Linear Parameter Varying Power Regulation of Variable Speed Pitch
Manipulated Wind Turbine in the Full Load Regime
T. SHAQARIN1, MAHMOUD M. S. AL-SUOD2
1Department of Mechanical Engineering, TafilaTechnical University, Tafila 11660, JORDAN
2Department of Electrical Power Engineering and Mechatronics, TafilaTechnical University, Tafila
11660, JORDAN
Abstract: - In a wind energy conversion system (WECS), changing the pitch angle of the wind turbine blades is
a typical practice to regulate the electrical power generation in the full-load regime. Due to the turbulent nature
of the wind and the large variations of the mean wind speed during the day, the rotary elements of the WECS
are subjected to significant mechanical stresses and fatigue, resulting in conceivably mechanical failures and
higher maintenance costs. Consequently, it is imperative to design a control system capable of handling
continuous wind changes. In this work, Linear Parameter Varying (LPV) H∞ controller is used to cope with
wind variations and turbulent winds with a turbulence intensity greater than ± 10%. The proposed controller is
designed to regulate the rotational rotor speed and generator torque, thus, regulating the output power via pitch
angle manipulations. In addition, a PI-Fuzzy control system is designed to be compared with the proposed
control system. The closed-loop simulations of both controllers established the robustness and stability of the
suggested LPV controller under large wind velocity variations, with minute power fluctuations compared to the
PI-Fuzzy controller. The results show that in the presence of turbulent wind speed variations, the proposed LPV
controller achieves improved transient and steady-state performance along with reduced mechanical loads in
the above-rated wind speed region.
Key-Words: - Wind energy; H∞ Control; Variable Pitch Wind Turbine; Fuzzy Control.
Received: November 9, 2021. Revised: October 30, 2022. Accepted: December 1, 2022. Published: December 13, 2022.
1 Introduction
During the past decades, the usage of wind turbine
plants increased and became more competitive
among other renewable energy forms. The current
trend is toward the advancement of green energy
production. On the contrary, more restrictions are
enforced to reduce the energy produced from
conventional generation sources, which produce
greenhouse gases that lead to global warming.
Recently, the wide adoption of wind farms in the
United States positively affects greenhouse
emissions and water consumption. For instance, in
2018, the United States’ wind capacity of 96 GW
annually lessened CO2 emissions and water
consumption by about 200 million metric tons, and
95 billion gallons, respectively, [1].
Due to the turbulent nature of the wind and the large
variations of the mean wind speed during the day,
the rotary elements of the WECS are subjected to
significant mechanical stresses and fatigue, resulting
in conceivably mechanical failures and higher
maintenance costs.
As a result, mechanical failures are unavoidable in
dynamic and turbulent wind speed conditions unless
proper power regulation control systems are
used, such as pitching controllers. Despite
decreasing mechanical stresses on the wind turbine's
rotary elements, a sophisticated control system helps
in stabilizing, maximizing, and restricting the
generated power at above-rated wind speeds, [2].
In wind energy generation systems, the effect of
turbulence variations and noise created by turbines
may cause fluctuations and a reduction in power
output. Wind turbines are designed and analyzed by
modeling and simulating the turbines using various
software and hardware techniques. The wind turbine
model in WECS was developed by Manyonge et al.,
[3], via examining the power coefficient parameter
needed to understand the wind turbine dynamics
over its operational regime, which contributes to
controlling the performance of wind turbines.
The work presented by Taher et al., [2],
introduced a Linear Quadratic Gaussian (LQG)
gain-scheduling controller to cope with the variable
wind velocity in the WECS. They introduced gain-
scheduling controllers (GSC), which aimed to
manipulate the blades’ pitch angle, regulate the
electrical torque, and maintain the rotor speed
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constant at their nominal values at the full load
region. Petru and Thiringer, [4], presented in their
work a dynamic model for a wind turbine in both
fixed-speed and stall-regulated systems. The main
objective of his work is to evaluate the dynamic
power quality effect on the electrical grid.
