Page 219 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 183

_
P ^ P 1 ^ P 1
x 2 x 1
2 1 1 2 1
2
R L C 2 R C 2 x 1 R L C 2 2 Þ
1
f L f f x 1 C f x C f
_
^
^ P 2 2 2
2 P 2 P 2 52 c 1 z 2 c 2 z 1 z 2
1 2
Γ 2

^ ^
_ x 2 2 c 1 P 2 c 1 P 1
2 c 2 1 z 1 1 c 1 1 c 2 Þz 2 2 1
1 ð
C f x 1 C f x 1 C f
_
x 2 x 1 ^ P 2 ^ P 1 ^ P 1
2 1 1 2 1 Þ
2
2
R L C 2 R C 2 x 1 R L C 2 x 1 C f x C f
f L f f 1
_ !
^ P 2
c 1 z 2 z 2
1 P 2 ^ P 2 2 2 1 2 (5.203)
Γ 2 x 1 C f x 1 R L C f
Analyzing (5.203), it can be concluded that stability is assured if the
terms in brackets and marked in blue are set to zero. Then it is guaran-
teed that Eq. (5.203) # 0 because the design parameters c 1 ,c 2 . 0. In
order to render the terms in brackets to zero, the variables d and the
^
update law for the second estimate P 2 are set to cancel out the rest of the
terms marked in blue:
!
L f C f 2 1 1
d 5 c 2 1 z 1 2 c 1 1 c 2 Þz 2 1 2 x 1
1
2
E L f C f R C 2 f
ð
L
! _
1 ^ ^ ^ P 2 ^ P 1 ^ P 1
R f c 1 P 2 c 1 P 1
1 1 2 x 2 1 2 2 2 1 2 2 Þ
L f C f R L C x 1 C f x 1 C f x 1 R L C x 1 C f x C f
f f 1
(5.204)
!
_ ^ c 1 z 2 z 2
P 2 5 Γ 2 2 1 2 (5.205)
x 1 C f x 1 R L C
f
An interesting observation can be made by analyzing the complete
system composed of the error variables (5.206) and the update laws for
the estimates (5.199) and (5.205).