0 0 0 0 0 0 0 0 sin cos cos sin cos sin sin cos od oq


   
   
(0) (0)
(0) (0)
(0) (0)
(0) (0)
sin cos
cos sin
,
cos sin sin cos
od oq
S C
od oq
I I
T T
I I
 
 
   
 
 
 

   
 
  
 
 
 
where
(0) 
is the initial value of

.
Similarly, the input signal to the inverter model is the bus voltage which is expressed on the
common reference frame. The bus voltage can be converted to the individual inverter reference
frame using reverse transformation, given by
 
1 1
bdq S bDQ V
v T v T   
   
    
   
   
   
where

28
   
   
(0) (0)
1
(0) (0)
sin cos
cos sin
bD bQ
V
bD bQ
V V
T
V V
 
 

 
 
  
 
 
 
It is noted that in case of multiple inverters, the inverter whose dq frame is taken as the
common reference frame has to provide its reference frequency change ∆ ω
com
to all the sub‐
modules. Also, care should be taken to apply this modification in the power controller model.
A complete state‐space model of the i‐th VSC (inverter) is given below by combining the state‐
space models of the power controller, voltage controller, current controller and output LCL
filter,
   
com
inv INV inv INV com i i i i bDQi i
x A x B v B


 
        
 
 
 
.
 
INVω
inv
INVc
i
i
i
oDQi
i
C
x
i
C
 
   
 
 
 

   
where
inv
T
i i i i dqi dqi ldqi odqi odqi
x P Q i v i               
 
and
 
1 2
1 1 1 1 2 2
INV
1 1 1
1
2 1 1 1
1 1 2 2
3
13 13
0 0
0 0
0
0 0
Pi Pi
V i Pvi V i
C i V i Pvi C i Vi C i V i C i
i
LCL i C i V i Pvi
LCLi
LCL i Vi LCL i C i Vi LCL i Ci
LCL i C i V i C i
LCL i P i
A B
B C B
B D C B C B D B
A
B D D C
A
B T B D C B C
B D D D
B C



 
 
 
 

 


 

 
  
 
 

 
 
com
com
INV
1
2
13 2 13 1
0
0
0
,
0
0
0
P
i i
LCL S
B
B B
B T



 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1 13
INVω
1 13
0 0 0 1
0 0 0 0 1
P
i
C i
C
i



 






   
INVc
2 13
0 0 0 0 0 0
i C S
C T T

  
 
5.8 State Space Representation of an Islanded Microgrid System
The islanded operation is realized by opening the isolating the PCC switch (shown in Figure 26)
which disconnects the microgrid from the main grid. Once the microgrid is isolated the

29
microsources feeding the system are responsible for maintaining the voltage and frequency
while sharing the power. During autonomous operation it is important to avoid over‐loading of
inverters and to ensure that the changes in load are taken by VSC converts in a well‐controlled
manner.