Sabzevari et al., [5], proposed a maximum power
point tracking (MPPT) approach emanating from a
neural network that is trained offline using particle
swarm optimization (PSO). The proposed MPPT
estimated the wind speed to adapt the fuzzy-PI
controller. The adaptive controller manipulated the
boost converter duty cycle for the driven permanent
magnet synchronous generator (PMSG). Macêdo
and Mota, [6], presented a comprehensive
description of the wind turbine system equipped
with an asynchronous induction generator. They
implemented a controller using a Fuzzy control
system by manipulating the pitch angle. The main
objective of their work was to reduce the
fluctuations in the generator output power. Salmi et
al., [7], designed an optimal backstepping controller
via particle swarm optimization (PSO) and an
artificial bee colony (ABC) algorithm for doubly fed
induction generator (DFIG) wind turbines. They
aimed at MPPT and to decrease transient loads by
controlling power transferred between the generator
and the electrical grid in the presence of uncertainty.
Aissaoui et al., [8], presented in their work a
comprehensive model of the WECS equipped with
PMSG; they designed a Fuzzy-PI controller to
maximize the extracted power with low power
fluctuation. They managed to control and regulate
the generator speed with low fluctuations.
The use of nonlinear control systems is
considered one of the prevalent control systems of
wind energy conversion systems. Thomsen, [9],
described and analyzed different nonlinear control
techniques for power and rotor speed regulation of
the wind turbine. Additionally, other nonlinear
control methods were implemented including gain
scheduling technique, feedback linearization, [10],
and sliding mode control, [11].
The work by Shao et al., [12] deals with the
restitution of the wind turbine pitch actuator system
by addressing the PI- and PID-based pitch control
methods. They sought to enhance the control system
by mitigating the effect of pitch delay perturbations
on the wind turbine output power.
Robust control theory tackled the control
problem of WECS due to its ability to deal with
external disturbances and model uncertainties.
Bakka and Karimi, [13], implemented a mixed H2-
H∞ control design for the WECS, based on state-
feedback control. They successfully regulated the
rotor rotational speed subject to disturbances in the
gearbox and wind turbine tower. Muhando et al.,
[14], described a design of multi-objective H∞
control of the WECS that incorporates a doubly-fed
induction generator. They designed a controller to
accomplish the dual purpose of energy capture
optimization and alleviating the cyclic load against
wind speed fluctuations.
Linear varying parameter (LPV) controller is
convenient for control problems that involve the
regulation of wind turbine output power. Initially,
the nonlinear model is transformed into an LPV
model that consists of a group of linear models by
assuming free-stream velocity, turbine shaft angular
velocity, and pitch angle as varying parameters.
Therefore, the control structure is reduced to an
LPV controller which is a convex combination of
linear controllers. Ying et al., [15], have presented
in their work a designed H∞ loop shaping torque and
LPV-based pitch controllers. They aimed to enhance
the performance of the pitch actuator system
through the region of transition around the nominal
wind speed. Gebraad et al., [16], presented an LPV
controller for WECS in the partial load region. They
designed a full model that aimed to control the rotor
vibration in the partial load region, and they used a
proportional controller to enhance the produced
power from the wind turbine. Inthamoussou et al.,
[17], proposed an LPV controller for regulating the
power of WECS above the rated wind speed. The
suggested controller was compared with gain-
scheduling PI and H∞ controllers. The work done by
Lescher et al., [18], adopted multivariable gain-
scheduling controllers for the wind turbine based on
a linear parameter-varying control approach. They
designed a two-bladed wind turbine by situating
smart micro-sensors hosted on the blades. They
aimed to alleviate wind turbine cyclic loads
addressed by the wind turbine in a full load regime.
The work presented here is mainly targeting the
development of a robust linear parameter-varying
H∞ controller for a WECS via pitch manipulation. It
is precisely aimed at regulating generator output
power via the regulation of the generator shaft
angular velocity and torque. The control problem in
hand lays down restrictions on the control design
spec. Initially, the acceptable power fluctuations are
limited to ± 5% of its nominal value regardless of
the incoming wind speed variations. Additionally,
the pitch actuator limitations introduce extra
restrictions on the pitch angle and its derivative.
These restrictions impose limitations on closed-loop
performance. Furthermore, the suggested LPV
controller is compared with the fuzzy logic
controller under the same operating conditions.
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2 Modelling of the Wind Turbine
The WECS operating region is dependent on the
wind velocity, and it has three main regions: the No-
Generation Region (NGR), the Partial-Load Region
(PLR), and the Full-Load Region (FLR) as shown in
Fig. 1.
