Network Topology and Graph

Transcription

1 Network Topology Network Topology and Graph EEE442 Computer Method in Power System Analysis Any lumped network obeys 3 basic laws KVL KCL linear algebraic constraints Ohm s law Anawach Sangswang Dept. of Electrical Engineering KMUTT 2 Graph Theory What is a graph? Complete descriptions Representation by points and lines Relationships between points Another way of depicting situations It tells us Is there a direct road from one intersection to another How the electrical network is wired up Which football teams have played which Note: Intersection between PS and QT is not a vertex 3 4

2 Definitions Definitions P, Q, R, S, and T are called vertices Lines are called edges The whole diagram is called a graph Degree of a vertex is the number of edges with that vertex as an end-point This is the same graph as the previous one Two graphs are the same if two vertices are joined by an edge in one graph iff the corresponding vertices are joined by an edge in the other Here straightness and length are not in our concerns 5 6 Definitions Definitions Suppose the roads joining Q-S and S-T have too much traffic Build extra roads joining them and a car park at P Q-S or S-T are called multiple edges An edge from P to itself is called a loop Graphs with no loops or multiple edges are called simple graphs 7 8

3 Directed Graph Making the roads into one-way streets Isomorphism Isomorphism G1 and G2 are isomorphic if there is a one-one correspondence between the vertices of G1 and G2 such that the number of edges joining any two vertices of G1 is the same as those of G2 G1 G2 Indicated by arrows What has happened at T?? 9 10 Isomorphism: Counting Labeled graphs Adjacency and Incidence Two vertices, v and w, are adjacent if there is an edge v-w joining them The vertices, v and w, are incident with such an edge Unlabeled graphs Two distinct edges e and f are adjacent if they have a vertex in common 11 12

4 Degree Isolated vertex: a vertex of degree 0 End vertex: a vertex of degree 1 Subgraph A subgraph of a graph G is a graph where each vertex belongs to G and each edge also belongs to G Handshaking lemma: In every graph, the sum of all vertex-degrees is an even number the number of vertices of odd degree is even Subgraph Obtaining subgraph Problem Place the letters A, B,C, D, E, F, G, H into the eight circles in the figure below in such a way that no letter is adjacent to a letter that is next to it in the alphabet Deleting {e} Deleting {v} Hint: trying all the possibilities is not a good idea! 15 16

5 Problem A possible solution Connectivity and A walk is a way of getting from one vertex to another Walk: a finite sequence of edges/vertices in which any 2 consecutive edges/vertices are adjacent v w x y z z y w is a walk of length 7 from v to w Connectivity Trail: A walk in which all the edges are distinct Path: A walk in which all the vertices are distinct (except v 0 = v m ) Closed path/trail: same beginning and end vertices (v 0 = v m ) Cycle: A closed path with at least 1 edge v w x y z z x = trail v w x y z = path v w x y z x v = closed trail v w x y v = cycle Connectivity A graph is connected if and only if there is a path between each pair of vertices 19 20

6 Connectivity A disconnecting set in a connected graph G is a set of edges whose removal disconnects G Connectivity Cutset: A set of branch of a connected graph G whose removal of the set of branches results in a disconnected graph After the removal of the set of branches, the restoration of any one branch from the set results in a connected graph If a cutset has only 1 edge e, we call e a bridge { e, e, e} { e, e, e, e} Connectivity Edge connectivity: λ(g) The size of the smallest cutset in G G is k-edge connected if λ(g) k Separating set: a connected graph G is a set of vertices whose deletion disconnects G Connectivity: κ(g) The size of the smallest separating set in G G is k-connected if κ(g) k Tree A graph G is called a tree G is connected G contains all nodes G has no loop (or cycle) { e, e, e, e} { e, e, e, e, e} { e, e, e, e, e}

7 Tree: Spanning Tree Given any connected graph G, spanning tree is a tree that connects all the vertices of G Obtaining a spanning tree Choose a cycle and remove any one of its edges and the resulting graph remains connected Keep doing this until there are no cycles left Tree Adding any edge of G not contained in T to obtain a unique cycle The set of all cycles formed this way is the fundamental set of cycles of G Electrical Networks Want to find the current in each wire KVL VYXV VWYV VWYXV i1r 1+ i2r2 = E i R + i R i R = i R + i R + i R = E Similarly, cycles VWYV, WZYW VWZYV Need to get rid of redundancy Redundant Electrical Networks Using a fundamental set of cycles KVL VYXV, VYZV, VWZV, VYWZV, KCL Vertex X, Vertex V, Vertex W, Vertex Z, i1r 1+ i2r2 = E i R + i R + i R = = 0 i R i R i R i R i R + i R + i R = i0 i1 = 0 i1 i2 i3+ i5 = 0 i3 i4 i7 = 0 i5 i6 i7 =

8 Matrix Representation Adjacency matrix A is an nxn matrix whose ij-th entry is the number of edges joining vertex i and vertex j Incidence matrix M is an nxm matrix whose ij-th entry is 1 if vertex i is incident to edge j Incidence Matrix: Directed Graph For a directed graph G with n nodes and b branches, the incidence matrix A= a ij a ij n b where = 1 if branch j is incident at node i, and the arrow is pointing away from node i = -1 if branch j is incident at node i, and the arrow is pointing toward node i = 0 if branch j is not incident at node i Example Example KCL i1 i i = i i 5 i 6 KVL v v v 2 v1 v3 v = 2 v 4 v 3 v

9 Eulerian graphs G is Eulerian if there exists a closed trail containing every edge of G A non-eulerian graph G is semi-eulerian if there exists a trail containing every edge of G Theorems Euler 1736: A connected graph G is Eulerian iff the degree of each vertex of G is even Corollary 1: A connected graph is Eulerian iff its set of edges can be split up into disjoint cycles Corollary 2: A connected graph is semi-eulerian iff it has exactly two vertices of odd degree Lemma: If G is a graph in which the degree of each vertex is at least 2, then G contains a cycle Konigsberg bridges problem Can you cross each of the seven bridges shown below exactly once and return to your starting point? Konigsberg bridges problem 35 36

10 Travelling Salesman Problem A travelling salesman wishes to visit every city and return to his starting point The shortest path problem Find the shortest path from A to L 37 38