Short Transmission Line (33kV)

A short transmission line is considered when line distance is less than 80km and voltage of up to 50kV. Since this is a short transmission line, capacitance effect distributed along the line can be neglected.

This problem example demonstrate how to find receiving end current, sending end current, sending end voltage, sending end power factor,voltage regulation and transmission efficiency.

INPUT DATA
3-Phase supply voltage	33kV
3-phase power to deliver	3.5 MW
Line impedance	4 + j5 Ohm

Let's calculate first single phase voltage Vr, set as reference phasor.

Total impedance per phase = 4 + j5 Ohm = 6.4 /_ 51.3° Volt
Receiving end voltage, Vr=33kV/1.732 =19/_0° kV

Note: Vr is phase voltage Red to neutral

Since loads are balanced, all line currents are the same ,thus I_R =1/3 x 3.5MW/( 19kV x 0.86 ) =72 A or

3-phase current = 3.5MW/(1.732 x 33kV x 0.86) = 72A

I_R lags V_R by an angle of cos ^-1 0.86 = 30.68° (see phasor diagram below)
I_R =72 /_ -30.68° Amps

Note that Sending End Current,Is = Receiving End Current,Ir

Single phase Sending End Voltage, Vs = Ir Z + V_r = (72 /_ -30.68° x 6.4 /_ 51.3°) + 19000/_0° = 455.7/_20.62 + 19000/_0° = 19426.5 + j 160.5 =19427.2 /_0.5° Volts

3-phase Sending End Voltage = 1.732 x 19427.2 V =  33.65 kV. This means that in order to keep 33kV 3-phase at receiving end, a voltage of 33.65 kV must be injected at sending end side, eg at secondary side of 132kV/33kV transformer.

Voltage regulation = (Vs - Vr )/Vr = (19427.2 -19000)/19000 x 100 = 2.2 %

angle of Is + angle of Vs = 30.68° + 0.5° =31.18°

Sending end power factor = cos 31.18°=0.855 lagging.

or alternatively, from phasor diagram above, sending end power factor ={ Vr Cos ø_r + Ir R } / Vs = { (19000 x 0.86 ) + (72 x 4) }/19427.2 = 0.855 lagging

Losses per 3-phase lines = 3 x Ir ² R = 3 x 72 ² x 4 =60.8kW

Efficiency of transmission line = Power _delivered /Power _{sent out} =3500/ (3500 + 60.8) =98.3%

OUTPUT DATA
3-phase sending end voltage	33.65 kV
Voltage regulation	2.2%
Sending end power factor	0.855 lagging
Losses in 3-phase lines	60.8 kW
Efficiency of transmission lines	98.3%

Note: Voltage regulation and efficiency of medium transmission line (end condensor method, PI-method and T-method are covered in e-book

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