# Optimal Design Parameters for Hybrid DC Circuit Breakers Using a Multi-Objective Genetic Algorithm
## 3 Parameters Optimization Using Multi-Objective Genetic Algorithm
### Process of the Proposed MOGA  
The MOGA mimics the process of natural selection to find the optimal solutions of optimal parameter design problems. Figure 3 shows the flowchart of the proposed MOGA for optimizing the parameters of the HCB. First, a set of input data (i.e., maximum generation; population size; constraint conditions; weight coefficients; and three operation probabilities of selection, crossover, and mutation) is initialized. Then, a population of individuals is normalized, where each individual consists of the values of parameters in the HCB topology (i.e., R, L, L\({}_{\text{P}}\), C\({}_{\text{P}}\)). These values are randomly generated using a uniform distribution with a lower and upper bound. For instance, the individual i is shown by the vector \(\text{X}_{\text{i}}=\left[\text{R}_{\text{i}},\text{L}_{\text{i}},\text{L}_{ \text{P},\text{i}},\text{C}_{\text{P},\text{i}}\right]\). The value of parameter \(\text{X}_{\text{i}}[\text{k}]\) is determined by Equation (14), where k is the index of the vector \(\text{X}_{\text{i}}\), \(\alpha\) is generated randomly between \([0,1]\), and \(\text{X}_{\text{i,min}}[\text{k}]\) and \(\text{X}_{\text{i,max}}[\text{k}]\) are the lower and upper bounds of parameter \(\text{k}^{\text{th}}\) of individual \(\text{X}_{\text{i}}\). \[\text{X}_{\text{i}}=\text{X}_{\text{i,min}}[\text{k}]+\alpha(\text{X}_{\text{ i,max}}[\text{k}]-\text{X}_{\text{i,min}}[\text{k}]) \tag{14}\]  
To evaluate an individual, the values of design parameters are implemented in the dynamic model of the HCB in the MATLAB/Simulink environment, and then the output signals for HCB operation, including the interruption time, recovery time, and peak fault current, are measured. These results are used to calculate the fitness function, as given in Equation (15). The value of the fitness function determines the chance of selecting an individual for the next generation (i.e., selection and crossover). \[\text{F}(\text{x})=\frac{1}{\text{f}^{\prime}(\text{x})} \tag{15}\]  
The individuals with higher fitness value have a higher probability of remaining in the next generation. This can be achieved via a selection operation with the roulette wheel approach, as expressed by Equation (16), in which the area of each segment is proportional to the fitness value of the individual. Then, the roulette wheel is spun a number of times that is equal to the population size to select the individuals with the highest fitness value. The selected individuals (parents) will go through crossover and mutation operations to produce new individuals (children). \[\text{P}_{\text{s}}=\frac{\text{F}(\text{x})}{\sum_{\text{i=1}}^{\text{n}} \text{F}(\text{x})} \tag{16}\]  
Crossover operation combines two parents (p\({}_{1}\) and p\({}_{2}\)) and produces children (c\({}_{1}\) and c\({}_{2}\)) for the next generation. This process is carried out by using a single-point crossover, and the crossover probability is the set of p\({}_{\text{c}}=0.8\). We often choose a random value (p\({}_{\text{r}}\)) between [0, 1] and compare it with the crossover probability. If p\({}_{\text{r}}\leq\text{p}_{\text{c}}\), the crossover is performed. The crossover point on both parents is randomly generated between [1, 4]. It determines the number of parameters that are taken from each parent. The parameters preceding the crossover point are copied from p\({}_{1}\) to c\({}_{1}\) and from p\({}_{2}\) to c\({}_{2}\). The parameters of p\({}_{1}\) following the crossover point are placed in the corresponding positions in c\({}_{2}\) and vice versa for the remaining parameters of p\({}_{2}\). Otherwise, if p\({}_{\text{r}}>\text{p}_{\text{c}}\), the parameters of p\({}_{1}\) and p\({}_{2}\) are directly copied into c\({}_{1}\) and c\({}_{2}\), respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter (Unit)** & **Range of Values** \\ \hline R (\(\Omega\)) & 40–80 \\ L (mH) & 2–40 \\ L\({}_{\text{P}}\) (mH) & 0.3–6.67 \\ C\({}_{\text{P}}\) (uF) & 1.8–24 \\ \hline \hline \end{tabular}
\end{table}