*4.1. Marine Predator Algorithm*

The marine predator algorithm (MPA) is a bioinspired, metaheuristic optimization technique [26] that has been applied to various optimization problems. A few of the applications of MPA are estimating the parameters of solar PV cells [27], MPPT for solar PV systems [28], and many more. In this section, the MPA is applied in MPPT in an optimized way to the optimal expected output for EVs.

The key points of MPA are (i) the Levy motion for a prey environment of low concentration given in Equation (13), (ii) the Brownian motion for a prey environment of high

concentration given by Equation (14), and (iii) the very decent memory in recalling their partners and the location of successful hunting shown in Figure 9. These features make the marine predator's technique more advanced compared to other bioinspired techniques. The population can be started by Equation (15). Dmin and Dmax are the lower and upper limits for the variables, and rand is the random number.

$$\text{Lévy } (\alpha) = 0.05 \times \frac{\text{x}}{|\mathbf{y}|^{\frac{1}{\alpha}}} \tag{13}$$

$$\mathbf{f}(\mathbf{x};\ \mu,\ \sigma) = \frac{1}{\sqrt{2\pi}} \mathbf{e}^{-\frac{\chi^2}{2}} \tag{14}$$

$$\mathbf{D}\_0 = \mathbf{D}\_{\rm min} + \text{rand} \left( \mathbf{D}\_{\rm max} - \mathbf{D}\_{\rm min} \right) \tag{15}$$

**Figure 9.** Three phases in marine predator algorithm (MPA) optimization.

An elite matrix is developed by the fittest solutions among the marine predators following the survival of the fittest idea. Naturally, the topmost predators (denoted by de) are brilliant in hunting and is given in Equation (16). The position of the predator gets updated from time to time. The prey matrix is developed in which di,j gives the jth position of the prey and is given by Equation (17).

$$\text{Elite} = \begin{bmatrix} \text{de}\_{1,1} & \cdots & \text{de}\_{1,n} \\ \vdots & \ddots & \vdots \\ \text{de}\_{m,1} & \cdots & \text{de}\_{m,n} \end{bmatrix}\_{\mathbf{m}\times\mathbf{n}} \tag{16}$$

$$\text{Pray} = \begin{bmatrix} \mathbf{d}\_{1,1} & \cdots & \mathbf{d}\_{1,\mathbf{n}} \\ \vdots & \ddots & \vdots \\ \mathbf{d}\_{\mathbf{m},1} & \cdots & \mathbf{d}\_{\mathbf{m},\mathbf{n}} \end{bmatrix}\_{\mathbf{m}\times\mathbf{n}} \tag{17}$$

Optimization Process of MPA

There are three phases in optimization, as shown in Figure 9. Depending upon the velocity ratio and time, the phases are classified. Phase 1: predator is traveling slower than the prey (increased velocity ratio). Phase 2: predator and prey are at the almost same pace (unity velocity ratio). Phase 3: predator is traveling faster than the prey (decreased velocity ratio).

The prey is traveling faster than the predator. This is called as exploration phase, and it happens only in the initial or starting iterations of the algorithm and is given by Equations (18) and (19). Here, R is a rand [0, 1]. Set in a high exploration phase, this phase happens for the first three of the iterations. Prey is accountable for the exploration, and it is given by the Equations (20)–(26). That CF is a step-size controlling parameter for a predator:

$$\xrightarrow[]{\text{Stepsize}\_{\text{a}}} = \overset{\rightarrow}{\text{R}\_{\text{B}}} \times \left( \xrightarrow{\text{\tiny\\_}} \overset{\rightarrow}{\text{R}\_{\text{B}}} - \overset{\rightarrow}{\text{R}\_{\text{B}}} \times \xrightarrow{\text{\tiny\\_}} \text{Prey}\_{\text{a}} \right); \text{ a} = \text{i} \dots \text{n} \tag{18}$$

$$
\xrightarrow[]{\text{Prey}\_{\text{a}}} = \xrightarrow[]{\text{Prey}\_{\text{a}}} + \text{P}\overset{\rightarrow}{\text{R}} \times \xrightarrow[]{} \xrightarrow[]{} \text{Stepsize}\_{\text{a}}\tag{19}
$$

