## 2 PSs Siting and Timing: Technical Background and Problem Formulation
### Perpendicular Bisector Rule and the Voronoi Diagram  
Given \(n\) sites in a two-dimension space \(s_{1}\), \(s_{2}\),..., \(s_{n}\) the easiest graphical way of dividing the space in a tiling of regular polygons is applying the so-called perpendicular bisector. As shown in Figure 3, the application of this rule consists of the two following steps: (i) draw a straight line (dashed red lines in Figure 3) between a site (the red square in Figure 3) and each of its neighbors (the blue squares in Figure 3); (ii) perpendicularly bisect each of those lines, i.e., divide it in two with a line that intersects it at a ninety-degree angle (green lines in Figure 3). The second step determines a set of geometrical lines equidistant among the sites and their neighbors. Then, the set of all such lines around a site encloses the area closer to it than any other site.  
The Voronoi diagram represents a computerized perpendicular bisector method [24; 25]. In mathematics, a Voronoi diagram is a way of dividing space into several regions. A set of points called sites is specified beforehand, and for each site, there will be a corresponding region consisting of all points closer to that site than any other. The regions are called Voronoi cells. Thus, given \(n\) sites \(s_{1}\), \(s_{2}\),..., \(s_{n}\) in a distance space \((P,d)\), partition \(P\) into regions \(S_{1}\), \(S_{2}\),..., \(S_{n}\), such that:  
\[S_{i}=\big{\{}p\in\ P\ \big{|}\ d(p,s_{i})<d\big{(}p,s_{j}\big{)},i\
eq j\big{\}} \tag{3}\]  
The Euclidean metric can be used as a distance parameter so that the distance \(d(A,B)\) between two points, \(A\) and \(B\), is given by the following Equation (4):  
\[d(A,B)=d[(a_{1},a_{2}),(b_{1},b_{2})]=\sqrt{(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}} \tag{4}\]  
In two dimensions, the Voronoi diagram partitions the plane into cells, edges, and vertices:  
* Voronoi edges: where the closest neighbor of a point changes. Every point on a Voronoi edge is equidistant from at least two sites.
* Voronoi cells: all points closer to a given site than any other site.
* Voronoi vertex: the intersection of at least three edges. It is equidistant from at least three sites.  
Figure 4 shows the Voronoi diagram, in two-dimension space, for fifteen sites. The sites are depicted using red squares, while the Voronoi edges are the green lines.  
Figure 3: Example of the perpendicular bisector rule.  
The Voronoi diagram could be used to create the PS service areas. The sites aforementioned are the PSs feeding the DN, and, as a result, Voronoi cells are the service areas of each PS. To respect the rule of supplying the loads with the closest PS, any secondary substation (SS) and the related customers should be supplied from the PS of the cell it belongs to.  
Although, defining service areas considering only the geometrical distance is as easy as incorrect. Electrical information must be taken into account as well. In this sense, it could be more reasonable to use a weighted Voronoi diagram [26; 27]. In mathematics, a weighted Voronoi diagram is a Voronoi map where cells are defined using a metric that considers the standard distances, e.g., Euclidean metric, modified by weights assigned to the site points. Therefore, given \(n\) sites \(s_{1}\), \(s_{2}\),..., \(s_{n}\) and given an associated weight \(w_{i}\) for each site \(s_{i}\), the weighted distance from a point \(p\) to a site \(s_{i}\) is defined as in (5):  
\[d_{w}(p,s_{i})=\frac{|p-s_{i}|}{w_{i}} \tag{5}\]  
Referring to this metric, we could select, for each PS, weight proportional to its capacity and the power density of its service area. Thus, installing a PS with large weight and power, located in an area with less power density, is closer to many SSs.