Formula Sheets: Characteristics & Performance of Transmission Lines | Power Systems - Electrical Engineering (EE) PDF Download

 Page 1


Power S ystems F ormula Sheet:
Char acteristics & Performance of
Tr ansmission Lines
1. Tr ansmission Line Models
• Short Line (< 80 km):
Z =R+jX
L
, Y ˜ 0
V
s
=V
r
+I
r
Z, I
s
=I
r
whereV
s
,I
s
are sending-end voltage and current,V
r
,I
r
are receiving-end volt-
age and current.
• Medium Line (80-250 km, Nominal -p model):
Z =R+jX
L
, Y =j?C
V
s
=V
r
+I
r
(
Z +
YZV
r
2
)
, I
s
=I
r
+
YV
r
2
+
YV
s
2
• Long Line (> 250 km):
? =
v
zy, Z
c
=
v
z
y
where? = a+jß is propagation constant,Z
c
is char acteristic impedance,z =
R+j?L (series impedance per unit length),y =G+j?C (shunt admittance per
unit length).
V
s
=V
r
cosh(?l)+I
r
Z
c
sinh(?l)
I
s
=I
r
cosh(?l)+
V
r
Z
c
sinh(?l)
2. ABCD Par ameters
• Gener al F orm :
[
V
s
I
s
]
=
[
A B
C D
][
V
r
I
r
]
• Short Line :
A = 1, B =Z, C = 0, D = 1
1
Page 2


Power S ystems F ormula Sheet:
Char acteristics & Performance of
Tr ansmission Lines
1. Tr ansmission Line Models
• Short Line (< 80 km):
Z =R+jX
L
, Y ˜ 0
V
s
=V
r
+I
r
Z, I
s
=I
r
whereV
s
,I
s
are sending-end voltage and current,V
r
,I
r
are receiving-end volt-
age and current.
• Medium Line (80-250 km, Nominal -p model):
Z =R+jX
L
, Y =j?C
V
s
=V
r
+I
r
(
Z +
YZV
r
2
)
, I
s
=I
r
+
YV
r
2
+
YV
s
2
• Long Line (> 250 km):
? =
v
zy, Z
c
=
v
z
y
where? = a+jß is propagation constant,Z
c
is char acteristic impedance,z =
R+j?L (series impedance per unit length),y =G+j?C (shunt admittance per
unit length).
V
s
=V
r
cosh(?l)+I
r
Z
c
sinh(?l)
I
s
=I
r
cosh(?l)+
V
r
Z
c
sinh(?l)
2. ABCD Par ameters
• Gener al F orm :
[
V
s
I
s
]
=
[
A B
C D
][
V
r
I
r
]
• Short Line :
A = 1, B =Z, C = 0, D = 1
1
• Medium Line (Nominal-p ) :
A = 1+
YZ
2
, B =Z, C =Y
(
1+
YZ
4
)
, D = 1+
YZ
2
• Long Line :
A = cosh(?l), B =Z
c
sinh(?l), C =
sinh(?l)
Z
c
, D = cosh(?l)
3. Performance Par ameters
• V oltage Regulation (VR) :
VR =
|V
s, no-load
|-|V
r, full-load
|
|V
r, full-load
|
×100 %
F or short line: V
s, no-load
=V
r
(whenI
r
= 0 ).
• Tr ansmission Efficiency :
? =
P
r
P
s
×100 %
whereP
r
=|V
r
||I
r
| cos?
r
(receiving-end power),P
s
=P
r
+P
loss
,P
loss
=|I
r
|
2
R .
• Power Flow (Short Line) :
P
r
=
|V
s
||V
r
|
X
L
sind
whered is the power angle (phase difference between V
s
andV
r
),X
L
=?L .
• Maximum Power Tr ansfer :
P
max
=
|V
s
||V
r
|
X
L
Occurs whend = 90
?
.
4. Surge Impedance and Loading
• Surge Impedance :
Z
c
=
v
L
C
?
F or lossless line: Z
c
=
v
L
C
.
• Surge Impedance Loading (SIL) :
SIL =
V
2
LL
Z
c
MW
whereV
LL
is line-to-line voltage (in kV , converted to V for calculation).
2
Page 3


Power S ystems F ormula Sheet:
Char acteristics & Performance of
Tr ansmission Lines
1. Tr ansmission Line Models
• Short Line (< 80 km):
Z =R+jX
L
, Y ˜ 0
V
s
=V
r
+I
r
Z, I
s
=I
r
whereV
s
,I
s
are sending-end voltage and current,V
r
,I
r
are receiving-end volt-
age and current.
• Medium Line (80-250 km, Nominal -p model):
Z =R+jX
L
, Y =j?C
V
s
=V
r
+I
r
(
Z +
YZV
r
2
)
, I
s
=I
r
+
YV
r
2
+
YV
s
2
• Long Line (> 250 km):
? =
v
zy, Z
c
=
v
z
y
where? = a+jß is propagation constant,Z
c
is char acteristic impedance,z =
R+j?L (series impedance per unit length),y =G+j?C (shunt admittance per
unit length).
V
s
=V
r
cosh(?l)+I
r
Z
c
sinh(?l)
I
s
=I
r
cosh(?l)+
V
r
Z
c
sinh(?l)
2. ABCD Par ameters
• Gener al F orm :
[
V
s
I
s
]
=
[
A B
C D
][
V
r
I
r
]
• Short Line :
A = 1, B =Z, C = 0, D = 1
1
• Medium Line (Nominal-p ) :
A = 1+
YZ
2
, B =Z, C =Y
(
1+
YZ
4
)
, D = 1+
YZ
2
• Long Line :
A = cosh(?l), B =Z
c
sinh(?l), C =
sinh(?l)
Z
c
, D = cosh(?l)
3. Performance Par ameters
• V oltage Regulation (VR) :
VR =
|V
s, no-load
|-|V
r, full-load
|
|V
r, full-load
|
×100 %
F or short line: V
s, no-load
=V
r
(whenI
r
= 0 ).
• Tr ansmission Efficiency :
? =
P
r
P
s
×100 %
whereP
r
=|V
r
||I
r
| cos?
r
(receiving-end power),P
s
=P
r
+P
loss
,P
loss
=|I
r
|
2
R .
• Power Flow (Short Line) :
P
r
=
|V
s
||V
r
|
X
L
sind
whered is the power angle (phase difference between V
s
andV
r
),X
L
=?L .
• Maximum Power Tr ansfer :
P
max
=
|V
s
||V
r
|
X
L
Occurs whend = 90
?
.
4. Surge Impedance and Loading
• Surge Impedance :
Z
c
=
v
L
C
?
F or lossless line: Z
c
=
v
L
C
.
• Surge Impedance Loading (SIL) :
SIL =
V
2
LL
Z
c
MW
whereV
LL
is line-to-line voltage (in kV , converted to V for calculation).
2
5. F err anti Effect
• Definition : Receiving-end voltage (|V
r
| ) exceeds sending-end voltage (|V
s
| ) un-
der no-load or light-load conditions for long lines.
• No-Load V oltage :
V
r
=
V
s
cosh(?l)
F or lossless line: |V
r
|>|V
s
| due to capacitive charging current.
6. Power Angle and Stability
• Power Angle Equation :
P =
|V
s
||V
r
|
B
sind
whereB is the imaginary part of the ABCD par ameterB .
• Steady-State Stability Limit : Maximum power tr ansfer without loss of syn-
chronism:
P
max
=
|V
s
||V
r
|
|B|
• Power Angle C urve : Power (P ) vs. angle (d ) shows stable oper ation for 0
?
<
d < 90
?
.
3