Fig. 1: Full operating region of variable speed
WECS
The schematic diagram of the variable speed WECS
is depicted in Fig. 2 (A), the figure shows four
subsystems: the mechanical, the aerodynamic, the
pitch actuator, and the generator subsystem. The
wind-captured aerodynamic power ( ) can be
obtained by the following equation, [19];
where is the air density, is the blade radius, and
is the wind velocity. The aerodynamic torque (
can be expressed by the following:
where is the rotor rotational speed. The power
coefficient ( ) can be given by Taher et al. [2]:
where:
where λ is the tip speed ratio and β is the pitch
angle.
Fig. 2: Schematic diagram of the wind turbine (A),
Two-mass WECS scheme (B).
The WECS model is considered a two-mass model
as shown in Fig. 2(B). In this model, the turbine
consists of two main components separated by the
transmission: the low-speed shaft (rotor side) and
the high-speed shaft (generator side). The gearbox
ratio of the system is defined by:
where is also defined by:
where and are the generator
speed, low-speed torque, high-speed torque, shaft
twist, spring constant, damping coefficient, and
gearbox ratio, respectively. The wind turbine
mechanical dynamic equations [19] are obtained
using Newton’s second law:
(A)
(B)
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where are rotor inertia and the generator
inertia. The generator model is given by:
where is the desired generator torque. This is a
simplified generator first-order model with a time
constant ( ). The generator power can be acquired
by:
The pitch actuator is designed to regulate the
rotational rotor speed at its rated value. It works by
controlling the input power aerodynamic flow at the
full load region. The turbine blades will turn in
when the power is too low and will turn out when
the power is too high. Generally, the power
coefficient in (3) is minimized by raising the blades’
pitch angle (β). The blade pitching process of the
WECS imposes a time delay to reach the desired
set-point value. Thus, the pitch actuator model is
first-order with a rate limiter
constrained to extreme values of ± 12 deg/s. The
pitch actuator model shown in Fig. 3 is described as
follows:
where and are the input blades’ pitch angle
and the time constant of the pitch actuator,
respectively.
Fig. 3: Pitch actuator model block diagram
2.1 Nonlinear WECS
The equations of the mechanical dynamics obtained
in Eq. (8) and (9) can be reformulated as:
The state-space model for the non-linear wind
turbine system is:
where the state vector , the
control action and the measured
output .
2.2 Linearized WECS
The wind turbine aerodynamic torque ( ) is a
nonlinear function of rotor speed, wind speed, and
pitch angle. The linearized model of the nonlinear
system is realized via the first-order Taylor
approximation approach. The nonlinear
aerodynamic torque in Eq. (2), can be linearized as
the following:
where and are the coefficients of
linearization operating points defined as:
.
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The linearized rotor rotational speed can be obtained
by substituting Eq. (17) in Eq. (13) to get the
following formula:
The linear state-space model can be given by the
following representation:
where the state vector
, the
control action , the
measured output and
is the exogenous input. The model transfer function
is
3 Controller Design
The robustness of a control system to disturbances
has always been the main issue in feedback control
systems. No need for feedback control if there are
no disturbances in control systems, [20]. The robust
control aims to achieve both robust performance and
robust stability of the closed-loop system. In this
section, LPV control based on mixed-sensitivity H∞
control is designed and presented above the rated
wind energy conversion system. The primary
objective of the proposed controller is to regulate
the rotational rotor speed and the generator torque.
Accordingly, maintaining the generator output
power of the WECS to the rated power. Moreover,
the turbulent wind velocity with large variations is
presented in this research to estimate the stability
and robustness of the suggested controller.
3.1 LPV Controller Design
The nonlinear WECS can be modeled as a linearized
state-space system whose parameters vary with their
states, [21], [22], [23]. The varying parameters ( )
of the model are presumed to be measurable,
bounded in a polytopic system, and slowly varying
in real-time:
The LPV model matrices are,
The LPV controller is intended to control the WECS
in the full-load regime, which covers wind speeds
ranging from 11 to 24 m.s-1. As a result, the wind
speed (v) is the scheduling parameter. The rotor
rotational speed of the wind turbine is held
constant at a rated value of 4.3 rad/s. Where the
main goal is to regulate the generator’s output
power around its rated value. The time-varying
parameter varies in a polytopic system whose
vertices are ) and ), in which the
parameters are assumed to be measurable and
slowly varying. The WECS LPV model can be
realized with two vertices as follows:
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The controller state space matrices at the
vertices can be given by:
where and
.
