Analysis and Design of Root Locus Based Controllers and
State Feedback Controllers for Wind turbine control
Nayab Shahid RP20-EE-425
Nayabs118@gmail.com
Abstract:
In this lab, a thorough analysis
of wind turbine control plant has been done
with the help of control system techniques.
Starting with the introduction the
importance and need for this plant has been
mentioned and the mathematical model of
the plant is created. After that the plant has
been simulated on MATLAB (2016a)
/Simulink as well as Multisim and the
responses have been observed. The results
are mentioned and discussed. The pole
locations are found for different values of
gear ratio using Root Locus technique In
the last part the Value of N (gear box ratio)
for complex poles are found having a
damping ratio of 0.5. (Nice, 2016)
1-Introduction
Wind energy is one of the environment
friendly and efficient source of electricity
as the wind is everlasting as well as
copious resource. Wind Turbines can be
controlled by controlling the speed of
generator, by angle adjustment and in
some cases rotation of the complete
turbine. Wind turbine control is important
for optimal performance ,physical stability
and safe operation. Wind turbines are
dynamic systems having complex designs.
(Ogata, 2010) Control of wind turbine is
quite difficult because of the non-linear
dynamics. Winds behaviour is stochastic
so its difficult to control and predict.For
effective power generation careful control
is necessary. The aim of this project is to
create and analyse the state feedback and
root locus based efficient controllers for
wind turbine control system. (Nice, 2016)
The goal is to design a controllers that can
change the angle of pitch, and speed
according to changing environmental
conditions. It is necessary to maintain
constant power output despite changes in
the environment
2- Mathematical Modelling:
Figure 1 Control loop for a constant speed pitch control
𝑷𝒐(𝒔)
𝑻𝒓(𝒔)
= 𝑮𝒅𝒕(𝒔)
=. (3.92𝐾𝑙𝑠𝑠𝐾ℎ𝑠𝑠𝐾𝑔𝑁^2 𝑠)
/{𝑁^2 𝐾ℎ𝑠𝑠(𝐽𝑟𝑠^2
+ 𝐾𝑙𝑠𝑠)(𝐽𝑔𝑠^2 [𝑡𝑒𝑙 𝑠 + 1]
+ 𝐾𝑔𝑠)
+ 𝐽𝑟𝑠^𝑠2 𝐾𝑙𝑠𝑠[(𝐽𝑔𝑠^2
+ 𝐾ℎ𝑠𝑠)(𝑡𝑒𝑙 𝑠 + 1)
+ 𝐾𝑔𝑠]}
where Po(s) is the Laplace transform of the
output power from the generator and Tr(s)
is the Laplace transform of the input torque
on the rotor. Substituting typical numerical
values into the transfer function yields
(Nice, 2016)
= ((𝟑. 𝟗𝟐)(𝟏𝟐. 𝟔 ∗ 𝟏𝟎𝟔 )(𝟑𝟎𝟏
∗ 𝟏𝟎𝟑 )(𝟔𝟖𝟖)𝑵𝟐 𝒔)
3-Design of Root Locus Base Controller:
3.1 Simulink Model:
𝟐
/{𝑵 (𝟑𝟎𝟏
∗ 𝟏𝟎^𝟑 )(𝟏𝟗𝟎, 𝟏𝟐𝟎𝒔𝟐
+ 𝟏𝟐. 𝟔
∗ 𝟏𝟎𝟔 )(𝟑. 𝟖𝒔^𝟐 [𝟐𝟎
∗ 𝟏𝟎−𝟑 𝒔 + 𝟏) + 𝟔𝟔𝟖𝒔]}
The transfer function of drive train is given
to us as:
Po(s)
Tr(s)
num=[2.3882*10^(10) 119.4*10^(10)
21152*10^(10) 7910.67*10^(10)
13.90*10^(15)];
den=[1 49.989 8.7974*10^(4)
3.99594*10^(6) 0];
gs=tf(num,den)
rlocus(gs)
zeta=0.455;
sgrid(zeta,0)
[k,p]=rlocfind(gs);
=
(3.92)(12.6∗106 )(301∗103 )(688)𝑁2 𝑠
(190120𝑠2 )(12.6×10−6 )(3.8𝑠2 +301000)
(1+0.02𝑠)+668𝑠
2
𝑁 (301×103 )(190120𝑠2 +12.6×10−6 )
(3.8𝑠2 (1+0.02𝑠)+668𝑠)
1+
(190120𝑠2 )(12.6×10−6 )(3.8𝑠2 +301000)
(1+0.02𝑠)+668𝑠
The open loop transfer function of the
drive train is given by:
𝐺(𝑠) =
𝑁 2 (301×103 )(190120𝑠2 +12.6×10−6 )
(3.8𝑠2 (1+0.02𝑠)+668𝑠)
(190120𝑠2 )(12.6×10−6 )(3.8𝑠2 +301000)
(1+0.02𝑠)+668𝑠
Figure 3.1.1 Root Locus of Open Loop
From the figure 3.1.1 we can see the root locus
(Poles and zeros ) of the open loop system.
