ASSESSMENT OF EXPLICIT MODELS BASED ON THE LAMBERT W-FUNCTION FOR MODELING AND SIMULATION OF DIFFERENT DYE-SENSITIZED SOLAR CELLS (DSSCs)

,


INTRODUCTION
Nowadays, renewable energy plays a great role in reducing fossil resources consumption [1] due to problems arising from the use of fossil resources such as global warming, climate change, and air pollution [2] to mention a few.Presently, among the various renewable energy sources, solar energy is likely the most applicable, as it is clean, safe, and unlimited [3,4].In one year, the amount of solar energy received from the sun is 10 4 times greater than the world's energy consumption [4].It has been revealed the installed photovoltaic power increased from 100.9 GW in 2012 to 230 GW in 2015, rising to 400 GW in 2017 [1,5].This rate of increase has been feasible due to a new brand of solar cells that permit production growth while reducing costs and environmental impact [6].
Modeling has become a crucial step for photovoltaic system design and development, as it permits appropriate and accurate energy production forecasts [7].The modeling of solar cells/panels is usually performed by using equivalent circuit models represented with mathematically implicit equations which are not easy to solve.However, the Lambert Wfunction has been identified as a useful tool to solve these equations.
The purpose of this paper is to present simplified model expressions in terms of the Lambert W-function, which is usually applied in photovoltaic devices, and depicts how this function is needed to solve equations connected to these systems.The desired model expressions were obtained by matching famous mathematical equations (exponential functions, polynomials, hyperbolic functions) to points on the Lambert W-function calculated numerically with the highest available accuracy.
The approach presented in this paper is to apply simplified model equations based on the Lambert W-function that can be solved easily with a pocket calculator, to model and simulate DSSC systems behavior.

Modeling and simulation of solar cells
A host of researchers have reported that ideal solar cells behave like a current source connected in parallel with a diode [7][8][9].This ideal model is achieved with resistors to represent the losses and sometimes with additional diodes that takes into account other phenomena [10,11].The most common circuit equivalent to a solar cell consists of a current source, one diode and two resistors; one in series and one in parallel [12][13][14][15][16][17][18][19].It is worth noting each of the element in the equivalent circuit one parameter has to be calculated except two in the case of the diode whose behavior is represented by the Shockley equation [20].Thus, five parameters are required to be determined when applying this method [21][22][23][24][25][26][27][28][29][30][31][32][33].This simple equivalent circuit has been used quite well to reproduce the current-voltage curve or simply I-V curve.Three important points of the I-V curve known as characteristic points namely: short circuit, maximum power, and open circuit points are used as input data.These representative points depend on temperature, irradiance of the photocurrent source, characteristic points and usually the normal information included in the manufacturer's datasheets.
The conventional equation (1) describes a simple diode with a distinctive I-V curve  =   − 1 , (1) where a is the modified ideality diode factor (quality factor or emission coefficient) which varies with the nature of diode is determined according to the fabrication process and the semiconductor material.
When the semiconductor is illuminated, it will produce a photo-generated current I ph , which will result in a vertical translation of the I-V curve of a quantity that is almost entirely related to the surface density of the incident energy.The equivalent circuit solar cell containing series resistance R s , shunt resistance R sh , photocurrent I ph , diode saturation current I o , modified diode ideality factor, a is depicted in Fig.The single-diode model assumes an ideal cell is pictured as a current generator that is linked to a parallel diode with an I-V characteristic which is mathematically defined by Schokley equation ( 2) where I and V are the terminal current and voltage respectively, I o the junction reverse current, a is the modified junction ideality factor, R s and R sh are the series and shunt resistance respectively.Equation ( 2) is transcendental in nature hence it is not possible to solve for V in terms of I and vice versa.However, explicit solutions can be obtained using the principal branch of the Lambert W-function Wo [21,[34][35][36][37].
One can directly find the current for a given value of voltage using equation (3) or the voltage via (4), which makes the computation easy and robust in contrast to (2).The Lambert W function is readily available in all computation procedures [21,35].Finally, for simulation purpose the current can be calculated for each model by plugging the appropriate model parameters for any given value of V into equation (3) and vice versa for V for any given value of I in equation ( 4).However, if the curve fit fails due to parameter irregularity, for example R sh negative or complex we neglect R sh =∞, the last term in equation ( 2) vanishes reducing the five-parameter model to four-parameter model.Therefore, equation (3) reduces to equations ( 5) Furthermore, if equation ( 5) fails to yield good curve fit then R s is neglected and equation ( 2) reduces to the ideal diode equation ( 1) representing a three-parameter model.Thus, equations ( 1), ( 3) and ( 5) can be used for simulation of three-, four-and five-parameter models respectively.

The explicit model equations based on the Lambert W-function
There are many explicit models to study the current-voltage behavior of a solar cell [38].Notwithstanding, the results do not sustain any of the physical appearance of the photovoltaic conversion process, they are attracting great attention and accurate enough to produce recent discoveries from time to time [39].Some of the explicit models with solutions based on the Lambert W-function include: I.The El-Tayyan model [40].The proposed El-Tayyan model equation for generating I-V characteristics of solar cell or PV module is in the form EEJP. 4 (2022) Jamu Benson Yerima, Dunama William, et al where C 1 and C 2 are coefficients of the model equation.These coefficients are given by [41]as and, if  / >> 1: However, Babangida [42] have shown that the relationships between the conventional model parameters (I o and a) and the El-Tayyan coefficients (C 1 , C 2 ) are given by equations ( 9) and ( 10) Thus, a and I o in equations ( 9) and ( 10) are the two model parameters for the El-Tayyan model.

