Chapter 2

Review of literature

2.1 Review of literature modelling

As described previously in a section on operation principles there are many components and

processes of photovoltaic cells. Theoretical models and equations of these processes are

discussed and presented.

Modelling of Conventional Solar Cells

Gerischer (1979) provides a thorough physical and theoretical description of the processes

involved in photo electrolysis. Gerischer discusses most aspects of solar cells, including the

semiconductor-electrolyte interface and associated electron transfer, the transfer of charge in

the bulk semiconductor and solution, thermodynamic and kinetic aspects, the various

materials that can make up a cell and general principles of cell operation such as illumination

effects. Gerischer makes simplifying assumptions concerning the concentrations of the redox

couple in solution, the electrons and holes in the conduction and valence bands and the

electrons in the bulk of the semiconductor.

Boudreaux, Williams & Nozik (1980) applied quantum mechanical approach to modelling

electron transfer at the semiconductor-electrolyte interface, and investigate conditions for

electron injection. The model gives rise to conditions for electron injection, but does not

compare numerical results with experimental results.

Orazem & Newman (1984) present a mathematical model for a gallium arsenide (GaAs)

liquid-junction photovoltaic cell. Macroscopic transport equations for the semiconductor and

the electrolyte bulk are coupled to a microscopic model of the semiconductor-electrolyte

interface. Their interface model is based on the diffuse double-layer theory, This model

differs from others in the literature in its completeness; it includes all possible reactions

involving ionic species in the electrolyte and all reactions involving charge transport in the

semiconductor. However, due to the difficulty in measuring kinetic parameters at the

interface, the authors have made simplifying assumptions concerning the form and value of

the rate constants involved in the model. In a second paper authors validate their model by

comparing their numerical results with experimental results. These comparisons show that

cell performance is influenced by the kinetics of the interfacial reactions, dopant

concentration, semiconductor thickness and conditions of illumination. The model was also

used to calculate optimal thickness and dopant concentration for the GaAs system. In a third

paper, Orazem & Newman use their previous one-dimensional model to discuss the optimal

design for a solar cell , investigated changes in cell efficiency due to changes in the surface

area of the semiconductor and the effects of varying the placement of the counter-electrode

and current collectors.

Bonham & Orazem (I992a) present a mathematical model to estimate the impedance

response of a semiconductor. Their aim is to investigate the existence and distribution of

interfacial and deep-level electronic states in semiconductors. They base their model on the

previous model of Orazem & Newman (1984). The model is intended to play a predictive

role, suggesting experimental designs that will be more sensitive to deep level states and that

can significantly affect charge transfer, hence increasing the efficiency of a photovoltaic

device. The authors also compare their model predictions with experimental results. Their

model appears to be consistent with equivalent electrical circuits that have been proposed to

model semiconductors with surface sites. In another paper by Bonham & Orazem they

compare their model with the Mott-Schottky theory to identify deep-level electronic states.

Smith & Nozik (1996) model electron transfer at a general semiconductor-liquid interface.

They present molecular dynamic simulations of the interface, modelling the full electronic

structure of the semiconductor and redox couple. The model addresses the adsorption of

redox species and contaminants at the interface and the effects of these processes upon

electron transfer. This work provides useful insight into the way in which a semiconductor-

liquid interface can be modelled.

Smith, Halley & Nozik (1996) review models of electron transfer at liquid-liquid and liquid-

semiconductor interfaces. They show that although the models under review are physically

plausible. The authors further suggest and implement some enhancements for more important

features of the interfaces like the potential structure of the interface; the coupling of reaction

coordinates to the solvent, and the electron transfer through a bridge formed by chemisorbed

species.

Fantoni, Vieira & Rodrigo (1999) present models of amorphous and microcrystalline

silicon based photo devices .The physical model is described and a simulation of the

electrical behaviour of such devices is presented. The authors implement a numerical

simulation of a silicon pn-junction and discuss the results.

Krueger, lung & Gajewski (1994) model a gallium arsenide-electrolyte junction to analyse

excitation intensity under flat band and depletion layer conditions. Their two-dimensional

model of the semiconductor-electrolyte interface identifies the effects of excitation intensity

and surface recombination on the produced current. An interesting result is that the reaction

rate for surface recombination may be a function of time rather than a constant.