For the predator population:

$$\begin{aligned} \stackrel{\textstyle \text{Stepsize}}{\text{Stepsize}\_{\text{a}}} = \stackrel{\textstyle \text{R}}{\text{R}}\_{\text{L}} \times \left( \stackrel{\textstyle \text{Elite}}{\text{Elite}\_{\text{a}}} - \stackrel{\textstyle \text{R}}{\text{R}}\_{\text{L}} \times \stackrel{\textstyle \text{Prey}}{\text{Prey}\_{\text{a}}} \right); \mathbf{a} = \mathbf{i} \ldots \frac{\mathbf{n}}{2} \end{aligned} \tag{20}$$

$$\begin{array}{c}\hline\hline\text{Prey}\_{\text{a}} = \stackrel{\textstyle \longrightarrow}{\text{Prey}\_{\text{a}}} + \stackrel{\textstyle \longrightarrow}{\text{P}} \stackrel{\textstyle \longrightarrow}{\text{Stepsize}\_{\text{a}}}\\\hline\end{array} \tag{21}$$

For the prey population:

$$\mathbf{\dot{\color{red}{Stepsize}}}\_{\mathbf{a}} = \mathbf{\overset{\longrightarrow}{\mathbf{R}}}\_{\mathbf{B}} \times \left( \mathbf{\overset{\longrightarrow}{\mathbf{R}}}\_{\mathbf{B}} \times \mathbf{\overset{\longrightarrow}{\mathbf{Elite}}}\_{\mathbf{a}} - \mathbf{\overset{\longrightarrow}{\mathbf{Prey}}}\_{\mathbf{a}} \right); \mathbf{\overset{\longrightarrow}{\mathbf{a}}} = \frac{\mathbf{n}}{2} \dots \mathbf{n} \tag{22}$$

$$\xrightarrow[\text{Prey}\_{\text{a}}]{} \xrightarrow[\text{Elite}\_{\text{a}}]{} + \text{P} \xrightarrow[\text{CF}]{} \times \xrightarrow[\text{Stepsize}\_{\text{a}}]{} \tag{23}$$

$$\overrightarrow{\text{Stepsize}}\_{\text{a}} = \overrightarrow{\text{R}}\_{\text{L}} \times \left( \overrightarrow{\text{R}}\_{\text{L}} \times \overrightarrow{\text{Elite}}\_{\text{a}} - \overrightarrow{\text{Prey}}\_{\text{a}} \right); \text{ a} = 1 \ldots \text{n} \tag{24}$$

$$\begin{array}{c}\hline\text{Prey}\_{\text{a}} = \stackrel{\text{\longrightarrow}}{\text{Elite}\_{\text{a}}} + \text{P}\,\stackrel{\text{\longrightarrow}}{\text{CF}} \times \stackrel{\text{\longrightarrow}}{\text{Stepsize}\_{\text{a}}}\end{array} \tag{25}$$

To avoid the eddy formation or fish aggregating devices (FADs), which may change the marine predators' behavior, marine predators may take a long jump, as is given in Equation (26):

$$\begin{aligned} \stackrel{\textstyle \text{Prey}\_{\text{a}}}{\text{Prey}\_{\text{a}}} = \left\{ \begin{aligned} \stackrel{\textstyle \text{Prey}\_{\text{a}}}{\text{Prey}\_{\text{a}}} + \text{CF} \left( \stackrel{\textstyle \text{D}\_{\text{min}}}{\text{D}\_{\text{min}}} + \stackrel{\textstyle \text{R}}{\text{R}} \times \left( \stackrel{\textstyle \text{D}\_{\text{max}}}{\text{D}\_{\text{max}}} - \stackrel{\textstyle \text{D}\_{\text{min}}}{\text{D}\_{\text{min}}} \right) \times \stackrel{\textstyle \text{U}}{\text{U}} \right) \text{if } \text{r} \le \text{FAD}\_{1} \\\ \stackrel{\textstyle \text{Pray}\_{\text{a}}}{\text{Prey}\_{\text{a}}} + \left( \text{FAD}\_{\text{s}} \times (1-\text{r}) + \text{r} \right) \left( \stackrel{\textstyle \text{Pray}\_{\text{r}1}}{\text{Pray}\_{\text{r}1}} - \stackrel{\textstyle \text{Pray}\_{\text{r}2}}{\text{Pray}\_{\text{r}2}} \right) \text{if } \text{r} \le \text{FAD}\_{\text{s}} \end{aligned} \right. \end{aligned} \tag{26}$$
