Network Topology and Graph

EEE442 Computer Method in Power System Analysis

Anawach Sangswang

Dept. of Electrical Engineering

KMUTT

2

Network Topology

� Any lumped network obeys 3 basic laws

� KVL

� KCL

� Ohm’s law

linear algebraic constraints

3

Graph Theory

� Complete descriptions

� Representation by points and lines

� Relationships between points

4

What is a graph?

� 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

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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

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Definitions

� 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

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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

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Definitions

� 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

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Directed Graph

� Making the roads into one-way streets

� Indicated by arrows

� What has happened at T ??

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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

G1G2

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Isomorphism: Counting

� Labeled graphs

� Unlabeled graphs

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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

� Two distinct edges e and f are adjacent if they

have a vertex in common

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Degree� Isolated vertex: a vertex of degree 0

� End vertex: a vertex of degree 1

� Handshaking lemma: In every graph,

� the sum of all vertex-degrees is an even number

� the number of vertices of odd degree is even

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Subgraph

� A subgraph of a graph G is a graph where each

vertex belongs to G and each edge also belongs

to G

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Subgraph

� Obtaining subgraph

� Deleting {e}

� Deleting {v}

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

� Hint: trying all the possibilities is not a good idea!

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Problem

� A possible solution

� and

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Connectivity

� 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

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Connectivity

� Trail: A walk in which all the edges are distinct

� Path: A walk in which all the vertices are distinct

(except v0 = vm)

� Closed path/trail: same beginning and end

vertices (v0 = vm)

� 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

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Connectivity

� A graph is connected if and only if there is a path

between each pair of vertices

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Connectivity

� A disconnecting set in a connected graph G is a

set of edges whose removal disconnects G

{ }3 6 7 8, , ,e e e e

{ }1 2 5, ,e e e

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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

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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

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Tree

� A graph G is called a tree

� G is connected

� G contains all nodes

� G has no loop (or cycle)

{ }1 5 6 3, , ,e e e e

{ }1 2 5 7 8, , , ,e e e e e

{ }2 4 7 6 3, , , ,e e e e e

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

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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

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Electrical Networks

� Want to find the current in each wire

� KVL

� VYXV –

� VWYV –

� VWYXV –

� Similarly, cycles VWYV, WZYW � VWZYV

� Need to get rid of redundancy27

1 1 2 2i R i R E+ =

3 3 4 4 2 20i R i R i R+ − =

1 1 3 3 4 4i R i R i R E+ + =

Redundant

Electrical Networks

� Using a fundamental set of cycles

� KVL

� VYXV,

� VYZV,

� VWZV,

� VYWZV,

� KCL

� Vertex X,

� Vertex V,

� Vertex W,

� Vertex Z,

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1 1 2 2i R i R E+ =

2 2 5 5 6 60i R i R i R+ + =

3 3 5 5 7 70i R i R i R+ + =

2 2 4 4 5 5 7 70i R i R i R i R− + + =

0 10i i− =

1 2 3 50i i i i− − + =

3 4 70i i i− − =

5 6 70i i i− − =

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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

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Incidence Matrix: Directed Graph

� For a directed graph G with n nodes and b

branches, the incidence matrix

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

ij n bA a

× =

ija

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Example

� KCL

1 2 3 4 5 6

1 1 0 0 1 1 0

2 0 1 0 0 1 1

3 0 0 1 1 0 1

− − −

1

2

3

4

5

6

1 0 0 1 1 0 0

0 1 0 0 1 1 0

0 0 1 1 0 1 0

i

i

i

i

i

i

− = − −

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Example

� KVL1

2

1

3

2

4

3

5

6

1 0 0

0 1 0

0 0 1

1 0 1

1 1 0

0 1 1

v

vv

vv

vv

v

v

= − −

−

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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

� Lemma: If G is a graph in which the degree of

each vertex is at least 2, then G contains a cycle

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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

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Konigsberg bridges problem

� Can you cross each of the seven bridges shown

below exactly once and return to your starting

point?

Konigsberg bridges problem

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Travelling Salesman Problem

� A travelling salesman wishes to visit every city and return to his starting point

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The shortest path problem

� Find the shortest path from A to L