Handbook of Electrical Engineering

70 HANDBOOK OF ELECTRICAL ENGINEERING

Qs=Imaginary part of(EI∗)

=I

(

Eqsin(δr+Ø)−Edcos(δr+Ø)

)

The active and reactive power losses are,

Ploss=Id^2 Rd+Iq^2 Rq

Qloss=Id^2 Xd+Iq^2 Xq

From which the summations of powers are,

PS=Pr+Ploss

QS=Qr+Qloss

The equations above are shown for the steady state. However they apply equally well for the
transient and sub-transient states provided the substitutions forEd′,Eq′,E′′d,Eq′′,Xd′,X′q,X′′dand
X′′qare made systematically. Such substitutions are necessary in the digital computation of transient
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 assumeRd=Rq=Rawhich is very practical. In addition the steady
state variablesEd andδr can be assumed to be zero. Hence the equations in sub-section 3.5.1
become.

Vd=Vsinδ

Vq=Vcosδ

Ed= 0

Eq=E

Id=−Isin(φ+δ)

Iq=Icos(φ+δ)

Id=

(Eq−Vq)Xq−VdRa

XdXq+Ra^2

Iq=

(Eq−Vq)Ra−VdXd

XdXq+Ra^2

Pr=

Pr 1 +Pr 2

DEN

Qr=

Qr 1 +Qr 2

DEN