Load Flow Studies
GRAPHS § § § § § § Elements – line segments in a graph that represent network components. Nodes – the terminals of the line segment. Incident node and element – if the node is terminal of the element. Graph – shows the geometrical interconnection of the elements of a network. Subgraph – any subset of elements of a graph. Path – a subgraph of connected elements with no more than two elements connected to any one node. § Connected graph – if and only if there is a path between every pair of nodes. § Oriented graph – if each element of a connected graph is assigned a direction.
Example :
Single-line diagram:
2
4
1 G
G
3
G
Oriented connected graph:
7
1
6
2
3
4
5
4
2 1
3
0
Bus Admittance Matrix 1 -- A.C. Nerves, U.P. Dept. of Electrical & Electronics Engineering, Nov. 13 2003
1
§ Tree – a connected subgraph containing all nodes of a graph but no closed path. § Branches – the elements of a tree § Number of branches b required to form a tree:
b = n −1
where n = no. of nodes in the graph
§ Links – elements of the connected graph that are not included in the tree. § Cotree – a subgraph formed by the links of a connected graph. § Number of links l of a connected graph: l =e−b It follows that l = e − n +1
where e = no. of elements of a connected graph
Example :
7
1
6
2
3
4
5
4
2 1 Branch Link
0
3
e=7 n=5 b=4 l=3
Bus Admittance Matrix 1 -- A.C. Nerves, U.P. Dept. of Electrical & Electronics Engineering, Nov. 13 2003
2
INCIDENCE M ATRICES
ˆ Element-Node Incidence Matrix A
The elements of the element-node incidence matrix of a connected graph are as follows:
aij = 1 aij = −1 aij = 0
if the ith element is incident to and oriented away from the jth node. if the ith element is incident to and oriented towards from the jth node. if the ith element is not incident to the jth node.
The dimension of the matrix is e × n.
Example :
For the previous
Example :
Single-line diagram:
2
4
1 G
G
3
G
Oriented connected graph:
7
1
6
2
3
4
5
4
2 1
3
0
Bus Admittance Matrix 1 -- A.C. Nerves, U.P. Dept. of Electrical & Electronics Engineering, Nov. 13 2003
1
§ Tree – a connected subgraph containing all nodes of a graph but no closed path. § Branches – the elements of a tree § Number of branches b required to form a tree:
b = n −1
where n = no. of nodes in the graph
§ Links – elements of the connected graph that are not included in the tree. § Cotree – a subgraph formed by the links of a connected graph. § Number of links l of a connected graph: l =e−b It follows that l = e − n +1
where e = no. of elements of a connected graph
Example :
7
1
6
2
3
4
5
4
2 1 Branch Link
0
3
e=7 n=5 b=4 l=3
Bus Admittance Matrix 1 -- A.C. Nerves, U.P. Dept. of Electrical & Electronics Engineering, Nov. 13 2003
2
INCIDENCE M ATRICES
ˆ Element-Node Incidence Matrix A
The elements of the element-node incidence matrix of a connected graph are as follows:
aij = 1 aij = −1 aij = 0
if the ith element is incident to and oriented away from the jth node. if the ith element is incident to and oriented towards from the jth node. if the ith element is not incident to the jth node.
The dimension of the matrix is e × n.
Example :
For the previous