## 2 Selection of Vertiports
### K-Means Algorithm  
The data collected earlier are the number of commuters in each administrative district, and this is closely related to the demand when the UAM is commercialized. Therefore, vertiports should be in appropriate locations to meet users' needs. For this purpose, the data was clustered using the MATLAB K-means algorithm. The K-means algorithm is most commonly used to cluster data into \(k\) clusters. This minimizes the average squared Euclidean distance between the allocated data and the cluster center during the clustering process. The cluster center is defined as the mean or centroid \(\mu\) of the data in a cluster \(\omega\), as shown in Equation (1) [17].  
\[\overset{\rightarrow}{\mu}(\omega)=\frac{1}{|\omega|}\sum_{\overset{\rightarrow }{x}\in\omega}\overset{\rightarrow}{x} \tag{1}\]  
The measure of how well the centroids represent the data in their clusters is the residual sum of the squares or \(RSS\), which is the squared distance of each vector from its centroid summed over all vectors, as shown in Equation (2). The objective of the K-means algorithm is to minimize the \(RSS\).  
\[RSS=\sum_{k=1}^{K}\sum_{\overset{\rightarrow}{x}\in\omega_{k}}\left|\overset{ \rightarrow}{x}-\overset{\rightarrow}{\mu}(\omega_{k})\right|^{2} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**District** & **Number of Commuters** & **Number of Data** \\ \hline Gangnam-gu & 609,119 & 122 \\ \hline Jung-gu & 352,261 & 70 \\ \hline Seecho-gu & 300,469 & 60 \\ \hline Jongno-gu & 282,783 & 57 \\ \hline Seongnam-si & 231,130 & 46 \\ \hline � & � & � \\ \hline Namdong-gu & 112,460 & 22 \\ \hline Total & 7,959,469 & 1388 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Number of commuters and data in major district.  
Figure 2: Administrative districts represented by squares.  
The most important thing in the K-means algorithm is that the parameter \(k\) should be initially specified, and the entire process of clustering is repeated until the location of the cluster centers, where the data is allocated, remains unchanged. The iterative process can be simplified using the K-means function built in MATLAB, which represents the data earlier, but it is somewhat difficult to specify the \(k\)-value. As mentioned earlier, \(k\) is the number of clusters and also the number of vertiports. In other words, the number of vertiports must be determined first, and the number is then used for clustering.  
An arbitrary \(k\) value can be assigned using a rule of thumb [18] to determine the approximate \(k\) value for clustering or an elbow method to compare the results while increasing the number of \(k\) values sequentially. However, according to Porsche Consulting in Germany, about 1050 eVTOLs are needed to replace various transportation types, such as private cars and public transportation for commuting in Sao Paulo, Brazil, which has 21 million people. In the case of vertiports, they announced they will initially start at existing transportation hubs, like airports and helipads. They initially planned five vertiports for 120 eVTOLs. They will then increase it to 40 and eventually to 100 vertiports [19]. Based on this information, it is assumed that 100 vertiports will be appropriate in the Seoul metropolitan area, which has a similar population to Sao Paulo.  
How clustered data were formed when the \(k\) value was 40 and 100 is presented in Figure 3. The data within the same cluster is presented as points of the same color, and the vertiports are symbolized as black stars.  
There are various ways to build vertiport infrastructures in the Seoul metropolitan area; however, huge costs will be required since they will be located in urban areas. Therefore, it is very important to specify the exact number of vertiports, i.e., the parameter \(k\).