Page 315 - Handbook of Electrical Engineering

FAULT CALCULATIONS AND STABILITY STUDIES 301

Typical changes would be transformer and generator reactances, limiting the maximum size
of the largest motors, providing special starters for large motors (e.g. Korndorfer method), provision
of special interlocks or inhibits on the switchgear. Occasionally, however, it is necessary to extend
the existing power system, e.g. extra load, more generators, adding an unusually large motor, or to
interconnect systems using long-distance cables or overhead lines. When this happens it is essential
to carry out a stability study to ensure that the existing equipment still performs satisfactorily and
that any new equipment is compatible in all respects.

11.11.1 Steady State Stability

Steady state stability relates to the ability of the synchronous source (generators) to transfer power
to the synchronous sink (motors and/or other generators). This may be explained by simplifying the
synchronous power system as a transmission link (cable or overhead line) of reactance X and zero
resistance, a synchronous source (generator at the sending end of the link) and a synchronous sink
(load at the receiving end).
The source has an internal emf E S and the sink has an internal emf E R ,
Where phasor
ˆ
E S =|E S | δ ◦
and
E R, =|E R | 0 ◦ (reference phasor)
ˆ
The current flowing between E S and E R is:

ˆ ˆ
E S − E R
ˆ
I =|I| −Ø =
X
Since the reactance X consumes no power, the receiving end power must equal the sending
end power. (If the end voltages are not in steady state synchronism then the system is regarded as
being unstable.)
Hence:-

ˆ ˆ
ˆ ˆ
Power transferred (P) = Real part of E R I or E S I
(E R 0 )(E S δ − E R 0 )
◦ ◦ ◦

= Real
X
2
E R (E S cos δ + jE S sin δ) − E R
= Real
jX
2
−jE R E S cos δ + E R E S sin δ + jE R
= Real
X
Therefore,
E R E S sin δ
P = (11.10)
X