제 5 장
– Transmission Lines : Steady –State Operation -
성균관대학교
김 철 환
l Previous lectures have covered
how to calculate the distributed inductance,
capacitance and resistance
of transmission lines.
l In this section we will use
these distributed parameters to develop the transmission line models
used in power system analysis.
© 2012 Cengage Learning Engineering. All Rights Reserved.
5.1 Medium and Short Line Approximations 5.2 Transmission-Line Differential Equations 5.3 Equivalent π-Circuit
5.4 Lossless Lines
5.5 Maximum Power Flow 5.6 Line Loadability
5.7 Reactive Compensation Techniques
▣ 단거리 및 중거리 송전선로(short and medium-length transmission-line) - ABCD parameters
▣ the two-port network
▣ 단거리 송전선로 - 80 km
▣ 중거리 송전선로 - 80 ~ 250 km
▣ 장거리 송전선로 - 250 km 이상 :
, s
s I
V sending-end voltage and current :
, R
R I
V receiving-end voltage and current
▣ The relationship between the sending-end and receiving-end quantities can be written as
Or, in matrix format, where A, B, C, and D are parameters the
depend on the transmission-line constant R, L, C, and G.
▣ Network theory texts [5] show that ABCD parameters apply to linear, passive, bilateral two-port networks, with the following general relation :
R R
S
R R
S
DI CV
I
BI AV
V
+
=
+
=
ú û ù ê ë
é ú û ù ê ë
= é ú û ù ê ë é
R R S
S
I V D
C
B A
I V
1
=
- BC
AD
▣ The circuit in Figure 5.2 represents a short transmission line, usually applied to overhead 60-Hz lines less than 80km long.
Only the series resistance and reactance are included.(shunt admittance is neglected.)
The circuit applies to either single-phase or completely transposed three-phase lines operating under balanced conditions
For a completely transposed three-phase line, Z is the series impedance,
are positive-sequence line-to-neutral voltages, and are positive-sequence line currents.
▣ Parameter’s notation is followings.
z = R+jωL Ω/m, series impedance per unit length y = G+jωC S/m, shunt admittance per unit length Z = zl Ω, total series impedance
Y= yl S, total shunt admittance l = line length m
R S and V
V IS and IR
▣ ABCD parameters for the short line is obtained by KVL, KCL.
▣ By comparing (5.1.7) and (5.1.3), (A,B,C,D) and (1,Z,0,1)
For medium-length lines, typically ranging from 80 to 250 km at 60Hz, it is
common to lump the total shunt capacitance and locate half at each end of the line.
R S
R R
S
I I
ZI V
V
=
+
=
ú û ù ê ë é ú û ù ê ë
= é ú û ù ê ë é
R R S
S
I V Z
I V
1 0 1
0
1
=
=
=
=
\
C
Z B
D
A
▣ 중거리 송전선로
- lump the total shunt capacitance - a nominal π-circuit
- Figure 5.3.
▣ ABCD parameter of π-circuit (using KVL, KCL)
R R
R R
R
S V Y YZ V ZI
I Z V
V = + + = + ) +
1 2 ( 2 )
(
R R
R R
R R
S R
R S
YZ I YZ V
Y
ZI Y YZ V
Y I V
Y V Y I V
I
2 ) 1 ( 4 )
1 (
] 2 2 )
1 2 [(
2 2
+ + +
=
+ +
+ +
= +
+
=
(Writing a KVL equation)
(Writing a KCL equation)
Writing these equations in matrix format,
Thus, comparing them
▣ ABCD parameters of common networks in Figure 5.4
Including a series impedance
network that approximates a short line and a π circuit that
approximates a medium-length line.
A medium-length line could also be approximated by the T circuit shown in Figure 5.4, lumping half of the series impedance at each end of the line.
▣ 전압변동(Voltage regulation )
The change in voltage at the receiving end of the line when the load varies from no-load to a specified full load at a specified power factor, while the sending-end voltage is held constant.
