Page 166 - Modern Control of DC-Based Power Systems

130 Modern Control of DC-Based Power Systems

€ F 3
V ψ 1 F 1 V ψ 1 F 2 V ψ 1 5 0 (5.54)
V ψ 1 V ref
3 3
1 X X
21
F 1 5 1 A ðÞa i2 (5.55)
j; i
R eq C eq
j51 i51
!
3 3
X X
F 2 5 η A ðÞa i1 2 δ (5.56)
21
j; i
j51 i51
!
3 m3
1 X X
21
F 3 5 1 A ðÞa in11 P eq (5.57)
j; i
C eq
j51 i51
Eq. (5.54) can also be rewritten to investigate the system’s internal
behavior.
ð
_
V ψ 52 F 1 V ψ 2 F 2 V ψ dt 5 0 (5.58)
This corresponds to a PI control law inside the manifold, which has as
a consequence that asymptotic stability can be achieved and a constant
disturbance can be successfully rejected.
The system of (5.54) can also be expressed by its characteristic equa-
tion and therefore the stability conditions of (5.60) apply.

2 F 3
s 1 F 1 s 1 F 2 s 1 5 0 (5.59)
s 1 V c;ref
det AðÞ 6¼ 0; F 3 5 0; F 1 . 0; F 2 . 0; T i . 0; i 5 1; m (5.60)

Without restricting the generality F 3 5 0 can be set in order to deter-
mine the stability condition easily. By selecting a specific choice of coeffi-
cients of (5.61), system asymptotic stability can be ensured and the
nonlinearity is compensated [25].

1
21
21
21
F 3 5 0- 1 A ð 1; 1Þa 16 1 A ð 1; 2Þa 16 1 A ð 1; 3Þa 16
C eq
21 21 21 21
3; 1Þa 36
1 A ð 2; 1Þa 26 1 A ð 2; 2Þa 26 1 A ð 2; 3Þa 26 1 A ð
!
21 21
3; 3Þa 36 5 0
1 A ð 3; 2Þa 36 1 A ð