethanol at ratio of 1:1 (10 ml), and 
stirred for 20 min, to prevent the agglomeration of the particles. Triton–X 0.5 ml (salonix) was 
added, the resultant mixture was grounded to facilitate coating of the colloid on the substrate in 
order to obtain homogenous paste. Doctor blade technique was adapted to coat the TiO2 paste 
on FTO glass substrate with active area of 0.16 cm2. TiO2 coated films were sintered at 450 

0C 
for 30 min. The films were cooled at room temperature. To attain sensitization of dye, films were 
dipped in dye like ethanolic solution of the plant dyes for 24 h. The sensitized electrodes were 
rinsed with ethanol to remove the unanchored dyes. Counter electrode was obtained by placing 
a thin layer of platisol which was squeezed printed using a polyester mesh sintered at 450 0C for 
30 min. A drop of Iodolyte TG – 50 was casted on the surface of sensitized photo anodes, to 
penetrate into the porous structure via capillary action. The Pt-coated FTO electrode was then 
clipped onto the top of dye absorbed TiO2 working electrode to form the complete solar cell. The 
photo electrochemical cell (PEC) was then mounted in a sample holder inside a metal box with 
an area of 1 cm2 opening to allow light from the source. The photoelectrode and the counter 
electrode were overlappingly placed in a holder so that the titanium dioxide covered area of the 
photoelectrode was the only part of the photoelectrode that was in contact with the counter 
electrode. The non-titanium dioxide covered area of the photoelectrode and the non- overlapping 
edge of the counter electrode were connected to the measuring equipment by means of cords 
and crocodile clips. All experiments were carried out at ambient temperature. The image of one 
of the assembled cells is shown in Fig. 2.  

 

Figure 2. Assembled DSSC   

3.2.3 Characterization of extracted dyes and solar cells 

The photovoltaic parameters of the fabricated DSSCs were measured using a computer 
controlled digital source meter (Keithley, 2400) and a solar simulator (AM 1.5 G, 100 mW cm− 2 

Oriel) as light source. The light intensity was adjusted with a reference silicon cell (PV 
measurement Co). The sample was masked with black opaque tape along the board of the active 
area. The photovoltaic parameters, open circuit voltage (Voc), short circuit current density (Isc), fill 
factor (FF) and overall efficiency (η) were obtained from I- V curve. 

 

 

 



  
 
 
 
 

 

121 

 

Computational and Experimental Research  
in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

4. Results and discussion 

Table 1. Characteristic parameters (Isc, Imp, Vmp, Voc, Pmax, FF, η) of the solar cells studied 

Source of natural dye Photovoltaic parameters 
English Name Scientific Name Isc 

(mA) 
Imp 
(mA) 

Vmp 
(V) 

Voc (V) Pmax  
(mW) 

FF η % 

Control  TiO2/N719  9.355 7.574 0.4 0.590 3.028 0.54 3.02 
Witch seed flower  Striga hermonthica 1.970 1.379 0.4 0.639 0.551 0.43 0.55 
Bitter gourd  Momordica charantia 9.244 6.450 0.4 0.536 2.580 0.51 2.57 
Bougainvillea Bougainvillea 3.450 2.783 0.3 0.484 0.834 0.50 0.83 
Flamboyant  Delonix regia 1.717 1.442 0.4 0.610 0.576 0.55 0.57 
Wild marigold Calendula arvensis 1.600 0.957 0.3 0.504 0.287 0.35 0.28 
Red cockscomb  Celosia cristata 1.580 1.290 0.3 0.490 0.387 0.49 0.38 
Lantana Lantana camera 1.530 1.262 0.4 0.600 0.504 0.54 0.50 
Hibiscus Hibiscus rosa sinensis 1.480 1.090 0.3 0.450 0.327 0.49 0.32 
Sun flower  Helianthus 1.590 1.081 0.4 0.530 0.432 0.51 0.43 
Rose flower Rosa 1.690 1.283 0.4 0.563 0.512 0.53 0.51 
Orange peel  Citrus aurantium 1.400 1.121 0.2 0.370 0.224 0.43 0.22 
Tomato Lycopersicon 

esculentum 
0.230 0.135 0.2 0.290 0.027 0.40 0.03 

Mango peel Mongifera indica 2.51 2.130 0.4 0.618 0.852 0.76 1.00 
Guava peel Psidium guajava 0.900 0.669 0.3 0.452 0.201 0.49 0.20 

 
Table 1 depicts the characteristic parameters of the solar cells studied. The results show that the 
DSSC with witch seed flower dye has the highest conversion efficiency 2.57 %, second mango 
peel 1 %, and third bougainvillea 0.83 % after the standard cell 3.02 %. In this paper, I-V curve 
matching is done only for measured and calculated model data from these four solar cells for 
clarity. Using the values of the characteristic points (Isc, Imp, Voc, Vmp) in Table 1, the El Tayyan 
coefficients or model parameters were computed (Table 2). 

