Electrical Power Systems

288 Electrical Power Systems
Note that Pa = Pi – Pe = accelerating power and d0 is the initial power angle before the rotor
begins to swing because of a disturbance. The stability criterion dd
= 0 (at some moment)
dt
implies that
d
z Pd a d
= 0 ...(11.42)
d0
This condition requires that, for stability, the area under the graph of accelerating power
P a versus d must be zero for some value of d; that is, the positive (or accelerating) area under the
graph must be equal to the negative (or decelerating) area. This criterion is therefore known as
the equal–area criterion for stability and it is shown in ig. 11.7.
ig. 11.7: Power angle characteristic.
In ig.11.7, point ‘a’ corresponding to the d0 is the initial steady-state operating point. At
this point, the input power to the machine, Pi0 = Pe0. Where Pe0 is the developed power. When
a sudden increase in shaft input power occurs to Pi, the accelerating power, Pa, becomes positive
and the rotor moves towards point ‘b’. We have assumed that the machine is connected to a
large power system so that |Vt| does not change and also that xd does not change and that a
constant field current maintains |Eg| constant. Consequently, the rotor accelerates and the
power angle begins to increase. At point ‘b’, Pi = Pe and d =d1. But dd
is still positive and d
dt
overshoots ‘b’, the final steady-state operating point. Now Pa is negative and d ultimately
reaches a maximum value d2 , or point ‘c’ and then swing back towards ‘b’. Therefore, the rotor
settles to the point ‘b’, which is ultimate steady-state stable operating point as shown in ig.
11.7. In accordance with eqn. (11.42), the equal-area criterian requires that, for stability,
Area A1 = Area A2
d1
d2
or zbPi - Pmax sin dgdd = zbPmax sin d - Pigdd ...(11.43)
d0
d1