Modelling of Dye-Sensitised Solar Cells

Gerischer & Willig (1976) discuss the physical processes and theoretical equations of redox

reactions of excited dye molecules at electrodes, and compare these with rate equations for

the same reactions of the molecule at ground state. They provide details of these reactions at

both semiconductor and insulator surfaces. This work provides a very good background to the

processes involved in electron transfer via excited dye molecules, and the model equations

that may form the basis of models for describing these processes.

Soedergren, Hagfeldt, Olsson & Lindquist (1994) develop a steady state theoretical model

for the current-voltage (I-V) characteristics of micro porous semiconductor films in

photovoltaic cells. Their model incorporates the assumption that charge transfer in the bulk

semiconductor occurs via diffusion, and the diffusion length of the electrons remains

constant which implies that recombination processes are important. Relevant experimental

results are compared with theoretically obtained I-V characteristic curves for a colloidal

TiO2 film electrode and a dye-sensitised TiO2 film . Their modelling results agree well with

experimental data.

Gerischer (1995) in his model for the kinetics of photo-induced processes on spherical TiO2

particles examined only Titania particles and is not specific to DSSC. However the processes

that are modelled do occur in the Titania particles in Titania DSSC. The photo-induced

processes include the excitation of an electron in a TiO2 particle due to photon adsorption, as

well as recombination reactions that occur between TiO2 particles and redox couples.

Electron-hole recombination reactions of the conduction and valence bands of a TiO2 particle

are also modelled.

Cao, Oskam, Meyer & Searson (1996) model electron transfer in the porous TiO2 in

photochemical cells, by extending the work of Soedergren et al. (1994) to include the

transient response of the cell. The paper presents photocurrent transient measurements, and

includes the transient response of the cell by modelling the diffusion coefficient of electrons

as a function of the light intensity .The predictions of this model are physically consistent

with an electron transport process controlled by thermal excitation. However, the model does

not take into account all the complex processes involved with charge transfer in the

semiconductor, but instead includes only a Beer-Lambert Law to account for the production

of electrons under illumination.

Matthews, Infelta & Gratzel (1996) present a detailed model of the steady state

photocurrent produced by sensitised semiconductor electrodes. Many of the charge transfer

reactions at the semiconductor-dye-electrolyte interface are considered. The model represents

the concentration of electrons via thermodynamics, and the steady state concentration of dye

species are considered to be dependent on the injection and recombination rate constants.

They observe a reasonable qualitative comparison to DSSC operation in terms of potential

distribution and I vs. V distributions.

Papageorgiou, Gratzel & Infelta (1996) discuss mass transport in photoelectrochemical

solar cells, specifically within the electrolyte. A model is presented for the electrolyte in thin

layer nanocrystalline solar cells, to investigate the effects various electrolyte parameters on

the overall cell current. This model predicts concentrations of redox couples in solution as a

function of current density, location and direction of illumination, but does not provide a

complete model of the solar cell since interfacial loss mechanisms and transport within the

semiconductor are not explicitly included. Papageorgiou et al. (1998) also developed a

theoretical model for experimental studies to determine diffusion coefficients of in porous

TiO2 films. They investigate the effect film porosity has on the limiting currents of the DSSC.

Ferber et al. (1998) present an electrical model of the DSSC that relates material parameters

to cell performance. Their model is a simplified, one-dimensional, complete electrical model

which accounts for charge transport in both bulk solid and solution, as well as the redox

processes at the counter electrode and the semiconductor-dye-electrolyte interface. The

transport of charge within the TiO2 conduction band and the electrolyte system is described

with typical continuity and mass transport equations. Only one loss mechanism is considered

in this model: relaxation from the TiO2 conduction band to the redox electrolyte. Generation

of electrons via the injection from excited dye molecules is accounted for using Beer-

Lambert’s law and the ratio of rates of reactions involving these electrons give a modified

Boltzmann's approximation (Bard & Faulkner 2001) for the electron density .The electric

field within the DSSC is modelled via Poisson’s equation, and an equivalent circuit model is

used to account for TCO resistances and any shunt resistances. This model does not account

for either electron trapping/detrapping in the solid, optical loss mechanisms in the electrolyte

or side-reactions. The model appears to compare well with experimental data.

