## 2 Distributed PV System and Proposed MPPT Algorithm
### Multi-Strategy Improved Tuna Swarm Optimization (ITSO)
#### 2.2.2 Spiral Foraging  
According to the standard TSO, the weight coefficients \(\alpha 1\) and \(\alpha 2\) control the global exploration and local development ability of the algorithm. Basic weight coefficients are represented mathematically as shown in Equations (14) and (15).  
\[\alpha_{1}=a+(1-a)\cdot\frac{t}{t_{\max}} \tag{14}\]  
\[\alpha_{2}=(1-a)-(1-a)\cdot\frac{t}{t_{\max}} \tag{15}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{**Type of Pattern**} & \multicolumn{2}{l}{**Irradiance**} & \multicolumn{2}{l}{**Maximum Peak**} \\  & \multicolumn{2}{l}{**Power Point**} \\  & \multicolumn{2}{l}{**W/m\({}^{2}\)**} & \multicolumn{2}{l}{**(MPP)/W**} \\ \hline \multirow{4}{*}{Pattern 1} & G11 = 600 & G12 = 600 & G13 = 600 \\  & G21 = 600 & G22 = 600 & G23 = 600 \\  & G31 = 600 & G32 = 600 & G3  
A larger weight coefficient is beneficial to the algorithm in obtaining a larger search space in the pre-iteration and mid-iteration. When the weight coefficient is smaller, the algorithm can better carry out local development in the later iteration and accelerate the convergence of the algorithm. The linearly changing weight coefficient will easily lead to the algorithm falling into the local optimum at the early stage of iteration, and the slow decline at the later stage will make the local development ability of tuna individuals insufficient.  
To balance the global and local search capabilities of TSO, \(\alpha\)1, and \(\alpha\)2 make nonlinear improvements, and the weight coefficient after improvement is shown in Formulas (16) and (17). The trend in the original weight coefficient and the improved weight coefficient with the number of iterations is shown in Figure 6. It can be seen in Figure 6 that the improved weight coefficient changes nonlinearly with \(\alpha\)1 and \(\alpha\)2. The range of change in the first half of \(\alpha\)2 enhances the global exploration ability of the algorithm. When the iteration reaches the later stage, \(\alpha\)1 and \(\alpha\)2 rapidly change to a stable value, making the individual quickly approach the optimal individual and fully complete the local development. Therefore, the improved weight coefficient can further balance the global and local search process [7].  
\[\alpha_{1}=1-0.3\sin\biggl{(}\frac{t}{t_{\max}}\times\frac{\pi}{2}\biggr{)}^{2} \tag{16}\]  
\[\alpha_{2}=0.3\sin\biggl{(}\frac{t}{t_{\max}}\times\frac{\pi}{2}\biggr{)}^{2} \tag{17}\]  
The mathematical equation of the final spiral feeding is as follows:  
\[X_{i}^{t+1}=\left\{\begin{array}{c}\alpha_{1}\cdot\left(X_{rand}^{t}+\beta \cdot\left|X_{rand}^{t}-X_{i}^{t}\right|\right)+\alpha_{2}\cdot X_{i}^{t},\;i=1,\\ \alpha_{1}\cdot\left(X_{rand}^{t}+\beta\cdot\left|X_{rand}^{t}-X_{i}^{t}\right| \right)+\alpha_{2}\cdot X_{i-1}^{t},\;i=2,3,\ldots,NP,\;,\;\text{if}\;rand< \frac{t}{t_{\max}}\\ \alpha_{1}\cdot\left(X_{best}^{t}+\beta\cdot\left|X_{best}^{t}-X_{i}^{t} \right|\right)+\alpha_{2}\cdot X_{i}^{t},\;i=1,\\ \alpha_{1}\cdot\left(X_{best}^{t}+\beta\cdot\left|X_{best}^{t}-X_{i}^{t} \right|\right)+\alpha_{2}\cdot X_{i-1}^{t},\;i=2,3,\ldots,NP\;\;,\;\text{if}\;rand \;\geq\frac{t}{t_{\max}}\end{array}\right. \tag{18}\]  
Figure 6: A comparison of weight coefficient improvements.  
where \(X_{\text{rand}}^{t}\) is a randomly generated reference point in the search space, \(X_{i}^{t+1}\) is the position after iteration, \(t_{\text{max}}\) is the maximum number of iterations, \(t\) is the current number of iterations, and \(X_{best}^{t}\) is the current best location.