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Bus Impedance Matrix-Z Bus
18
Power Systems I The Bus Impedance Matrix l Definition l Direct formation of the matrix u inversion of the bus admittance matrix is a n 3 effort u for small and medium size networks, direct building of the matrix is less effort u for large size networks, sparse matrix programming with gaussian elimination technique is preferred 1 - = bus bus Y Z
description
z bus
Transcript of Bus Impedance Matrix-Z Bus
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s I
Th
e B
us
Impe
dan
ce M
atrix
lD
efin
itio
n
lD
irect
fo
rmat
ion
o
f the
m
atrix
uin
vers
ion
of
th
e bu
s ad
mitt
ance
m
atrix
is
a
n3
effo
rtu
for
sma
ll a
nd
med
ium
si
ze n
etw
ork
s, di
rect
bu
ildin
g of
th
e m
atrix
is le
ss e
ffort
ufo
r la
rge
size
n
etw
ork
s, sp
ars
e m
atrix
pr
ogr
amm
ing
with
gau
ssia
n el
imin
atio
n te
chn
iqu
e is
pr
efe
rre
d
1
=bu
sbu
sY
Z
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s I
bus
no
de =
gr
aph
ver
tex
line
bran
ch =
ed
ge
Form
ing
the
Bu
s Im
peda
nce
M
atrix
lG
raph
th
eory
te
chn
iqu
es he
lps
expl
ain
th
e bu
ildin
gpr
oce
ss
0
12
3
12
3
4
5
co-tr
ee
0
12
3
12
3
4
5
sele
cted
tr
ee
j0.2
j0.4
j 0.4
j0.4
j0.8
12
3ex
tendi
ng
tree
br
anch
loop
closin
g br
anch
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s I
Part
ial
Net
work
m bus
Z
1 2 i j 0 Refe
rence
bus
bus
bus
IZ
V=
Form
ing
the
Bu
s Im
peda
nce
M
atrix
lB
asic
co
nst
ruct
ion
o
f the
n
etw
ork
an
d th
e m
atrix
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s I
Part
ial
Net
work
m bus
Z
1 2 p m 0 Ref
eren
ce
Part
ial
Net
wo
rkm bu
sZ
1 2 p m 0 Ref
eren
ce
q
q
qqp
pq
Iz
VV
+=
qq
qI
zV
00+
=
Add
ing
a Li
ne
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s I
+=
==
==
====
=
qmp
pqpp
pmpp
pp
mp
mm
mp
mm
pppm
ppp
p
pm
p
pm
p
qmp
IIIII
zZ
ZZ
ZZ
ZZ
ZZ
Z
ZZ
ZZ
Z
ZZ
ZZ
ZZ
ZZ
ZZ
VVVVV
MM
LL
LL
MM
OM
OM
M
LL
MM
OM
OM
M
LL
LL
MM
21
21
21
21
22
222
21
11
121
11
21
Add
ing
a Li
ne
to an
Ex
istin
g Li
ne
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s I
==
==
=
====
=
qmp
q
mm
mp
mm
pmpp
pp
mp
mp
qmp
IIIII
z
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
VVVVV
MM
LL
LL
MM
OM
OM
M
LL
MM
OM
OM
M
LL
LL
MM
21
0
21
21
22
2221
11
2111
21
00
00
0000
Add
ing
a Li
ne
from
R
efer
ence
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s I
Part
ial
Net
wo
rkm bu
sZ
1 p q m 0 Ref
eren
ce
Part
ial
Net
work
m bus
Z
1 p i m 0 Ref
eren
ce
0=
+
=
pq
lpq
qp
lpq
VV
Iz
VV
Iz
00
00
=
=
pl
p
pl
p
VI
z
VI
z
Clo
sin
g a
Loo
p
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s I
Elim
inat
ing
a n
ode
fro
m th
e sy
stem
bus
ll
Told bus
nbu
sll
nT
n
nbu
sold
nn
bus
nbu
s
nbu
sllnT
ll
lln
bus
nT
ln
nbu
sold
nn
bus
nbu
s
l
nbu
s
lln
Tn
old
nn
bus
nbu
s
IZ
ZZ
ZI
ZZ
ZI
ZV
IZZ
II
ZI
Z
IZ
IZ
V
II
ZZ
ZZ
V
=
=
=
+
=
+
=
=
]1[
]1[
]1[
]1[
][
]1[
]1[
]1[
]11[
]1[
]1[
]1[
]1[
][
]1[
]1[
]11[
]1[
]1[
][
]1[ 00
Kro
n R
edu
ctio
n
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s I
pqqq
pppq
ll
lmqp
llpm
qmpq
qqpp
qpp
q
mp
mq
mm
mq
mp
m
qpqq
qmqq
qpq
pppq
pmpq
ppp
pq
mq
p
mqp
ZZ
Zz
Z
IIIII
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
Z
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
VVVV
20
1
11
111
11
11
111
1
+
+=
=
MM
LL
LL
MM
OM
MO
M
LL
OL
MM
LM
MO
M
LL
MM
Then
ex
ecute
K
ron re
duct
ion on
Z l
l
Add
ing
a Li
ne
betw
een
tw
o Li
nes
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s I
ppp
ll
lmqp
llpm
pipp
p
mp
mm
mi
mp