According to Eq. (26), the LPV controller is
designed as a convex combination of the vertices ѱ1
and ѱ2.
3.2 Mixed-weight H∞ Control Design
The mixed-weight H∞ controller’s technique
provides a closed-loop response of the system by
shaping the frequency responses for noise
attenuation and disturbance rejection. The controller
design involves incorporating additional weighting
functions in the original system, carefully chosen to
demonstrate the system’s performance and
robustness specifications, [24]. As mentioned in
Shaqarin et al., [25], [26], the generalized form of
the mixed-weight H∞ problem can be elicited as
explained in Fig. 4. Where , and are the
external input, the system-measured output,
performance output, system input, and the control
input, respectively. The general expanded plant
can be provided by:
where and are weighting functions. The
closed-loop transfer function from to can be
formulated using Eq. (28) and (29) as follows:
where is the sensitivity function, is the
complementary sensitivity function. The primary
goal is to define a controller ) that reduces the
infinity norm of in the polytope ( ), such
that , where γ is the upper bound of
. The selection of frequency-
dependent weights (W1, W2, and W3) substantially
enhances the control design. Generally, at low
frequencies, the sensitivity function is made small.
This results in excellent disturbance rejection and a
low tracking error. The complementary sensitivity
function is also reduced in the high-frequency
domain. As a result, there is high noise rejection and
a broad stability margin. Further details on the
weight selection can be found in Shaqarin et al. [25,
26].
Fig. 4: Mixed-weight closed-loop system
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3.3 PI-Fuzzy Logic Control (PIFLC) Design
The Fuzzy Logic Control (FLC) system has become
one of the most common intelligent techniques
utilized in many applications in current control
systems. FLC can cope with various types of
systems, ranging from linear processes to highly
complex systems, such as nonlinear processes or
time-varying systems. FLC versatility is attributed
to its parameter tunability, such as the membership
function type and number, rule base, scaling factor,
inference techniques, fuzzification, and
defuzzification. The process of fuzzy logic is shown
in Fig. 5(A), which consists of a fuzzifier, inference
engine, rule base, and defuzzifier. The process can
be explained along these lines: the crisp inputs from
the input data are initially fuzzified as fuzzy inputs.
Hence, they trigger the inference engine and the rule
base to generate the fuzzy output. The inference
engine provides an input/output map just after
blending the activated rules. Then, the inference
engine outputs are sent to the defuzzifier, which
produces the crisp outputs.
Fig. 5: Internal structure of the fuzzy controller(A),
PIFLC control structure (B).
In the developed controller shown in Fig. 5(B), two
input fuzzy variables: error (e) and change in error
(Δe) with the output Δu are shown. With a sampling
period Ts, the signal e is sampled and Δe is
calculated as:
where k denotes the sample number, and z-1 denotes
the unit time delay. As depicted in Fig. 5(B), the
PIFLC controller output u(k) can be found as:
It is worth mentioning that the continuous control
output u(t) is obtained by assuming a zero-order
hold between samples.
The membership functions of the inputs and the
output are depicted in Fig. 6(A), where μ is the
membership value. The performance of the PIFLC
can be tuned via error gain (Ke), the change in the
error (Kde), and the change in control output (Ku), as
shown in Fig. 5(B).
Fig. 6: Membership functions of the inputs e and Δe
and the output Δu (A), Input-output surface
relationship (B).
The FLC used in this work is a Mamdani type,
where the structure of fuzzy rule is formulated as:
(B)
(A)
(A)
(B)
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where are fuzzy sets. The system has 49
fuzzy rules, and the surface input-output
relationship is shown in Fig. 6 (B).
