Page 127 - Hybrid-Renewable Energy Systems in Microgrids

Study of control strategies of power electronics during faults in microgrids 111

where α and β represent the two phases in stationary frame. Distributed VSIs are gen-
erally interfaced with the network through a three-wire connection, which does not
provide zero-sequence current. Therefore, zero-sequence components are ignored in
the following analysis. Then, the instantaneous active power p and the instantaneous
reactive power q are defined according to instantaneous power theory [2]:

vi
p =⋅ = v i + v i (7.3) p=v⋅i=vαiα+vβiβ
αα
ββ
q = v ⋅ = v i − v i (7.4) q=v⊥⋅i=vβiα-vαiβ
i
βα
⊥
αβ
T
T
where v = [v α v β ] and i = [i α i β ] are the voltage and current vectors, respectively. The
operator “·” represents the dot product of vectors and the subscript “ ⊥ ” denotes an ⊥
orthogonal version of the original vector, which can be obtained by:
 0 1   0 1   v α   v β 
v =   v =     =   (7.5) v⊥=01−10v=01−10 vαvβ=vβ
⊥
  − 10     − 10     v β     − v α   −vα

As the voltage and current can be regarded a superposition of symmetrical compo-
nents, the instantaneous power shown in Eqs. (7.3) and (7.4) are rewritten as:

−
v +
+
−
+
−
+
p = v i + v i ( α + v ) ( + i ) + v ( β + + v ) ( + i ) (7.6) p=vαiα+vβiβ=vα++vα−iα++
+
=
−
i
i
α
ββ
β
α
α
αα
β
β
iα−+vβ++vβ−iβ++iβ−
−
+
v +
+
=
v +
−
−
−
−
+
q = v i − v i ( β + v ) ( + i ) ( α + v ) + i ( + i ) (7.7) q=vβiα−vαiβ=vβ++vβ−iα++iα−−vα++v
i
α
β
β
β
α
α
αβ
βα
α−iβ++iβ−
where superscripts ‘+’ and ‘−’ represent positive- and negative-sequence compo-
nents, respectively. By expanding Eqs. (7.6) and (7.7) and rearranging different terms
according to sequence components, there are:
−+
+−
++
++
+−
−−
−+
p = v i + v i + v i + v i + v i + v i + v i + v i
−−
αα
ββ
ββ
ββ
ββ
αα
αα αα (7.8) p=vα+iα++vβ+iβ++vα−iα−+vβ−iβ−⊥v

-
+
+
+
vi ⋅+ vi ⋅ -- vi ⋅+ vi ⋅ - +
+⋅i++v-⋅i-+vα+iα−+vα−iα++vβ+iβ−+vβ-
−iβ+⊥v+⋅i-+v-⋅i+
q = v i − v i + v i − v i + v i + v i − v i − v i
++
++
+−
−+
−+
−−
+−
−−
αβ
αβ
βα
αβ
βα
αβ
βα βα (7.9) q=vβ+iα+−vα+iβ++vβ−iα−−vα−iβ−⊥v⊥

+ + vi ⋅ - + - vi ⋅ +
−
-
⊥
⊥
vi ⋅+ ⊥ vi ⋅+ ⊥ +⋅i++v⊥−⋅i-+vβ+iα−+vβ−iα+−vα+iβ−-
where voltage and current vectors expressed in stationary frame are: −vα−iβ+⊥v⊥+⋅i-+v⊥-⋅i+
 v +   v +   v −   v − 
v + =  α  ; v + =  β  ; v − =  α  ; v − =  β  (7.10) v+=vα+vβ+; v⊥+=vβ+−vα+; v− = vα−vβ−; v⊥− = vβ
 v +  ⊥  −v +   v −  ⊥  −v − 
 β   α   β   α  −−vα−