Sudden Change in Mechanical Input

Applications of Equal Area Criterion

Below figure shows the transient model of a single machine connected to infinite bus bar.

Let the mechanical input to the rotor be suddenly increased to P_m1. Due to the accelerating power  P_a = (P_m1-P_e) rotor accelerates and its speed increases (ω > ωs) as a result δ also increases as shown below in figure.

At angle δ1, P_a = (P_m1-P_e) = Pmax Sin δ1 = 0 (at point b), but the rotor angle continues to increase as ω > ωs. Now, P_a becomes negative (decelerating) hence, the rotor speed starts reducing but, the angle continues to increase till at angle δ2, ω = ωs (at point C). At C, the decelerating area A₂ equals the accelerating area A₁, i.e.,

Since the rotor is deaccelerating, the speed reduces below ωs and the rotor angle begins to reduce. The state point now traverses the P_e – δ curve in opposite direction as shown below in figure (power angle curve). The system oscillates about the new steady-state point b (δ = δ₁) with angle upto δ₀ and δ₂ on the two sides. When the oscillations decay out, the system settles to the new steady state where

P_m1-P_e = P_max Sin δ₁

Now, using figure (power angle curve), the two areas are

Now, any further increase in 1 m P means that the area available for A₂ is less than A₁, so that the excess kinetic energy causes δ to increase beyond point c and the decelerating power changes over to accelerating power  and hence, the system becomes unstable. The system will remain stable even though the rotor oscillate beyond δ = 90°, as long as the equal area criterion is met. The oscillations for δ = 90° is shown below in figure which is a stable system as A₂ (decelerating area) is greater than A₁ (accelerating area).

Note:

• For the transient stability of a synchronous generator connected to an infinite bus, the decelerating area must be greater than accelerating area. (For figure (power angle curve), A₁ is accelerating and A₂ is decelerating area).
• For the transient stability of a synchronous motor connected to an infinite bus, the decelerating area must be less than the accelerating area. (For motor in figure (power angle curve) and figure (transient stability with sudden increase), A₁ will be decelerating area while A₂ will be accelerating area).
• More the decelerating area, more the system stable (for generator action).