Ferber & Luther (1998) discuss the importance of light scattering in TiO2 particles to

increase current, and how to optimise light scattering. This optimisation is important for cell

efficiency; due to the effect particle size has on current flow. Too large particles will increase

scattering but decrease the surface area of the electrode, and hence decrease efficiency Ferber

& Luther present computer simulations showing how absorption can be increased by

optimising the size of TiO2 particles, and predict an optimal size to maximise efficiency

However, the proposed model and resulting predictions have not been validated against

experimental results.

Papageorgiou, Liska, Kay & Gratzel (1999) develop a model of electrolyte mass transport

in thin layer nanocrystalline photoelectrochemical solar cells. The governing equations in

their model describe the mass transport of electrolyte species between the electrodes in a

porous medium. Interfacial reactions are described via a Butler Volmer expression to

represent the oxidation-reduction reaction. The electron production from illumination is

described by Beer-Lambert’s law. The model takes into account geometrical and structural

properties of the porous counter electrode.

Ferber & Luther (2001) propose a two-dimensional mathematical model of the DSSC

based on the model developed by Ferber et al. (1998). The porous semiconductor is

represented by a laminar structure, where each column consists of TiO2 particles sintered

together. Similar to Ferber et al. (1998), Poisson's equation is used to describe the electric

field within the DSC and Beer-Lambert's law is used to describe the production of electrons.

Loss of electrons is modelled via one reaction equation, dependent on the concentration of

and the electron concentration. Helmholtz models are used to describe the TiO2-TCO and

electrolyte-platinum contacts. The model is used to investigate the potential distribution

within the DSSC. Although this is a detailed model of the DSC, important loss mechanisms

have still not been included; for example, interfacial side reactions and electron trapping and

detrapping in the solid.

Usami & Ozaki (2001) present a model of charge transport in a DSSC. This model considers

two processes of charge transfer from the semiconductor: transport of electrons to the current

collecting electrode and the loss mechanism of transport of electrons into the electrolyte.

Their results show that thermal release of electrons trapped in shallow traps is important for

effective electron transport. The authors also propose methods to improve the recombination

processes involved at the TCO.

Nelson, Eppler & Ballard (2002) measure and investigate photoconductivity and the

trapping of charge within the porous film of a DSSC. Additionally a model is proposed to

explain the observed charge movement. They model electrons within the conduction band of

the semiconductor. The rate of change in electron concentration with respect to time is

described by the rate of reactions that involve the electrons, namely reactions with the dye

and production due to photon absorption. Their model, composed of differential equations, is

fit to experimental data to analyse the conductivity of electrons in the film under illumination.

Kern, Sastrawan, Ferrere, Stangl & Luther (2002) use a theoretical model of a DSSC to

analyse experimental results from electrical impedance spectroscopy (EIS), studying the

influence that TiO2 thickness, cell thickness, charge transfer at the platinum electrode and the

lifetime of injected electrons has on overall cell efficiency They show that EIS is a suitable

characterisation tool to monitor changes in cell parameters and design.

Fabregat-Santiago et al. (2003) present a model of the semiconductor porous electrode

within a DSC to examine characteristics of charge accumulation, charge transport, and

interfacial charge transfer. They use an equivalent circuit to model these characteristics, and

thermodynamic equations are used to describe electron density Capacitance, potential

distribution and electronic surface states are investigated by comparing the model with

experimental data.

Asbury et al. (2003) propose a two state model to capture the effect of the relation- ships

between the injection and relaxation of the excited dye molecules. Additionally, their model

describes the concentration of excited dye molecules, singlet and triplet, by the use of rate

equations. For example, the rate of change in the concentration of the singlet dye state is

given by the rate equation .A similar rate equation exists for the triplet state. The solutions of

the coupled differential equations are investigated by varying the rate constants of injection

and relaxation. Authors find the magnitude of the injection rates from both singlet and triplet

state increases with the ratio of the rate constants . The rate of the injection component from

the triplet states also depends on the relative energies of the dye, compared with to the energy

of the conduction band edge.

Bisquert, Zaban, Greenshtein & Mora-Sero (2004) present a model to determine electron

lifetime, rate constants for charge transfer and energy levels within the DSSC. Their model is

based on Marcus-Gerischer transition rates and models of redox potentials.

