Page 165 - Modern Control of DC-Based Power Systems

Control Approaches for Parallel Source Converter Systems 129

T T
1
3
with Ψ 5 ðψ .. . ψ Þ and X 5 ðI L1 .. . I L3 Þ
0 1
0 1
a 13 a 14 a 15
b 1
B C B 5 ^
a 23 a 24 a 25 A @ A
A 5 @
b m (5.50)
a 33 a 34 a 35
a i;n11
i 5 1; m
b i 52 a i1 M 2 a i2 V 2
V 1 V ref
21
The control law can be calculated with U 5 A G and results in:
21 21 21
1; 3Þg 3
u 1 5 A ð 1; 1Þg 1 1 A ð 1; 2Þg 2 1 A ð
21 21 21
2; 3Þg 3 (5.51)
u 2 5 A ð 2; 1Þg 1 1 A ð 2; 2Þg 2 1 A ð
21 21 21
3; 3Þg 3
u 3 5 A ð 3; 1Þg 1 1 A ð 3; 2Þg 2 1 A ð
T
Where G is a vector G 5 ðg 1 g 2 .. . g m Þ , and each element corresponds
to:
0 1
ψ X
3
g i 52 i 1 @ a ij12 2 a i2 a in11 A V 1 V ref
2
1
T i L j R eq C eq
j51 V 1V ref R eq :C eq
0 1
!
3
a in11 X i j P eq
2ηa i1 V 2 @ A
a i2 2
2 1 δM 2
V 1V ref j51 C eq V 1 V ref C eq
(5.52)
Substituting (5.52) in (5.51)
m
X
21
u i 5 A ðÞd j i 5 1; m (5.53)
i; j
j51
As it is mentioned in [12], control laws (5.53) ensure stability towards
the manifolds (5.46). After reaching the manifold, the coefficients in
(5.48) define the current sharing. In other words, after passing the tran-
sient time and reaching steady state, by choosing the proper coefficients
in (5.48), the current sharing is provided [25]. As the system was forced
by the control laws (5.53) to hit the manifold, the stability on the mani-
fold has to be guaranteed as part of the synergetic control design. The sta-
bility conditions (F 1 ,F 2 ,F 3 ) can be analyzed by (5.54), including
(5.55) (5.57).