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384 Examples, problems and exercises
12. (i) Write an equation for the phasor voltage E s at the sending end of a lossless
cable, in terms of the voltage V r , power P r and reactive power Q r at the
receiving end, if the electrical length is y radians. Use this equation to derive
an expression for the reactance X required to make the no-load voltage at the
receiving end of a radial transmission cable equal to the sending-end voltage.
(ii) Using the theory of part (i) and any necessary development thereof, deter-
mine the values of four reactors equally spaced along a 500 kV, 80-km
symmetrical line such that the no-load voltage profile is substantially flat.
Of the four reactors, one is at the sending end and one at the receiving end,
and the synchronous machines at the two ends contribute no reactive power.
The line series inductive reactance is 0:60
/km and shunt capacitive
susceptance is 50:0 mS/km.
(iii) What is the maximum voltage in the compensated line of part (b) at no-load,
and where does it occur?
(iv) What is the total combined reactive power of the four reactors at no-load
and rated voltage?
Z o
(i) E s V r cos y jZ o I r sin y V r cos y sin y
X
so that for E s V r
sin y
X Z o
1 cos y
p 6
(ii) y 80 (0:60 50 10 ) 0:43818 radians 25:1
p 6
Z 0 (0:60/(50 10 )) 109:545
(iii) 2X at the ends and X at two intermediate locations (26.7 km from each end).
Fig. 9.7
Z 0 sin (y/n) 109:545 sin (25:1/3)
X 748
2 1 cos (y/n) 2 1 cos (25:1/3)
Maximum voltage is at the mid-point and at 80/6 13:3 km from each end:
V m E s /cos (y/2n) with n 3; i.e. V m 500/ cos (25:1/6) 500
1:00267 kV line±line or 1.00267 p.u.
p
2
(iv) Totalcompensatingreactivepower (500/ 3) /748 3 (1 11/2 1/2)
1:003MVAr.
13. A 500-kV cable is 80 km long and has a mid-point dynamic shunt compensator
that maintains the voltage at its terminals equal to 1.0 p.u. under all loading
