Page 128 - Hybrid-Renewable Energy Systems in Microgrids

112 Hybrid-Renewable Energy Systems in Microgrids

 i +   i − 
i + =  α  ; i − =  α  (7.11)
i+=iα+iβ+; i−=iα−iβ−
 i +   i − 
 β   β 
With only fundamental frequency components considered, the voltage and current
expressed in stationary reference frame can be transformed into synchronous refer-
ence frame using Park transformation:

vd+vq+=coswtsinwt−sinwtcoswtvα+v  v +   cos ω ( t ) sin ω ( t )   v + 
β+; vd−vq− = cos−wtsin−wt−sin−w-  d  =    α  ;
 v +   −sin ω ( t ) cos ω ( t )   v + 
tcos−wtvα−vβ−  q     β  (7.12)
 v −   cos (− ω ) t sin (− ω ) t   v − 
 d     α 
=
 v −   −sin (− ω ) t cos (− t v − 
ω )  
 q     β 
 i +   cos ω ( t ) sin ω ( t )   i + 
 d     α  ;
=
 i q +   −sin ω ( t ) cos ω ( t )   i + β 
id+iq+ = coswtsinwt− sinwtcoswtiα-       (7.13)
+iβ+; id−iq−=cos−wtsin−wt−sin−wt   i d −    cos (− ω ) t sin (− ω ) t   i α −  
 
ω )  
cos−wtiα−iβ−  i −  =   −sin (− ω ) t cos (− t i − 
 q     β 

where subscripts ‘d’ and ‘q’ respectively denote d- and q-axes in synchronous refer-
ence frame. By substituting Eqs. (7.12) and (7.13) into Eqs. (7.6) and (7.7), respec-
tively, the instantaneous power can be further expressed in synchronous reference
frame as:

t)
(
(
p = P + P cos2ω + P sin2ω t)

s2
c2
p=P¯⊥v+⋅i++v−⋅i− + Pc2cos vi⋅+ vi ⋅ − (7.14)
+
−
+
−
+
−
+
2wt+Ps2sin2wt⊥v+⋅i−+v−⋅i+ vi⋅+ vi ⋅
t)
(
t)
(
q = Q + Q cos2ω + Q sin2ω

s2
c2
q = Q ¯ ⊥ v⊥+ ⋅ i++v⊥ − ⋅ i − + + − − (7.15)
vi
+
−
vi
⊥
vi⋅+ ⊥ ⋅ vi⋅+ ⊥ − ⋅ +
⊥
+ Q c 2 cos 2 w t + Q s 2 sin -
2wt⊥v⊥+⋅i−+v⊥−⋅i+ where
3
−−
−−
++
P¯=32vd+id++vq+iq++vd−id P = v i ( ++ + v i + v i + v i ) (7.16)
2 dd qq dd qq
−+vq−iq−
3
−+
+−
+−
Pc2=32vd−id++vq−iq++vd+ P = v i ( −+ + v i + v i + v i ) (7.17)
d d
c2
d d
q q
q q
id−+vq+iq− 2