## 2 Presentation and Modelling of the Solar Cell  
The electrical behavior of the solar cell is modelled by its outputs current versus voltage characteristic. Further, a solar cell is mathematically modelled in two common methods [24; 70], single diode (SDM) and double diode (DDM), with consider parasitic phenomena by series and shunt resistances. Moreover, the flexible hydrogenated amorphous silicon a-Si:H solar cell with loss current \(I_{rec}\) is paralleled with the original photo-generated current source and the current sink representing the recombination current in the \(i\)-layer of a P-I-N solar cell [71; 72; 73; 74; 75]. The two models are given in Figure 1.  
The current-voltage behavior of a solar cell is described according to the electrical intrinsic parameters and nonlinear implicit equation, for a given illumination and temperature.  
\[I=I_{ph}-I_{SD1}\left[\exp(\frac{V+IR_{s}}{a_{1}V_{T}})-1\right]-SI_{SD2} \left[\exp(\frac{V+IR_{s}}{a_{2}V_{T}})-1\right]-\frac{V+IR_{s}}{R_{P}} \tag{1}\]  
where, \(I_{ph}\) is the photocurrent, \(I_{SD1}\) and \(I_{SD2}\) are the saturation currents, \(a_{1}\) and \(a_{2}\) are the diffusion and recombination diode quality factors; \(R_{s}\) and \(R_{p}\) are the resistances in series and parallel, respectively, \(V_{T}\) is the thermal voltage (which will be defined in the followings), and:  
\[S=\left\{\begin{array}{l l}SDM&\text{for}&Open=0\\ SDDM&\text{for}&Close=1\end{array}\right. \tag{2}\]  
Figure 1: Equivalent circuit solar cell model: (a) single and double diode, (b) flexible hydrogenated amorphous silicon a-Si:H.  
The current-voltage characteristic of a flexible solar cell is:  
\[I=I_{ph}\Bigg{(}1-\frac{d_{i}^{2}}{(\mu\tau)_{eff}[V_{bi}-(V+IR_{s})]}\Bigg{)}-I_{ s}\bigg{[}\exp\bigg{(}\frac{(V+IR_{s})}{aV_{T}}\bigg{)}-1\bigg{]}-\frac{V+IR_{s}}{R_{P}} \tag{3}\]  
where, the voltage \(V_{bi}\) represents the built-in field voltage over the \(i\)-layer, in single junction amorphous silicon solar cells, and in [76] it has been determined to be in the range 0.9 V; \(d_{i}\) represents the thickness of the \(i\)-layer, the effective \(\mu\tau\)-product _(\(\mu\tau\))\({}_{eff}\)_ represents average mobility-lifetime product for election and hole, and quantifies the quality of the active layer in terms of recombination of photo-generated carriers. The thermal voltage is \(V_{T}=KT/q\) where \(K\) is Boltzmann's constant, \(T\) is the cell absolute temperature in Kelvin and \(q\) is the electronic charge, \(a\) is the diode quality factor.  
The photocurrent \(I_{ph}\) describes the irradiation dependent recombination in \(i\)-layer and reduced by the recombination current, as follows:  
\[I_{rec}=I_{ph}\Bigg{[}\Bigg{(}\frac{\mu\tau}{d_{i}^{2}}\bigg{)}(V_{bi}-(V+IR_{ s}))\Bigg{]}^{-1} \tag{4}\]  
where, \(I_{rec}\) is the current sink and it represents the recombination current in the \(i\)-layer of a P-I-N; the current through the diode represents the diffusion process of charge carriers and the last term represents the shunt leakage current \(I_{p}\) and is modelled as a space charge limited current [77; 78].  
In Equations (1) and (3), the five, seven and eight parameters which define the current versus voltage relation of solar cell and photovoltaic module, vary in accordance with solar irradiance, cell temperature and depend on reference values reported on datasheet.