Predictive Control of Three Phase AC/DC Converters

56 CHAPTER 4. PREDICTIVE DIRECT POWER CONTROL
( ))
1
Q err = Q ref − Q − 2
(f q1 t 1 + f q2 t 2 + f q3
2 T s − t 1 − t 2
(4.39)
To minimize power errors, least square optimization method has been introduced.
The cost function J has been defined as a sum of squared errors:
and after replacing (4.38) and (4.39) into (4.40):
J = P 2 err + Q 2 err (4.40)
J = [ P ref − P − 2 ( f p1 t 1 + f p2 t 2 + f p3
( 1
2 T s − t 1 − t 2
))] 2
+ [ Q ref − Q − 2 ( f q1 t 1 + f q2 t 2 + f q3
( 1
2 T s − t 1 − t 2
))] 2 (4.41)
The optimal application times t 1 , t 2 , t 3 , which minimize cost function value J,
during sampling time T s , satisfies minimum value condition:
⎧
⎪⎨
⎪⎩
∂J
∂t 1
= 0
∂J
∂t 2
= 0
(4.42)
Partial differences of J function are expressed as:
∂J
∂t 1
= 4 [ (P ref − P − f p3 T s ) (f p3 − f p1 ) + (Q ref − Q − f q3 T s ) (f q3 − f q1 ) ]
+ 8 [ (f p3 − f p1 ) 2 + (f q3 − f q1 ) 2] t 1
+ 8 [ (f p3 − f p2 )(f p3 − f p1 ) + (f q3 − f q2 )(f q3 − f q1 ) ] (4.43)
t 2
∂J
∂t 2
= 4 [(P ref − P − f p3 T s ) (f p3 − f p2 ) + (Q ref − Q − f q3 T s ) (f q3 − f q2 )]
+ 8 [(f p3 − f p1 )(f p3 − f p2 ) + (f q3 − f q1 )(f q3 − f q2 )] t 1
+ 8 [ (4.44)
(f p3 − f p2 ) 2 + (f q3 − f q2 ) 2] t 2
The J gets minimum value when (4.43), (4.44) are equal to zero (4.42). The solution
of sets of conditions are application times given by:
[
t 1 = (P ref − P )(f q2 − f q3 ) + (Q ref − Q)(f p3 − f p2 )
+ (f p2 f q3 − f p3 f q2 )T s
]/
[
]
2((f q2 − f q3 )f p1 + (f q3 − f q1 )f p2 + (f q1 − f q2 )f p3 )
(4.45)
t 2 =
[
(P ref − P )(f q3 − f q1 ) + (Q ref − Q)(f p1 − f p3 )
+ (f q1 f p3 − f q3 f p1 )T s
]/
[
]
2((f q2 − f q3 )f p1 + (f q3 − f q1 )f p2 + (f q1 − f q2 )f p3 )
(4.46)
t 3 = 1 2 T s − t 1 − t 2 (4.47)
Appendix C shows an example operation of presented control method for
specified conditions.