Page 189 - Hybrid-Renewable Energy Systems in Microgrids

Vsq=RsIsq+wsψsd+1wb.dψsqdt

170 Hybrid-Renewable Energy Systems in Microgrids

1 d ψ sq
V sq = R I + ωψ sd + . (9.10)
ssq
s
Vrd=RrIrd−w 2 ψrq+1wb.dψrddt ω b dt
1 d ψ
V rd = R I − ωψ + . rd (9.11)
rrd
ω
rq
2
b dt
1 d ψ rq
Vrq=RrIrq+w 2 ψrd+1wb.dψrqdt V rq = R I + ωψ rd + . (9.12)
2
ω
rrq
b dt
ψs=LsIs+LmIr ψ = LI + LI (9.13)
s s
s
m r
ψr=LmIs+LrIr ψ = LI + LI (9.14)
r
m s
r r
⊥ r w b d I r d d t = − R ´ r I r d + R s L m L s 2 ψ s d − L
mLswrψsq+L´rw Irq−LmLsVsd
2
L
Te=LmLsψsqIrd−ψsdIrq T = m ψ ( I − ψ I ) (9.15)
e sq rd sd rq
L s
Using Eqs. (9.9–9.15), state space representation of IG can be written as a function
of state variables:
1 d ψ sd R s RL m
s
1wbdψsddt=−RsLsψsd+RsLmLmIrd+wsψsq+Vsd ω b dt =− L s ψ sd + L m I rd + ω ψ sq + V sd (9.16)
s
1wbdψsqdt=−RsLsψsq+RsLmLmIrq−wsψsd 1 d ψ sq =− R s ψ + RL m I − ω ψ + V (9.17)
s
+Vsq ω b dt L s sq L m rq s sd sq

´
´ dI rd =− RI + RL m ψ − L m ω ψ ´ ω I − L m V (9.18)
L r
s
ω b dt r rd L 2 s sd L s r sq + Lr 2 rq L s sd
L ´ r dI rq ´ RL m L m ´ L m
s
⊥ r w b d I r q d t = − R ´ r I r q + R s L m L s 2 ψ s q + L ω dt =− RI rq + L 2 ψ + L ω ψ − Lr ω I rd − L V sq (9.19)
r
sd
2
r
sq
mLswrψsd−L´rw Ird−LmLsVsq b s s s
2
2
⊥´r=Rr+Lm2Rs/Ls2 here, Ŕ = R r + L R / L 2 s
r
m
s
2
⊥r=Lr−Lm2/Ls Ĺ r = L r − L / L s
m
where V s and V r are stator and rotor voltage, L m is the mutual inductance, I rd and I rq
are the d-axis and q-axis rotor current, L s and L r are stator and rotor self-inductance,