Stevenson J. Power system analysis
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3
CHAER 8 THE IMPEDANCE MODEL AND NE
ORK
CALCULATIONS
8.5 CALCUO
N
OF
Zbus
ELEME
N
TS
O
M
Yb
us
When the full numerical form of
Z
b
us is not explicitly required for an applica
tion, we can readily calculate elements of
Z
b
us
as needed if the upper- and
lower-triangular factors of
Y
bus
are available. To see how this can be done,
consider postmultipiying
Z
h
u
s
by a vector with only one nonzero element 1 m
= 1
in row m and all other elements cqual to zero. When
Z
o
u
s
IS an N X N matrix,
we have
@
®
-
Z
l
1
Z
12
Z
i
m
Z
I
N
0
Z
i
m
@
Z
21
Z
22
Z
2m
Z
2
N
0
0
Z
2
m
@
Z
m
l
Z
m
2
Z
mm
Z
m
N
-
Z
mm
(8.
46)
1
m
®
Z
N
l
Z
N
2
Z
Nm
Z
NN
0
®
Z
N
m
-
-
Zb
us
z
(
m
)
bus
Thus, postmultiplying
Z
bus
by the vector shown extracts the mth column, which
we have called the vector
Z
h� ;; that is
-
Z
i
m
0
Z
2
m
z
(m) £
column m
b
u
s
of
®
Z
m
m
Z
bus
®
Z
Nm
Since the product of
Y
b
us
and
Z
bu
s
equals the unit matrix, we have
o
o
o
o
Y
Z
-
Y
z(m)-
bus bus 1m - bus bus - 1
m
o
o
I
(8.
4
7)
If the lower-triangular matrix L and the upper-triangular matrix U of
Y
b
us
are
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