II. The Karmalkar and Haneefa model [43]
. This model presents the current-voltage relation as where the model parameters are: III.The Das model [44].The current-voltage for this model is given by where the coefficients are: IV.The Saetre [45] and Das model [44].This model was proposed independently by Das [41] and Saetre [42] given by the following equation where the model parameters f and g are estimated with output current measurements at V=0.8V oc and V=0.9V oc .
Using the maximum power point conditions, ,  = ,   | = − , such that  =   = , the following equations are obtained: Assuming α f <<1, then Therefore, plugging equation ( 21) into equation (20), the equations for f and g are finally given by V. The 1-diode/2-resistors equivalent circuit model.The mathematical form of this model is already defined by equation ( 2) whose solution for I or V in terms of Lambert W-function is given by equation ( 3) or ( 4) respectively.Many researchers like [35] have published a solution of equation ( 2) based on the Lambert W-function which requires the diode ideality factor n as an input, say n=1.1 for the silicon cells studied and R s is determined via equation (24).
where W -1 is the lower branch of the Lambert W-function and A, B, C, and D auxiliary parameters defined as: Most often the modified diode ideality factor a in terms of n and the thermal voltage V T is defined by equation ( 27) such that V T is also defined by equation ( 27) where k is the Boltzmann constant, T is the absolute temperature and q is the electron charge.In another vein, [43] avoided the assumption of the value of n instead he deduced that the modified diode ideality factor a is equal to the second El Tayyan coefficient C 2 i.e he set a=C 2 given by equation ( 29) Furthermore, the parameter R sh is calculated via [21] equation ( 30) Finally, the remaining parameters I o and I ph are found by equations ( 31) and ( 32) respectively In this paper, equations ( 24) and (29-32) are used to extract the five model parameters (a, R s , R sh , I o , and I ph ) to study the performance of DSSCs.In Table 1, the characteristic or representative points namely the short circuit point (I sc , 0), open circuit point (0, V oc ) and the maximum power point (I mp , V mp ) were obtained from the I-V curves of measured currents and voltages for three DSSCs are included.These points were used as input data for the modeling and simulation of the DSSCs studied.values for all DSSCs while h negative value for DSSC with bitter gourd dye and positive values for DSSCs with bougainvillea and mango peel dyes.This means k is regular parameter for all dyes whereas h is regular for bitter gourd dye and irregular for bougainvillea and mango dyes.The parameters are directly proportional to one another.In Table 5, the Saetre and Das model parameters f and g are included.Both parameters are positive and therefore they are regular.Also, f and g are inversely proportional.However, the DSSCs with bougainvillea and mango dyes exhibit parameter irregularity in R sh and I ph .In Table 6, the single-diode model parameters (a, R s , R sh , I o , and I ph ) are included.The DSSC with bitter gourd dye have all the parameters positive and hence they are regular.Similarly, the other DSSCs show parameter irregularity in R sh and I ph for DSSC with bougainvillea dye and only R sh for DSSC with mango dye.Also, a and I ph are inversely proportional to R sh and R s respectively.In all cases, the model parameters were used in appropriate model equations for the simulation of the DSSCs investigated.In this work, the five-parameter model was used to simulate DSSC with bitter gourd dye with regular parameters whereas the four-parameter model for the remaining DSSCs with irregular parameters yielded good curve fits

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

Figure 2 .Figure 3 .
Figure 2. El Tayyan model (a) characteristic curves and (b) differences between measured and simulated currents and powers

Figure 4 .Figure 5 .Figure 6 .
Figure 4. Das model (a) characteristic curves and (b) differences between measured and simulated currents and powers

Table 1 .
The characteristic points for three DSSCs

Table 2 .
The El Tayyan model parameter for three DSSCs

Table 2
contains the two parameters for the 2-parameter El Tayyan model and both parameters are positive and less than unity.This means the parameters are regular parameters.The two parameters are inversely proportional to each other.

Table 3 .
The Karmalkar and Haneefa model parameter for three DSSCs

Table 3
depicts the three parameters of the Karmalkar and Haneefa 3-parameter model.Two of the parameters,   , have positive values whereas the parameter K has all values negative.This implies that    are regular parameters and K is irregular parameter.The three parameters are inversely proportional to one another.

Table 4 .
The Das model parameter for three DSSCs

Table 4
contains the two parameters (k and h) for the 2-parameter Das model.The parameter k has positive

Table 5 .
The Saetre and Das model parameter for three DSSCs

Table 6 .
The Single diode circuit 5-parameter model for three DSSCs

Table 7 .
Some common features of the five models studied

Table 7
depicts the 2wnumber of model parameters (MP), observations (OB), parameter irregularities (PI), and simulation model (SM) that produced good curve match for all the DSSCs studied.