Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 8, N. 4
319 where Kp
d
is a proportional constant for the d-axis current controller and t
i d
is the constant of integration time for the d-axis current controller.
The output of the q-axis current controller V
q
can be expressed by:
= − +
+ −
+ +
+ 16
where Kp
q
is the proportional constant for the q-axis current controller and t
i q
is the constant of integration timefor the q-axis current controller.
a d-axis current controller
Iq +
- PI
Controller +
Iqref
r d d
r r
Vqt +
b q-axis current controller Figs. 6. PI control of a d-axis and b q-axis currents
The PI controller design is determined by pole assignment then the proportional gain and the integration
time constant for d-axis and q-axis current are calculated as follows:
Kp
d
= 2 ζL
d
– Rs 17
= 2
− 18
Kp
q
= 2 ζWnL
q
– Rs 19
= 2
− 20
where ζ is the damping coefficient selected to be 0.707 and Wn is the natural frequency which was selected
based on the desired closed loop settling time. The larger the value of Wn, the shorter the closed loop settling time.
V. Results and Discussion
For the developed model based on a variable-speed wind turbine system we assumed an 8.5kW PMSG
system. To test the performance of the PSO-PI controller
method, we used a Simulink simulation in MATLAB with sampling time is 20 μs, where simulations were
performed with both constant and variable wind speeds. Fig. 7 shows the generator speed response at constant
wind conditions of 10 ms, where the PSO-PI and PI controllers are compared. Both controllers are able to
follow the set-point but the PSO-PI controller produced no overshoot and had a lower settling time.
Figs. 8 show the d-axis and q-axis current controller responses for the PI and PSO-PI controllers. Both
controllers were able to follow the d-axis current reference at zero. However the PSO-PI controller
generated a closer response. Fig. 9 shows the error rate in the speed control. The PSO-PI controller showed a lower
error compared to the PI controller based on pole assignment.
Fig. 7. Generator speed response
a PI controller
b PSO-PI controller Figs. 8. q-axis and d-axis current responses
Fig. 9. Error rate of the speed controllers over time
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
5 10
15 20
25 30
X: 0.0959 Y: 26.72
Times G
e n
e ra
to r
A n
g u
la r
S p
e e
d ra
d s
PID Reference
Optimized PSO
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
-1 1
x 10
-15
Id A
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
100 200
300 Iq
_ re
f A
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
100 200
300 Times
Iq A
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
-5 5
10 x 10
-16
Id A
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
100 200
300 Iq
_ re
f A
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
100 200
300 Times
Iq A
0.05 0.1
0.15 0.2
0.25 0.3
0.35 0.4
0.45 0.5
-5 5
10 15
20 25
30
Time s E
rr o
r r
a d
s PSO-PI
PI
Copyright © 2015 Praise Worthy Prize S.r.l. - All rights reserved International Review of Automatic Control, Vol. 8, N. 4
320 Fig. 10 shows the response speed with a change in
wind speed and Figures 11 show the response current. The PSO-PI controller followed changes in wind
speed. Table I compares the performance of the PSO-PI and PI controllers.
TABLE I P
ERFORMANCE
O
F
PSO-PI A
ND
PI C
ONTROLLER
PSO-PI PI
P 8.4969
7.1 I
324.3317 460
Mp 3.5
Ess 0.1
0.18 Tr ms
0.018 0.018
Tsms 0.048
0.0719
Fig. 10. Generator speed response with changes in wind speed
a PI controller
b PSO-PI controller Figs. 11. q-axis and d-axis current response with changes
in wind speed
VI. Conclusion
