Page 137 - Modern Control of DC-Based Power Systems

Background 101

R 1 I 1 I 2 R 2

V 1 I L V 2

Figure 3.10 A simple circuit controlled by a droop control approach.
It is then necessary to express the line current as a function of the
forcing functions. By applying simple circuit solution methods, the fol-
lowing relation is extracted:

I 1 1 R 2 21 I L
5 (3.28)
I 2 R 1 1 R 2 R 1 1 V 2 2 V 1
Substituting the expression of the current in the droop relation, the
overall system equations are obtained

R d1 R 2 R d1
V 1ref 5 V 1n 2 I L 1 ð V 2 2 V 1 Þ
R 1 1 R 2 R 1 1 R 2
(3.29)
R d2 R 1 R d2
V 2ref 5 V 2n 2 I L 1 ð V 1 2 V 2 Þ
R 1 1 R 2 R 1 1 R 2
By applying the hypothesis that the relation between the reference
and the actual voltage can be described by means of a simple dominant
pole, the following relations hold:

dV 1
V 1 1 τ 1 5 V 1ref
dt
(3.30)
dV 2
V 2 1 τ 2 5 V 2ref
dt
By substituting these equations in the two main equations the final
form of the dynamic model is obtained. It is then possible to extract the
state matrix:

! !
2 3
1 1
R d1 R d1
2 1 2
6 7
R 1 1 R 2 τ 1 R 1 1 R 2 τ 1
6 7
A 5 6 ! ! 7 (3.31)
6 1 7
R d2 R d2
2 1 2
4 1 5
R 1 1 R 2 τ 2 R 1 1 R 2 τ 2