**4. Modeling equations of photovoltaic cell**

Reference to the circuit of figure 1c, then show all equations needed to obtain all the parame‐ ters that define the model in standard conditions of measurement (*SCM*). In equation 1 shows the intensity value generated by the photovoltaic cell, [9]: *I* is output current of pho‐ tovoltaic cell, *V* is output voltage of photovoltaic cell, *IL* is the photogenerated current, *I0* is the saturation current of diode, *RS* is series resistance due to the junction between the semi‐ conductor and the metal contacts (interconnects), *RP* is parallel resistance due to no linearity of union *PN*, *m* is ideal factor of diode and *Vt* is thermovoltage shown in equation 2 (where: *k* is the Boltzmann constant, *q* is the electron charge and *T* is temperature in degree Kelvin).

$$\mathbf{I} = \mathbf{I}\_{\mathbf{L}} - \mathbf{I}\_{0} \left[ \mathbf{e}^{\left(\frac{\mathbf{V} + \mathbf{I} \cdot \mathbf{R}\_{\mathbf{S}}}{\mathbf{m} \cdot \mathbf{V} t}\right)} - \mathbf{I}\right] - \left[\frac{\mathbf{V} + \mathbf{I} \cdot \mathbf{R}\_{\mathbf{S}}}{\mathbf{R}\_{\mathbf{P}}}\right] \tag{1}$$

$$\mathbf{Vt} = \frac{\mathbf{k} \cdot \mathbf{T}}{\mathbf{q}} \tag{2}$$

\_ 0 \_ 1 · *OC SMC*

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(10)

and an

*FF V*

*FF I* æ öæ ö =- ç ÷ ç ÷ç ÷ è øè ø

Parameters Affect to… Units

β *VOC mV/ºC* or *%/ºC* α *ISC mA/ºC* or *%/ºC* δ *PMAX mW/ºC* or *%/ºC*

Also, changes in irradiance (*G*) affect the value of *ISC* and *VOC*. Therefore, using approxima‐ tion of Luque-Sala and Duffie & Beckman, [10], we obtain the equation 9 for *IL*. Besides em‐ pirical tests simulation to study the effects of temperature on *I0*, used for get equation 10,

\_ \_ ( )

a

( )

· ·

*mkT*

\_

*C SCM*

1

· · *C A <sup>A</sup> T T C G Tc C G T* -= Þ= + (11)


(9)

\_ · · 1 ·( *L SC SCM C C SCM C SCM <sup>G</sup> I I T T <sup>G</sup>* <sup>=</sup> + -

( \_ \_ ( ))

b

*OC SCM C SCM*

<sup>0</sup> · ·

*<sup>G</sup> <sup>I</sup>*

*e*

\_ \_

æ ö + - ç ÷ è ø

<sup>=</sup> æ ö ç ÷

è ø

temperature of equilibrium within a photovoltaic cell to an irradiance of *800W/m2*

The relationship between temperature ambient (*TA*) and cell (*TC*), can used equations 1*1 and 12*, [11]. It is based on normal operating temperature cell (*TNOC*), is defined as the average

outside temperature of *20ºC*. Use this approximation is interesting because there are statis‐ tics of temperature ambient on geographic situation but temperature cell depends to PV cell

*SC SCM C C SCM SCM q V Tc T*

a

+ -

· · 1 ·(

*<sup>G</sup> <sup>I</sup> T T*

*SC SMC*

Changes in temperature affect the values of *ISC, voc* and *PMAX*, when cell temperature increase the *VOC* decrease, same with the *PMAX*, and when irradiation increase the *ISC* also increase. The datasheet used parameters of *table 1*, establishing the relationship between units (volt‐ age, current and power) and temperature. Temperature can be expressed on degree Celsius

*S*

*R*

**Table 1.** Parameters that include temperature variations on photovoltaic cell

based on the approximation Duffie&Beckman.

and module.

or Kelvin, depends of manufacturer.

