Electrical Power Systems

Power System Stability 287
system. However, solution of swing equation is not always necessary to investigate the system
stability. Rather, in some cases, a direct approach may be taken. Such an approach is based on
the equal-area criterion.
Now consider eqn. (11.18),
\
Md
dt
2
d
2 = Pi – Pe 2
Md d
2 = Pa
dt
2
d d pa
\
= ...(11.39)
2
dt M
As is shown in ig.11.6, in an unstable system, d increases indefinitely with time and
machine loses synchranism. In a stable system, d undergoes oscillations, which eventually
die out due to damping. rom ig. 11.6, it is clear that, for a system to be stable, it must be that
dd
dd
= 0 at some instant. This criterion = 0 can simply be obtained from eqn. (11.39).
dt
dt
HG I KJ
Multiplying eqn. (11.39) by 2dd
, we have
dt
2
2dd
d d 2P
. 2 =
dt dt M
ig. 11.6: A plot of d (t).
d
dt
a d
...(11.40)
which, upon integration with respect to time, gives
dd
dt
= 2
M Pd
d
z a d
...(11.41)
HG I KJ 2
d0