Bisquert, Zaban & Salvador (2002) present a model for the recombination process for

nanoporous semiconductor electrodes in contact with a redox electrolyte within a DSSC. The

governing equations are based on non-equilibrium steady-state thermodynamics for

interfacial electron transfer via surface states. Their analysis investigates the influence of the

distribution of electrolyte levels and surface states on the electron recombination kinetics

under open circuit conditions.

Bisquert (2004) also presents a theoretical study of charge transfer and transport within the

DSC. The modelling and experimental Work focuses on identifying components of the

diffusion coefficient associated with electron density lifetime, and trap states. They observed

that diffusion coefficients depend on electron density, and can be attributed to diffusion

through traps, that is the coefficients depend on average hopping distances and rates.

Construction of theoretical models and performance simulations are necessary to improve

the properties of DSSC. As with the modelling of conventional pn-junction solar cells, a

number of approaches based on the continuity equation, the transport equation, and

Poisson’s equation have been attempted [7–17] However, because the relation between the

models and devices has not been clear, none of these efforts led to an instructive and effective

way of improving the efficiency of DSSC. The mechanism of conventional solar cells is well

understood by way of equivalent circuits which are considered to be useful tools to analyze

cell devices and improve cell performance [6]. It is necessary to obtain DSSC equivalent

circuits to accelerate the development of practical DSSC based photovoltaic modules.

Electrochemical impedance spectroscopy (EIS) has been widely used to correlate device

structure with a suitable model for the study of the kinetics of electrochemical and

photoelectrochemical processes occurring in DSSC [18–30]. EIS is a well-known technique

used for the study of electrochemical systems and its results are usually discussed in terms of

equivalent circuits [18]. Boddy has proposed a number of equivalent circuits for the

semiconductor–electrolyte interface consisting of combinations of resistors and capacitors

[19]. It has also been found that the electrochemical properties of electrodes are strongly

affected by their morphology [20–22]. These studies discuss the electrochemical impedance

of the TiO2/dye/electrolyte interface under static conditions.

2.2 Review of literature of equivalent circuit and Electrochemical impedance

spectroscopy (EIS)

Liyuan Han (2004), (2006) in their publication on modelling of an equivalent circuit for dye-

sensitized solar cell investigated internal resistance in a dye-sensitized solar cell (DSSC)

using electrochemical impedance spectroscopy measurements. Four resistance elements were

observed in the impedance spectra, and their dependencies on the applied bias voltage were

characterized. It is found that the resistance element related to charge transport at the

TiO2/dye/electrolyte interface displays behaviour like that of a diode, and the series resistance

elements largely correspond to the sum of the other resistance elements. To minimize the

internal resistance in DSSC, the influence of cell parameters such as sheet resistance of TCO

glass substrate, roughness factor of platinum counter electrode and cell thickness, on the

impedance spectra were studied An equivalent circuit for DSCs is proposed based on these

results. Equivalent circuit of DSSC is similar to that of Si solar cells when working in DC

condition.

Han 2004

FIG. 1. Electrochemical impedance spectrum of a DSC. The three semicircular shapes are assigned to impedances related to charge transport at the Pt counter electrode (Z1) in the high-frequency region, at the TiO2 /dye/electrolyte interface (Z2) in the middle-frequency region, and inNernstian diffusion within the electrolyte (Z3) in the low-frequency region,respectively. R1 , R2 , and R3 are described as the real parts of Z1 , Z2, and Z3 , respectively. Rh is defined as a resistance in the high-frequency range over 106 Hz.

FIG. 4. Equivalent circuit obtained from EIS and I –V characteristics of DSCs. R2 is equated with the resistance of a diode. The sum of R1 , R3~Warburg impedance!, and Rh largely corresponds to the series resistance of DSCs. A constant-current source is parallel with Rsh . C1 and C2 are capacitance elements of Z1 and Z2 ~see Fig. 1!, respectively

Koide (2006) used DSSC with high energy conversion efficiency of over 8% to characterize

the dependence of each internal resistance element on applied bias voltage. Three semicircles

were observed in the measured frequency range of 0.1 Hz–1 MHz In their analysis , the three

semicircles were attributed to the redox reaction at the platinum counter electrode (Z1), the

electron transfer at the TiO2/dye/electrolyte interface (Z2), and carrier transport by ions

within the electrolyte (Z3). The resistance elements R1, R2, and R3 are described as the real

part of Z1, Z2 and Z3, respectively. It was found that the resistance element Rh in the high

frequency range over 1MHz is influenced by the sheet resistance of TCO and the contact

resistance between the TCO and TiO2. The former is the main factor in Rh, as the value of Rh

increases in direct proportion to the sheet resistance of TCO.

