1. Introduction

2. PV Modeling and Problem Formulation

3. Proposed Optimization Algorithm

3.1. Overview of CO Algorithm

Akbari et al. [32] recently developed the CO algorithm as a powerful optimization algorithm for mimicking specific cheetahs’ hunting strategies. This algorithm utilizes three important strategies: searching for prey, sitting and waiting, and attacking. The algorithm introduces leaving the prey and returning home to avoid getting stuck in local optimal points. In this section, the mathematical model of the CO algorithm is explained, then the ICO algorithm is presented.

Based on these strategies, as shown in Figure 2, cheetah populations are formed in different arrangements. The probable hunting arrangements of each cheetah are considered equivalent to the solution to the problem. It is assumed that the best position among the population determines the prey (best solution). Cheetahs adjust their possible arrangements to optimize their performance during the hunting period.

3.1.1. Searching Strategy

A cheetah scans its surroundings or searches for suitable prey based on environmental conditions and hunting behavior. A mathematical model's searching phase looks like this [32]:

$$X_{i,j}^{t+1}=X_{i,j}^t+{\widehat{r}}_{i,j}^{-1}.{\alpha }_{i,j}^t$$

(18)

So that $X_{i,j}^t$ represents the current arrangement and $X_{i,j}^{t+1}$ represents the new arrangement of cheetah $i$ at hunting time $t$. The inverse of a normally distributed random number ${\widehat{r}}_{i,j}$ represents the randomization parameter. Besides, the random step length is defined by ${\alpha }_{i,j}^t$, where is expressed for the leader as follows [32]:

$${\alpha }_{i,j}^t=0.001\times t/T\times (U_j-L_j)$$

(19)

Where $U_j$ and $L_j$ are the upper and lower limits of the variable $j$. The length of a hunting time is represented by $T$. For other members of a group of cheetahs, the random step length is expressed based on the distance of the cheetah $i$ and arbitrarily selected cheetah $k$ in a group as follows [32]:

$${\alpha }_{i,j}^t=0.001\times t/T\times (X_{i,j}^t-X_{k,j}^t)$$

(20)

3.1.2. Sitting-and-Waiting Strategy

Cheetahs are swift hunters. During the chase, speed and flexibility require much energy. Therefore, the duration of the attack and chase cannot be long. As a result, one of the important strategies of cheetahs during the hunting process is to wait until the prey is close enough to them. Then, they start the attack. Hunting success can be increased by this behavior, which is modeled as follows [32]:

$$X_{i,j}^{t+1}=X_{i,j}^t$$

(21)

3.1.3. Attacking Strategy

At the appropriate time, cheetahs attack their prey. Speed and flexibility are two critical factors that the cheetah exploits during its attack. Cheetahs attack with maximum speed so that the cheetah must reach a close distance from their prey in the shortest possible time. In this case, the prey notices the cheetah's attack and starts to run away. Because of the cheetah's high speed and short distance from the prey, the prey prefers to escape by changing directions suddenly. Therefore, the cheetah uses its high flexibility to place the prey in unstable conditions and catch it. Attacks may take place individually or in groups. In solo attack mode, the cheetah's position change is adjusted based on the position of the prey. This can be done interactively in a group attack based on the status of other members of the group and the prey. This strategy can be expressed using the following mathematical model [32]:

$$X_{i,j}^{t+1}=X_{B,j}^t+{\check{r}}_{i,j}.{\beta }_{i,j}^t$$

(22)

$${\check{r}}_{i,j}=\left|r_{i,j}{|}^{\mathrm{e}\mathrm{x}\mathrm{p}(r_{i,j}/2)}\mathrm{s}\mathrm{i}\mathrm{n}\right(2\pi r_{i,j})$$

(23)

Where $X_{B,j}^t$ is the prey position; ${\check{r}}_{i,j}$ is the turning factor which reflects the sudden changes of the prey while fleeing; and $r_{i,j}$ is a randomly chosen value from a normal distribution. The interaction factor is defined by ${\beta }_{i,j}^t$ in (22) which is expressed as follows [32]:

$${\beta }_{i,j}^t=X_{k,j}^t-X_{i,j}^t$$

(24)

