## 4 Applications in Optimization
### Non-Evolutionary Chaotic Optimization Algorithms
#### 4.5.2 Improved Lozi Map-Based Chaotic Optimization Algorithm (ILCOA)  
The COLM algorithm (called the Lozi Chaotic Optimization Algorithm (LCOA)) was improved by Alizadeh et al. in [103]. ILCOA was employed to solve a single-machine Power System Stabilizer (PSS) design problem, and its promising features in comparison with LCA were highlighted. In ILCOA, the chaotic variable interval is partitioned into three areas, namely, \(A_{1}=[0,C_{1}]\), \(A_{2}=[C_{1},C_{2}]\) and \(A_{3}=[C_{2},1]\), in which \(0<C_{1}<0.5\) and \(0.5<C_{2}<1\) are adjustable parameters. During the chaotic global search, each time a value of the chaotic variable \(z_{i}\) is generated from the Lozi map and its validity is evaluated. If the chaotic variable \(z_{i}\) belongs to \(A_{1}\) or \(A_{3}\), it must be regenerated; otherwise, it is reserved and used for calculating the objective function and checking its optimum, as it certainly belongs to the area \(A_{2}\). The lower (\(L_{i}\)) and upper (\(U_{i}\)) limits of the solution variables have been reformed as well. Thus, in the chaotic global search subroutine of ILCOA, \(U_{i}\) and \(L_{i}\) are replaced by \(U_{i}^{new}\) and \(L_{i}^{new}\), respectively, where  
\[U_{i}^{new}=\frac{1-C_{1}}{C_{2}-C_{1}}U_{i}+\frac{1-C_{2}}{C_{2}-C_{1}}L_{i}, \tag{65}\]  
\[L_{i}^{new}=\frac{-C_{1}}{C_{2}-C_{1}}U_{i}+\frac{C_{2}}{C_{2}-C_{1}}L_{i}, \tag{66}\]  
\[z_{i}^{new}=\frac{1}{C_{2}-C_{1}}\times(z_{i}-C_{1}). \tag{67}\]  
and where, similar to the chaotic global search subroutine, \(C_{1}\leq z_{i}\leq C_{2}\); therefore, \(0\leq z_{i}^{new}\leq 1\). There is another parameter in the chaotic local search subroutine, namely, the step size \(\lambda\), which is an important parameter in the convergence behavior of optimization methods. It adjusts small ergodic ranges around \(\hat{X}\). The selection of the step size \(\lambda\) has a direct effect on the process of generating a new trial solution. Balance between the global and local exploration abilities is provided by an appropriate value of \(\lambda\). In [103], the authors introduced a new selection strategy for step size \(\lambda\), namely, the semi-exponential step size (SESS), which focuses more on smaller values of \(\lambda\) and contributes to better exploitation for refining the results:  
\[\left\{ \begin{array}{l}\lambda=\lambda_{max}-\frac{\lambda_{max}-\lambda_{mid}}{ iter_{max}^{2}}\times iteration,\quad 1\leq iteration\leq iter_{max}^{1}\\ \lambda=\left[\lambda_{max}-\frac{\lambda_{max}-\lambda_{mid}}{ iter_{max}^{2}}\right]\times e^{\frac{(inq_{max}-iteration)}{k}},\quad iter_{max}^{1}\leq iteration \leq iter_{max}^{2}\end{array} \right. \tag{68}\]  
where \(\lambda_{m}ax\) is the initial step size, \(\lambda_{mid}\) is the step size of the linear section ending, \(iter_{max}^{2}\) is the total number of searching generations in the local search subroutine, and \((iter_{max}^{2}=M_{L})\) represents the generations in which the step size is linearly reduced. Different ending values of \(\lambda\) can be achieved by adjusting \(k\). The results of the tests to solve a single-machine PSS design problem were promising, and showed that the ILCOA has features such as higher stability, better ability to escape from local minima, more robustness to search space size, and finer performance in comparison to LCOA, especially in high-dimensional problems with large search spaces. In [104], Pourmousa et al. used ILCOA to evaluate the performance of photovoltaic panels. This evaluation inevitably involves having the exact model of solar cells. Different approaches to solar cell modeling have been proposed in the literature, and can generally be classified as either traditional or intelligent methods. To obtain an accurate model of such highly nonlinear systems, however, is a challenging task that continues to defy researchers.