Han 2006

Fig. 7. Equivalent circuits obtained from EIS and I–V characteristics of DSCs. Z1, Z2, Z3 are the impedances in DSCs. Z2 is the impedanceof a diode. The sum of R1, R3 and Rh largely corresponds to the series resistance of DSCs.A constant-current source is in parallel with Rsh

Koide 2006

Fig. 1. Simple equivalent circuit model for conventional pn-junction solar cells.This model consists of a constant current source (Iph), a diode, series resistance (Rs) and shunt resistance (Rsh).

Ref important

As described previously the initial steps of charge separation in a DSSC are the injection of

an electron from a photo excited dye to the conduction band of the TiO2 and subsequently,

the transfer of an electron from the hole transport molecule to the dye .The first process is

usually completed within 200 ps, and the latter, the regeneration of the oxidized dye, is

completed within the nanosecond time scale for liquid electrolyte DSSC containing an

redox couple (Bisquert & Quiñones, 2006). It is very important to study this

phenomenon with an appropriate analytical technique. Electrochemical impedance

spectroscopy (EIS) is an experimental method of analyzing electrochemical systems; this

method can be used to measure the internal impedances for the electrochemical system over a

range of frequencies between mHz-MHz (Wang et al, 2005); additionally EIS allows

obtaining equivalent circuits for the different electrochemical systems studied.

Figure 6(a) shows a typical equivalent circuit for DSSC; this model has four internal

impedances. The first impedance signal (Z1) related to the charge transfer at the platinum

counter electrode in the high-frequency peak (in kHz range) and the sheet resistance (Rh) of

the TCO in the high frequency range (over 1 MHz); the second signal (Z2) related to the

electron transport in the TiO2/dye/electrolyte interface in the middle-frequency peak (in the

1–100 Hz), and the third signal (Z3) related to the Nernstian diffusion within the electrolyte in

the low-frequency peak (in the mHz range)

In figure 6(b) is shown Nyquist diagram of a DSSC from the result of a typical EIS analysis.

Finally, the total internal impedance of the DSSCs is expressed as the sum of the resistance

components (R1, R2, R3, and Rh). High performance of the DSSC is achieved when this total

internal resistance is small (Shing et al, 2010).

Fig.

Fig. 6. Scheme of: (a) Equivalent circuit model for DSSCs and (b) Nyquist plot of the DSSCs

from EIS analysis (adapted from Shin, 2010).

Additionally Rs can be described as (Islam & Han, 2006):

Rs = Rh + R1 + R3

The proposed equivalent circuit of a DSC is thus similar to that of a conventional solar cell

except for its having large capacitances C1 (in 10μF/Cm2 range) and C3 (in 1F/Cm2 range).

However, the capacitances can be omitted since the solar cells work under direct current

(DC) conditions. Therefore the equivalent electrical circuit of DSSC is composed of a diode

(D), a series resistance (Rs), a shunt resistance (Rsh) and a constant-current source (Iph) similar

to that of a conventional solar cell. Hence the abundance of experience gained through the

development of high-efficiency conventional solar cells 11,12 can be applied to DSSC. It is

considered that this equivalent circuit may facilitate the realization of high-performance

DSSC.

2.3 Review of literature of Parameter analysis

[The photovoltaic (PV) industry has achieved a durative development at an

annual average rate of 40% since 2000 [1]. In 2009, the consolidated figure of

world solar cell production had increased up to 9.34 gigawatt (GW). The market

installations reached a record of 7.3 GW [2]. For the mass production of solar

cells, it has become a crucial issue to have access to the detailed information of

the solar cell parameters. These parameters are used not o n l y to evaluate solar

cells performance and quality control, but also for fabrication process

optimization and scientific research [3]. ] { r e f 1 1 } The accurate values of these

parameters could provide the precise quantitative relationship between DSSC performance

and DSSC physical structure, chemical composition or manufacture processing.

