# Optimal Design Parameters for Hybrid DC Circuit Breakers Using a Multi-Objective Genetic Algorithm
## 2 HCB Topology and Problem Statement
### Problem Statement  
Based on the explanation of the configuration and working principles of the HCB, it is obvious that the performance of the HCB depends directly on the parameters design of the commutation branch with the L\({}_{\text{P}}\) and C\({}_{\text{P}}\) elements, the discharging branch with the R element, and the reversing branch with the L element. These elements should be optimal parameters to satisfy the operation requirements of the HCB for HVDC grid application. Consequently, the HCB should have a low amplitude of fault current as well as the operation speed of fault current interruption and recovery time. Hence, in order to optimize parameters of the HCB, the operation of the HCB needs to minimize the peak fault current, interruption time, and recovery time. Each of the above characteristics is considered an objective function to determine the optimal values of HCB parameters (i.e., R, L, L\({}_{\text{P}}\), C\({}_{\text{P}}\)) [21; 28], and can be calculated as. \begin{table}
\begin{tabular}{c c c} \hline \hline
**States** & **Definition and Operation** & **Time** \\ \hline \multirow{4}{*}{Normal operation} & \multicolumn{2}{c}{Breaker operates in normal conduction} \\  & \multicolumn{2}{c}{condition:} \\  & \multicolumn{2}{c}{\(\bullet\)} & \(S_{1}\) and \(S_{2}\) have closed. \\  & \multicolumn{2}{c}{\(\bullet\)} & System current flows through \(S_{1}\) \\  & \multicolumn{2}{c}{and brings a nominal current.} \\ \hline \multirow{4}{*}{Fault inception} & \multicolumn{2}{c}{Fault current arrives at the circuit breaker} \\  & \multicolumn{2}{c}{location:} \\  & \multicolumn{2}{c}{\(\bullet\)} & Current increases. \\  & \multicolumn{2}{c}{\(\bullet\)} & DC side voltage starts to drop. \\ \hline \multirow{4}{*}{Relay time} & \multicolumn{2}{c}{Time for fault detection:} \\  & \multicolumn{2}{c}{\(\bullet\)} & Breaker receives trip signal sent \\  & \multicolumn{2}{c}{from system.} \\  & \multicolumn{2}{c}{\(\bullet\)} & Switch \(S_{1}\) begins to open first. \\ \hline \multirow{4}{*}{Fault interruption} & \multicolumn{2}{c}{After a delay time of switch \(S_{1}\):} \\  & \multicolumn{2}{c}{\(\bullet\)} & Switch \(S_{1}\) has opened. \\  & \multicolumn{2}{c}{\(\bullet\)} & Thyristors \(T_{1a-1b}\) or \(T_{2a-2b}\) \\  & \multicolumn{2}{c}{have closed.} \\  & \multicolumn{2}{c}{\(\bullet\)} & A current zero is generated in \(S_{1}\) \\  & \multicolumn{2}{c}{from the L\({}_{\text{P}}\)C\({}_{\text{P}}\) resonant circuit.} \\  & \multicolumn{2}{c}{\(\bullet\)} & The capacitor is charged until it \\  & \multicolumn{2}{c}{reaches the MOV clamping voltage.} \\  & \multicolumn{2}{c}{\(\bullet\)} & Current is then commutated into \\  & \multicolumn{2}{c}{the MOV.} \\  & \multicolumn{2}{c}{\(\bullet\)} & MOV absorbs the energy stored in \\  & \multicolumn{2}{c}{the system.} \\ \hline \multirow{4}{*}{Residual switch open} & \multicolumn{2}{c}{Residual current circuit breaker} \\  & \multicolumn{2}{c}{(S\({}_{2}\)) opens:} \\ \cline{1-1}  & \multicolumn{2}{c}{\(\bullet\)} & Current has reached leakage level \\ \cline{1-1}  & \multicolumn{2}{c}{(several mA).} \\ \cline{1-1}  & \multicolumn{2}{c}{\(\bullet\)} & Residual current is removed by \(S_{2}\). \\ \hline \multirow{2}{*}{Discharging operation} & Capacitor is discharged to the \\ \cline{1-1}  & \multicolumn{2}{c}{system voltage.} \\ \hline \multirow{2}{*}{Reversing operation} & \multicolumn{2}{c}{Polarity of the capacitor is changed and} \\ \cline{1-1}  & \multicolumn{2}{c}{opposite to the system voltage.} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summarizing sequence of operation of the HCB. * Peak fault current (I\({}_{\text{pk}}\)) is the maximum value of the fault current flowing through the HCB during fault current interruption. The fault current is computed by tracking the fault current, as given in (1). \[\text{f}_{1}=\text{I}_{\text{pk}}=\text{Peak}\{\text{i}_{\text{dc}}(\text{t})\}\] (1)
* Interruption time (T\({}_{\text{int}}\)) is the total time to clear the short-circuit fault from the grid.