(4.349×109 )𝑠5 +(2.175×1011 )𝑠4 +3.852×
=𝑁 2 (
=
1013 𝑠3 +1.441×1013 𝑠2 +2.533×1015 𝑠)
0.1821𝑠5 +9.103𝑠4 +(1.602×104 )𝑠3 +
(7.21×105 )𝑠2
2.3882×1010 (𝑠4 +50.012𝑠3 +8857.2𝑠2
+3313.4𝑠+(5.8243×105 ))
𝑁 2 ( 𝑠4 +49.989𝑠3 +(8.7974×104 )𝑠2 + )
(3.9594×106 )𝑠
Thus Open loop poles:
0, -45.12, -2.435±𝑗296.22
And Open loop zeros:
0, -45.12, -2.435±𝑗296.22
Figure 3.1.2 Simulation of Open Loop
In the figure 3.1.2 we see the Matlab
simulation of the open loop system in
which we can see the system didn’t
achieved the desired output in order to
achieve the desired output we have to
add zeros and poles in the open loop
transfer function.
Find state space using the transfer
function given above:
5- Controllability:
Figure 3.1.3 Simulation Output
Form the figure 3.1.3 we can see the output of
plant our system didn’t reach to our desired
output so we have to design a controller to
remove the error.
4- Design of Feedback Controller:
A feedback controller is a control
system component that uses feedback
information to adjust the system's
output and maintain a desired
behaviour or set point. It continuously
measures the system's output and
compares it to a reference value or
desired set point. The difference
between the measured output and the
set point is called the error signal. The
feedback controller then generates a
control signal based on the error signal
and sends it back to the system to
adjust its operation. The goal of a
feedback controller is to minimize the
error between the system's output and
the desired set point, thereby improving
the system's performance and stability .
controllability of the system based on the
provided transfer function.


Settling time .5 sec.
%Overshoot 20%
Desired Equation
Hence it’s a controllable matrix as it is a full
rank matrix
5.1 Matlab Code:
A=[0 1;-14.7 -7.15];
B=[0;1];
C=[119.7 28.4];
D=0;
pos=input('Type desired %OS ');
Ts=input('Type desired settling time
');
z=(log(pos/100))/(sqrt(pi^2+log(pos/100)
^2));
wn=4/(z*Ts);
[num,den]=ord2(wn,z);
r=roots(den);
poles=[r(1) r(2)];
K=acker(A,B,poles)
Figure 6.1.1 State-space Modelling
Figure 6.1.2 Graphical representation of statespace modelling
Conclusion:
Concluding this report we can say that we
have successfully designed root locus and
state feedback controller for wind turbine
control system. The simulation has been
done on MATLAB and Simulink and the
output results show that both of the
designed controllers are able to regulate
speed and pitch angle of wind turbine
despite the environmental changes. In this
way optimal output is obtained from the
system thus minimizing the losses. In these
kind of systems controllers play a very
important role in regulating the systems
output. Without controllers the system
would be inefficient. Moreover Harvard
style referencing is used to support our
statements. (Ogata, 2010)
.
References:
[1] Norman S. Nise, Control Systems
Engineering, Sixth Edition, John Wiley and
Sons, Inc, 2016.
[2] Umar Farooq, Lecture notes, Control
Systems, 2023.