If the generator load is suddenly increased in one step from P1 to P2, and the mechanical input from the prime mover is also changed from
P1 to P2 to match with the load, the generator power angle (rotor lead angle) will start changing from δ1 to δ2 under the accelerating power
Pa = P2 – P1
The rotor with large mechanical inertia will take a relatively long time to reach from δ1 to δ2, overshoot δ2, and reach δ3, which can be beyond the Pmax point at δ = 90°.
Above δ2, the generator supplies more power than the prime mover input P2, and hence the rotor decelerates back to δ2 and undershoots below δ2. The generator now has more mechanical input than electrical output, accelerating the rotor again. Such swings around δ2 continue until damped out by the power loss in the damper bars. For simplicity, if we ignore the damper bar effect during the first swing, the machine will be stable if the accelerating energy equals the decelerating energy.
In a torsional system, since the energy is equal to (torque × angle δ) and the torque equals (power ÷ angular speed), we can write the condition for the transient stability, that is, the
accelerating energy = decelerating energy
The generator will be stable during the transient oscillations if the two areas are equal, and hence the name equal area criteria of transient stability.
Impact tripping breaker on ship generator stability
The generator can also lose stability if the circuit breaker trips under a short circuit, delivering no electrical power to the load, rotor
will accelerate under the prime mover input power, accelerating to increase the rotor angle to δ2. If the fault is cleared in time and the circuit breaker is closed under the automatic reclosure scheme (common on land-based systems), the generator starts delivering electrical power in excess of the prime mover input and starts decelerating to its original power angle δ1 after the inertial overshoot to δ3.
The mechanical transients, which are much slower than electrical transients, occur in a synchronous generator supplied by pulsating power prime movers (e.g., diesel engine) or synchronous motor driving pulsating loads (e.g., compressor loads). If the transient oscillations are small around small δ, sinδ ≈ δ in radians, and the torque becomes linear with δ, that is,
T = Tmax × δ
The machine appears as a torsional spring with springs constant
K = T/δ = Tmax n-m/rad
For large oscillations, the linearity no longer holds, and the solution must be obtained by a step-by-step numerical method on the computer.
Moreover, as δ approaches 90°, the spring becomes softer and the
machine may lose synchronism.
For a large turbine-generator, typical tcritical is less than one second for automatic reclosure of the circuit breaker after a fault for maintaining
transient stability, and the mechanical oscillation frequency is generally less than 1 Hz.