Wolf and Rauschenbach 1963 extracted the lumped series resistance in []. Agarwal et al

1982 [] determined series resistance of solar cell using the non-linearity in the plot of short

circuit current versus light intensity, same was observed by Singh and Singh in 1983[] at

moderate light intensity. Rajkanan et al 1979[] determined series resistance using

experimental data in dark and illuminated condition. Araujo and Sanchez [] studied area

under (І-V) curve and determined low series resistance under high illuminations.

Fill Factor (FF) which is another important parameter of solar cell have been studied by

various researchers. Pulfery in 1978[] studied the dependence of FF on dark saturation

current and diode ideality factor. Mitchell et al. (1977) calculated the FF for a solar cell

using a parametric approach. Green (1981, 1982-83)[] made two significant contributions to

techniques for evaluating solar cell FFs. First was a graphical technique and secondly an

empirical expression was provided to determine FFs. In 1983 [] an attempt was made to

determine FF using open circuit voltage.

Schechter et al. in 1982 [] proposed an analytical method to determine solar cell array

parameters. Another analytical expression was derived by Singal et al in 1981 [] to

determine the series resistance dependent photocurrent and voltage at the maximum power

point and consequently the equations of the maximum power and curve factor for solar cells.

Charles et al. in 1981 [] proposed a numerical method (also known as the exact five-point

parameter method) based on fitting a theoretical curve to the experimental curve at open

circuit voltage, short circuit current, maximum power point and slopes at open circuit and

short circuit points.

Single diode equivalent circuit of solar cell was used by Phang et al. (1984) to analytically

determine solar cell parameters and compared them with those of Newton-Raphson

(iterative) method proposed by Kennerud (1969) and Charles et al. (1981). Phang et al.

(1984) also provided error contours to show the significance of the method.

Chan and Phang (1984) determined shunt resistance of solar cell by measuring the open

circuit voltage and short circuit current at very low illumination. Chan and Phang (1987)

also described two analytical methods for the extraction of solar cell single- and double-

diode model parameters from (І-V) characteristics. It has a wide range of validity provided

the series resistance and illuminations are not both high.

Temperature dependence of solar cell parameters for materials with different energy gaps was

presented by Fan 1986[]. Deb et al 1985 [] determined maximum power point based on

geometrical consideration. This enabled the measurement of FF, (Iph+I0), Rs and n from a

single set of connected observations around the maximum power point for a fixed intensity of

illumination.

Junction ideality factor which amounts due to various physical phenomenon occurring in

diode is another crucial parameter for study of solar cell. An extensive investigation has been

carried out in past to estimate its value [].

Datta et al. 1992 [] proposed a computer aided fitting technique for the simultaneous

determination of different solar cell parameters. Enebish et. al 1993[] provided a numerical

technique based on Newton-Raphson method to estimate parameters of solar cell using

double diode equivalent circuit model. Girardini and Jacobsen 1990 propose a numerically

intensive model of a silicon cell with an optimization algorithm. A technique for

simultaneous optimization of various input parameters leading to maximization of solar cell

efficiency and open circuit voltage technique was established by Pelanchon and Mialhe in

1990. Conde et al.presented a generalized model for a two terminal device including the

effects of a series resistance. This model was used for the particular case of diode and it can

be extended for solar cells also.

Sanchez et al. (1996) evaluated semiconductor device’s intrinsic model parameters from its

experimental extrinsic forward (І-V) characteristics, independently of the parasitic resistance

that might be present in series within the real device. The Lateral Optimization method

proposed by Ranuarez et al. (2000) extracted diode parameters under the presence of

parasitic series and shunt resistance. It is an accurate, efficient, and robust method for

extracting semiconductor device parameters. Though they hadn’t used it for solar cells but it

can be.

A quadratic two-dimensional fitting process which is robust but accurate, fast and applicable

for parameter extraction of illuminated solar cells enabled the extraction of the intrinsic and

extrinsic model parameters of solar cells containing parasitic series resistance and shunt

conductance from the explicit analytic solutions of their illuminated (І-V) characteristics

[Conde et al., 2005, 2006]. It is based on the Co-content function CC which is applied to the

exact explicit analytical solutions of the cell’s illuminated (І-V) characteristics in terms of

Lambert W-function.

