BUS IMPEDANCE MATRIX

• The impedance matrix is given by

• Zbus = Ybus-1

• Since the bus admittance matrix is symmetrical, the bus impedance matrix is also symmetrical around the principal diagonal.

• In bus impedance matrix, the elements on the main diagonal are called driving point impedance of the buses or nodes and the off-diagonal elements are called the transfer impedances of the buses or nodes.

• The bus impedance matrix is very useful in fault analysis or calculations.

The bus impedance matrix can be determined by

Taking the inverse of the bus admittance matrix formed.

Bus building algorithm.

Bus building algorithm

• Let us denote the original Zbus of a system with n-number of independent buses as Zorig.

• When a branch of impedance Zb is added to the system, the Zorig gets modified.

The branch impedance Zb can be added to the original system in the following four different ways.

• Case 1 : Adding a branch of impedance Zb from a new-p to the reference bus.

• Case 2: Adding a branch of impedance Zb from a new-p to an existing bus-q.

• Case 3 : Adding a branch of impedance Zb from an existing bus-1 to the reference bus.

• Case 4 : Adding a branch of impedance Zb between two existing buses h and q.

Case 1 : Adding Zb from a new bus-p to the reference bus.

• Consider a n-bus system as shown in fig. Let us add a bus-p through an impedance Zb to the reference bus.

• The addition of a bus will increase the order of the bus impedance matrix by one.

• In this case the elements of (n+1)th column and row all zeros except the diagonal.

• The diagonal element is the added branch impedance Zb.

• The elements of original Zbus matrix are not altered.

• The new bus impedance matrix will be as shown in equation

b

orig

newbus

Z

Z

Z

|000

|

0|

0|

0|

,

Case 2 : Adding Zb from a new bus–p to existing bus-q

• Consider a n-bus system as shown in fig. in which a new bus-p is added through an impedance Zb to an existing bus-q.

• The addition of a bus will increase the order of the bus impedance matrix by one.

• In this the element of (n+1)th column are the elements of qth column and elements of (n+1)th row are the elements of qth row.

• The diagonal element is given by sum of Zqq and Zb.

• The elements of original Zbus matrix are not altered.

bqqqqqq

q

q

newbus

ZZZZZ

Zbus

Z

Z

Z

|.

|

.|

.|

|

|

21

2

1

,

Case 3 : Adding Zb from an existing bus-q to the reference bus

• Consider a n-bus system as shown in fig.c in which an impedance Zb is added from an existing bus-q to the reference bus.

• Let us consider as if the impedance Zb is connected from a new bus-p and existing bus-q. Now it will be an addition as that of case-2.

• Then we can short-circuit the bus-q to reference bus. • This is equivalent to eliminating (n+1)th bus (i.e. bus-p in this case) and so the bus

impedance matrix has to be modified by eliminating (n+1)th row and (n+1)th column.

• The reduced bus impedance matrix can be formed by a procedure similar to that of bus elimination in bus admittance matrix.

• This reduced bus impedance matrix is the actual new bus impedance matrix. Every element of actual new bus impedance matrix can be determined using the equation below..

)1)(1(

)1()1(,

nn

knnjjkactjk Z

ZZZZ

Case 4 : Adding Zb between two existing buses h and q

• Consider a n-bus system shown in fig. d, in which an impedance Zb is added between two existing buses h and q.

• In this case the bus impedance matrix is formed as shown in equation .

• Here the elements of (n+1)th column is the difference between the elements of column-h and column-q.

• • The elements of (n+1)th row is the difference between

the elements of row-h and row-q.

• The diagonal element is given by equation .

bqqqnhnqheqh

nqnh

qh

qh

newbus

ZZZZZZZ

ZZ

Zbus

ZZ

ZZ

Z

|.

|

|

.|

|

|

211

22

11

,

hqqqhhbnn ZZZZZ 2)1)(1(

• Since the modification does not involve addition of new bus, the order of new bus impedance matrix has to be reduced to n x n by eliminating the (n+1)th column and (n+1)th row.

• This reduced bus impedance matrix is the actual new bus impedance matrix.

• Every element of the actual new bus impedance matrix can be determined using equation given below.

)1)(1(

)1()1(,

nn

knnjjkactjk Z

ZZZZ