3.1.4. Strategy Selection Mechanism

Choosing the right strategy in the CO algorithm is done randomly [32]. Let $r_2$ and $r_3$ be two random numbers from a uniform distribution. If $r_2$ is greater than $r_3$, the sit-and-wait strategy is selected; otherwise, one of the search or attack strategies takes place. There is a condition between the two strategies of search and attack, which is controlled based on the H factor (see Figure 3). This factor decreases with time, which is expressed as follows [32]:

$$H={\mathrm{e}}^{2(1-t/T)}(2r_1-1)$$

(25)

where $r_1$ is a random value from [0, 1]. A condition has been set between these two strategies so that searching is the most likely choice at the start of hunting season. An attack will likely occur as the time of hunting progresses.

The pseudo-code of CO is summarized in Algorithm 1 [32].

Algorithm 1: The CO Algorithm

3.2. Improved Cheetah Optimizer (ICO) Algorithm

The CO algorithm has shown good capabilities in solving large-scale problems. However, as will show in the experimental results, it needs improvement in terms of convergence speed and computing time in identifying the parameters of photovoltaic models. For this purpose, a modified version of the CO algorithm is presented to cover these shortcomings.

3.2.1. Searching Strategy

In the search mode of the CO algorithm, each cheetah updates its position based on its previous position. This is when cheetahs usually follow the leader of the group and follow him. On this basis, the searching strategy in (18) is modified based on the leader position (second-best cheetah’s position in the population), $X_{L,j}^t$, as follows:

$$X_{i,j}^{t+1}=X_{L,j}^t+{\widehat{r}}^t.{\alpha }_{i,j}^t$$

(26)

Where, the randomization parameter (${\widehat{r}}^t$), and random step length (${\alpha }_{i,j}^t$) are modified as follows:

$${\widehat{r}}^t=r\prime /r″$$

(27)

$${\alpha }_{i,j}^t=X_{k\prime ,j}^t-X_{i\prime ,j}^t$$

(28)

Here, $r\prime$ and $r″$ are random values of the normal distribution function. $X_{k\prime ,j}^t$ and $X_{i\prime ,j}^t$ are the position of k’th and i'th cheetahs in the sorted population.

It is worth noting that updating the position of each cheetah around the position of the group leader can help the local search phase. In addition, the second term on the right side of the relationship (26) causes the diversity of solutions and thus contributes to the global search phase (exploitation phase). Also, by creating long steps during the hunting period, the random parameter will cause the solution to go out of the range of variables and thus be replaced with the new random solution in the population. Consequently, in addition to diversifying the solutions, it can prevent the algorithm from getting stuck in local optimal points.

3.2.2. Attacking Strategy

Moreover, the attacking strategy in the ICO algorithm is reformulated as follows:

$$X_{i,j}^{t+1}=X_{B,j}^t+{\check{r}}^t.{\beta }_{i,j}^t$$

(29)

Where ${\check{r}}^t$ is a random value from [0, 1].

In the CO algorithm, the interaction factor is expressed using the position of the adjacent cheetah (see Eq. (24)). While cheetahs usually attack their prey singly. Therefore, their position should be adjusted based on the position of the prey. Hence, in this proposed attack strategy, each cheetah updates his/her position relative to the prey during the attack mode and moves towards it, which is defined as follows:

$${\beta }_{i,j}^t=X_{B,j}^t-X_{i,j}^t$$

(30)

This proposed attack strategy helps the CO algorithm to find the near-optimal solution faster. Therefore, the local search capability (exploitation phase) of the CO algorithm is enhanced and its convergence speed will be increased.

The pseudo-code of the proposed ICO is summarized in Algorithm 2.

Algorithm 1: The ICO Algorithm

4. Experimental Results

5. Conclusion

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