Analysis of Multi-Agent-Based Adaptive Droop-Controlled … 457
Fig. 2. Power flow and vector diagram between the main grid
and the inverter interfaced DG.
and the
transformation is shown in Eq. (2).
( )
( ) ( )
( )
( ) ( )
/
2 2
2
3 3
3
2 2
3 3
abc dq
cos wt cos wt cos wt
C
sin wt sin wt sin wt
p p
p p
é ù
- +
ê ú
= ×
ê ú
- - - - +
ê ú
ë û
(1)
( ) ( )
( ) ( )
( ) ( )
/
2 2
3 3
2 2
3 3
dq abc
cos wt sin wt
C cos wt sin wt
cos wt sin wt
p p
p p
é ù
-
ê ú
ê ú
= - - -
ê ú
ê ú
ê + - + ú
ë û
(2)
By using the former
transformation, the
mathematical equation of the inverter circuit can be obtained
from Fig. 1 as Eqns. (3)-(6). In this paper, the subscripts
and
represent the
component and the
component of some variable after the
transformation.
df d od f d f q
L u u R w L ii i= - - × + × ××
&
(3)
qf q oq f q f d
L u u R w L ii i= - - × - × ××
&
(4)
f od d od f oq
C u i i w C u× = - + × ×
&
(5)
f oq q oq f od
C u i i w C u× = - - × ×
&
(6)
Eqns. (3)-(4) demonstrate that
are not only
affected by
, but that they are affected by the
coupling currents
and
. Reference [29]
shows that the feedforward quantities
and cross
decoupled quantities
and
are used to
achieve independent current control in the
axis.
Thus, the inner current control loop is designed as Eqns.
(7)-(8) [29] to eliminate the current coupling.
( )
*
Ii
d pi d d f q od
K
u K i i w L i u
s
æ ö
= + × - - × × +
ç ÷
è ø
(7)
( )
*
Ii
q pi q q f d oq
K
u K i i w L i u
s
æ ö
= + × - + × × +
ç ÷
è ø
(8)
Applying Eqns. (7)-(8) into Eqns. (3)-(4) the following are
obtained:
( )
( )
*
Ii
f f d pi d d
K
s L R i K i i
s
æ ö
× + = + × -
ç ÷
è ø
(9)
( )
( )
*
Ii
f f q pi q q
K
s L R i K i i
s
æ ö
× + = + × -
ç ÷
è ø
(10)
As a result,
and
can be controlled separately without
coupling.
Eqns. (5)-(6) indicate that
are affected by
and the coupling voltages
and
.
Reference [29] indicates that the feedforward quantities
and cross decoupled quantities
and
are used to achieve independent voltage
control in the
axis. To eliminate the voltage coupling,
the outer voltage control loop is designed as Eqns. (11)-(12)
[29].
( )
*
Iu
d pu od od f oq od
K
i K u u w C u i
s
æ ö
= + × - - × × +
ç ÷
è ø
(11)
( )
*
Iu
q pu oq oq f od oq
K
i K u u w C u i
s
æ ö
= + × - + × × +
ç ÷
è ø
(12)
Applying Eqns. (11)-(12) into Eqns. (5)-(6), the following
are obtained:
( )
*
Iu
f od pu od od
K
s C u K u u
s
æ ö
× × = + × -
ç ÷
è ø
(13)
( )
*
Iu
f oq pu oq oq
K
s C u K u u
s
æ ö
× = + -
ç ÷
è ø
× ×
(14)
and
can be controlled separately without
coupling.
B. Adaptive Droop Control
The values of the active and reactive powers flowing
between two AC voltage sources, which are connected in
parallel through the line impedance as shown in Fig. 2, can be
calculated as Eqns. (15)-(16) [30], [31]. In Fig. 2,
and
are the line impedance and inductance, respectively.
and
are the Root-Mean-Square (RMS) value of the AC
voltage sources, and their phases are
and
,
respectively. If
and
is small, then
Eqns. (15)-(16) can be simplified as Eqns. (17)-(18).
( )
1
1 2 2
2 2
U
P R U U cos XU sin
R X
d d
= - +
é ù
ë û
+
(15)
( )
1
2 1 2
2 2
U
Q RU sin X U U cos
R X
d d
= - + -
é ù
ë û
+
(16)
(17)
(18)
As shown by Eqns. (17)-(18), the active power, flowing
from voltage source 1 to 2 through a highly inductive line
impedance, can be controlled by varying the phase
. The
reactive power supplied by source 1 can be controlled by
controlling the magnitude of source 1 (
). This is the basis of
the conventional
and
droop controls. As