Equation 1 can simplify the last term with a high value of *RP* (for example 100kΩ). Further‐ more, *IL* is considered equal to the short circuit current in *SCM* (*ISC\_SCM*), [1]. Then we obtain the equation 3 at *SCM*.

$$\mathbf{I} = \mathbf{I}\_{\text{SC\\_SCCM}} - \mathbf{I}\_{\text{O\\_SCM}} \left[ \mathbf{e}^{\left(\frac{\mathbf{V} + \mathbf{I} \cdot \mathbf{R}\_5}{\mathbf{m} \cdot \mathbf{V} t}\right)} - 1 \right] \tag{3}$$

The value of *I0* is obtained for *SMC* (*I0\_SMC*) using equation 4, based on [10], considering open voltage circuit in *SCM* (*VOC\_SCM*) and cell temperature in *SCM* (*TC\_SCM*).

$$I\_{0\\_SCM} = \frac{I\_{\text{SC\\_SCM}}}{\left(\left(^{q\cdot V\_{\text{OC\\_SCM}}} \sum\_{\mathbf{w}\in\mathbf{k}\cdot\mathbf{T}\_{\text{C\\_SCM}}} \right)} - 1\right) \tag{4}$$

There is an empirical relationship between the value of *VOC\_SCM* and *ISC\_SCM* with *RS*, [9]. Then, needs calculate fill factor of ideal device (*FF0*) at equation 5, using parameter *voc* of equation 6, and calculate *RS* using equation 8. The fill factor (*FF*) of photovoltaic cell shows at equa‐ tion 7. This approach only use when *RP* is high, therefore fill factor depends of *RS* value.

$$FF\_0 = \frac{\text{voc} - \ln\left(\text{voc} + 0, 72\right)}{\text{voc} + 1} \tag{5}$$

$$\text{twoc} = \frac{V\_{\text{OC\\_SCM}}}{Vt} \tag{6}$$

$$FF = \frac{V\_{MAX\\_SCM} \cdot I\_{MAX\\_SCM}}{I\_{SC\\_SCM} \cdot V\_{OC\\_SCM}} \tag{7}$$

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$$R\_S = \left(1 - \frac{FF}{FF\_0}\right) \left(\frac{V\_{\text{OC\\_SMC}}}{I\_{\text{SC\\_SMC}}}\right) \tag{8}$$

Changes in temperature affect the values of *ISC, voc* and *PMAX*, when cell temperature increase the *VOC* decrease, same with the *PMAX*, and when irradiation increase the *ISC* also increase. The datasheet used parameters of *table 1*, establishing the relationship between units (volt‐ age, current and power) and temperature. Temperature can be expressed on degree Celsius or Kelvin, depends of manufacturer.


**Table 1.** Parameters that include temperature variations on photovoltaic cell

conductor and the metal contacts (interconnects), *RP* is parallel resistance due to no linearity of union *PN*, *m* is ideal factor of diode and *Vt* is thermovoltage shown in equation 2 (where: *k* is the Boltzmann constant, *q* is the electron charge and *T* is temperature in degree Kelvin).

m·Vt S

Equation 1 can simplify the last term with a high value of *RP* (for example 100kΩ). Further‐ more, *IL* is considered equal to the short circuit current in *SCM* (*ISC\_SCM*), [1]. Then we obtain

V I·RS

æ+ ö ç ÷ è ø é ù

ê ú ë û

\_

*C SCM*

· · 1

P

<sup>q</sup> <sup>=</sup> (2)

(1)

(3)

(4)

R

V I·RS

æ+ ö ç ÷ è ø é ù ê ú é ù <sup>+</sup> =- -- ê ú ê ú ë û ê ú ë û

V I·R II Ie 1

k·T Vt

m·Vt II I e 1 SC\_SCM 0\_SCM

\_

*OC SCM*

*I*

voltage circuit in *SCM* (*VOC\_SCM*) and cell temperature in *SCM* (*TC\_SCM*).

0 \_ ·

0

*FF*

*voc voc FF voc*

*I*

*SCM q V*

*e*

ê ú =- - ê ú

The value of *I0* is obtained for *SMC* (*I0\_SMC*) using equation 4, based on [10], considering open

\_

æ ö ç ÷ è ø <sup>=</sup> æ ö ç ÷ è ø

*SC SCM*

*mkT*

There is an empirical relationship between the value of *VOC\_SCM* and *ISC\_SCM* with *RS*, [9]. Then, needs calculate fill factor of ideal device (*FF0*) at equation 5, using parameter *voc* of equation 6, and calculate *RS* using equation 8. The fill factor (*FF*) of photovoltaic cell shows at equa‐ tion 7. This approach only use when *RP* is high, therefore fill factor depends of *RS* value.