 
Table 2. El Tayyan model parameters of the solar cells studied 

Source of natural dye El Tayyan model parameters 
English Name Scientific Name C1 C2 a=C2 

 
n Io (A) 

Control  TiO2/N719  0.009838 0.195796 0.195796 7.5626 4.833×10-4 
Witch seed flower  Striga hermonthica 0.009700 0.062867 0.062867 2.4282 1.872×10-4 
Bitter gourd  Momordica charantia 0.009245 0.060393 0.060393 2.3327 1.293×10-6 
Bougainvillea Bougainvillea 0.003927 0.229684 0.229684 8.8715 4.775×10-4 
Flamboyant  Delonix regia 0.001862 0.238962 0.238962 9.2299 1.450×10-4 
Wild marigold Calendula arvensis 0.001777 0.218650 0.218650 8.4453 1.773×10-4 
Red cockscomb  Celosia cristata 0.001830 0.246337 0.246337 9.5148 2.504×10-4 
Lantana Lantana camera 0.001632 0.216544 0.216544 8.3640 1.022×10-4 
Hibiscus Hibiscus rosa 

sinensis 
0.001558 0.150090 0.150090 5.7972 7.769×10-5 

Sun flower  Helianthus 0.001590 0.054753 0.054753 2.1148 9.943×10-8 
Rose flower Rosa 0.001695 0.095372 0.095372 3.6837 4.628×10-6 
Orange peel  Citrus aurantium 0.001946 0.291046 0.291046 11.241

6 
5.457×10-4 

Tomato Lycopersicon 
esculentum 

0.000230 0.043932 0.043932 1.6969 3.130×10-7 

Mango peel Mongifera indica 0.002760 0.257501 0.257501 9.9459 2.504×10-4 
Guava peel Psidium guajava 0.000951 0.154409 0.154409 5.9640 5.091×10-5 



  
 
 
 
 

 

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in Materials and Renewable Energy (CERiMRE)         
Volume 5, Issue 2, page 114-132 
eISSN : 2747-173X 
 
 

Submitted : August 14, 2022 
Accepted : October 24, 2022 
Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

 
Using the values of the characteristic points in equations (6) and (7), the numerical values of the 
El Tayyan coefficients or model parameters (C1, C2) were computed (Table 2). The emergence 
of the two coefficients suggests that the El Tayyan model is a two-parameter model, the 
parameters being the modified ideal diode factor a and the diode saturation current Io which are 
related to C1 and C2 via equations (8) and (9). Equation (8) shows Io is function of both C1 and C2 
whereas equation (9) reveals a is equal to C2. The values of Io and a were computed using 
equations (8) and (9) respectively. The results show all the parameters are regular i.e. they have 
positive values. Furthermore, a depends on the type of dye used to fabricate the solar cell. Thus, 
in our model, the El Tayyan modified ideal factor C2=a, is used to generate a set of four model 
parameters (Io, Iph, Rs, Rsh) for systems where previous researchers assumed the diode ideal 
factor constant, n=1.1 [39] and n=1.2 [40] for comparison purpose (Table 3). 
 

Table 3. El Tayyan-Cubas model parameters for the solar cells studied 

Natural dye Cubas/El Tayyan model parameters, n or a not 
constant 

Cubas model parameters 
n=1.1 or a=0.028479 

English Name a Rs (𝞨) Rsh (𝞨) Io (A) Iph (mA) Rs (𝞨) Rsh (𝞨) Io (A) Iph 
(mA) 

Control  0.1958 -0.8 -109.9 7.27×10-4 9.42 15.8 339.5 8.10×10-12 9.79 
Witch seed 
flower  

0.0629 42.0 -245.8 3.67×10-4 1.63 124.3 711.0 2.55×10-13 2.31 

Bitter gourd  0.0604 -0.4 184.8 8.84×10-7 9.22 8.3 152.7 4.18×10-11 9.75 
Bougainvillea 0.2297 30.0 -22.0 2.52×10-3 -1.25 45.8 1160.0 1.32×10-10 3.59 
Flamboyant  0.2390 19.1 -239.7 3.21×10-4 1.58 99.4 2993.6 7.83×10-13 1.77 
Wild marigold 0.2187 94.8 57.1 7.75×10-4 -1.06 146.8 375.2 1.82×10-11 2.23 
Red 
cockscomb  

