Page 90 - Handbook of Electrical Engineering

70 HANDBOOK OF ELECTRICAL ENGINEERING

∗
Q s = Imaginary part of (EI )

= I E q sin(δ r + Ø) − E d cos(δ r + Ø)
The active and reactive power losses are,

2 2
P loss = I d R d + I q R q
2 2
Q loss = I d X d + I q X q

From which the summations of powers are,

P S = P r + P loss
Q S = Q r + Q loss

The equations above are shown for the steady state. However they apply equally well for the

transient and sub-transient states provided the substitutions for E , E , E , E , X , X , X and

d q d q d q d
X are made systematically. Such substitutions are necessary in the digital computation of transient

q
disturbances in power systems, those that are often called ‘transient stability studies’.
3.5.2 The Particular Case of a Salient Pole Generator
The first simplification is to assume R d = R q = R a which is very practical. In addition the steady
state variables E d and δ r can be assumed to be zero. Hence the equations in sub-section 3.5.1
become.
V d = V sin δ
V q = V cos δ
E d = 0

E q = E
I d =−I sin(φ + δ)
I q = I cos(φ + δ)
(E q − V q )X q − V d R a
I d =
2
X d X q + R a
(E q − V q )R a − V d X d
I q =
2
X d X q + R a
P r1 + P r2
P r =
DEN
Q r1 + Q r2
Q r =
DEN