Page 106 - Solutions Manual to accompany Electric Machinery Fundamentals

4-11. A paper mill has installed three steam generators (boilers) to provide process steam and also to use some
its waste products as an energy source. Since there is extra capacity, the mill has installed three 10-MW
turbine generators to take advantage of the situation. Each generator is a 4160-V, 12.5 MVA, 60 Hz, 0.8-
PF-lagging, two-pole, Y-connected synchronous generator with a synchronous reactance of 1.10  and an
armature resistance of 0.03 . Generators 1 and 2 have a characteristic power-frequency slope s P of 5
MW/Hz, and generator 3 has a slope of 6 MW/Hz.
(a) If the no-load frequency of each of the three generators is adjusted to 61 Hz, how much power
will the three machines be supplying when actual system frequency is 60 Hz?
(b) What is the maximum power the three generators can supply in this condition without the ratings
of one of them being exceeded? At what frequency does this limit occur? How much power does
each generator supply at that point?
(c) What would have to be done to get all three generators to supply their rated real and reactive
powers at an overall operating frequency of 60 Hz?
(d) What would the internal generated voltages of the three generators be under this condition?
SOLUTION

(a) If the system frequency is 60 Hz and the no-load frequencies of the generators are 61 Hz, then the
power supplied by the generators will be

P  1 s P 1  f nl1 f sys     5 MW/Hz 61 Hz 60 Hz  5.0 MW

P  2 s P 2  f nl2 f sys     5 MW/Hz 61 Hz 60 Hz   5.0 MW

P  3 s P 3  f nl3  f sys  6 MW/Hz 61 Hz 60 Hz    6.0 MW

Therefore the total power supplied by the generators is 16 MW.
(b) The maximum power supplied by any one generator is (12.5 MVA)(0.8) = 10 MW. Generator 3
will be the first machine to reach that limit. Generator 3 will supply this power at a frequency of
10 MW  6 MW/Hz 61 Hz   f sys 

f sys  59.33 Hz

At this point the power supplied by Generators 1 and 2 is

P  1 P  2 s P 1  f nl1 f sys     5 MW/Hz 61 Hz 59.33 Hz  8.35 MW
The total power supplied by the generators at this condition is 26.7 MW.

(c) To get each of the generators to supply 10 MW at 60 Hz, the no-load frequencies of Generator 1 and
Generator 2 would have to be adjusted to 62 Hz, and the no-load frequency of Generator 3 would have to
be adjusted to 61.67 Hz. The field currents of the three generators must then be adjusted to get them
supplying a power factor of 0.80 lagging. At that point, each generator will be supplying its rated real and
reactive power.
(d) Under the conditions of part (c), which are the rated conditions of the generators, the internal
generated voltage would be given by
E  A V   R I  A A jX I
S A

The phase voltage of the generators is 4160 V / 3 = 2402 V, and since the generators are Y-connected,
their rated current is

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