0.2463 87.7 -20.0 2.62×10-3 -5.34 104.5 3568.7 5.02×10-11 1.63 

Lantana 0.2165 7.3 -420.7 1.83×10-4 1.50 104.3 2562.1 9.62×10-13 1.59 
Hibiscus 0.1501 -5.6 -1052.3 9.55×10-5 1.49 77.7 981.3 1.56×10-10 1.60 
Sun flower  0.0548 -3.5 957.6 6.44×10-8 1.58 42.0 829.4 8.53×10-12 1.67 
Rose flower 0.0954 0.7 3032.4 4.11×10-6 1.69 68.0 1185.0 3.41×10-12 1.79 
Orange peel  0.2910 131.6 21.5 -2.03×10-3 9.97 123.0 -584.4 3.96×10-9 1.11 
Tomato 0.0839 -92.1 2457.0 1.40×10-7 2.21 107.5 2171.9 4.08×10-9 0.24 
Mango peel 0.2575 20.7 -115.0 6.71×10-4 2.06 71.6 2552.7 8.80×10-13 2.58 
Guava peel 0.1544 -3.0 -1317.4 6.67×10-5 9.02 130.7 1707.1 9.01×10-11 0.97 

 
Table 3 contains two model parameters generated from Cubas model for n fixed or constant 
(n=1.1) for all solar cells and Cubas-El Tayyan model for n varying with dyes or photosensitizers 
used in fabricating the solar cells. For the Cubas-El Tayyan model, all the model parameters 
except a manifest parameters’ irregularities i.e. the parameters have both negative and positive 
values.  The number of negatives values are 0, 6, 9, 1, 3 for the parameters a, RS, Rsh, Io, Iph 
respectively. Also, the two models produce different values for all the model parameters except 
Iph have same values and Io values for Cubas model have smaller values compared to those of 
Cubas-El Tayyan values. Furthermore, the values of Iph in Table 3 are not equal to the values of 
Isc in Table 1 but they are of the same order of magnitude.  
 

Table 4. El Tayyan-Senturk model parameters for the solar cells studied. 

Natural dye Senturk/El Tayyan model parameters, n or a not 
constant 

Senturk model parameters 
n=1.2,  a=0.031068 



  
 
 
 
 

 

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Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

English Name a Rs (𝞨) Rsh (𝞨) Io (A) Iph 
(mA) 

Rs 
(𝞨) 

Rsh (𝞨) Io (A) Iph (mA) 

Control  0.1958 -13.1 319.8 3.74×10-4 9.877 8.3 211.8 4.10×10-11 9.877 
Witch seed 
flower  

0.0629 82.1 21.5 5.38×10-7 2.222 84.4 20.2 1.62×10-11 27.186 

Bitter gourd  0.0604 0.8 145.1 8.63×10-7 9.930 5.5 134.0 1.98×10-10 9.930 
Bougainvillea 0.2297 -43.7 361.0 3.64×10-4 3.704 21.2 411.1 4.51×10-10 3.704 
Flamboyant  0.2390 -86.5 2550.0 1.17×10-4 1.803 50.4 1372.6 4.11×10-12 1.803 
Wild marigold 0.2187 -115.8 216.6 9.82×10-5 1.966 71.5 369.4 7.98×10-11 1.966 
Red 
cockscomb  

0.2463 -100.8 807.3 1.93×10-4 1.692 48.2 946.5 1.72×10-10 1.692 

Lantana 0.2165 -88.6 2316.8 8.07×10-5 1.611 53.5 1407.5 4.96×10-12 1.611 
Hibiscus 0.1501 -70.0 664.0 5.39×10-5 1.612 38.6 697.2 5.26×10-10 1.612 
Sun flower  0.0548 7.3 785.9 6.54×10-8 1.712 30.2 738.9 4.05×10-11 1.712 
Rose flower 0.0954 -14.1 1139.4 3.36×10-6 1.799 38.2 925.8 1.65×10-11 1.799 
Orange peel  0.2910 -127.1 169.6 4.02×10-4 1.548 45.7 621.1 6.94×10-9 1.548 
Tomato 0.0839 -7.4 1724.8 1.74×10-7 0.266 91.9 1757.7 1.14×10-8 0.266 
Mango peel 0.2575 -63.3 1966.7 2.04×10-4 2.632 36.0 992.1 4.70×10-12 2.632 
Guava peel 0.1544 -118.1 1126.6 3.57×10-5 0.979 64.3 1179.6 3.03×10-10 0.979 

 
Table 4 depicts the values of model parameters generated from two models, Senturk model for n 
fixed (n=1.2) and Senturk-El Tayyan model for n varying for all solar cells. For the Senturk-El 
Tayyan model, all the model parameters are regular except RS has 13 values negative i.e. only 
Rs manifests parameter irregularity. Also, all the El Tayyan model parameters differ from that of 
Cubas model parameters except Iph values. However, the Iph values differ from Isc values (Table 
1) and the Io values for both models (Table 4) differ from those of El Tayyan in Table 2. Thus, the 
overall results show that the number of parameter irregularity vary with model. The acceptance 
of the significance of the parameter irregularity lies in the validity of the fitness between the model 
data and measured data I-V curves. However, in this paper, the fitting of I-V curves is done for 
only the first four solar cells with high efficiencies for clarity.  



