Handbook of Electrical Engineering

FAULT CALCULATIONS AND STABILITY STUDIES 305

and Pa= 2 πM

df

dt

Pfw=Ffw·f

Pem=fem(X′′d,X′′q,Rd,Rq,Xf,Rf,[fo−f])

Pelec=felec(V , E,sinδc,Xdg,Xq,X′d,X′q,Ra)

Pmech=Gpm(p)

[

Pref+A

(

f−fo

fo

)]

Where: M = polar moment of inertia of the generator and its prime-mover.
f = generator shaft speed (i.e. frequency).
fo = reference frequency of the system, e.g. 50 Hz or 60 Hz.
Ffω = friction and windage coefficient.
V = terminal voltage of the generator.
E = fe(If)=internal emf of the generator as created by the field currentIf.
δ = rotor angle between the terminal voltage and the rotor direct axis.
Xd = direct axis synchronous reactance.
Xqg = quadrature axis synchronous reactance.
X′d = direct axis transient reactance.
X′q = quadrature axis transient reactance.
X′′d = direct axis sub-transient reactance.
X′′q = quadrature axis sub-transient reactance.
Xfg = rotor field leakage reactance.
Rd = direct axis rotor damper bar resistance.
Rq = quadrature axis rotor damper bar resistance.
Rf = rotor field circuit resistance.
Ra = stator resistance.
Gmp(p) = transfer function for the dynamics of the prime mover.
(p) = general differential operator

d()

dt

Pref = power set-point of the prime mover.

A = governor droop setting.

fe,fem andfelecare functions of the variables shown.

In some situations, the rate of change of shaft frequency is equal to the second rate of change
of rotor angle, e.g. when the system frequency remains almost constant or changes slowly. Hence:

df

dt

=

1

2 π

d^2 δ

dt^2

11.11.2.2 Multi-generator situations

The equations of sub-section 11.11.2.1 can be applied to all the generators in an interconnected
system. At steady state stable conditions all the generator shaft frequenciesf must be equal. During
disturbed conditions, the average frequency of rotation of each generator shaft will be equal, otherwise