Explicación del método de eliminación sucesiva de kron para transformar sistemas eléctricos de potencia enormes a sistemas de eléctricos de potencia de menor tamaño equivalentes eliminando barras del sistema.
2. Content
Modification of Ybus matrix
Incidence Matrix and Ybus matrix
Example
Kron’s reduction
Example
Reference:
Análisis s de sistemas de potencia Grainger y Stevenson, Capítulo 7 2
3. Ybus Modification
To add or subtract an admittance between two nodes, it is enough to
add or subtract the corresponding building block from the original
Ybus matrix,
* *
* *
+
-
m n
m
n
Original Ybus
a a
a a
Y Y
Y Y
Building block of the branch to
be added or subtracted.
m n
m
n
3
4. Ybus Modification
When there are coupled branches:
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
M
a
M b
Y Y
Y Y
m n
m
n
p q
p
q
* * * *
* * * *
* * * *
* * * *
Original Ybus
Building block of the coupled
branches to be added or
subtracted
+
-
m n p q
m
n
p
q
4
5. Incidence Matrix and Ybus Matrix
Ia
Ib
Ic
Id Ie
5
6 4
4 6
2
10
a a
b b
c c
d d
e e
V I
j
V I
j j
V I
j j
j
V I
j
V I
The elementary admittance
matrix or Y primitive relates the
branch voltages and currents.
.
pr pr pr
Y V I
Equation 1
Network with data in admitance
5
6. Incidence Matrix and Ybus Matrix
a
b
c
d e
A directed graph can be associated with the
network where the arrows indicate the
direction of the currents.
0 0 1
1 0 1
0 1 1
1 0 0
1 1 0
A
This directed graph has a branch node
incidence matrix associated with the following
convention:
1 If the current leaves the node
-1 If the current reaches the node
a
b
c
d
e
6
7. Incidence Matrix and Ybus Matrix
1
2
1
3
3
3
1
2
ref
ref
a
b
c
d
e
V V V
V V V
V V V
V V V
V V V
pr
V AV
Equation 2
1
2
3
0 0 1
1 0 1
0 1 1
1 0 0
1 1 0
a
b
c
d
e
V
V
V
V V
V
V
V
Va
+
_ Vb
Vc
Vd Ve
+ +
+ +
_
_
_
_
The relationship
between branch
voltages and node
voltages is given by:
Expressing the
equations in
matrix form we
have:
7
8. Incidence Matrix and Ybus Matrix
1
2
3
0 1 0 1 1
0 0 1 0 1
1 1 1 0 0
a
b
c
d
e
I
I
I
I I
I
I
i
Similarly, applying
Kirchhoff's current law
at each node, we
have:
Ia
Ib
Ic
Id Ie
I1
I2
I3
1
2
3
e d
c e
c
b
b
a
I I I I
I I I
I I I I
Expressing the
equations in
matrix form we
have:
T
pr
A I I
Ecuación 3
8
9. Incidence Matrix and Ybus Matrix
From equation 1 we have: .
pr pr pr
Y V I
From equation 2 we have: pr
V AV
Replacing we have: pr pr
Y AV I
Premultiplying by: T
A T T
pr pr
A Y A V A I
T
pr
A Y A V I
T
pr
A I I
We have:
Replacing equation 3:
It yields: T
bus
pr
A Y A Y
9
10. Example
For the following network:
1- Obtain Ybus by observation
2-Obtain Ybus using the incidence matrix and
Y primitive
3- Get the new Ybus if branch 4-2 is deleted
4-Obtain the new Ybus if a branch of z = 0,2j
is added in parallel with the existing branch
in nodes 4-3
10
11. Successive Elimination Method
Real power systems are normally made up of hundreds of bars. To avoid inverting large
matrices, and thus making calculations with less computational effort, the technique of
successive elimination or Gaussian elimination is used.
This method consists of successively eliminating the unknowns of the problem until only one of
them remains, to later make an inverse substitution that allows finding the other unknowns.
Example: solve the system of equations using Gaussian elimination
1
2
3
1
2 0 1
4 2 6 24
6 2 4 2
x
x
x
11
12. Successive Elimination Method
1
2
3
1
1
1 0 2
2
0 2 8 22
0 2 1 1
x
x
x
F1/2
F2 – 4F1
F3 – 6F1
1
2
3
1
1
1 0 2
2
0 1 4 11
0 0 7 21
x
x
x
The element at position (1,1) is
selected as the pivot
F2/-2
F3 -2F2
The element at position (1,1) is
selected as the pivot
From the last equation we have: 3
7 21
x 3 3
x
Replacing in the second equation: 2 4(3) 11
x 2 1
x
Replacing in the tird equation: 1
1 1
(3)
2 2
x 1 2
x
12
13. Successive Elimination Method
The same concept can be applied to reduce the Ybus matrix.