Chegaar et al 2001 used nonlinear least-squares optimization algorithm based on Newton

method to evaluate five solar cell parameters (series resistance, ideality factor, photocurrent,

shunt conductance and diode saturation current). Auxiliary function and computer-fitting

routine were used to extract parameters from current-voltage characteristics in 2006[].

Bouzidi (2007) , Chegaar(2008) and Kaminski et al ( ) used different techniques to

evaluate solar cell parameters from (І-V) characteristics.

Optimum load, another important parameter of solar cell had been a challenge for

researchers. Kothari et al. in 1982 use lagrange method of undetermined multipliers to

determine optimum load for a given silicon solar cell. The technique was further used by

Shvets et al. in 2009 to determine series and shunt resistance.

All the methods discussed till now were approximate and not explicit due to transcendental

nature of current-voltage relation of solar cell. Several attempts have been made to determine

exact analytical solution for the same.

Jain and Kapoor 2004 used Lambert W-function technique to solve transcendental (І-V)

relation of solar cell explicitly. They extended the use of W-function in studying different

parameters of organic solar cells as well as solar cell array parameters[].Co-content

function (CC) and Lambert W function to extract all the parameters needed for

single-diode model was p r e s e n t e d by Ortiz-Conde et al. [11].

Lambert W function and special transfunction theory (STFT) were applied for

the determination of the ideality factor by Kapoor et al [12,13]. The comparison

between Lambert W function integral method a n d different ia l method

was presented [14]. Habibe Bayhan et al 2007 proposed that the analytical method

based on the dark current-voltage data and the Lambert W-function could be useful

for carrying out highly accurate computations for the diode ideality factor of p-n

junction devices modelled with relatively high series resistance values. Nchimunya

Mwiinga et al 2008 his paper on the Dynamics of Solar Cell Optoelectronic Device

concluded that the conventional approximations that are aimed at simplifying the analysis of

solar cell parameters often lead to inconsistent expressions. Further problems involving the

extraction of parameters based on the one-diode real solar cell model can be analyzed using

Lambert W function with great ease. F.Ghani et al (2011) used Lambert W-function for

Numerical determination of parasitic resistances of a solar cell and in (2012) they used the

Lambert W function for extraction of solar cell modelling parameters.

Requirement of pure and defect free semiconductor and high cost are the major drawback of

conventional solar cell. Researchers across the globe were in search of a material or

phenomenon that can provide an alternate to conventional solar cell. The Dye Sensitized solar

cell (DSSC) developed by Gratzel [ref 20 Seo] proves to be a promising alternate because of

its transparency, simple fabrication process, low cost and short energy payback time

Feber et al (1998) proposed an electric model of DSSC for calculations of internal steady-

state characteristics. By applying continuity equation, transport equation , Poisson, s equation

to all electroactive species and by assuming linear Boltzmann relaxation approximation for

back reaction differential equations were derived. This numerical algorithm requires 10 input

parameters of DSSC

Berginc et al presented a simple model of DSSC based on diffusion and recombination

processes. The model enables steady-state calculation of complete I-V characteristics,

concentration of electroactive species and current densities in DSSC.

Kern et al (2002) revolutionised DSSC research by conceptualizing EIS for determination of

cell parameters and developed a model for interpretation and analysis of electrochemical

impedance spectra. It was shown that EIS is a suitable characterization tool.

Bay, West et al (2004) proposed a transmission line model for IMVS,IMPS and EIS

response of a photo electrode in DSSC. This model gives graphical description of the way

reaction and transport processes are coupled. In analytical expressions influence of

electrochemical impedance is separated from photo-response.

