Predictive Control of Three Phase AC/DC Converters

132 APPENDIX C. PREDICTIVE DIRECT POWER CONTROL
Next, from (4.40) cost function J is described as:
[ (−15.45481322062509
J =
∗ 10
6 ) t 1 + ( −4.141104721640333 ∗ 10 6) t 2
2 [ (15.45481322062509
+ 721.1485808337910]
+ ∗ 10
6 ) t 1
(C.2)
+ ( 26.76852171960985 ∗ 10 6) ] 2
t 2 − 971.4140632912274
In order to find minimum value of J function, partial derivatives (4.43), (4.44)
are calculated:
∂J
∂t 1
= 0 ⇒ (955405006.7376324
∗ 10
6 ) t 1 + ( 955405006.7376324 ∗ 10 6) t 2 (C.3)
= 52316.47905831899 ∗ 10 6
∂J
∂t 2
= 0 ⇒ (955405006.7376324
∗ 10
6 ) t 1 + ( 1467405006.737632 ∗ 10 6) t 2 (C.4)
= 57979.34049008143 ∗ 10 6
Figure C.1 (a), (b) show cost function J diagram versus voltage vector application
time t 1 , t 2 for conditions listed in Tab. C.1. The minimum value of J
function is reached by (4.45), (4.46):
t 1 = 43.69815468216835 µs
t 2 = 11.06027623391102 µs
t 3 = 45.24156908392062 µs
(C.5)
where t 3 is calculated from (4.47).