**3. Modeling and problem formulation**

There are several electrical models, used by researchers, to describe the physical behaviors of PV cells. The Single Diode Model, containing the five unknown parameters, used in this paper is represented in **Figure 3**. By the cause of compromise between accuracy and simplicity, the SDM is selected herein.

The mathematical expressions related to the current-voltage, (I-V) relationship of the PV cell is as follow.

$$I = I\_L - I\_D - I\_{sh} \tag{1}$$

$$I = I\_L - I\_{ds} \left( e^{\left(\frac{V \ast R\_s \cdot I}{n \cdot V\_i}\right)} - \mathbf{1} \right) - \frac{V + R\_s \cdot I}{R\_{sh}} \tag{2}$$

The overhead mathematical equation is in a nonlinear form and has a set of five unknown parameters (*I*L*, I*ds*, n, R*s*, R*sh). The main challenge is to get the accurate values of all the PV model's parameters values while keeping a reasonable computational effort.

Several approaches permit the formulation of the optimal nonlinear PV parameters determination problem, using the error (between real and simulated data) [10].

Our focus is to estimate the PV parameters values of the SDM model using RTC France data at the conditions of irradiance about 1000 W/m2 and of temperature about 300°C. We do not review the identification process as detailed on our previous work [20]; our focus is restricted on the third part of identification process, which is the estimation of PV parameters values. The big focus is to optimize the damping factor of LM through GWO. The characteristics of RTC France Silicon-cell data from datasheet are presented on the following **Table 1**.

The real experimental data used of RTC France are presented on the following **Table 2**.

**Figure 3.** *PV cell's electrical equivalent circuit (SDM) [12].*

#### *Solar Cells - Theory, Materials and Recent Advances*


#### **Table 1.**

*Characteristic data from R.T.C. France (Si solar cell).*


**37**

*Study of a New Hybrid Optimization-Based Method for Obtaining Parameter Values of Solar Cells*

Hybrid optimization-based algorithms have become the modern choice for resolving challenging problems [41–43]. A compromise is gotten in this work, from a combination of a traditional numeric optimization-based with a metaheuristic

The estimation/identification process can be gotten in three major steps, such as the initial step of prediction through the use of least-squares mean (LSM), the getting of optimal PV parameters values through Levenberg-Marquardt (LM), and

Prediction of initial PV parameters values using LSM [44, 45] for the two parts of the introduced real experimental points of I-V curve characteristics as described

The prediction in the linear part [46, 47] of the model can be obtained simply

( ) = −+ ( ) ( )

where *a* and *b* are constants depending on a determinant and others constants

The prediction in the nonlinear part [19, 48] of the model can be obtained with a

( ) ( ) ( ) =+∗ +∗ −

( += + ) ( ) ( )

where *C*0*, C*1, C2 and *b* are constants depending on a determinant, on the hessian

Once obtaining initial values of PV parameters values, we introduce them on the

LM in order to optimize their values, as explained in the following subsection.

0 1 <sup>2</sup> log 1 *al*

logarithmic way through the use of the following logarithmic expression.

*Model Model*

and other constants introduced by the user.

*I iCCI iC*

*I i aV i b Model* ( ) =∗ + *Model* ( ) (3)

*Error i I i I i* ( ) = − Re *al Model* ( ) ( ) (4)

Re

*Error i I i I i* ( ) = − Re *al Model* ( ) ( ) (7)

2 *J i J i error i* 1 (8)

*I i*

*<sup>b</sup>* (6)

2 *J i J i error i* 1 (5)

the optimization of a dominant factor through GWO as detailed below.

**4.1 Least squares mean (initial phase of prediction)**

through the use of the following expressions.

*DOI: http://dx.doi.org/10.5772/intechopen.93324*

swarm-based method.

• For the linear part:

introduced by user.

• For the nonlinear part:

below.

**4. Hybrid optimization-based method**

**Table 2.** *Real data from RTC [38].* *Study of a New Hybrid Optimization-Based Method for Obtaining Parameter Values of Solar Cells DOI: http://dx.doi.org/10.5772/intechopen.93324*
