PARÁMETRO CARACTERÍSTICAS  Variación discreta

Graph theory can be used to model networks. In Section 2.4 we also specified that a design criterion is to have a radially operated network. To check whether this constraint holds, we can use graph theory. In addition, graph theory can be used for load flow calculations, which is discussed in Section 3.5. In Section 3.4.1 we consider some concepts in graph theory and in Section 3.4.2 we discuss how we check the radial configuration constraint in an MV network.

3.4.1

Concepts in graph theory

This section is based on the work of Fritschy (2018) and Bondy & Murty (1976). In graph theory a graph is denoted as 𝐺(𝑉, 𝐸), Where 𝑉 is a set of nodes and 𝐸 a set the edges as shown below.

𝑉 = {𝑣1, 𝑣2, … , 𝑣𝑛}, 𝐸 = {𝑒1, 𝑒2, … , 𝑒𝑚} (3.4)

Figure 3.2 shows an example of an undirected graph 𝐺({𝑣1, 𝑣2, … , 𝑣5}, {𝑒1, 𝑒2, … , 𝑒6}).

26 This graph can be described using an adjacency matrix, which is a symmetrical matrix with

dimensions (𝑛 ∗ 𝑛). It describes which nodes are directly connected by edges. The adjacency 𝐴 matrix of the example above looks as following:

𝐴(G) = [ 0 1 0 1 0 1 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 1 0] 𝐴(G)𝑖𝑗 = {1,_0, 𝑖𝑓 𝑛𝑜𝑑𝑒 𝑖 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑜 𝑛𝑜𝑑𝑒 𝑗_{𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒} (3.5)

The graph has a one-on-one connection with its (undirected) incidence matrix 𝐶(𝐺). The incidence matrix shows which node is connected to which edge. Usually, the incidence matrix is denoted by 𝐼, but later we use 𝐼 for a vector of currents, which is why we use 𝐶 instead. The dimensions are

(𝑛 ∗ 𝑚), where 𝑛 is the total number of nodes and 𝑚 the total number of edges. The following

matrix is the incidence matrix of the example:

𝐶(G) = [ 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 0 1 1] 𝐶(G)𝑖𝑗= { 1, 𝑖𝑓 𝑛𝑜𝑑𝑒 𝑖 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑜 𝑒𝑑𝑔𝑒 𝑗 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (3.6)

The graph and incidence can be either a directed or an undirected graph. If the graph is directed, the direction of an edge is expressed by denoting:

𝐶(G)𝑖𝑗= { 1, 𝑖𝑓 𝑒𝑑𝑔𝑒 𝑗 𝑠𝑡𝑎𝑟𝑡𝑠 𝑖𝑛 𝑣𝑖 −1, 𝑖𝑓 𝑒𝑑𝑔𝑒 𝑗 𝑒𝑛𝑑𝑠 𝑖𝑛 𝑣𝑖 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (3.7) Connectivity

A graph is connected if and only if for each pair (𝑣, 𝑤) ∈ 𝑉 there exists a path (Fritschy, 2018). A

vertex cut of G is a subset 𝑉′ of 𝑉 such that G(V’) is disconnected. A k-vertex cut is a vertex cut of k elements. The connectivity 𝐾(𝐺) is the minimum k for which 𝐺 has a k-vertex cut, otherwise it is 𝑛 − 1. The connectivity of a graph 𝐾(𝐺) = 0 if the graph is either trivial or disconnected. A graph is said to be k-connected if 𝐾(𝐺) ≥ 𝑘.

Degree

The degree of a node is the number of edges that are connected to the node. If there is a node with a degree of 0, we know that this node is unconnected to the graph.

Cycles

A cycle is a (sub)graph whose nodes can be arranged in a cyclic sequence such that two nodes are adjacent if and only if they are consecutive in the sequence (Fritschy, 2018).

3.4.2

Checking the radial configuration criterion

In graph theory a radial configuration of a network is also referred to as a spanning tree or just a tree. (Fritschy, 2018) formulated four properties of a spanning tree. When 𝐺 is a graph with n nodes, the following properties are equivalent:

1. 𝐺 is a tree.

2. 𝐺 is connected and has 𝑛 − 1 edges. 3. 𝐺 has no cycles and has 𝑛 − 1 edges.

27 We can see the configuration as a subset of edges from the whole network. We know that we need a subset of in total 𝑛 − 1 edges from the original network to have a possibility of being a radial

configuration. In addition, the graph should be connected, such that we meet the second property. These characteristics are easily checked for a network and seem a logical way to check the radial configuration constraint. Rstudio offers a package ‘igraph’ in which a function ‘vertex.connectivity’ determines the connectivity of a graph.

We can also calculate the number of open edges that are present in a network that is radially operated. The following equation holds (Andrei & Chicco, 2008):

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑁𝑂𝑃𝑠 = 𝑞 = 𝑚 − 𝑛 + 𝑠 (3.8)

Where 𝑠 is the number of nodes that supply the network, which relate to the number of substations in our problem. The 𝑠 is added to the equation, because we would need one less edge to be able to have separate unconnected networks. Note that two substations should not be in direct contact with each other, otherwise we create a loop in the network and violate the radial operation constraint. The fact that we need separate unconnected networks in case of multiple substation is counteracting our strategy in checking the radial configuration criterion, as we stated that we check the

connectivity of the graph in combination with the 𝑛 − 1 edges. A way to solve this, is to aggregate all substations to one substation. When doing this, the property should hold again on the network that we create by doing this.