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Lecture 13
Transmission Lines: Steady-State Operation
Reading: 5.1 – 5.5
Homework 4 is due on March 1st
Dr. Lei Wu
Department of Electrical and Computer Engineering
EE 333
POWER SYSTEMS ENGINEERING
Outline
Distributed line model
ABCD parameters
Equivalent PI model
Lossless line
2
c
-1
zwhere characteristic impedance Z = Ω
y
propagation constaint γ= zy m
( )( )
( ) ( )( ) ( )
( )( )
cosh sinh 01
sinh cosh 0
c
c
x Z xV x V
x xI x IZ
γ γ
γ γ
=
Outline
Distributed line model
ABCD parameters
Equivalent PI model
Lossless line
3
( )( )
( ) ( )( ) ( )
( )( )
cosh sinh 01
sinh cosh 0
c
c
x Z xV x V
x xI x IZ
γ γ
γ γ
=
S R R
S R R
V V VA BT
I C D I I
= =
Outline
Distributed line model
ABCD parameters
Equivalent PI model
Lossless line
4
( )'
2 1'
Z B
AY
B
=−
=
' '1
2'
Y ZA
B Z
= +
=
' '1
2' '
' 14
Y ZD
Y ZY C
+ =
+ =
Lossless Transmission Lines
For a lossless line, R=G=0, thus, z=jωL and y=jwC.
Characteristic (Surge) impedance – pure real number
Propagation constant – pure imaginary number
ABCD parameters
5
( )( )zy j L j C j LCγ ω ω ω= = =
c
z j L LZ
y j C C
ωω
= = =
( ) ( ) ( )cosh cosh cos2
j LCl j LCle eA D l j LCl LCl
ω ω
γ ω ω−+= = = = =
( ) ( ) ( )sinh sinh sin2
j LCl j LCl
c
L L e e LB Z l j LCl j LCl
C C C
ω ω
γ ω ω−−= = = =
( ) ( )sinhsin /
c
l LC j LCl
Z C
γω= =
Lossless Transmission Lines
Surge impedance loading (SIL)
The power delivered by a lossless line to a load resistance equal
to the surge impedance.
The voltage profile remains constant
6
( ) ( )
( ) ( )
cos sin
cos sin
RS R R R
j LClR R
VLV AV BI LCl V j LCl
C L C
LCl j LCl V e Vω
ω ω
ω ω
= + = +
= + =
S RV V=
Lossless Transmission Lines
Surge impedance loading (SIL)
7
( ) ( )
( ) ( )
sinos
cos sin
RS R R R
j LClR R
j LCl VI CV DI V c LCl
L C L C
V VLCl j LCl e
L C L Cω
ωω
ω ω
= + = +
= + =
( )* 2
** Rj LCl j LCl RS S S R
VVS V I e V e
L C L Cω ω
= = =
No reactive power flow across the line
The real power flow along a lossless line at SIL remains constant
Power Transfer in Lines
V1 V2
+ +
- -
I1 I1TransmissionLine with
ABCDS12 S21
( )
1 1 1 2 2 2
1 22
2* 1 2 2
1 2 '
'
2 2 ' 12
c
with ,
osh ; ' '
Z
Z
A
A
Z
A
V V V V
V AVI
B
V V AV
x A B Z
Z
Z
S V IZ
γθ θ
θ θ
θ
θ θ
θ
→
== ∠ = ∠
−=
= ∠ = = ∠
= = ∠ − − ∠ −
8
Often we want to know the maximum power that could be
transferred through a transmission line
Power Transfer in Lossless Lines
9
If we assume a line is lossless (i.e., with the impedance of jX )
and are just interested in real power transfer.
Thus, the maximum real power transfer is
'
21 2 2
12 12 12
0 ; ' , 90
90 90
A ZB Z jX
V V AVP jQ
Z Z
θ θ
θ
= ° = = = °
+ = ∠ ° − − ∠ °
( ) ( )1 2 1 212 12 12cos 90 sin
V V V VP
Z Zθ θ= − ° − =
1 212Max V V
PX
=
Lossless Line Example
10
For a 765kV lossless transmission line with receiving end line-
to-line voltage of 765kV and surge impedance loading .
z=j0.535 Ω/mile, y=j 7.75* 10-6 mho/mile. Calculate the ABCD
parameter, the equivalent Π circuit, and the theoretical
maximum real power that a 200-mile line can deliver.
( ) ( )( ) ( )
( ) ( )( ) ( )
-3 oc
200
-3 -3
-3 -3
Z = 262.74Ω γ= =2.036*10 90
cosh sinh
1sinh cosh
cosh 2.036*10 * 200 262.74sinh 2.036*10 * 2000
1sinh 2.036*10 * 200 cosh 2.036*10 * 200
262.74
c
c x
zzy
y
x Z xA B
Tx xC D
Z
j j
j j
γ γ
γ γ=
= ∠
= =
= =
.9182 104.06
0.0015 0.9182
j
j
Lossless Line Example
11
For a 765kV lossless transmission line with receiving end line-
to-line voltage of 765kV and surge impedance loading .
z=j0.535 Ω/mile, y=j 7.75* 10-6 mho/mile. Calculate the ABCD
parameter, the equivalent Π circuit, and the theoretical
maximum real power that that a 200-mile line can deliver.
( ) ( )' 104.06
2 1 2 0.9182 1'0.0016
2 104.06
Z B j
AYj
B j
= =− −
= = =
max
765*7655623.92
' 104.06s RV V
P MWX
= = =
12
Thermal limits
Thermal limit is due to heating of conductor and hence depends
heavily on ambient conditions.
These lines can operate at 200 degrees C.
For many lines, sagging is the limiting constraint. Trees grow,
and will eventually hit lines if they are planted under the line.
Angle limits
While the maximum power transfer occurs when line angle
difference is 90 degrees, actual limit is substantially less due to
multiple lines in the system.
Voltage stability limits
As power transfers increases, reactive losses increase as I2X.
This would cause the voltage drops, resulting in a potentially
cascading voltage collapse.
Limits Affecting Power Transfer
Midterm Exam
Is scheduled on March 8th during the class.
Covers all materials we have discussed in chapters 2-5.
Close book, close note, you can bring one letter-size double-
sided information sheet.
Calculator.
Problems
4 true/false
3 multiple choices
3 calculations
13
14
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