Page 152 - Modern Control of DC-Based Power Systems

116 Modern Control of DC-Based Power Systems

After the compensation a linear system is present, where the classical
pole placement technique can be performed so that the natural frequency
and damping coefficient are shifted to the desired values. Additionally, a
PI controller is added in parallel, which drives the system output to the
reference value.
It is possible to make an analogy to terrestrial power systems where
the fast stabilizing action of the LSF corresponds to the primary control
action, while the slower dynamics where the PI control bring the system
to the reference value corresponds to a secondary control action in terres-
trial power systems.
In the following, it is shown how to achieve LSF using the general
approach of [6]. Furthermore, it is explained how the control law can be
developed from conditions on the natural frequency and damping ratio as
it was done in [10]. Additionally, the derivation of the control law incor-
porates the Lie derivative.
As explained in Section 5.2, the cable parameters can be neglected
and the CPL capacitances and output filter capacitances can be replaced
by an equivalent component: C eq 5 C CPL 1 C f 1 1 C f 2 1 C f 3
The system can be derived from the model in Fig. 5.2 as follows:

1 V P eq
_ V 5 I 1 1 I 2 1 I 3 Þ 2 2
ð
C eq R L C eq C eq V
(5.14)
V d k UE k
_
R fk
I k 52 I k 2 1 fork 5 1; 2; 3
L fk L fk L fk
To shorten the calculations a time constant T f 5 L fk is defined. This is
R fk
only valid if the ratios are equal for all filters. Besides, the equations are
simplified by summing up the inductances 1 5 1 1 1 1 1 and
L eq L f 1 L f 2 L f 3
currents i 5 I 1 1 I 2 1 I 3 .
I V P eq
_ V 5 2 2
C eq R L C eq C eq V
(5.15)
1 V d 1 E 1 d 2 E 2 d 3 E 3
_ I k 52 I 2 1 1 1
T f L eq L f 1 L f 2 L f 3

Still, the system description (5.15) is not ideal for the LSF design as it
is described in [6]. Therefore, the following state-space representation
will be used to show the transformation of the system:

_ x 5 AðxÞ 1 BxðÞUu
(5.16)
y 5 c xðÞ