Han L et al (2004) measured an efficient DSC sensitized with black dye .A short circuit

photocurrent density of 20.1 mA/cm2, an open-circuit voltage of 0.71V, a fill factor of 0.71

and an overall conversion efficiency of 10.1% was obtained when measured under standard

AM 1.5 sunlight.{6}

Masaki Murayama et al (2006) Carboxylic acid treatment, especially using acetic acid, was

effective in increasing the short-circuit current (Isc) of the cell. As a result of the equivalent

circuit analysis, the shunt resistance (Rsh) value tended to increase due to carboxylic acid

treatment. In contrast, the series resistance (Rs) value of the treated cell was smaller than that

of the non treated cell. Carboxylic acid treatment was effective not only in improving the

electrical contact between TiO2 particles but also in blocking the leakage current from the

TiO2 surface to electrolyte. Thus, this analytical method using a one-diode equivalent circuit

is useful for evaluating the effect of dye-sensitized solar cell treatment.{8}

Ishibashi K et al (2008) proposed an extensively valid and stable method for derivation of

all parameters of a solar cell from a single current-voltage characteristic which requires only

the valid assumption for a general solar cell. The experimental data were approximated by a

ninth-degree polynomial expression. {5}

Murayama M, Mori T. Et al (2008) as a novel type of photovoltaic device, dye-sensitized

solar cell using plastic substrate was fabricated at low-temperature by sol-gel method. The

internal electrical characteristics of the cells were evaluated by an equivalent circuit analysis.

Mixing Ti-precursor sol reduced the internal series resistance of cell. This improved cell

efficiency when the cell is fabricated at low-temperature. {7}

{Hanmin T et al (2009) found that there could be several different groups of equivalent

circuit parameters fitting well to the same one experiment-measured I–V curve,. In order to

overcome the uncertainty of circuit parameters estimation, an improved method, based on

extending the scanning voltage range and thus getting the I–V linear function when V is

applied between 0 and reverse stopping voltage, is introduced. The further experiment, by

connecting resistance in the outside circuit, demonstrates that the improved method can

confirm uniquely the values of DSSCs equivalent circuit parameters}1

{Sophie Wenger et al (2010) have developed an experimentally validated and accurate

optical model for dye sensitized solar cells. This model allows us to correctly compute the

dye absorption function for any incident spectrum, illumination direction, or stack assembly

(e.g., including antireflective coatings or back reflectors). Hence, the internal quantum

efficiency of test devices—an important cell characteristic that was difficult to assess so far—

can be accurately computed. By coupling the results of the optical model to an electrical

model for charge generation, transport, and recombination, intrinsic parameters (such as the

diffusion length and the injection efficiency) can be extracted from steady-state

measurements. With the coupled model, the different optical and electric losses can be

quantified. This comprehensive loss analysis paves the way for a systematic, model-assisted,

optimization of dye-sensitized solar cells.}{4}

{Ahmed. A. El Tayyan et al (2011) in their paper on dye sensitized solar cell: parameters

calculation and model integration describe a method to calculate internal parameters of DSSC

(L, α, m, D, η0, τ). Their approach provides estimation of the internal parameters and the

effect of voltage loss at the counter electrode/electrolyte interface on the total voltage and the

behaviour of the cell theoretically. Their approach is based on the electron diffusion

differential model and the values of the short circuit current density Jsc, open circuit voltage

Voc, and the current density and voltage at the maximum power point i.e. Jmp and Vmp,

respectively. Parametric analyses were conducted to study the effect of temperature, and

electrode thickness on various cell parameters}3

{Hafez HS et al (2012) paper on Extraction of the DSSC parameters under dark and

illumination conditions, Current-voltage characteristics in dark and illumination were

measured using a high internal impedance electrometer (Keithly 6517A). The electronic

parameters such as saturation current, the ideality factor, barrier height and the mean value of

the series resistance of DSSC were 3.892x10-5 and 4.037, 0.728 and 5.014 k Ω }2

Otakwa R. V. M.*, et al (2012) Effects of air mass on the performance of a dye-sensitized

solar module (DSSM) have been investigated in a tropical area in Nairobi, Kenya. Outdoor

measurements were performed at different times on different days and a series of current-

voltage (I-V) characterizations carried out at different air mass values and module tilt angles.

The module performance parameters: Short circuit current density, (Jsc), Open circuit

voltage, (Voc), Fill factor, (FF) and solar-to-electricity conversion efficiency, (η) were

extracted from the I-V curves. The module’s η and FF increased with air mass while Voc and

Jsc decreased with increase in AM. The module performed better in the afternoon hours than

in the morning hours. The results may be useful in tuning dye-sensitized solar cells for use in

the tropics as well as in the design of Net Zero Energy buildings{9}