4 Results and Discussions
This section aims to assess the performance and
stability of the suggested LPV-based H∞ control
system of the WECS. This is accomplished via
simulating the closed-loop response of the suggested
controller using MATLAB/SIMULINK. Moreover,
the suggested controller is compared with the
response of the PIFLC and the WECS. In this work,
the nominal parameters of the Vestas V29-225 kW
WECS shown in Table 1 were used in the
simulation. The linearization coefficient values
( and ) for both vertices are presented in
Table 2. The weights are selected as follows:
was chosen to yield better disturbance rejection
through the shaping of the sensitivity function,
which leads to a small tracking error. The weighting
function was designed to shape the control
sensitivity function, aiming at limiting the actuator
effort. More precisely, is responsible for
limiting the pitch actuator effort to cope with the
limited actuator bandwidth.
Table 1. Parameters of the Vestas WECS
Parameters
Value
225 kW
4.3 rad/s
105.78 rad/s
0.00655 rad
10 kg.m2
90000 kg.m2
24.6
14.3 m
80000 s-1
8000000 N/m
0.15 s
0.1 s
1.225 kg.m3
Table 2. Coefficients of the linearization for the two
vertices
Vertex
Using the above-mentioned control technique, the
proposed LPV system based on an H∞ controller is
intended to regulate the outputs of the WECS to the
rated values without imposing large variations. The
simulation of the closed-loop system of the WECS
with both LPV controller and Fuzzy logic controller
is discussed in the following sections.
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Fig. 7: Comparison of the nonlinear and linearized
WECS open-loop step responses.
4.1 Nonlinear Versus Linearized WECS
Simulation
The dynamic model of the WECS was obtained for
both nonlinear and linearized cases as shown in
sections (2.2 and 2.3). The linearization is carried
out around operating points, which vary in a
polytopic system whose vertices are
and
. The rotational
speed of the rotor ( ) is assumed constant at the
rated value.
Figure 7 shows the step response of the non-linear
and the linearized systems under four wind speed
values. The figure shows that the steady-state
responses of the non-linear and linearized systems
are identical at the linearization operating points
with slight differences in the transients. However,
the discrepancy between the two systems increases
as the operating points change and move away from
the linearization points.
4.2 Open-loop Response of WECS Subjected
Wind Speed change
The open-loop response of WECS is simulated to
evaluate the wind turbine performance when
exposed to various wind speeds at a fixed pitch
angle. The variable-speed wind turbine in Fig. 8
started with a smooth wind speed ranging from 11
m/s to 24 m/s at a minimum pitch angle of zero
degrees. The figure shows that the rotor rotational
speed and the speed of the generator increase as the
free-stream velocity increases.
Fig. 8: Open-loop response of WECS subjected to a
smoothly varying wind speed.
As a result, the generator's output power is increased
to up to four times its nominal value. This motivates
the need for closed-loop control of the WECS for
power regulation. This is due to the fact that the
generator's speed is also increased four times above
its rated value, which Jeopardizes the wind turbine's
safety and complicates the connection with the grid.
4.3 Controlled WECS Subjected to a Step-
Ramp Change in Wind Speed
To evaluate the closed-loop system of the suggested
LPV controller with variable speed WECS, step-
ramp changes in free-stream velocity with a white
noise variance of 0.0102 are introduced. These steps
force the controller to modify the blade’s pitch
angle, and consequently the generator’s rotational
speed. Figure 9 illustrates the simulation of the
WECS that started with a noisy free-stream velocity
of 24 m/s, ranging from 0 to 25 sec. Then, the free-
stream velocity stepped down with a slope of -0.6
m/s2, until it reached a mean free-stream velocity of
17.5 m/s after 35 sec. This free-stream velocity was
maintained constantly from t = 35 sec to 60 sec.
Another stepped-down free-stream velocity
occurred with the same previous slope until it
reached the minimum free-stream of 11 m/s after 70
sec, then it remained constant. The figure shows that
the response of the closed-loop system does not
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introduce high oscillations over the entire operating
region.
Fig. 9: Closed-loop simulation of the WECS with
LPV controller subjected to step-ramp change in
wind speed with white noise.
The response of the generator speed slightly
changes around the nominal value, which is 105.78
rad/s. This indicates the LPV-based H∞ controller’s
effectiveness in maintaining the generator’s output
power very close to the rated value of 225 kW, in
presence of noisy varying free-stream velocity, as
seen in Fig. 9. Regarding the mechanical safety
aspects, it is depicted in the figure that slight
oscillations occurred in rotor shaft ( ),
as seen in the response of the shaft twist angle over
the whole operating range.