( )


*VOC SCM* \_ *voc Vt* <sup>=</sup> (6)

*I V* <sup>=</sup> (7)

ln 0,72 1

\_ \_ \_ \_ · · *MAX SCM MAX SCM SC SCM OC SCM*

*V I*

L 0

the equation 3 at *SCM*.

124 New Developments in Renewable Energy

Also, changes in irradiance (*G*) affect the value of *ISC* and *VOC*. Therefore, using approxima‐ tion of Luque-Sala and Duffie & Beckman, [10], we obtain the equation 9 for *IL*. Besides em‐ pirical tests simulation to study the effects of temperature on *I0*, used for get equation 10, based on the approximation Duffie&Beckman.

$$I\_L = I\_{\text{SC\\_SCM}} \cdot \frac{G}{G\_{\text{C\\_SCM}}} \left(1 + \alpha \cdot (T\_{\text{C\\_CSM}})\right) \tag{9}$$

$$I\_0 = \frac{I\_{SC\_u,SCM} \cdot \frac{G}{G\_{SCM}} \left(1 + \alpha \cdot (T\_C - T\_{C\_u,SCM})\right)}{\left(\omega \cdot \left(\sqrt{\left(V\_{OC\_u,SCM} + \rho \cdot \left(T\_C - T\_{C\_u,SCM}\right)\right)} \sqrt{\sum\_{m \in \mathcal{X}\_{C\_u,SCM}}}\right)}\right)} \tag{10}$$

The relationship between temperature ambient (*TA*) and cell (*TC*), can used equations 1*1 and 12*, [11]. It is based on normal operating temperature cell (*TNOC*), is defined as the average temperature of equilibrium within a photovoltaic cell to an irradiance of *800W/m2* and an outside temperature of *20ºC*. Use this approximation is interesting because there are statis‐ tics of temperature ambient on geographic situation but temperature cell depends to PV cell and module.

$$T\_{\mathbb{C}} - T\_A = \mathbb{C} \cdot \mathbb{G} \implies \mathbb{T}\mathfrak{c} = \mathbb{C} \cdot \mathbb{G} + T\_A \tag{11}$$

$$C = \frac{T\_{\text{NCC}} - 20}{800 \text{ W} / m^2} \tag{12}$$

resistence (*Rs*) and zero bias junction capacitance (*Cjo*), to adjust to PV cell model must

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Current measurement of PV cell model used current probe (*Icell*) on probe library, for voltage meas‐ urement used a wire label (*Vcell*) to get voltage on node. Power generate (*Pcell*) of PV cell is calculates using equ4 as the product of current (*Icell.I*) and voltage (*Vcell.V*) measurement. To represent a *IV curve* is needs to get a variable *I* using function *PlotVS()* on equ4. The results on simulation show on

In *QUCS* is a used component of library simulations for configuring simulation, for example to get *IV curve* need components: *dc simulation* and *parameter sweep* (figure 4). The configura‐ tion of simulation that show on figure 4, changes value of variable *Rl* from *0,01Ω* to *10Ω*,

Also, if changes value of *Irradiance* variable on eqn2 (figure 3), changes the solar condition and current generate of PV cell. For example on figure 5 shows *IV curve* to different values of solar irradiance at the same value of cell temperature (*25ºC*): *1000W/m2* (*G\_1000*), *750W/m2*

graphical (*Cartesian* on *diagrams* library) and table (*Tabular* on *diagrams* library).

variable *Rl* is used to change value of resistor *R3* (figure 3).

(*G\_500*).

**Figure 5.** IV Curve for different values of irradiation

changue to *0Ω* and *0F* respectively.

**Figure 4.** Simulation configuration

(*G\_750*) and *500W/m2*

Figure 3 shows the simulation window with all the necessary equations and the visualiza‐ tion of results using *QUCS*. Photovoltaic cell simulated in figure 3 is *C3ISF200SB* of Isofoton, [12]. QUCS allows represented on the same page a circuit and results of simulation, for ex‐ ample in figure 3 included: *IV Curve*, output power curve, output current curve, output volt‐ age curve and a table with numerical results. For functions used in *QUCS* see on [13], in figure 3 used following equations: eqn1 for parameters of photovoltaic cell, eqn2 for change temperature ambient to cell temperature, eqn3 for parameters adjust to equivalent circuit and eqn4 to calculate variables to represented results on graphical depends to output meas‐ urement of equivalent circuit.

