Page 164 - Modern Control of DC-Based Power Systems

128 Modern Control of DC-Based Power Systems

Special attention should be given to the newly introduced term MðtÞ,
which is a dependent currents source and represents the changes of the
system load [25]. This is approximated by a piecewise constant function
which acts as an integral action:

dM V 2 V ref V
5 5 (5.45)
dt T I R d T I R d
To design the synergetic controller, it is necessary to define macrovari-
ables which are based on system states. Recall that the number of macro-
variables is equal to number of control channels [25]. The result of the
synergetic control design is a control law which ensures stable movement
to and along the manifolds towards the equilibrium point [15].
Þ 5 0; i 5 1 .. . 3 (5.46)
i
ψ I L1 ; I L2 ; .. . ; Vð
The following macrovariables were chosen to obtain the desired
dynamic properties.
_
T i :ψ 52 ψ ;T i . 0 (5.47)
i
i
The convergence speed is defined by the vector T i whose size corre-
sponds to the number of participating converters. There are three control
channels, therefore three macrovariables ψ .. . ψ :
1 3
a 16
1
ψ 5 a 11 M 1 a 12 V 1 a 13 I L1 1 a 14 I L2 1 a 15 I L3 1
V 1 V ref C eq
a 26
2
ψ 5 a 21 M 1 a 22 V 1 a 23 I L1 1 a 24 I L2 1 a 25 I L3 1 (5.48)
V 1 V ref C eq
a 36
3
ψ 5 a 31 M 1 a 32 V 1 a 33 I L1 1 a 34 I L2 1 a 35 I L3 1
V 1 V ref C eq
An interesting finding according to [25] of the synergetic control law
in parallel power converter system is that coefficients a x3 , a x4 , and a x5 allo-
cate the current sharing ratings. The coefficients a x6 provide the ability to
compensate the impact of the CPL. A detailed explanation of the proper-
ties of the coefficients can be found in [15] and [25]. Defining ψ as a vec-
tor enables us to generalize the system:

ψ 5 AX 2 B (5.49)