## 7 Conclusions
  
where \(t_{ST}\) is the switching time of the teleccontrolled sectionalizers, and \(t_{S,msc}\) is the switching time of the manual sectionalizer _msc_. _A3. Scenario C (e.g., i = 30, k = 28, and msc = 29 in Figure 1):_  
Considering a network where islanding operation is not permitted, the following equations must be applied:  
\[ \begin{array}{l}\lambda_{i,k}=f_{k}\\ U_{i,k}=f_{k}\,t_{R,k}\end{array} ; \tag{10}\]  
otherwise, when islanding operation is permitted:  
\[ \begin{array}{l}\lambda_{i,k}=f_{k}\\ U_{i,k}=f_{k}\left[\begin{array}{c}t_{ST}+t_{S,msc}+t_{AV,msc}+\\ \left(1-\rho_{A,msc}\right)\,\left(t_{R,k}-t_{ST}-t_{S,msc}-t_{AV,msc}\right) \end{array}\right]\end{array} , \tag{11}\]  
where \(\rho_{A,msc}\) is the PoA of the island _msc_ that is created by manually opening the sectionalizer _msc_, and \(t_{AV,msc}\) is the time to be again available of the DGs in the island _msc_. _A4. Scenario D (e.g., i = 30, k = 35, and CBS j = 34 in Figure 1):_  
Considering a network where islanding operation is either permitted or not, the following equations must be applied:  
\[ \begin{array}{l}\lambda_{i,k}=0\\ U_{i,k}=0\end{array} . \tag{12}\]  
_A5. Scenario E (e.g., i = 30, k = 7 and j = 8 in Figure 1):_  
Considering a network where islanding operation is not permitted, the following equations must be applied:  
\[ \begin{array}{l}\lambda_{i,k}=f_{k}\\ U_{i,k}=f_{k}\,t_{R,k}\end{array} ; \tag{13}\]  
otherwise, when islanding operation is permitted:  
\[ \begin{array}{l}\lambda_{i,k}=f_{k}\,\left(1-\rho_{A,j}\right)\\ U_{i,k}=f_{k}\,\left(1-\rho_{A,j}\right)\,t_{R,k}\end{array} . \tag{14}\]  
_A6. Scenario F (e.g., i = 30, k = 5, j = 8, and msc = 6 in Figure 1):_  
Considering a network where islanding operation is not permitted, the following equations must be applied:  
\[ \begin{array}{l}\lambda_{i,k}=f_{k}\\ U_{i,k}=f_{k}\,t_{R,k}\end{array} ; \tag{15}\]  
otherwise, when islanding operation is permitted:  
\[ \begin{array}{l}\lambda_{i,k}=f_{k}\left[\begin{array}{c}\left(1-\rho_{A, j}\right)+\rho_{A,j}\left(1-\rho_{A,msc}\right)\end{array}\right]\\ U_{i,k}=f_{k}\,\left[\begin{array}{c}\left(1-\rho_{A,j}\right)\left(t_{ST}+ t_{S,msc}+t_{AV,msc}\right)+\\ \left(1-\rho_{A,msc}\right)\left(t_{R,k}-t_{ST}-t_{S,msc}-t_{AV,msc}\right) \end{array}\right]\end{array} . \tag{16}\]JA7. Scenario G (e.g., i = 30, k = 20, j = 8, and msc = 19 in Figure 1):  
Considering a network where islanding operation is not permitted, the following equations must be applied:  
\[ \begin{array}{c}\lambda_{i,k}=f_{k}\\ U_{i,k}=f_{k}\,\left(t_{ST}+t_{S,msc}\right)\end{array} ; \tag{11}\]  
otherwise, when islanding operation is permitted:  
\[ \begin{array}{c}\lambda_{i,k}=f_{k}\,\left(1-\rho_{A,j}\right)\\ U_{i,k}=f_{k}\,\left(1-\rho_{A,j}\right)\,\left(t_{ST}+t_{S,msc}\right)\end{array} . \tag{12}\]  
_A8. Scenario H (sub-scenario H1, e.g., i = 30, k = 3, j = 8, and telecontrolled sectionalizer tsc = 4- or -sub-scenario H2, e.g., i = 30, k = 1, j = 8, tsc = 4, and msc = 2 in Figure 1):_  
Considering a network where islanding operation is not permitted, the following equations must be applied in both sub-scenarios:  
\[ \begin{array}{c}\lambda_{i,k}=f_{k}\\ U_{i,k}=f_{k}\,t_{R,k}\end{array} ; \tag{13}\]  
otherwise, when islanding operation is permitted and sub-scenario H1 is considered:  
\[ \begin{array}{c}\lambda_{i,k}=f_{k}\left[\left(1-\rho_{A,j}\right)+\rho_{A, j}\left(1-\rho_{A,tsc}\right)\right]\\ U_{i,k}=f_{k}\left[\left(1-\rho_{A,j}\right)\left(t_{ST}+t_{AV,tsc}\right)+ \left(1-\rho_{A,tsc}\right)\left(t_{R,k}-t_{ST}-t_{AV,tsc}\right)\right]\end{array} , \tag{14}\]  
or, if sub-scenario H2 is considered:  
\[