Bus Impedance Matrix-Z Bus

u

u

u

inversion of the bus admittance matrix is a n3 effort for small and medium size networks, direct building of the matrix is less effort for large size networks, sparse matrix programming with gaussian elimination technique is preferred

Direct formation of the matrix

−1 Z bus = Ybus

Definition

Power Systems I

l

l

The Bus Impedance Matrix

1

j 0.4

j0.2

3

j0.8

2 j0.4

j0.4

1

3

4 5

2

selected tree

3

1

0 1 3 extending tree branch

2

1

3

4

0

5

2 2

loop closing co-tree branch

bus node = graph vertex line branch = edge

Graph theory techniques helps explain the building process

Power Systems I

l

Forming the Bus Impedance Matrix

Z

m bus

Partial Network 0 Reference

2 i j

1

Vbus = Z bus I bus

Basic construction of the network and the matrix

Power Systems I

l

Forming the Bus Impedance Matrix

q

0 Reference

Z

q

0 Reference

Vq = 0 + z q 0 I q

m bus

2 p m

Partial Network

2 p m

Vq = V p + z qp I q

Power Systems I

Z

m bus

Partial Network

1

1

Adding a Line

Power Systems I

 V1   Z11 V   Z  2   21 M  M    V p  =  Z p1 M  M    Vm   Z m1 V  = Z p1  q  Z m2 = Z p2

M

Z 21 Z 22 M Z p2

Z1 p Z2 p M Z pp

L L O L Z1m Z 2m M Z pm

O M O M L Z mp L Z mm L = Z pp L = Z pm

L L O L

= Z1 p = Z2 p M = Z pp

  I1   I    2 M    I p  M M   = Z mp   I m  = Z pp + z pq   I q 

Adding a Line to an Existing Line

Power Systems I

 V1   Z11 V   Z  2   21 M  M    V p  =  Z p1 M  M    Vm   Z m1 V   = 0  q  L Z1 p L Z1m L Z 2 p L Z 2m O M O M

=0 L =0 L

=0

Z p 2 L Z pp L Z pm M O M O M Z m 2 L Z mp L Z mm

Z 21 Z 22 M

Adding a Line from Reference   I1   I    2 M    = 0  I p  M M    = 0  I m  = z0 q   I q 

=0 =0 M

Power Systems I

z p0 Il − Vp = 0

z pq I l + Vq − V p = 0

Z

m bus

1 p

0 Reference

i m

z p0 Il = Vp − 0 →

0 Reference

q m

Partial Network

z pq I l = V p − Vq →

Z

m bus

Partial Network

1 p

Closing a Loop

T

I

[ n×1] bus [1×n ]

Power Systems I

Z ll

I bus [1×n ]

 old ∆Z∆Z T  =  Z bus −  I bus Z ll  

∆Z T [ n×1] → Il = − I bus [1×n ] Z ll

∆Z[1×n ]∆Z T [ n×1]

+ Z ll [1×1] I l

old Vbus [1×n ] = Z bus [ n×n ] I bus [1×n ] −

0 = ∆Z

Vbus [1×n ]

old Vbus [1×n ]   Z bus [ n×n ]  0  = ∆Z T [ n×1]   

∆Z[1×n ]   I bus [1×n ]   Z ll [1×1]   I l  old = Z bus [ n×n ] I bus [1×n ] + ∆Z [1×n ] I l

Eliminating a node from the system

Kron Reduction

M Z pp Z qp M Z mp

O L L O L

Power Systems I

L

L O

O

L

L

Z mm

Z qm M

Z pm

M

Z1m

Z qq −Z pq L Z qm −Z pm

Z mq

Z qq M

Z pq

M

Z1q

Z ll = z pq + Z pp + Z qq − 2 Z pq

L Z qp −Z pp

Z1 p

L

Then execute Kron reduction on Zll

 V1   Z11 M  M    V p   Z p1    Vq  =  Z q1 M  M    Vm   Z m1  0   Z q1−Z p1   

Adding a Line between two Lines Z1q −Z1 p   I1    M M Z pq −Z pp   I p    Z qq −Z qp   I q  M M   Z mq −Z mp   I m  Z ll   I l 