The suggested controller adequately maintained the
generator speed and output power at their rated
levels while maintaining the shaft's angle of twist
nearly constant. It's worth noting that for the
mechanical loads to be in the acceptable range, the
less the peak twist angle, the less mechanical stress.
4.4 Closed-loop Response of the WECS with
LPV and PI-Fuzzy Controllers in Presence of
Turbulent Wind Speed
In this work, the Von Karman turbulence spectrum
model is used with an average velocity of 17.5 m/s,
with a turbulence intensity of the incoming wind
flow greater than ± 10%, a turbulence length scale
of 170 m, and a 2 m/s standard deviation. The wind
speed used in the turbulence model shown in Fig. 10
varies in a range between 11 and 24 m/s, which lies
in the full load region.
Fig. 10: Closed-loop simulation of the WECS with
both LPV and PIFLC Controllers Subjected to
Turbulent Wind Speed.
The proposed control system was able to handle
these wind speed variations with high efficiency, as
depicted in Fig. 10. The generator speed and the
electromagnetic torque were controlled in the full
load region of the WECS around their nominal
values, which lead to stabilizing and maintaining the
generator output power around its rated value,
without violating the pitch angle ranges;
and the pitch actuator constraints;
. A PIFLC is designed and
implemented in this paper to be compared with the
proposed LPV control system. The gains were
selected as presented in Table 3.
Table 3. PIFLC Gains of the pitch controller
Gain
Value
2
2
-2
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The closed-loop simulations for both controllers are
presented in Fig. 10. The figure shows the
robustness of the proposed controller in enhancing
the performance and stability of the WECS against
free stream velocity and wind variations. The
proposed controller response showed much fewer
fluctuations in the generator outputs. This proves
that the suggested LPV controller was superior at
maintaining the output power of the generator at
around its nominal value without causing large
fluctuations in the output power of the WECS. On
the other hand, there were significant power spikes
and fluctuations in the Fuzzy controller response
that exceeded the permissible design limitations.
Though it was clear that the suggested LPV
controller in this work was assigned to only one
varying parameter (v) capable of satisfying the
required control objectives.
The severe wind turbulence conditions with large
mean wind speed variations implemented on the
wind turbine, can undeniably conclude the main
benefits of the LPV controller over the PIFLC
controller, as shown in Fig. 10. The closed-loop
response from the LPV controller indicates that the
generator speed has very few fluctuations ( 4%),
whereas the PIFLC controller has significantly large
peaks in the generator speed ( 24%). This is
translated to 4% and 24% peak fluctuations
around the mean in the generated power for LPV
and PIFLC controller cases, respectively. The
fluctuations in the twist angle of the wind turbine
shaft are comparable for both cases, whereas the
variance of the fluctuation in the twist angle is two
times less for the LPV controller case. This is highly
beneficial in reducing the destructive mechanical
loads on the wind turbine shaft. It is worth noting
that the aforementioned analysis neglects the startup
conditions.
5 Conclusion
In this paper, the suggested LPV based on an H∞
controller was employed to control a WECS via,
manipulating the blades’ pitch angle in the full load
regime. The proposed controller was able to
maintain and regulate the turbine shaft angular
velocity, the electromagnetic torque, and thus the
generator output power of the WECS to their
nominal values. The proposed control design
demonstrated proper performance and robustness
when applied to a 225-kW WECS under turbulent
free-stream velocity conditions. In comparison with
the PIFLC, the suggested LPV controller was more
effective in coping with the turbulent wind speed
with a turbulence intensity of ~ ±10%, which
improved the wind turbine performance in terms of
minimizing the fluctuations and smoothing the
generator power. When the proposed LPV controller
was assigned to only a single varying parameter (v),
it showed its capability of meeting the desired
control objectives of regulating and stabilizing the
desired output power around its rated value, while
complying with the pitch angle range; ,
and the pitch actuator constraints; .
References:
[1] Center for Sustainable Systems, University of
Michigan (2019) Wind Energy Factsheet, Pub.
(No. CSS07-09), USA.
[2] Taher, S. A., Farshadnia, M., & Mozdianfard, M.
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