Ocean Engineering 228 (2021) 108897
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Individual/collective blade pitch control of floating wind turbine based on
adaptive second order sliding mode
Cheng Zhang, Franck Plestan ∗
Ecole Centrale de Nantes-LS2N, UMR CNRS 6004, 44321 Nantes, France
ARTICLE
INFO
Keywords:
Second order sliding mode
Adaptive control
Floating wind turbine
Blade pitch control
ABSTRACT
A new control strategy based on adaptive second order sliding mode approach is applied to a floating wind
turbine system in the above rated region. This adaptive controller is well adapted to a highly nonlinear
system as floating wind turbine, and can be easily implemented with very reduced knowledge of modeling.
The proposed controller partially based on multi-blade coordinates transformation combines collective and
individual collective blade pitch control, for power regulation, platform pitch motion reduction and reduction
of blades fatigue load. The proposed controller is implemented on FAST simulator and shows high level of
performances.
1. Introduction
Floating wind turbines (FWTs) allow the use of the huge wind
resource in ocean area and are considered as a promising solution
of renewable energy. However, some issues arise: first-of-all, unlike
the onshore wind turbine, the floating platform introduces additional
degrees of freedoms (DOFs) that have negative impacts, especially on
the platform pitch motion. They also induce the issue of negative
damping (Nielsen et al., 2007) that leads to system instability and
degrades the power production. Furthermore, with the increasing capacity and flexibility of wind turbines, fatigue loads of the structure
became more and more important especially for floating systems and
affect the service life. Therefore, reducing the fatigue loads is a keypoint (Bossanyi, 2003; Menezes et al., 2018) for large scale wind
turbines. The control strategy must provide an efficient solution for
such problems and appears crucial for wind turbine systems.
The main control objectives of FWT in the above rated region consist
in maintaining the power output at rated value meanwhile avoiding the
negative damping, i.e. reducing the platform pitch motion (Jonkman
et al., 2009). Many works have been made based on the collective
blade pitch (CBP) control strategy; in this case, all the blades of the
wind turbine are controlled by a similar way. Among existing results,
in (Jonkman, 2008), the famous baseline gain scheduling proportional
integral (GSPI) control is proposed: platform pitch motion is successfully reduced but with a large power fluctuation. In (Wakui et al.,
2017), a novel gain scheduling control strategy is developed improving
the power regulation while keeping the same platform pitch motion
as GSPI. Linear quadratic regulator control, model predictive control
and feed-forward control (Namik et al., 2008; Schlipf et al., 2012,
2015) have been also applied to FWT systems. However, fatigue load
reduction is not considered in those works. In terms of the fatigue
load reduction, individual blade pitch (IBP) control (Bossanyi, 2003;
Lio et al., 2018; Ossmann et al., 2017; Selvam et al., 2009) has been
introduced: in this case, the blades are independently controlled. This
approach has also been extended to floating ones (Cunha et al., 2014;
Suemoto et al., 2017; Namik and Stol, 2014).
Nonetheless, in all these works, the control design is based on linearized models around an equilibrium point obtained by the FAST (Fatigue, Aerodynamics, Structures and Turbulence) software (Jonkman
et al., 2005). These approaches based on linearization around an operating point, are in opposition with the use of FWTs in a large operating
domain. Hence, in order to avoid this drawback, different models
for different equilibrium points are required that induces design of
different controllers producing a large effort of parameter tuning.
An other solution could be the use of nonlinear control strategies
based on nonlinear models. However, given that the nonlinear models
of FWT (Homer and Nagamune, 2018; Jonkman et al., 2005; Sandner
et al., 2012) are not well adapted to the control design, there are few
studies on the nonlinear control of FWT. As detailed just below, a
solution consists in designing nonlinear control laws that are efficient
on a large operating domain, but without precise models in order to
reduce as much as possible the modeling effort. Sliding mode control
(SMC) (Utkin, 1977) is a well-adapted solution, given its robustness
versus uncertainties and perturbations.
Then, in the current work, SMC is applied for the floating wind
turbines control. SMC requires very limited knowledge of system model
∗ Corresponding author.
E-mail addresses: cheng.zhang@ec-nantes.fr (C. Zhang), franck.plestan@ec-nantes.fr (F. Plestan).
https://doi.org/10.1016/j.oceaneng.2021.108897
Received 30 October 2020; Received in revised form 19 January 2021; Accepted 13 March 2021
Available online 7 April 2021
0029-8018/© 2021 Elsevier Ltd. All rights reserved.
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
with 𝑥 the system state vector depending on the DOFs enabled during
the linearization process, 𝑢 = [𝛽1 𝛽2 𝛽3 ]𝑇 the control input vector (i.e.
the pitch angle of each of the three blades) and 𝛥 the wind perturbation
input. Matrices 𝐴, 𝐵 and 𝐵𝑑 depend on the considered DOFs and are
obtained from FAST depending on the operating conditions. Considering that the above rated wind speed of NREL 5MW wind turbine is
varying between 11.4 m/s and 25 m/s, in such large operating domain,
different linear models, i.e. different matrices 𝐴, 𝐵 and 𝐵𝑑 , must be
carried out depending on the variation of the wind speed.
Remark 1. As detailed in the sequel, only 2 DOFs are considered for
the control design: in this case, the state vector is composed by the
platform pitch angle, its velocity and the rotor speed. The matrices 𝐴, 𝐵
and 𝐵𝑑 are provided by FAST software for each operating point. As an
example, considering a wind speed equal to 16 m∕s and a rotor speed
equal to 12.1 rpm, they are
Fig. 1. OC3 spar-buoy floating wind turbine.