Z1 p M Z pp Z ip

O M L Z mp L − Z pp

L O L L

L L O L

Power Systems I

Z1m M Z pm Z im

M O M Z mi L Z mm − Z pi L − Z pm

Z1i M Z pi Z ii

Then execute Kron reduction on Zll

Z ll = z p 0 + Z pp

 V1   Z11 M  M    V p   Z p1     Vi  =  Z i1 M  M    Vm   Z m1  0  − Z p1   

− Z1 p   I 1  M   M    −Z pp   I p    − Z ip   I q  M M   − Z mp   I m  Z ll   I l 

Adding a Line from a Line to Reference

 V1   Z11 M  M    V p   Z p1     Vi  =  Z i1 M  M    Vm   Z m1  0   Z l1    Z

new jk

=Z

M L Z mp L Z lp

O

M

O Z pm L Z im

L

L Z1m

old jk

−

Z ll

Z jl Z lk

M O M Z mi L Z mm Z li L Z lm

Z ii

M Z pi

O

M L Z pp L Z ip

Z1i

L Z1 p

Z1l   I1  M   M    Z pl   I p    Z il   I q  M M   Z ml   I m  Z ll   I l 

Kron reduction removes an axis (row & column) from a matrix while retaining the axis’s numerical influence

Power Systems I

l

Kron Reduction

u

u

n

n

zq0

the new off-diagonal row and column filled with (0) the diagonal element (m+1),(m+1) filled with the element impedance

start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with

Rule 1: Addition of a branch to the reference

Power Systems I

l

Z-Bus Building Rules

u

u

u

n

n

the new off-diagonal row and column filled with a copy of row p and column p the diagonal element (m+1),(m+1) filled with the element impedance zpq plus the diagonal impedance Zpp

connecting to existing bus p start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with

Rule 2: Addition of a branch to an existing bus

Power Systems I

l

Z-Bus Building Rules

u

u

u

u

the new off-diagonal row and column filled with a copy of row q minus row p and column q minus column p the diagonal element (m+1),(m+1) filled with

zpq + Zpp + Zqq - 2 Zpq perform Kron reduction on the m+1 row and column

n

n

connecting to existing buses p and q start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with

Rule 3: Addition of a linking branch

Power Systems I

l

Z-Bus Building Rules

u

u

u

u

the new off-diagonal row and column filled with a copy of the negative of row p and the negative of column p the diagonal element (m+1),(m+1) filled with zp0 + Zpp

perform Kron reduction on the m+1 row and column

n

n

connecting to existing bus p and reference start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with

Rule 4: Addition of a linking branch

Power Systems I

l

Z-Bus Building Rules

Network

j0.8 j0.4

j0.4

2

1 3

4

2

3 Graph

1

0

Line adding order: 1-0, 2-0, 1-3, 1-2, then 2-3

3

Power Systems I

1

j 0.4

j0.2

Example

5

2

[] [ j 0.2]

0 j 0. 4 0

0  j 0.4

Power Systems I

 j 0.2 2.   0  j 0.2 3.  0  j 0.2

0. 1.

Example

j 0.2 0  j 0.6

 j 0.171  j 0.057   j 0.171

0.285 j 0.057

j 0.057

j 0.2

j 0.171 j 0.057  j 0.571

j 0.2  − j 0.4 0 j 0.6 j 0.2   j 0.2 j1.4  ( j 0.2 )( j 0.2) = j 0.17 Z11 = j 0.2 − j1.4

0  j 0.2  0 j 0.4  4.  j 0.2 0   j 0.2 − j 0.4

Power Systems I

 j 0.171  j 0.057 5.   j 0.171   j 0.114  j 0.16  j 0.08   j 0.12

Example j 0.171

j 0.114  j 0.285 j 0.057 − j 0.228 j 0.057 j 0.571 j 0.514   j1.14  − j 0.228 j 0.514 j 0.08 j 0.12 j 0.24 j 0.16 j 0.16 j 0.34 j 0.057