Transmission Lines: Example Problem A 220-kV, 150 MVA, 60-Hz, three-phase transmission line is 140 km long. The characteristic parameters of the transmission line are:

r = 0.09 Ω/km; x = 0.88 Ω/km; y =4.1×10-6 S/km where, r is the resistance per kilometer, x is the reactance per kilometer, y is the shunt admittance per kilometer. The voltage at the receiving end of the transmission line is 210 kV. Although this transmission line would normally be considered a medium-length transmission line, we will treat it as short line:

a) What is series impedance and shunt impedance of the transmission line? b) What is the sending end voltage if the line is supplying rated voltage and apparent

power at 0.85 PF lagging? At unity PF? At 0.85 PF leading? c) What is the voltage regulation of the transmission line for each of the cases in (b)? d) What is the efficiency of the transmission line when it is supplying rated apparent

power at 0.85 PF lagging?

Solution:

• The series resistance, series reactance, and shunt admittance of the transmission line:

( )( )( )( )( )( ) S 1074.5km 140 101.4

2.123km 140 Ω/km 88.0 8.16km 140 Ω/km 12.0

46 −− ×=×==

Ω===Ω===

ydY

xdXrdR

• The current out of this transmission line is given by. Note that the per-phase

equivalent circuit implicitly assumes a wye (Y) connection, so the current is the same in phase or line configuration.

Generator Sending

end

Load

Receiving end

Transmission line

A 412kV 2103

MVA 1503

3

=×

==

=

L

outR

RLout

VSI

IVS

The phase voltage of the transmission line is

kV 1213kV 210

==RV

Since the transmission line is considered as “short”, the admittance (or shunt capacitance) may be ignored. This produces in a per phase transmission line model consisting of a series resistance and inductance only. The phase voltage at the sending end of the line when the power factor is 0.85 lagging will be

kV 4.146.158)2.1238.16)(8.31412(0121 o∠=+−∠+∠=

++=

j

XIRIVVoo

LRRRs

The resulting line voltage at the sending end (0.85 PF lagging) is

kV 2756.1583 =×=LV The phase voltage at the sending end of the line when the power factor is unity will be

kV 6.216.137)2.1238.16)(0412(0121 o∠=+∠+∠=

++=

j

XIRIVRVoo

LRRs

The resulting line voltage at the sending end (unity PF) is

kV 2381373 =×=LV The phase voltage at the sending end of the line when the power factor is 0.85 leading

kV 0.255.110)2.1238.16)(8.31412(0121 o∠=+∠+∠=

++=

j

XIRIVRVoo

LRRs

The resulting line voltage at the sending end (0.85 leading) is

kV 1915.1103 =×=LV The voltage regulation of a transmission line is given by

100×−

=fl

flnlR V

VVV

• The voltage regulation at 0.85 PF lagging; PF unity; and 8.5 PF leading:

%7.8100210

210191

%7.13100210

210238

%1.31100210

210275

−=×−

=

=×−

=

=×−

=

R

R

R

V

V

V

• The output power from the transmission line at 0.85 PF lagging

kW 12785.04121213cos3

=×××== RRRout IVP θ

The input power from the transmission line

( ) kW 7.135)8.31(4.14cos412kV 6.1583cos3

=−−×××== SSSin IVP θ

The transmission line efficiency at full load and 0.85 PF lagging is

%6.93%100kW 7.135

kW 127%100 =×=×=in

outPPη