  
 
 
 
 

 

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Online : November 11, 2022 

DOI: 10.19184/cerimre.v5i2.33499 

 
(a) Standard 

 

 
     (b) Bitter gourd 
 



  
 
 
 
 

 

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DOI: 10.19184/cerimre.v5i2.33499 

 
(c) Bougainvillea 

 
(d) Mango Peel 

Figure 3. Curves for the explicit models analyzed, fitted to the (a) standard (control) (b) bitter gourd 
(c) bougainvillea and (d) mango solar cells data using the analytical methods proposed. 

 



  
 
 
 
 

 

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DOI: 10.19184/cerimre.v5i2.33499 

Figures 3(a-d) depict the I-V curves of solar cells (a) control (b) bitter gourd (c) bougainvillea and 
(d) mango for measured data and five models: El Tayyan, El Tayyan-Cubas, El Tayyan-Senturk, 
Cubas, and Senturk. The figures show the I-V curves are at constant current source at low 
voltages with a current approximately equal to the short-circuit current, Isc. With increasing voltage 
at a certain point, the current begins to drop off exponentially to zero at open-circuit voltage, Voc. 
The values of Voc varies with DSSC as well as model which is distinctly shown in (a) El Tayyan 
model and in (a/d) El Tayyan-Cubas model.  Over the entire voltage range, there is one point 
where the cell operates at the highest efficiency; this is the maximum power point, MPP (Vmp, Imp). 
The system design is to operate the cell at that point. However, the system design is complicated 
by the fact that the maximum power point varies with irradiance and temperature. It is worth noting 
the MPP changes with model.  

 
(e) control         

 

(f) Bitter gourd 



  
 
 
 
 

 

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DOI: 10.19184/cerimre.v5i2.33499 

 
(g) Bougainvillea 

 

 
(h) Mango Peel 

 

Figure 4. Curves for the explicit models analyzed taking error into account, fitted to the (e) standard 
(control) (f) bitter gourd (g) bougainvillea and (h) mango solar cells data using the analytical methods 

proposed. 

 
Figs. 4(e-h) depict best I-V curve fits after eliminating the error between output current model and 
measured current. The values 𝜀 of the normalized root mean squared error (NRMSE) of the output 
currents were generated using equation (24) which lie in the range 0.015<< 𝜀<<0.536 for all 
DSSCs and models. Based on the fact that the best model has the smallest value of 𝜀, our results 



  
 
 
 
 

 

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DOI: 10.19184/cerimre.v5i2.33499 

show that DSSC/best model as follows: DSSC control/El Tayyan model 𝜀=0.078, DSSC bitter 
gourd/El Tayyan model 𝜀=0.109, DSSC bougainvillea/Senturk model 𝜀=0.129, and DSSC 
mango/El Tayyan-Senturk model 𝜀=0.015.  

    

 

(i) Control 

 

 

(j) Bitter Gourd 

 



  
 
 
 
 

 

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DOI: 10.19184/cerimre.v5i2.33499 

 

(k) Bougainvillea 

 

(l) Mango Peel 
 

Figure 5. Fig. 5 Curves for output current errors in relation to short circuit current for the explicit 
models (i) control (j) bitter gourd (k) bougainvillea and (l) mango solar cells data.  

The differences between the output currents calculated with models using equation (25) mand 
output currents measured in relation to the short-circuit current and Figs. 5(i-k) depict current 
differences for the models and DSSCs studied.  

 



  
 
 
 
 

 

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DOI: 10.19184/cerimre.v5i2.33499 

Conclusion 

In this paper, the El Tayyan empirical formula in conjunction with other exponential PV model 
equations were used to obtain the value of the modified ideal diode factor instead of assuming its 
value a constant as proposed by some authors [39, 40]. Five models, El tayyan, El Tayyan-Cubas, 
El Tayyan-Senturk, Cubas, and Senturk were employed to extract the five model parameters for 
fifteen DSSCs. The results show that in terms of conversion efficiency, next to the standard solar 
cell, the most efficient DSSCs are the ones with photosensitizers namely bitter gourd, 
bougainvillea, and mango. Also, the results reveal all the model parameters are regular for El 
Tayyan model while they are irregular with varying degrees for the rest of the models. In addition, 
despite the parameter irregularities, there is good match between the model data and measured 
data I-V curves meaning presence of irregular parameters may not be undesirable for some 
applications. Thus, the irregularities associated with the extracted parameters revealed by these 
methods can serve as a useful assessment criterion and tool for researchers and engineers in 
deciding the proper extraction method for their application. 

 
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