3. Graphical representation of the coal-fired generating station

In general, networks are any systems which admit abstract mathematical representations as graphs. The nodes (vertices) of these networks indicate these systems' components [2, 3]. The occurrence of a relation or intercommunication in the midst of the components in these networks is represented by the set of connecting links (edges). It can be noted that the preceding high level of abstraction can, in general, be applied to wide-ranging systems. Consequently, within that sense, a theoretical framework is provided by networks. This theoretical framework enables convenience of conceptual representation of system relations in which characterization at system level provides for component interactions mapping.

The thermal power plant system as shown in Figure 1 is represented in the form of a graph G ¼ ðN, LÞ of Figure 2, where N is the set of nodes (or vertices) and L the set of links (or edges) [2, 3]. Let each of the six sub-systems of the generating station be denoted by nodes Ni 0 sði ¼ 1, 2, …: 6Þ, and the interconnection between the systems ðNi, NjÞ is represented by links Lij 0 s ði, j ¼ 1, 2, …: 6 and i 6¼ jÞ joining nodes Ni and Nj. The flow of heat and energy, steam, water, pulverized fuel, and pre-heated air connects all the six sub-systems. Nodes and links aid in illustrating this flow in Figure 2. When the thermal power plant is graphically represented, this then is termed as the system structure function (i.e., as discussed in Section2) [2–4].

When the links (arcs/edges) can be traversed in both directions, the graph is undirected. On the other hand, the graph is directed if the links (arcs/edges) can be traversed only in one direction indicated by an arrow. If an undirected graph has no self-loops the presence of at least one link per node guarantees that all the nodes are connected [2, 3]. Practical structures are in general substantially more connected as compared to this minimal threshold. Consequently, there exist numerous paths between any node pair. For directed graphs, the connectivity property is unwieldy because the nodes can relate to any of the following three of categories: (1) the nodes that are strongly connected (i.e., for this subset the nodes can be arrived at from any other node that is a member of the subset. The access is through following the direction of the links); (2) the transient nodes that only have outgoing links. Therefore, transient nodes cannot be accessed from any other node; and (3) the absorbing nodes that only have ingoing links. Thus, once reached, the absorbing nodes cannot be left [2, 3].

The main intuitive and illustrative tool to be used, for the coal-fired generating station system structure function, is the directed graph [2, 3]. Intuitively the directed graph can be considered as a set of points (or vertices/nodes) with arrows (or arcs/links/edges) joining some of the points [7]. An arc may be labelled. Conventionally, a digraph consists of a set of vertices (nodes/points), V(N) and a subset of ordered pairs of arrows called the arcs (links). The labelling of the digraph is a function from the arcs (links) to the real numbers. One can visualize a labelled digraph by considering the vertices (nodes) as points with arcs (links) as arrows going from vertex (node) i to vertex (node) j whenever ði, jÞ belongs to the sets of arcs (links). The ith vertex (node) of the arc (link) ði, jÞ is called its initial vertex (node), while the jth vertex (node) is called its terminal vertex (node). The arc (link) is then given a label which is the representation of that arc (link) under the labelling function. When the initial and terminal vertices (nodes) are identical, the arc (link) is called a loop. It can be noted that sometimes an arc (link) originates from its initial vertex (node) and that it terminates into its terminal vertex (node). The number of arcs (links) that originate from a vertex (node) is called its out degree, and the number of arcs (links) that terminates into that vertex is called its in degree.
