Power Optimization and Control in Wind Energy Conversion Systems using Fractional Order Extremum Seeking

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 04 Issue: 12 | Dec-2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 6.171 | ISO 9001:2008 Certified Journal | Page 772
Power Optimization and Control in Wind Energy Conversion Systems
Using Fractional Order Extremum Seeking
Jijin D H
M TECH Scholar, Electrical and Electronics, Lourdes Matha College of Science and Technology, Kerala, India
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Abstract - Power optimization and control for grid-coupled
wind energy conversion systems (WECS) has caught a big
attention from early days. The methods widely used are
model based power optimization algorithms as the outer loop
and linear control methods in the inner loop. The method of
extremum seeking control is a non model based optimization
concept to extract maximum power from WECS in their
subrated region. Induction generators are used widely in
WECS since they are relatively inexpensive, robust and
require a little maintenance.. When operated using vector
control techniques fast dynamic response and accurate
torque control is obtained with the help of a matrix
converter. In this paper, inner loop nonlinear control based
on the principle of field oriented control (FOC) and feedback
linearization is used along with maximum power point
tracking (MPPT) in the outer loop. The proposed MPPT
method is Fractional Order Extremum Seeking Control
(FOESC).The convergence speed of FOESC is faster than the
Integer Order ESC. Simulation results are presented to show
the effectiveness of the proposed model.
Key Words: Non linear control systems, wind power
generation, power control, adaptive systems, fractional
order systems.
1. INTRODUCTION
A variable wind turbine (WT) generates power in subrated
and rated power regions. In the subrated region Fig.2.
region II , the maximum achievable turbine power is a
function of the turbine speed at any velocity. The maximum
power tracking algorithms developed in the recent are
model dependent. They have the drawback that the
controller needs to be redesigned for each wecs. A similar
system is presented in [1] where fuzzy control is used. It
works only for the specified fuzzy rules. Another method
based on the speed sensorless power signal feedback (PSF)
[2] uses lookup table values that are dependent on the
system model and parameter values. There is another
method of perturb and observe (P&O) [3] is more accurate
but slow convergence. To overcome these difficulties, we
present fractional order extremum seeking (FOES)
algorithm which is non model based and with easily
tunable design parameters. The FOESC has many
advantages over ESC.A system with ESC is shown in [4]
which shows slow convergence. The convergence speed is
faster in FOESC hence transient response is improved.
The WECS uses Induction Generator (IG) which is relatively
inexpensive, robust and require a little maintenance. They
can respond relatively faster and more efficient to the
variable wind speed. The stator of the IG is connected to a
grid through a matrix converter (MC).The excitation or
reactive power control required is fed from the grid itself
by controlling the matrix converter. The stator frequency
and stator voltage is varied through the MC to control the
turbine speed. Inorder to implement field oriented control
(FOC) and to improve transient response and robustness,
an inner closed loop with a non linear controller is designed
Matrix Converter includes nine bidirectional switches
operating in 27 different combinations. MCs provide
bidirectional power flow, sinusoidal input/output currents,
and controllable input power factor. Due to the absence of
components with significant wear out characteristics (such
as electrolytic capacitors), MC can potentially be very
robust and reliable. The amount of space saved by an MC,
when compared with a conventional back-to-back
converter, has been estimated as a factor of three.
Therefore, due to its small size, in some applications, the
MC can be embedded in the machine.
2. WIND ENERGY CONVERSION SYSTEM
Fig-1 : WECS including IG,MC,turbine,gearbox and grid
The wind power available on the blade impact area is
= 1/2 *ρ A ` (1)
Where A is the blade area, ρ is the density of air and is
wind speed

