Predictive Control of Three Phase AC/DC Converters

4.3. VARIABLE SWITCHING FREQUENCY PREDICTIVE DIRECT POWER
CONTROL 43
Replacing (4.3), (4.4) and (4.5) in (4.1) and (4.2) it is possible to get functions
describing instantaneous active and reactive power time derivatives.
dP
dt = 3 ( )
1
2 U Lα
L (U Lα − U P α − RI Lα ) + ω L I Lβ +
( )
3 1
2 U Lβ
L (U Lβ − U P β − RI Lβ ) − ω L I Lα (4.6)
dQ
dt = 3 (
2 U Lα ω L I Lα − 1 )
L (U Lβ − U P β − RI Lβ )
3
2 U Lβ
+
( )
1
L (U Lα − U P α − RI Lα ) + ω L I Lβ
or it might be rewritten as difference equation:
(P + ∆P ) − P
= 3 ( )
1
∆t 2 U Lα
L (U Lα − U P α − RI Lα ) + ω L I Lβ +
( )
3 1
2 U Lβ
L (U Lβ − U P β − RI Lβ ) − ω L I Lα
(4.7)
(4.8)
(Q + ∆Q) − Q
∆t
= 3 (
2 U Lα ω L I Lα − 1 )
L (U Lβ − U P β − RI Lβ ) +
( )
3 1
2 U Lβ
L (U Lα − U P α − RI Lα ) + ω L I Lβ
(4.9)
where: P , Q are active and reactive power, ∆P , ∆Q are power differentials,
and sum of this components gives P P (4.10) and Q P (4.11) predicted active
and reactive power, ∆t is time difference and can be noted as a sampling time
(4.12).
P + ∆P = P P (4.10)
Q + ∆Q = Q P (4.11)
∆t = T s (4.12)
The predicted power values can be expressed as:
P P = 3 ( )
1
2 T s
[U Lα
L (U Lα − U P α − RI Lα ) + ω L I Lβ +
U Lβ
( 1
L (U Lβ − U P β − RI Lβ ) − ω L I Lα
) ]
+ P (4.13)
Q P = 3 (
2 T s
[U Lα ω L I Lα − 1 )
L (U Lβ − U P β − RI Lβ ) +
( ) ] 1
U Lβ
L (U Lα − U P α − RI Lα ) + ω L I Lβ + Q (4.14)