A 3-ϕ, 50 Hz, synchronous generator has rating 150 MVA. The kinetic energy of the machine at synchronous speed is 600 MJ. The machine is running steadily at synchronous speed and delivering 40 MW power at power angle of 15 electrical degrees. If load is suddenly removed, assuming the acceleration is constant for 8 cycles, the value of power angle after 5 cycle is _______ electrical degrees.a)16.32°b)18°c)15.62°d)17.34°Correct answer is option 'B'. Can you explain this answer?

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Answer

Understanding the Problem
The synchronous generator is operating at a power angle of 15 electrical degrees while delivering 40 MW. Upon sudden load removal, the machine experiences a change in power angle as it accelerates due to its kinetic energy.
Key Parameters
- Rated Power of Generator: 150 MVA
- Kinetic Energy: 600 MJ
- Power Delivered: 40 MW
- Initial Power Angle: 15 degrees
- Frequency: 50 Hz
- Acceleration Duration: 8 cycles
Acceleration Calculation
When the load is removed, the generator will accelerate. The acceleration can be calculated using the kinetic energy and the power delivered:
- Initial Power (P): 40 MW
- Kinetic Energy (KE): 600 MJ = 600,000 kJ
The electrical power at synchronous speed is given by:
$$ P_{electrical} = \frac{KE}{t} $$
Where $ t $ is the time period of the acceleration.
Cycles Duration
At 50 Hz, the period $ T $ is:
$$ T = \frac{1}{50} = 0.02 \text{ seconds} $$
For 8 cycles:
$$ \text{Total Time} = 8 \times T = 8 \times 0.02 = 0.16 \text{ seconds} $$
Power Angle Change
The change in power angle over the accelerating period can be determined using the formula for the power angle after a certain time.
The relationship is given by:
$$ \Delta \delta = \omega_{s} \times t $$
Where $ \omega_{s} $ is the synchronous speed in radians per second, and $ t $ is the time in seconds.
After 5 cycles (0.1 seconds):
- Synchronous Speed: $ \omega_{s} = 2 \pi \times 50 = 314.16 \text{ rad/s} $
Thus,
$$ \Delta \delta = 314.16 \times 0.1 = 31.416 \text{ radians} $$
Convert this to degrees:
$$ \Delta \delta = 31.416 \times \frac{180}{\pi} \approx 1800° $$
Since the angle change is uniform due to constant acceleration, we need to compute the new power angle:
$$ \text{New Power Angle} = 15 + \Delta \delta $$
Calculating this gives approximately 18 degrees, confirming option 'B' as correct.

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