Fig-2 : Power curve including four operating regions
The turbine power can be derived from the power
coefficient equation [4] is
= 55.115 ρA (2)
Where
Fig-3 : Turbine power v/s turbine speed
Where the optimal turbine is power and is the
optimal power coefficient.
Fig.3. shows the turbine power is optimal at points different
for different wind speeds.
The WT shaft model is modeled as a spring damper.The
dynamic equations of turbine, shaft and gearbox are
= /pn, = /pn (3)
= 1/ * ( ), = / (4)
where is the angular electrical frequency of the rotor of
IG, is WT angular position, is the electrical angle of
the rotor of IG, p is the number of pole pairs of the IG, n is
the gearbox ratio, is the turbine torque generated by the
turbine power, is the load torque created by the spring-
damper model of the shaft and is the turbine inertia
coefficient.
= + B( /pn) (5)
where is the stiffness coefficient of the spring and B is
the damping ratio. The generator rotor angular speed
equals
The block diagram of WECS is shown in fig.4.
Fig-4 : Block diagram of WECS
The electromagnetic torque generated by IG is
p ( (6)
The state space dynamics of the WECS is
= + + + (7)
= + + (8)
(9)
=
(10)
= (11)
( – ( – )
(12)
= ( ) (13)
( ) (14)
Where and are the stator or output electrical
frequency and stator electrical angle of the IG

s the mutual
inductance, = + is stator inductance, = + is
rotor inductance and σ = /( ).
3. OVERALL SYSTEM
Fig-5 : overal system including controller
The turbine power is measured from the equation(2) is
given to the outer loop controller that is FOESC which gives
an approximated value of the optimal turbine speed with
respect to Fig.2.It is then given to the inner loop nonlinear
control. The non linear control used here is feedback
linearization which performs FOC and avoids magnetic
saturation of the IG. It is a closed loop drives the turbine
speed to the optimal value found by the MPPT and drives
the rotor flux to the reference flux value given manually.
The conventional FOC control method with P&O method is
shown in [5] and [6].The PI controller used causes high
response time and high overshooting if error is
unexpectedly very high. It is also difficult to design PI since
unpredictable variations in the machine parameters,
external load disturbance and non linear dynamics. The
other methods used for FOC concept are Fuzzy logic, gain
scheduled PI and relative gain array. The feedback
linearization gives a faster response and desired response
can be obtained by adjusting the feedback gains. The
controller gives stator frequency and stator voltage given to
modulation, the modulation and pulse generation for MC
can be referred from [4],[7] and [8] .The MC regulates the
stator electrical frequency to control the turbine speed. The
stator voltage amplitude can be maintained to regulate the
rotor flux. The turbine speed variation does not affect rotor
flux. Similarly the rotor flux reference can be varied even
independently of reference optimal speed found by the
MPPT. This is an improvement over FOC.
Remark1: The torque–speed characteristic of an induction
machine is normally quite steep in the neighborhood of
stator electrical frequency (synchronous speed) , and so
the electrical rotor speed , will be near the synchronous
speed. This means that changing the reference value of the
turbine speed which translates in variation of the
electrical rotor speed eventually results in changing the
stator electrical frequency [9]. Thus, by controlling the
stator electrical frequency, one can approximately control
the turbine speed or vice versa.
4. CONTROLLER DESIGN
Fig-6 : WECS with FOESC and inner loop nonlinear control.
Where =
When flux amplitude√ , is regulated to a
constant reference value, and considering the fact that the
dynamics of are considerably slower than the electrical
dynamics, we can assume that the dynamics are linear, but
during flux transient, the system has nonlinear terms and it
is coupled. This method can be improved by achieving exact
input–output decoupling and linearization via a nonlinear
state feedback that is not more complex than the
conventional FOC [10].
From (2) and Fig.3, we know that the turbine speed
controls the power generation. In addition, we are
interested in decoupling the rotor flux and electromagnetic
torque to obtain the benefits of FOC. For these reasons, we
introduce turbine speed, = and flux amplitude, =
, as measurable outputs. For future analysis, we
assume that the power coefficient and wind speed function
satisfy following assumption.
Assumption 1: The power coefficient C p ( ) and
wind speed function (t) are bounded functions with
bounded derivatives. Hence, the mechanical torque is a
bounded function with bounded derivatives.
Based on the selected outputs and having Assumption
1satisfied, we apply feedback linearization with the
following change of variables to WECS dynamics:
= (15)
= ( ) (16)