**Figure 3.** Final result of the simulation Qucs photovoltaic cell

The equivalent circuit is formed by following components: current source (*dc current source* on *source* library), diode (*diode* on *non linear components* library) and resistors (*resistor* on *lumped components* library). The value of current source is calculate on variable *IL* (current generate) on equ3 based on equation 9, the value of current saturation on diode is calculate on variable *I0* on equ3 based on equation 10, the value of resistor series and parallel its cal‐ culate manually and indicate on eqn3. The model of diode simulation includes ohmic series resistence (*Rs*) and zero bias junction capacitance (*Cjo*), to adjust to PV cell model must changue to *0Ω* and *0F* respectively.

Current measurement of PV cell model used current probe (*Icell*) on probe library, for voltage meas‐ urement used a wire label (*Vcell*) to get voltage on node. Power generate (*Pcell*) of PV cell is calculates using equ4 as the product of current (*Icell.I*) and voltage (*Vcell.V*) measurement. To represent a *IV curve* is needs to get a variable *I* using function *PlotVS()* on equ4. The results on simulation show on graphical (*Cartesian* on *diagrams* library) and table (*Tabular* on *diagrams* library).

In *QUCS* is a used component of library simulations for configuring simulation, for example to get *IV curve* need components: *dc simulation* and *parameter sweep* (figure 4). The configura‐ tion of simulation that show on figure 4, changes value of variable *Rl* from *0,01Ω* to *10Ω*, variable *Rl* is used to change value of resistor *R3* (figure 3).

**Figure 4.** Simulation configuration

2 20


800 / *NOC <sup>T</sup> <sup>C</sup>*

urement of equivalent circuit.

126 New Developments in Renewable Energy

**Figure 3.** Final result of the simulation Qucs photovoltaic cell

*W m*

Figure 3 shows the simulation window with all the necessary equations and the visualiza‐ tion of results using *QUCS*. Photovoltaic cell simulated in figure 3 is *C3ISF200SB* of Isofoton, [12]. QUCS allows represented on the same page a circuit and results of simulation, for ex‐ ample in figure 3 included: *IV Curve*, output power curve, output current curve, output volt‐ age curve and a table with numerical results. For functions used in *QUCS* see on [13], in figure 3 used following equations: eqn1 for parameters of photovoltaic cell, eqn2 for change temperature ambient to cell temperature, eqn3 for parameters adjust to equivalent circuit and eqn4 to calculate variables to represented results on graphical depends to output meas‐

The equivalent circuit is formed by following components: current source (*dc current source* on *source* library), diode (*diode* on *non linear components* library) and resistors (*resistor* on *lumped components* library). The value of current source is calculate on variable *IL* (current generate) on equ3 based on equation 9, the value of current saturation on diode is calculate on variable *I0* on equ3 based on equation 10, the value of resistor series and parallel its cal‐ culate manually and indicate on eqn3. The model of diode simulation includes ohmic series Also, if changes value of *Irradiance* variable on eqn2 (figure 3), changes the solar condition and current generate of PV cell. For example on figure 5 shows *IV curve* to different values of solar irradiance at the same value of cell temperature (*25ºC*): *1000W/m2* (*G\_1000*), *750W/m2* (*G\_750*) and *500W/m2* (*G\_500*).

**Figure 5.** IV Curve for different values of irradiation

Further, if changes value *Tamb* variable on eqn2 (figure3), change ambient temperature and therefore the cell temperature condition based on equations 1*0 and* 11. For example on figure 6 shows *IV curve* to different values of cell temperature at the same value of irradiance (*1000W/m2* ): *50ºC* (*T\_50*), *25ºC* (*T\_25*) and *0ºC* (*T\_0*). Then, combining the two variables can adjust weather conditions.

and module shows on figure 7, after can be used subcircuit of PV cell or module on different

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practices.

**Figure 7.** Steps modeling subcircuit

**Figure 8.** Insert connection to subcircuit

**Figure 6.** IV Curve for different values of cell temperature
