Page 77 - Modern Control of DC-Based Power Systems

42 Modern Control of DC-Based Power Systems

However, this PID controller has a transfer function with an increas-
ing asymptote at high frequencies due to the derivative action [22].
This in practice has a disruptive operation of the PWM due to the
propagation of the switching ripple causing failure of the closed-loop
converter even though the voltage controller has been correctly
designed to have the required closed-loop performance. To avoid this
malfunction, high frequency poles ω P1 and ω P2 must be present in the
real voltage controller to reduce the gain of the controller transfer func-
tion at high frequencies. This can be obtained with the following trans-
fer function.

!
K i K d s 1
G c sðÞ 5 K p 1 1 (2.74)
s 1 1 s 1 1 s
ω P1 ω P2
The first pole ω P1 flattens the gain of the derivative component, while
the second one ω P1 allows the gain to drop. As a good practical design
rule, ω sw ω P1 {ω P2 10 ω P1 , where ω sw is the switching frequency
in rad/s. Solving (2.70) and (2.71) does not lead to the solution because
three coefficients need to be found, i.e., K p , K i , and K d , from two equa-
tions. Therefore, to solve (2.70) and (2.71), the designer needs to set one
of the coefficients and solve for the others. Typically, it is convenient to
set the value of the integral coefficient K i . This is chosen relatively low,
let us say 1{K i {10, to prevent antiwindup [23]. Finally, taking into
account the voltage control transfer function (2.74), its control coeffi-
cients are calculated as follows.

(2.75)
K p 5 cos arg G c ðjω c VM Þ = G vd ðjω c VM Þ

K i
K d 5 1 (2.76)
sin arg G c I ðjω c VM Þ

2
ω c VM

ω c VM G i L d OL ðjω c PICM Þ
2.5.1.3 Simulation Example
A buck converter is now considered as an example. The buck con-
verter switching frequency is 20 kHz, its input voltage is V g 5 400V,
output voltage is V 5 200V, and circuit parameters are L 5 3.5 mH,
C 5 50 μF, and R 5 30 Ω. A MATLAB script is provided in the
Appendix that is able to perform the design of the controllers in VMC
and PICM_FB.