Page 424 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 403
3 2
in phase ca, Y ca ˆ (170 ‡ j105:537) 10 /560 ˆ 0:542 ‡ j0:336 S ˆ G ca ‡ jB ca
p
B gab ˆ 0 ‡ (0:542 0:542)/ 3 ˆ 0S
p
B gbc ˆ ( 0:336) ‡ (0:638 0:542)/ 3 ˆ 0:391 S
p
B gca ˆ 0:336 ‡ (0:542 0:638)/ 3 ˆ 0:391 S
The resultant G ph ˆ G ab ‡ G bc ‡ G ca ˆ 0:638 ‡ 0:542 ‡ 0:542 ˆ 1:722 S
p
I ph ˆ I L ˆ (560/ 3) 1:722 ˆ 556:75 A
p
and the total power is 3 560 556:75 ˆ 540 kW (same as original unbal-
anced load).
28. (i) Write down the equations for the voltage and current profiles of a lossless
power transmission line in terms of the receiving-end voltage E r , the receiving-
end current I r , the surge impedance Z 0 , and the wavenumber b. Also give
formulas for b and Z 0 in terms of x L , the series inductive reactance per unit
length, and x C the shunt capacitive reactance per unit length.
(ii) Using the equations in part (a), prove that the current at the sending-end of a
symmetrical line is given by

E s sin y
I s ˆ j
Z 0 1 ‡ cos y
where y ˆ ba is the electrical length of the line and a is the actual length. Also
derive an equation for the mid-point voltage V m in terms of E s and y. Sketch
the profiles of jV(x)j and jI(x)j for 0 x a, where x is distance along the
line measured from the sending end.
(iii) A symmetrical cable line operates at no load with terminal voltages of 345 kV
line±line. It has an inductive reactance of 0:56
/km and a capacitive admit-
tance of 50:0 mS/km at 50 Hz. The cable length is a ˆ 19:8 km. Calculate and
draw to scale the phasor diagrams of line-to-neutral voltage and current
(a) at the sending end;
(b) at the mid-point; and
(c) at the receiving end.
Use the sending-end voltage E s as the reference phasor for all three cases, in order
toshowtherelativephaseanglesofthevoltagesandcurrentsatthethreepositions.

(i) V(x) ˆ V r cos b(a x) ‡ jZ 0 I r sin b(a x)
V r
I(x) ˆ j sin b(a x) ‡ I r cos b(a x)
Z 0
p p p p
where b ˆ o (lc) ˆ (x L /x C) and Z 0 ˆ (l/c) ˆ (x L x C ).