**4.3 Grey Wolf optimizer (optimization of damping factor's value)**

In this subsection, our focus is on the evolution of the function *f(I,V,ϴ*,λ*)* indicated by *f(*λ*)* for *ϴ* fixed at *ϴ*k, as regards with various varied values of the damping factor, at each iteration of the LM. As it is observed that at each iteration different local minimums values of *f(*λ*)* exist. So, for obtaining the global minimum of *f(*λ*),* which correspond to the best minimal value of the objective function *f(I,V,ϴ)*, we suggest using the swarm-based meta-heuristic GWO method.

The meta-heuristic methods are known for their simplicity, flexibility, derivation free process and the ability to find the global optimal solution. They are also appropriate for a diversity of problems without changing on their main structure. These methods can be based on a single solution or on population of solutions. The basic concepts can be obtained through exploration (exploring all of the search space and thus avoiding local optimum) and exploitation (investigating process in detail of the promising search space area).

Swarm-based intelligence (SI) methods, which derive from meta-heuristics, are based on the smart collective behavior of decentralized and self-organized swarms to ensure some biological needing such as food or security. A detailed discussion about the recent smart swarm-based algorithm, known as GWO is presented as follow.

Grey Wolf optimizer (GWO) algorithm, developed by Mirjalili in 2014, is a recent smart swarm-based meta-heuristic approach [50–52]. This algorithm mimics the leadership hierarchy and hunting process of Grey wolves in the wildlife. The following points represent the hierarchy in a wolf's group, which is about 5 to 12 members.


**Figure 4.** *The social hierarchical structure of Grey wolves (dominance decreases from the top-down) [51].*


When a pack of wolves sees a prey such as (gazelle, rabbit or a buffalo) they attack it in three steps and do not recede, **Figure 5**.

These three steps of the hunting process can be mentioned as follows.


**41**

stops moving.

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The mentioned above social hierarchy and hunting process of Grey wolves have

• The first, second and third best solutions are considered as α, β and δ wolves,

The following equations are used to model the encircling first step of Grey

where *i* represents the current iteration. *X* and *X*p represent the position vectors of the wolves and the prey, respectively. *A* and *C* are the coefficients and are

=∗∗ −

= ∗

where *a* is linearly decreasing from *2* to *0* throughout iterations, and *r*1, *r*2 are random values in an interval from *0* to *1*. In GWO, decreasing the values of *A*, from *2* to *0* during the optimization process, simulates the prey approach and provides the exploration ability of the algorithm. Besides, the exploitation ability of the

To mathematically simulate the second step of the Grey wolves hunting process, we suppose that the alpha (best candidate solution), beta and delta have a better knowledge about the potential location of the prey [53]. Therefore, the first three best solutions obtained so far are saved and oblige the other search agents (including the omegas) to update their positions according to the position of the best

=∗ − *D CX X*

= −∗ *X X AD* 1,2,3 , , 1,2,3 , , αβδ

> ( ) + + + =

The final third step is the hunting process as attacking the prey as soon as it

<sup>123</sup> <sup>1</sup>

αβδ

3

αβδ

*XXX X i* (22)

, , 1,2,3 , ,

=∗ − ( ) ( ) *D C X i Xi <sup>P</sup>* (16)

( + = −∗ ) ( ) *Xi X i A D* <sup>1</sup> *<sup>p</sup>* (17)

*A ar a* <sup>2</sup> <sup>1</sup> (18)

*C r* <sup>2</sup> <sup>2</sup> (19)

(20)

(21)

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

respectively.

wolves hunting process:

calculated as follows:

GWO comes from the random value of *C*.

search agents. In this regard, the following formulas are used.

αβδ

been mathematically modeled in GWO, as follows [51, 52]:

• The rest of the candidate solutions are considered as ω.

**Figure 5.** *The process of hunting prey by a group of wolves [51].*

The mentioned above social hierarchy and hunting process of Grey wolves have been mathematically modeled in GWO, as follows [51, 52]:


The following equations are used to model the encircling first step of Grey wolves hunting process:

$$\vec{D} = \left| \vec{C} \* \overrightarrow{X\_P}(i) - \vec{X}(i) \right| \tag{16}$$

$$
\overrightarrow{X}\left(i+1\right) = \overrightarrow{X\_p}\left(i\right) - \overrightarrow{A} \* \overrightarrow{D} \tag{17}
$$

where *i* represents the current iteration. *X* and *X*p represent the position vectors of the wolves and the prey, respectively. *A* and *C* are the coefficients and are calculated as follows:

$$
\vec{A} = 2 \,\, \* \, \vec{a} \,\, \* \, \overleftarrow{r\_1} - \vec{a} \tag{18}
$$

$$\vec{C} = \mathbf{2} \ast \overleftarrow{\mathbf{r}\_2} \tag{19}$$

where *a* is linearly decreasing from *2* to *0* throughout iterations, and *r*1, *r*2 are random values in an interval from *0* to *1*. In GWO, decreasing the values of *A*, from *2* to *0* during the optimization process, simulates the prey approach and provides the exploration ability of the algorithm. Besides, the exploitation ability of the GWO comes from the random value of *C*.

To mathematically simulate the second step of the Grey wolves hunting process, we suppose that the alpha (best candidate solution), beta and delta have a better knowledge about the potential location of the prey [53]. Therefore, the first three best solutions obtained so far are saved and oblige the other search agents (including the omegas) to update their positions according to the position of the best search agents. In this regard, the following formulas are used.

$$\overrightarrow{D\_{\alpha,\beta,\delta}} = \left| \overrightarrow{C\_{1,2,3}} \* \overrightarrow{X\_{\alpha,\beta,\delta}} - \vec{X} \right| \tag{20}$$

$$
\overrightarrow{X\_{1,2,3}} = \overrightarrow{X\_{\alpha,\beta,\delta}} - \overrightarrow{A\_{1,2,3}} \* \overrightarrow{D\_{\alpha,\beta,\delta}} \tag{21}
$$

$$
\vec{X} \left( i + \mathbf{1} \right) = \frac{\vec{X}\_1 + \vec{X}\_2 + \vec{X}\_3}{3} \tag{22}
$$

The final third step is the hunting process as attacking the prey as soon as it stops moving.

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**Figure 6.**

*PV parameters identification steps using the hybrid LM approach with GWO approach.*