1 1
11 12 13 14
2 2
21 22 23 24
31 32 33 34 3 3
41 42 43 44
3 4
V I
Y Y Y Y
V I
Y Y Y Y
Y Y Y Y V I
Y Y Y Y
V I
I2 I3
I1
I4
The current injections at the nodes are assumed to be known. A node of the system is
eliminated at each iteration
13
14. Successive Elimination Method
11 1 12 2 13 3 14 4 1
21 1 22 2 23 3 24 4 2
31 1 32 2 33 3 34 4 3
41 1 42 2 43 3 44 4 4
Y V Y V Y V Y V I
Y V Y V Y V Y V I
Y V Y V Y V Y V I
Y V Y V Y V Y V I
Dividing Ec4 by the pivot Y11
13 3
12 2 14 4 1
1
11 11 11 11
Y V
Y V Y V I
V
Y Y Y Y
Equation Ec8 is multied by Y21 and substracted from Ec5 :
Ec4
Ec5
Ec6
Ec7
Ec8
21 13
21 12 21 14 21 1
22 2 23 3 24 4 2
11 11 11 11
Y Y
Y Y Y Y Y I
Y V Y V Y V I
Y Y Y Y
Equation Ec8 is multiplied by Y31 and substracted from Ec6 :
31 12 31 13 31 14 31 1
32 2 33 3 34 4 3
11 11 11 11
Y Y Y Y Y Y Y I
Y V Y V Y V I
Y Y Y Y
Ec9
Ec10
14
15. Successive Elimination Method
Equation Ec8 is multiplied by Y41 and substracted from Ec7 :
31 12 31 13 31 14 31 1
32 2 33 3 34 4 3
11 11 11 11
Y Y Y Y Y Y Y I
Y V Y V Y V I
Y Y Y Y
Ec11
Equations Ec9-Ec11 can be rewritten as follows:
(1) (1) (1) (1)
22 2 23 3 24 4 2
(1) (1) (1) (1)
32 2 33 3 34 4 3
(1) (1) (1) (1)
42 2 43 3 44 4 4
Y V Y V Y V I
Y V Y V Y V I
Y V Y V Y V I
1 1
(1)
11
2,3,4
j
j j
Y I
I I para j
Y
1 1
(1)
11
2,3,4
j k
jk jk
Y Y
Y Y para j y k
Y
Donde:
Ec14
Ec13
Ec12
15
16. Successive Elimination Method
The procedure is repeated to eliminate V2
(1) (1) (1)
23 24 2
2 3 4
(1) (1) (1)
22 22 22
Y Y I
V V V
Y Y Y
(1)
22
Y
Ec12 is divided by
Then is multiplied by
(1)
32
Y y
(1)
42
Y
The result is substracted from Ec13 and Ec14 which yields:
(1) (1) (1) (1) (1) (1)
(1) (1)
32 23 32 24 32 2
33 3 34 4 3
(1) (1) (1)
22 22 22
(1) (1) (1) (1) (1) (1)
(1) (1)
42 23 42 24 42 2
43 3 44 4 4
(1) (1) (1)
22 22 22
Y Y Y Y Y I
Y V Y V I
Y Y Y
Y Y Y Y Y I
Y V Y V I
Y Y Y
Ec15
Ec16
16
17. Successive Elimination Method
Equations Ec15 and Ec16 can be rewitten in compact form as:
(2) (2) (2)
33 3 34 4 3
(2) (2) (2)
43 3 44 4 4
Y V Y V I
Y V Y V I
Ec17
Ec18
Again, one can divide equation Ec17 by and then multply by
To then substract from equation Ec18 and eliminate V3 as shown in equation Ec19.
(2) (2)
(2)
34 4 3
3 43
(2) (2)
33 33
Y V I
V Y
Y Y
(2)
33
Y (2)
43
Y
(2) (2) (2)
(2) (2) (2)
43 34 43
44 4
(2) (2)
33 33
4 3
Y Y Y
Y V I I
Y Y
(3) (3)
44 4 4
Y V I
At this point we can calculate V4 from Ec19 and start replacing back to find the
other voltages
Ec19
17
18. Reducción de Kron
The same concept of Gaussian elimination can be used to eliminate buses from a
power system and obtain reduced equivalent systems.
Example: Remove node 2 from the following network (data is given in admittances)
1
2
3
18 10 6 0
10 16 6 0
6 6 17 0
j j j V
j j j V
j j j V
18
19. Kron’s reduction
Example: Remove node 2 from the next network which has a current injection at node 2
(data is given in admittances)
−18𝑗 10𝑗 6𝑗
10𝑗 −16𝑗 6𝑗
6𝑗 6𝑗 −17𝑗
𝑉1
𝑉2
𝑉3
=
0
1∠20°
0
−11.75𝑗 0 9.75𝑗
− 5
8 1 − 3
8
9.75𝑗 0 −14.75𝑗
𝑉1
𝑉2
𝑉3
=
5
8 ∠20°
1
16 ∠110°
3
8 ∠20°
19