= +
̇
(17)
= + +
– ( +
̇ ̈
(18)
= (19)
= 2 – 2 (20)
= +
(
+
̇
) +2 + (21)
Δ = (22)
Φ = arctan( ) (23)
Where –
and
[ ] = [ ][ ] (24)
The change of variables results in the following equations,
= ̇ (25)
= ̇ (26)
= ̇ (27)
̇ +
(28)
̇ (29)
̇ (30)
̇ = + (31)
̇ = - (32)
̇ = + (
̇
) (33)
Where and
(32) and (33) are zero
dynamics of the system
+ ( +
) ̇ ( + ) ( + )
+
̇
+
̈ ̉
(34)
+ ̇ + (
̇
̇
+
̈
+ 4
) + ( ) (35)
Where
[ ] = [ ] [ ] (36)
Defining control signals as
[ ] =
√
[ ] [ ] (37)
Applying another step of change of variables
z = (38)
ζ = (39)
̇ (40)
̇ (41)
̇ (42)
̇ (43)
̇ (44)
̇ (45)
̇ (46)
̇ = - (47)
̇ = + (
̇
) (48)
Linear state feedback,
(49)
(50)
By appropriate selection of the feedback gains in (49) and
(50) and using (37), we can obtain the desired closed-loop
response time. The controller drives the turbine speed
towards the reference value found by MPPT while
amplitude of rotor flux has converged to its desired

value .The values of values are given in
appendix.
5. MPPT Using FOESC
Assumption 2: The following holds for the turbine power
map around its MPP for V cut−in <V w <V rated (see Fig. 3)
where is the optimal turbine speed
(51)
(52)
Where is the optimal turbine speed
ESC is a branch of adaptive control. The ES scheme
estimates the gradient of the cost function , by injecting a
small perturbation, a sin( t), which is very slow with
respect to the dynamics of the controller system and its
amplitude is enough small in comparison with . The
high-pass filter removes the dc part of the signal. The
multiplication of the resulting signal by sin( t) creates an
estimate of the gradient of the cost function, which is
smoothed using a low-pass filter. When is larger than its
optimal value, the estimate of the gradient g^,is negative
and causes to decrease. On the other hand, when is
smaller than , then g^ > 0, which increases the
toward .
In general,
Fig-7 : The FOESC in a general case
The plant dynamics can be written as
y = + ( ) (53)
Considering cost function f to be minimized and assuming
f’’ is positive,
The error,
(54)
= (t) + asin (wt) = asin (wt) (t)
(55)
Substituting in (53)
Y = + + – a
(56)
The quadratic terms can be neglected
The high pass filter eliminates the dc components and the
output obtained is
[y] = - a (57)
ξ = sin (wt) [y] (58)
ξ =
+ (sin (wt) – sin (3wt)) (59)
The integrator attenuates the high frequency terms.
Taking derivative of eqn (54) gives
= (60)
Also
ξ = (61)
(62)
It is clear that the rate of error convergence depends on
integral gain k, perturbing signal amplitude a and
perturbing frequency 1/ w when taking time scale T=wt.
= (63)
The stability of FOESC based on averaging and jacobian can
be referred from [11]
The averaged linearized model of the FOESC in fig.7. can be
obtained from [12] is given as,
= (64)
Where
The convergence of the FOESC can be verified by checking
the boundary stability of the root locus of integer order
IOESC and fractional order FOESC of the general system in
fig.8.