▣ Phasor diagram for short transmission line
| 100
|
|
|
|
| - ´
=
RFL RFL RNL
V V VR V
percent VRNL = no-load receiving-end voltage (=Vs/A) VRFL = full-load receiving-end voltage
▣ 송전능력(line loadability) : 선로가 얼마만큼의 전력을 송전할 수 있는가 하는 능력
▣ 3가지 line-loading limits
열적 한계(Thermal limit)- 도체의 최대 허용온도(단거리 선로)
전압강하 한계(Voltage-drop limit) - (중거리 선로)
정상상태 안정도 한계(Steady-state stability limit) – 동기 발전기의 동기 유지 능력 (장거리 선로)
▣ Example 5.1
Q) A three-phase, 60-Hz, completely transposed 345-kV, 200-km line has two 795,000-cmil 26/2 ACSR conductors per bundle and the following positive- sequence line constants :
z = 0.032+j0.35 Ω/km y = j4.2 [s/km]
Full load at the receiving end of the line is 700MW at 0.99 p.f. leading and at 95% of rated voltage. Assuming a medium-length line, determine the
following :
a. ABCD parameters of the nominal π circuit
b. Sending-end voltage , current , and real power c. Percent voltage regulation
d. Thermal limit, based on the approximate current-carrying capacity listed in Table A.4
e. Transmission-line efficiency at full load
Sol) a. The total series impedance and shunt admittance values are
Vs
I
s PsS C
Z B
pu D
A
S j
yl Y
j j
zl Z
° Ð
´
=
° Ð
+
° Ð
´
=
W
° Ð
=
=
° Ð
=
° Ð
° Ð
´ +
=
=
° Ð
´
=
´
=
=
W
° Ð
= +
= +
=
=
- -
-
- -
08 . 90 10
277 . 8 ) 78 . 174 01476
. 0 1 )(
90 10
4 . 8 (
78 . 84 29 . 70
159 . 0 9706 .
0 ) 5 . 0 )(
78 . 84 29 . 70 )(
90 10
4 . 8 ( 1
90 10
4 . 8 ) 200 )(
10 2 . 4 (
78 . 84 29 . 70 70
4 . 6 ) 200 )(
35 . 0 032 . 0 (
4 4
4
4 6
b. The receiving-end voltage and current quantities are
° Ð
= +
=
° Ð
´ =
= Ð
° Ð
=
° Ð
=
=
=
-
14 . 26 6
. 199 )
)(
( ) )(
(
11 . 8 246 . ) 1
99 . 0 )(
345 95
. 0 )(
3 (
99 . 0 cos 700
0 2 . 189 3 0
8 . 327
8 . 327 )
345 )(
95 . 0 (
1
R R
S R R R
I Z V
A V
I V
V kVLL
kVLN
kA
kVLN
현재 이 이미지를 표시할 수 없습니다 .
c. The no-load receiving-end voltage is
MW P
I
pu kV
V
S S
LL S
5 . 730
) 5 . 15 14
. 26 cos(
) 241 . 1 )(
8 . 345 )(
3 (
5 . 15 241 . 1
) 11 . 8 246 . 1 )(
159 . 0 9706 .
0 ( ) 0 2 . 189 )(
08 . 90 10
277 . 8 (
00 . 1 8
. 345 3
6 . 199
4
=
° -
°
=
° Ð
=
° Ð
° Ð
+
° Ð
° Ð
´
=
»
=
=
-
% 7 . 8
% 8 100
. 327
8 . 327 3
. 356
3 . 356
= - ´
=
=
=
VR percent
A kV
VRNL VS LL
d. From Table A.4, the approximate current-carrying capacity of two 795,000- cmil 26/2 ACSR conductors is
e. The full-load line losses are
and the full-load transmission efficiency is kA 8 . 1 9 . 0 2´ =
MW P
PS - R =730.5-700 =30.5
% 8 . 95 100 =
´
=
S R
P EFF P
percent