For a transient stability analysis, as long as equal area criterion is satisfied, the maximum angle to which rotor angle can oscillate is:  

Explanation:

Transient Stability Analysis and Equal Area Criterion:

Definition: Transient stability analysis is a crucial aspect of power system stability that evaluates the ability of a power system to maintain synchronism when subjected to a large disturbance, such as a short circuit or sudden load change. The rotor angle stability is an essential parameter in this analysis, as it represents the relative angular displacement of the synchronous machine rotors.

Equal Area Criterion: The equal area criterion is a graphical method used to assess transient stability. It states that for a system to remain stable following a disturbance, the area representing accelerating power (the excess mechanical input power over electrical output power) should be equal to the area representing decelerating power (the excess electrical output power over mechanical input power). These areas are plotted on the power-angle curve, which depicts the relationship between electrical power and rotor angle.

Correct Option Analysis:

The correct option is:

Option 3: Greater than 90˚

In transient stability analysis, the maximum rotor angle to which the rotor can oscillate is determined by the equal area criterion. The rotor angle can exceed 90˚ depending on the disturbance magnitude, system parameters, and power-angle characteristics. While small disturbances generally result in rotor angles below 90˚, large disturbances may cause the rotor angle to oscillate beyond 90˚ without losing synchronism, provided the accelerating and decelerating areas on the power-angle curve are equal.

This behavior occurs due to the nonlinear nature of the power-angle relationship, where stability depends not only on the magnitude of the rotor angle but also on the system's ability to balance accelerating and decelerating powers. Therefore, when the equal area criterion is satisfied, the rotor angle can oscillate to values greater than 90˚ without losing stability.

Additional Considerations:

It is important to note that the rotor angle exceeding 90˚ does not necessarily imply instability. Stability is determined by the ability of the system to return to synchronism after the disturbance. As long as the equal area criterion is satisfied, the system remains stable even if the rotor angle exceeds 90˚ momentarily during oscillations.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: 0˚ to 20˚

This option is incorrect because it severely underestimates the range of rotor angle oscillation during transient stability analysis. Rotor angles typically oscillate to values much higher than 20˚, especially for large disturbances. The power-angle curve and equal area criterion allow for a much broader range of oscillation, including values exceeding 90˚.

Option 2: 45˚ to 50˚

While rotor angles may oscillate within this range under specific conditions, this option is not universally applicable. The transient stability analysis considers a wide range of rotor angle oscillations, and the maximum angle depends on the system parameters and disturbance magnitude. In many cases, the rotor angle can exceed 50˚ or even 90˚ without losing stability.

Option 4: 65˚ to 85˚

This option is closer to the typical range observed in transient stability analysis but is still not universally applicable. The rotor angle can exceed 85˚ in scenarios involving large disturbances, provided the equal area criterion is satisfied. Therefore, limiting the maximum angle to this range does not encompass all possible stable conditions.

Conclusion:

The transient stability analysis is a complex evaluation of power system behavior under disturbances. The correct understanding of rotor angle oscillations, as governed by the equal area criterion, is crucial for ensuring system stability. While rotor angles within certain ranges may be typical, the ability of the system to remain stable depends on the balance of accelerating and decelerating powers, not the specific value of the rotor angle. As explained, the rotor angle can oscillate beyond 90˚ without losing stability, making Option 3 the correct answer.