## 4 Applications in Optimization
### Non-Evolutionary Chaotic Optimization Algorithms
#### 4.5.1 Chaotic Optimization Algorithms  
The tuning of PID controller (see Sections 4.1.2 and 4.3.2) was addressed by Coelho [95] using a non-evolutionary chaotic optimization approach based on the Lozi map. He considered an automatic regulator voltage (AVR) system. The problem of the dynamic stability of power systems has challenged power system engineers for decades. In a synchronous generator, the electromechanical coupling between the rotor and the rest of the system causes it to behave in a manner similar to a spring mass damper system, which exhibits oscillatory behavior around the equilibrium state following any disturbance, such as a sudden change in load, change in transmission line parameters, or fluctuations in the turbine output. Synchronous generator excitation control is one of the most important measures to enhance power system stability and to guarantee the quality of electrical power. The transfer function of such a system is provided by  
\[G(s)=\frac{0.07s+7}{0.0004s^{4}+0.0454s^{3}+0555 sr+1.51s+8}. \tag{63}\]  
The proposed chaotic optimization algorithm (CAO) (called the Chaotic Optimization Method based on Lozi map (COLM) by Coelho) is described as follows. The Lozi map for this CAO is defined by  
\[\left\{ \begin{array}{l}y_{1}(k)=1-a|y_{1}(k-1)|+y(k-1),\\ y(k)=by_{1}(k-1),\\ z(k)=\frac{y(k)-\alpha}{p-\alpha},\end{array} \right. \tag{64}\]  
where \(k\) is the iteration number. Similar to (30), the values of \(y\) are normalized in the range \([0,1]\) to each decision variable in the \(n\)-dimensional space of the optimization problem. Thus, we have \(y_{1}\in[-0.6418,0.6716]\) and \([\alpha,\beta]=[-0.6418,0.6716]\) for \(a=1.7\) and \(b=0.5\). Many unconstrained optimization problems with continuous variables can be formulated as the following functional optimization problem. Find \(X\) to minimize \(f(X)\), \(X=[x_{1},x_{2},\ldots,x_{n}]\), where \(f\) is the objective function and \(X\) is the decision solution vector consisting of \(n\) variables. Here, \(x_{i}\) bounded by lower limits \((L_{i})\) and upper limits \((U_{i})\). The COLM can be illustrated as follows. _Step 1:_ Initialization of variables and initial conditions. Set \(k=1\), \(y_{1}(0)\), \(y(0)\) and set the  
initial best objective function \(\tilde{f}\). _Step 2:_ Algorithm for chaotic global search. Begin  
_While \(k\leq M_{g}\) do_  
\(x_{i}(k)=L_{i}+z_{i}(k)\times(U_{i}-L_{i})\)  
_If \(f(X(k))<\tilde{f}\) Then_  
\(\tilde{X}=X(k)\)  
\(\tilde{f}=f(X(k))\)  
_End if_  
\(k=k+1\)  
_End while_  
End  
_Step 3:_ Algorithm for chaotic local search. Begin  
_While \(k\leq(M_{g}+M_{L})\) do_  
_for \(i=1\) to \(n\)_  
_If \(r<0.5\) then (where \(r\) is a uniformly distributed random variable with_  
\(\text{range}\left[0,1\right]\)_  
\(x_{i}(k)=\bar{x}_{i}+\lambda z_{i}(k)\times\left|U_{i}-\bar{X}_{i}\right|\)  
_Else_  
\(x_{i}(k)=\bar{x}_{i}-\lambda z_{i}(k)\times\left|\bar{X}_{i}-L_{1}\right|\)  
_End if_  
_End for_  
_If \(f(X(k))<\tilde{f}\) then_  
\(\tilde{X}=X(k)\)  
\(\tilde{f}=f(X(k))\)  
_End if_  
\(k=k+1\)  
_End while_  
End  
Here, \(M_{g}\) and \(M_{L}\) are the maximum number of iterations of the chaotic global and local searches, respectively, while \(M_{g}+M_{L}\) is the stopping criterion in iterations. The step size in chaotic local search is \(\lambda\). The best objective function is \(\tilde{f}\), and \(\bar{X}\) is the best solution from the current run of the chaotic search. A suitable value for the step size usually provides a balance between global and local exploration abilities, and consequently a reduction in the number of iterations required to locate the optimum solution.