## 3 Decomposition Method for Countermeasures against Excessive Fault Currents
### Main Problem  
Instead of direct dealing with discrete main decision variables for bus splitting and CLRs, this paper adopts a way of modeling them by inserting continuous or intermediate reactances as shown in Figure 2. At the final stage of the main problem process, the solution needs to be discretized. The concept of the inserted reactance for bus splitting was originally proposed in [16], and it can be extended for current limiting reactors. The formulation of the main problem can be expressed as follows:  
\[min\sum_{k=1}^{N_{\mathcal{C}AN}}X_{k} \tag{8a}\] \[s.t.\sum_{k=1}^{N_{\mathcal{C}AN}}S_{i,k}^{(1)}\left(X_{k}^{(1)}-X_{k}^{(0)}\right) \leq\alpha\left(I_{\text{CB}i}-I_{fi}^{(0)}\right)\] (8b) \[\sum_{k=1}^{N_{\mathcal{C}AN}}S_{i,k}^{(n)}\left(X_{k}^{(n)}-X_{k}^{(n-1)}\right) \leq\alpha\left(I_{\text{CB}i}-I_{fi}^{(0)}\right)\] (8c) \[X_{k}^{(min)}\leq X_{k}\leq X_{k}^{(max)}\] (8d) \[i=1,\ldots,N_{\text{F}LC}\quad k=1,\ldots,N_{\text{C}AN}\;n=1,\ldots,iter \tag{8e}\]  
where \(X_{k}\) and \(X_{k,0}\) denote the reactance at the \(k\)-th candidate location and its initial value. In Equation (8), \(N_{\mathcal{C}AN}\) and \(N_{\text{F}LC}\) are the total number of candidate locations and fault level constraints, respectively; \(S_{i,k}\) stands for the sensitivity of the fault current at bus \(i\) with respect to the change of the value of \(X_{k}\); \(I_{f,i}\) is the fault current level at bus \(i\) for every change in \(X;I_{\text{CB},i}\) is the breaking duty of CB at bus \(i\); and \(X_{k}^{min}\) and \(X_{k}^{max}\) are the lower and upper limit of \(X_{k}\). Lastly, \(\alpha\) is the acceleration coefficient. From the above formulation, one can notice that in the main problem the \(X\) variables are determined to satisfy the CB limits of the FLCs, minimizing the deviation from their initial values. As mentioned above, the aim of this paper is to deal with the two types of fault current countermeasures at one formulation and they are initially modeled with continuous reactances at candidate locations. In the case that the decision of main problem is made only by sensitivity information, it might cause infeasibility of constraints or a large increase in objective function values  
Figure 2: Inserted X variable for bus splitting and current limiting reactor installations. when applying the new X settings by main problem. The reason is that there exist nonlinear effects of transmission branch parameters. The nonlinearity in terms of branch parameter variations was investigated in [25]. In the preliminary simulation results of this paper, it was also noticed that there were some cases where inadequate settings of \(X\)s from the main problem caused infeasibility such as divergence of the subproblem with FFLC-OPF. Therefore, such a way to prevent those infeasible situations need to be implemented, and it is done by modifying the sensitivity of the reactance causing a rather large change in the generation cost or infeasibility. The modified formulation of the sensitivity of \(i\)-th FLC with respect to the change in the \(k\)-th reactance (\(\Delta X_{k}\)) for the corresponding candidate countermeasure is as follows:  
\[S_{i,k}=\widetilde{S}_{i,k}U_{k} \tag{9a}\] \[S_{i,k}\equiv\frac{\Delta I_{f,i}}{\Delta X_{k}} \tag{9b}\]  
where \(\Delta I_{f,i}\) denotes the change in fault current with respect to \(\Delta X_{k}\) and \(U_{k}\) is the binary variable representing whether to apply the \(k\)-th candidate as a countermeasure. In addition, the binary variable, \(U_{k}\), is chosen by observing the change of the generation cost by OPF before solving the main problem as follows:  
\[U_{k}= \begin{cases}0,&\Delta\sum_{j=1}^{NG}C\big{(}P_{G,j}\big{)}>\beta\text{ \emph{infinite feasible}}\\ 1,&\text{\emph{otherwise}}\end{cases}  \tag{10}\]  
where \(\beta\) is the threshold value of generation cost increase and \(NG\) is the total number of generators.