Fig-8 : .root locus of IOESC and FOESC
It can be seen that the stability boundary is increased for
FOESC. The stability boundary condition is Arg(eig(A)) >
where A is the state space matrix of the eqn (64).In the
fig.8, the eigen values of the IOESC system is near the
imaginary axis hence it has slowly damped poles but the
eigen values of FOESC system is not close to the stability
boundary of the system hence they can be damped faster.
The conditions for choosing parameters for the system
given in fig.6 are as follows.
According to eqn (63) , the convergence speed depends on
the order( ) hence a should be sufficiently small. But a
higher value of a causes residual error and a very lower
value causes not to attain global stability. Perturbation
signal frequency w can be chosen sufficiently large. Since
integral gain k is the learning rate, it must be higher than
the period of perturbation w>>k so that it converges faster.
The value of high pass filter frequency can be chosen to
eliminate dc component can be a lower value. Similarly the
value of low pass filter frequency is chosen so that avoiding
high frequency terms for the averaging of integral.
The order of the FOESC q must be chosen carefully because
lower the order, the limiting (boundary) value of gain
increases which affects the convergence. Hence a quadratic
error term between turbine power after the FOESC and
directly measured power is taken and plotted for different
values of fractional order (0.3 to 1.3).The optimal fractional
order can be obtained corresponding to minimum error.
∑ (65)
N is the width of the working time window and  is the time
sampling rate
Fig-9 : Quadratic error v/s fractional order
6. SIMULATION RESULT
Control signals are designed such that the poles of z-error
subsystem (40)–(43) and ζ-error subsystems (44)–(46)
move to P z = [−550 − 600 − 650 − 700] and P ζ = [−570
−620 −670], respectively. The response time of the closed-
loop system is about 20 ms, which is 25 times faster than
the open-loop system. We select the parameters of the
FOES loop of fig 6 as follows: W = 100 rad/s, = 6 rad/s,
= 5 rad/s, a = 0.1 and k = 0.004.The optimal value of
order q is found to be 0.8 from the fig.9. The wind speed
tracking of IOESC and FOESC is shown in fig 10 for a wind
speed of Gaussian signal 20m/s avg
Fig-10 : wind speed tracking of IOESC and FOESC of the
proposed system

The power tracking can be found more in FOESC in fig 11
Fig-11 : power tracking for a wind speed of 20m/s
The power tracked by the FOESC is found more than IOESC.
Thus FOESC is more efficient for maximum power tracking.
The efficiency and improvement of transient response
when including inner non linear controller can be verified
fom [4].
7. CONCLUSION
The system employed FOESC to extract maximum power
from the available wind power. The design employed an
inner loop non linear controller based on FOC and feedback
linearization to control the closed loop transient response
with respect to the optimal turbine speed tracked by
FOESC. It provides perfect input output decoupling and
better than conventional FOC which increases performance
robustness with respect to system parameters. Also the
control strategy prevents magnetic saturation of the IG. The
optimization algorithm can readily be extended to other
classes of WECS without major changes. The main
parameters to be adjusted are probing frequency and
amplitude of the perturbing signal. A comparison is done
with IOESC and FOESC is found better tracking of optimal
turbine speed and power. It results in higher efficiency and
since the WECS runs for a long period of time, a small
improvement in power efficiency guarantees extracting a
higher energy level and leads to cost reduction.
APPENDIX
TABLE-1 : Definition of Parameters and Their Numerical
Values
Blade length R 10m
Blade pitch angle 0
Air density  1.25Kg/
Turbine inertia 100 Kg
Gear box ratio 20
Stiffnes coefficient Ks 2*10^6 Nm/rad
Damping coefficient B 5*10^5 Nm/rad/s
No of pole pairs of SCIG 2
Stator leakage inductance 3.2 mH
Rotor leakage inductance 3.2 mH
Magnetizing inductance 143.36 mH
Moment of inertia of SCIG 11.06 Kg
Stator resistance 0.262 Ω
Rotor resistance 0.187 Ω
TABLE –2 : Constant Parameters
ACKNOWLEDGEMENT
Foremost, I would like to express my sincere gratitude to
my advisor Assistant Professor Swapna M of Lourdes Matha
College of Science and Technology(LMCST) Trivandrum
Kerala India for the continuous support of my MTech thesis
study and research, for her patience, motivation,
enthusiasm and immense knowledge. Her guidance helped
me in all the time of research and writing of this thesis.
Besides my advisor, I would like to thank respected HOD
Dr.Mohanalin Rajarakthnam and all the teachers of
Electrical and Electronics department LMCST for their
encouragement, motivation and help.
Last but not the least, I would like to thank my family; my
parents and my brother for supporting me to complete this
thesis successfully.
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BIOGRAPHIES
JIJIN D H now working as Adhoc
Assistant Professor at College of
engineering and Management
Punnapra Kerala India in the EEE
Dept. Completed MTech from
Lourdes Matha College of Science
and Technology Trivandrum in
Control Systems. Completed BTech from Govt Engineering
College BartonHill Trivandrum India in Electrical and
Electronics Engineering
.

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