lecture3-network matrices, the y-bus matrix; tap changing transformers
TRANSCRIPT
Po
wer
Sys
tem
s I
Th
e B
us
Ad
mit
tan
ce M
atri
x
lT
he
mat
rix
equ
atio
n f
or
rela
tin
g t
he
no
dal
vo
ltag
es t
o t
he
curr
ents
th
at f
low
into
an
d o
ut
of
a n
etw
ork
usi
ng
th
ead
mit
tan
ce v
alu
es o
f ci
rcu
it b
ran
ches
lU
sed
to
fo
rm t
he
net
wo
rk m
od
el o
f an
inte
rco
nn
ecte
dp
ow
er s
yste
mu
Nod
es r
epre
sent
sub
stat
ion
bus
bars
uB
ranc
hes
repr
esen
t tra
nsm
issi
on li
nes
and
tran
sfor
mer
su
Inje
cted
cur
rent
s ar
e th
e flo
ws
from
gen
erat
or a
nd lo
ads
node
bus
inj
VY
I⋅
=N
etw
ork
I k
Vk
Po
wer
Sys
tem
s I
Th
e B
us
Ad
mit
tan
ce M
atri
x
lC
on
stru
ctin
g t
he
Bu
s A
dm
itta
nce
Mat
rix
(or
the
Y b
us
mat
rix)
ufo
rm th
e no
dal s
olut
ion
base
d up
on K
irchh
off’s
cur
rent
law
uim
peda
nces
are
con
vert
ed to
adm
ittan
ces
ijij
ijij
xj
rz
y+
==
11
()
()
()
nk
knk
kk
kk
kin
jk
VV
yV
Vy
VV
yV
yI
−+
+−
+−
+=
−K
22
11
0
Po
wer
Sys
tem
s I
Mat
rix
Fo
rmat
ion
Exa
mp
le
j1.0
j0.8
j0.4
j0.2
j0.2
j0.0
8
4
312
Impe
danc
e D
iagr
am
gene
rato
r 1
z =
j1.0
line
12z
= j0
.4
line
13z
= j0
.2lin
e 23
z =
j0.2
line
34z
= j0
.08
43
12
Net
wor
k D
iagr
am
gene
rato
r 2
z =
j0.8
V2
V1
Po
wer
Sys
tem
s I
Mat
rix
Fo
rmat
ion
Exa
mp
le
y 10=
-j1
.0
y 20=
-j1
.25
y 12
= -
j2.5
y 13=
-j5
y 23=
-j5
y 34
= -
j12.
5
12
Adm
ittan
ce D
iagr
am
I 2I 1
4
3
()
()
()
()
()
()
()
()
34
43
43
342
332
13
31
32
231
221
220
2
31
132
112
110
1 00
VV
y
VV
yV
Vy
VV
y
VV
yV
Vy
Vy
I
VV
yV
Vy
Vy
I
−=
−+
−+
−=
−+
−+
=−
+−
+=
KC
L E
quat
ions
Po
wer
Sys
tem
s I
Mat
rix
Fo
rmat
ion
Exa
mp
le
()
()
()
443
343
434
334
3231
232
131
323
223
2120
121
2
313
212
113
1210
1 00
Vy
Vy
Vy
Vy
yy
Vy
Vy
Vy
Vy
yy
Vy
I
Vy
Vy
Vy
yy
I
+−
=−
++
+−
−=
−+
++
−=
−−
++
=Rea
rran
ging
the
KC
L E
quat
ions
()
()
()
⋅
−−
++
−−
−+
+−
−−
++
=
4321
4343
3434
3231
3231
2323
2120
21
1312
1312
10
21
00
00
00
VVVV
yy
yy
yy
yy
yy
yy
y
yy
yy
y
II
Mat
rix
For
mat
ion
of th
e E
quat
ions
Po
wer
Sys
tem
s I
Mat
rix
Fo
rmat
ion
Exa
mp
le
()
()
()
⋅
−−
−−
=
−=
=−
=+
+=
=−
==
=−
==
−=
++
==
−=
==
−=
=−
=+
+=
4321
21
3444
2321
2022
3443
3413
3113
3432
3133
1221
12
2332
2313
1210
11
50.12
50.12
00
50.12
50.22
00.500.5
000.5
75.850.2
000.5
50.250.8
00
50.12
75.8
50.12
00.5
50.22
50.2
00.550.8
VVVV
jj
jj
jj
jj
j
jj
j
II
jy
Yj
yy
yY
jy
YY
jy
YY
jy
yy
Yj
yY
Y
jy
YY
jy
yy
Y
Com
plet
ed M
atri
x E
quat
ion
Po
wer
Sys
tem
s I
Y-B
us
Mat
rix
Bu
ildin
g R
ule
s
lS
qu
are
mat
rix
wit
h d
imen
sio
ns
equ
al t
o t
he
nu
mb
er o
fb
use
sl
Co
nve
rt a
ll n
etw
ork
imp
edan
ces
into
ad
mit
tan
ces
lD
iag
on
al e
lem
ents
:
lO
ff-d
iag
on
al e
lem
ents
:
lM
atri
x is
sym
met
rica
l alo
ng
th
e le
adin
g d
iag
on
al
ij
yY
n jij
ii≠
=∑ =0
ijji
ijy
YY
−=
=
Po
wer
Sys
tem
s I
Exa
mp
le
Sys
tem
Dat
aL
ine
Sta
rtE
nd
X v
alu
eg
11
01.