m
ipim
iiip
i
pppm
pipp
p
pm
ip
mip
Zz
Z
IIIII
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
Z
VVVV
+=
=
0
1
1111
11
11
111 0
MM
LL
LL
MM
OM
MO
M
LL
OL
MM
LM
MO
M
LL
MM The
n ex
ecute
K
ron
re
duct
ion on
Z l
l
Add
ing
a Li
ne
from
a
Lin
e to
R
efer
ence
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s I
K
ron
R
edu
ctio
n
lK
ron
re
duct
ion
re
mo
ves
an
ax
is (ro
w &
colu
mn
) fro
m a
mat
rix w
hile
re
tain
ing
the
axis
s
nu
mer
ical
in
fluen
ce
ll
lkjl
old jk
new jk
lmqp
lllm
lilp
l
ml
mm
mi
mp
m
ilim
iiip
i
plpm
pipp
p
lm
ip
mip
ZZZZ
Z
IIIII
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
ZZ
Z
VVVV
=
=
MM
LL
LL
MM
OM
MO
M
LL
OL
MM
LM
MO
M
LL
MM
1
1111
11
11
111 0
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s I
Z-
Bu
s B
uild
ing
Ru
les
lR
ule
1:
A
dditi
on
o
f a br
anc
h to
th
e re
fere
nce
ust
art
with
e
xist
ing
net
wo
rk m
atrix
[m
m
]u
crea
te a
n
ew n
etw
ork
m
atrix
[(m
+1)
(m
+1)]
w
ithn
the ne
w off-
diago
na
l row
an
d co
lum
n fill
ed
with
(0)
nth
e di
ago
nal e
lem
ent (m
+1),
(m+
1) fill
ed
with
th
e ele
ment i
mpe
dan
cez q
0
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s I
Z-
Bu
s B
uild
ing
Ru
les
lR
ule
2:
A
dditi
on
o
f a br
anc
h to
an
ex
istin
g bu
su
con
ne
ctin
g to
e
xist
ing
bus
pu
sta
rt w
ith e
xist
ing
net
wo
rk m
atrix
[m
m
]u
crea
te a
n
ew n
etw
ork
m
atrix
[(m
+1)
(m
+1)]
w
ithn
the ne
w off-
diago
na
l row
an
d co
lum
n fill
ed
with
a co
py o
f row
p
and
colu
mn p
nth
e di
ago
nal e
lem
ent (m
+1),
(m+
1) fill
ed
with
th
e ele
ment i
mpe
dan
cez p
q pl
us
the di
ago
nal i
mpe
dance
Z p
p
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s I
Z-
Bu
s B
uild
ing
Ru
les
lR
ule
3:
A
dditi
on
o
f a lin
kin
g br
anch
uco
nne
ctin
g to
e
xist
ing
buse
s p
an
d q
ust
art
with
e
xist
ing
net
wo
rk m
atrix
[m
m
]u
crea
te a
n
ew n
etw
ork
m
atrix
[(m
+1)
(m
+1)]
w
ithn
the ne
w off-
diago
na
l row
an
d co
lum
n fill
ed
with
a co
py o
f row
q
min
us
row
p
an
d co
lum
n q
min
us
colu
mn
p
nth
e di
ago
nal e
lem
ent (m
+1),
(m+
1) fill
ed
with
z pq
+ Z p
p +
Z q
q - 2
Z pq
upe
rform
Kr
on
re
duct
ion
o
n th
e m
+1
row
a
nd
colu
mn
-
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stem
s I
Z-
Bu
s B
uild
ing
Ru
les
lR
ule
4:
A
dditi
on
o
f a lin
kin
g br
anch
uco
nne
ctin
g to
e
xist
ing
bus
p a
nd
refe
ren
ce
ust
art
with
e
xist
ing
net
wo
rk m
atrix
[m
m
]u
crea
te a
n
ew n
etw
ork
m
atrix
[(m
+1)
(m
+1)]
w
ithn
the ne
w off-
diago
na
l row
an
d co
lum
n fill
ed
with
a co
py o
f the
ne
gativ
e o
f row
p
and
the ne
gativ
e o
f colu
mn p
nth
e di
ago
nal e
lem
ent (m
+1),
(m+
1) fill
ed
with
z p0
+ Z p
pu
perfo
rm Kr
on
re
duct
ion
o
n th
e m
+1
row
a
nd
colu
mn
-
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stem
s I
j0.2
j0.4
j 0.4
j0.4
j0.8
12
3
0
12
3
12
35
4
Net
wo
rkG
raph
Lin
e ad
din
g ord
er:
1-
0, 2-
0, 1-
3, 1-
2, th
en 2-
3
Exam
ple
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s I
Ex
ampl
e
[] []
6.0
02
.00
4.0
02.0
02
.0.3
4.0
00
2.0
.2
2.0
.1.0
jj
jj
jj
jj
()(
)
=
=
571
.005
7.0
171
.005
7.0
285
.005
7.0
171
.005
7.0
171
.0
17.0
4.1
2.0
2.0
2.0
4.1
2.0
4.0
2.0
2.0
6.0
02
.04.0
04.0
02.0
2.0
02
.0
.4
11
jj
jj
jj
jj
jj
jj
jZ
jj
jj
jj
jj
jj
jj
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s I
Ex
ampl
e
34.0
16.0
12.0
16.0
24.0
08.0
12.0
08.0
16.0
14.1
514
.022
8.0
114
.051
4.0
571
.005
7.0
171
.022
8.0
057
.028
5.0
057
.011
4.0
171
.005
7.0
171
.0
.5
jj
jj
jj
jj
jj
jj
jj
jj
jj
jj
jj
jj
j
