## 4 Description of PODESCA and SBDMT
### Structure of SBDMT
#### 4.2.1 Sensitivity-Based Indices  
The SBIs are generally used to identify the most candidate buses for SC placement, where they are determined by the base-case load flow. Four indices are used in this paper; namely: VSI, PLI, VSenI, and VSM. Those indices are calculated as follows:  
Voltage Stability Index (VSI)  
Taking a distribution line between two buses \(k\) and \(k+1\), VSI at bus \(k+1\) is calculated by [35]  
\[VSI_{k+1}=\frac{4*X_{l}}{V_{k}^{2}}*\left(\frac{P_{k+1}^{2}}{Q_{k+1}}+Q_{k+1} \right), \tag{15}\]  
where \(X_{l}\) denotes the line reactance, \(V_{k}\) is the voltage at bus \(k\), \(P_{k+1}\) and \(Q_{k+1}\) are the active and reactive powers at bus \(k+1\), respectively. The candidate buses for SC placement are selected according to the highest VSI values [35]. Figure 2: Flowchart of the proposed permutated oppositional differential evolution sine cosine algorithm (PODESCA) and sensitivity-based decision-making technique (SBDMT). Power Loss Index (PLI)  
Considering one bus at a time (the slack bus is excluded), PLI is calculated by removing the reactive part of the load at that bus and then finding the active loss reduction (LR). Hence, it is given at bus \(k\) by [30]  
\[PLI_{k}=\frac{LR_{k}-LR_{min}}{LR_{max}-LR_{min}}. \tag{16}\]  
The buses are then sorted from the highest to the lowest values, where the first sorted buses are taken as the candidates for SC placement [30]. Voltage Sensitivity Index (V SenI)  
According to this index, the most voltage-sensitive buses are identified, which is a significant issue since those critical buses might put the system under the risk of collapse. By adding a SC with a size equal to 25% of the total system capacity and considering one bus at a time, V SenI at bus \(k\) is defined by [32]  
\[V SenI_{k}=\sqrt{\frac{\sum_{i=1}^{n}(1-V_{i})^{2}}{n}}. \tag{17}\]  
The candidate buses for SC placement are selected here according to the lowest V SenI values [32]. Voltage Stability Margin (VSM)  
This index also defines the \"near-collapse\" or \"weak\" buses in the system, where those buses might threaten the system's stability. It is determined at bus \(k+1\) by [26; 32]  
\[VSM_{k+1}=|V_{k}|^{4}-4*(P_{k+1}X_{l}-Q_{k+1}R_{l})^{2}-4*(P_{k+1}R_{l}+Q_{k+ 1}X_{l})*|V_{k}|^{2}, \tag{18}\]  
where \(R_{l}\) denotes the line resistance. According to the lowest VSM values, the candidate buses for SC placement are selected [26; 32]. After determining the indices' values, they are all normalized to be in the same domain between 0 and 1. Then, they are arranged in a (N \(\times\) M) matrix; (N is the total number of buses and M is the number of indices). This matrix is the input of TOPSIS. 4.2.1.5. TOPSIS  
TOPSIS was originally presented by [PERSON] and [PERSON] [43]. This MCDM technique selects two sets of solutions (the positive and negative ideal solutions: \(IS^{+}\), \(IS^{-}\)) from a decision matrix with N number of alternatives (total number of the RDS's buses) and M number of attributes (number of indices). From all the alternatives, \(IS^{+}\) has the best data; while \(IS^{-}\) has the worst data. By calculating the Euclidean distance of all points (taking a pair of positive and negative solutions at a time) and comparing them, the best point is determined which has the least distance from \(IS^{+}\) and the longest distance from \(IS^{-}\). The steps of TOPSIS are given as follows [43]:  
Step 1: set the N \(\times\) M input matrix (\(S_{l}\)). Step 2: find the normalized decision matrix (NDM) (\(S_{Nij}\)) from the input matrix in step 1 so that the dimensional problem is converted into nondimensional  
\[S_{Nij}=\frac{S_{Nij}}{\sqrt{\sum_{i=1}^{N}x_{ij}^{2}}}, \tag{19}\]  
where \(i=1,\dots,\)\(N\) and \(j=1,\dots,\)\(M\). Step 3: get the weighted NDM (\(WS_{Nij}\)) by  
\[WS_{Nij}=w_{j}\,\times\,S_{Nij}, \tag{20}\]  
where \(\sum\limits_{j=1}^{M}w_{j}=1\) represent the assumed weights. Step 4: define the positive and negative ideal solutions (\(IS^{+}\), \(IS^{-}\)) by  
\[