00g
25
01.
25L
11
20.
40L
21
30.
50L
32
30.
25L
42
50.
20L
53
40.
125
L6
45
0.50
Po
wer
Sys
tem
s I
Tap
-Ch
ang
ing
Tra
nsf
orm
ers
lT
he
tap
-ch
ang
ing
tra
nsf
orm
giv
es s
om
e co
ntr
ol o
f th
ep
ow
er n
etw
ork
by
chan
gin
g t
he
volt
ages
an
d c
urr
ent
mag
nit
ud
es a
nd
an
gle
s b
y sm
all a
mo
un
tsu
The
flow
of r
eal p
ower
alo
ng a
net
wor
k br
anch
is c
ontr
olle
d by
the
angu
lar
diffe
renc
e of
the
term
inal
vol
tage
su
The
flow
of r
eact
ive
pow
er a
long
a n
etw
ork
bran
ch is
con
trol
led
by th
e m
agni
tude
diff
eren
ce o
f the
term
inal
vol
tage
su
Rea
l and
rea
ctiv
e po
wer
s ca
n be
adj
uste
d by
vol
tage
-reg
ulat
ing
tran
sfor
mer
s an
d by
pha
se-s
hifti
ng tr
ansf
orm
ers
bus
i1:
abu
s j
a ca
n be
a
com
plex
num
ber
Po
wer
Sys
tem
s I
Mo
del
ing
of
Tap
-Ch
ang
ers
uth
e of
f-no
min
al ta
p ra
tio is
giv
en a
s 1:
a
uth
e no
min
al tu
rns-
ratio
(N
1/N
2) w
as a
ddre
ssed
with
the
conv
ersi
on o
f the
net
wor
k to
per
uni
tu
the
tran
sfor
mer
is m
odel
ed a
s tw
o el
emen
ts jo
ined
toge
ther
at a
fictit
ious
bus
x
uba
sic
circ
uit e
quat
ions
:
Vi
I iV
xy t
Vj
I j
1:a
()
xi
ti
ji
ja
xV
Vy
II
aI
VV
−=
⋅−
==
*1
Po
wer
Sys
tem
s I
Mo
del
ing
of
Tap
-Ch
ang
ers
lM
akin
g s
ub
stit
uti
on
s
()
()
()
jt
it
ja
it
j
ia
j
ji
ja
it
i
xi
ti
ja
x
Vay
Vay
VV
ayI
II
Ia
I
VV
yI
VV
yI
VV
2*
1*
1*
1
1
*
+−
=−
−=
−=
⋅−
=
−=
−=
=
Po
wer
Sys
tem
s I
YB
us
Fo
rmat
ion
of
Tap
-Ch
ang
ers
lM
atri
x fo
rmat
ion
{}
⋅
−−
=
+
−
=
−
+=
ji
tt
tt
ji
jt
it
j
jt
it
i
VV
ay
ay
ay
y
II
Vay
Vay
I
Vay
Vy
I
2*
2*
Po
wer
Sys
tem
s I
Pi-
Cir
cuit
Mo
del
of
Tap
-Ch
ang
ers
lV
alid
fo
r re
al v
alu
es o
f a
lT
akin
g t
he
y-b
us
form
atio
n, b
reak
th
e d
iag
on
al e
lem
ents
into
tw
o c
om
po
nen
tsu
the
off-
diag
onal
ele
men
t rep
rese
nt th
e im
peda
nce
acro
ss th
e tw
obu
ses
uth
e re
mai
nder
form
the
shun
t ele
men
t
no
n-t
ap s
ide
tap
sid
ey t
/ a
(1 -
a)
y t /
a2(a
- 1
) y t
/ a
ji