(especially in its adaptive version) while keeping robust versus uncertainties and perturbations. Such control algorithms ensure that the
sliding variable (that is defined from the control objectives) converges
to a vicinity of the origin in a finite time. However, due to the discontinuous feature of ‘‘standard’’ SMC control, chattering phenomenon
(i.e. high frequency oscillations of control input) appears. In order to
reduce the chattering effect, super-twisting (STW) (Shtessel et al., 2014)
control combined with gain adaptation (Plestan et al., 2010; Shtessel
et al., 2012) is applied in this study. The STW is one of most famous
second order sliding mode control that generates continuous control
input and thereby, reduce the chattering. Furthermore, STW only requires the knowledge of sliding variable; then, it can be viewed as an
output feedback control and is very simple to implement. Furthermore,
adaptive gain allows to keep control accuracy versus perturbations and
uncertainties, even in the case that the information of system model
is very reduced. In fact, only the relative degree (Isidori, 1999) of
the sliding variable is required. Hence, such algorithm is very welladapted to the FWT control problem. Notice that, in authors’ previous
works (Zhang et al., 2019a,b; Zhang and Plestan, 2020), super twisting
control with gain adaptation based on CBP control technology has been
successfully applied to FWT system. Here, the first novelty is based on
the fact that IBP and CBP control structures are combined to control the
power and to reduce the fatigue load of the wind turbine, especially
the blade load reduction (among the structure loads, the blade root
reduction is the most important, being the source of the loads for the
rest of the structures (Jelavić et al., 2010)). An other novelty is the
use of an adaptive second order sliding mode controller in the frame of
IBP/CBP control strategy.
Section 2 introduces the model of the FWT. Section 3 states the
control problem. Section 4 describes the STW control laws with adaptation laws based on both CBP and IBP control approaches, and their
application to the FWT. Section 5 displays the results obtained by
FAST/Simulink co-simulations, and their analysis.
1
−0.0402
−2.0615
⎡ 0 ⎤
𝐵𝑑 = ⎢0.0001⎥
⎢
⎥
⎣0.0253⎦
■
0 ⎤
⎡ 0 ⎤
−0.0003⎥ , 𝐵 = ⎢−0.0033⎥ ,
⎥
⎢
⎥
−0.1624⎦
⎣−0.9479⎦
(2)
Thus, the FWT modeling on the whole above rated region reads as
𝑥̇ = 𝐴(𝑡)𝑥 + 𝐵(𝑡)𝑢 + 𝐵𝑑 (𝑡)𝛥
(3)
By a more general point-of-view, system (3) can be represented as a
class of nonlinear system
𝑥̇ = 𝑓 (𝑥, 𝑡) + 𝑔(𝑥, 𝑡)𝑢
(4)
with 𝑓 (𝑥, 𝑡) including the term 𝐴(𝑡)𝑥 and the associated perturbations
𝐵𝑑 (𝑡)𝛥 and uncertainties, as 𝑔(𝑥, 𝑡) including the term 𝐵(𝑡). It is important to notice that, in the sequel, 𝑓 and 𝑔 are viewed as unknown but
bounded functions.
The main purposes of the controllers designed in this paper are the
limitation of the power at its rated value, the reduction of the platform
pitch motion and the attenuation the blade flap-wise root moment. The
two first objectives can be achieved by the CBP control while the third
one is fulfilled by IBP control. Since the CBP and IBP control can be
separately designed as two independent control loops (see details in
the sequel), two models are carried out.
Notice that FAST provides dozens of degree of freedoms (DOFs)
including the motions of tower, blades and platform, the rotation of
rotor, the yaw motion, . . . These models are too complex and large for
control design. Then, in order to simplify the control design, reduced
models are considered. However, in the sequel, notice that all the
simulations will be made by applying the controllers designed on reduce
models, to the full DOFs model running in FAST.
2. System modeling
The National Renewable Energy Laboratory (NREL) 5MW OC3Hywind floating wind turbine (see Fig. 1) is selected in this study. This
wind turbine is simulated by the well-known wind turbine simulation
software FAST (Jonkman et al., 2005), the detailed parameters and
the properties being given in Jonkman et al. (2009), Jonkman (2010).
However, the wind turbine model used in FAST is composed by a
large number of complex nonlinear functions and cannot be adopted
for control design.
FAST software can provide a linear state model of the FWT, around
an operating point that depends on the wind speed and the rotor speed.
In this case, for a given operating point, one can obtain the following
state–space model
𝑥̇ = 𝐴𝑥 + 𝐵𝑢 + 𝐵𝑑 𝛥
⎡ 0
𝐴 = ⎢−0.0141
⎢
⎣−0.0525
2.1. Reduced CBP control model
This model is focused on the platform pitch and the rotor. Concerning the control objectives of CBP controller, only two DOFs, the
rotor rotation and the platform pitch are considered. Based on a similar
writing as (4), the model for CBP control reads as
𝑥̇ 𝐶
=
𝑓𝐶 (𝑥𝐶 , 𝑡) + 𝑔𝐶 (𝑥𝐶 , 𝑡)𝑢𝐶
(5)
with 𝑥𝐶 = [𝜑 𝜑̇ 𝛺]𝑇 , 𝜑 being the platform pitch angle, 𝜑̇ the platform
pitch velocity and 𝛺 the rotor speed. The CBP control adjusts each blade
pitch angle by the same amount simultaneously. Hence, the control input
is defined as 𝑢𝐶 = 𝛽𝑐𝑜𝑙 , this angle being applied to each blade.
(1)
2
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
motion and reducing the flap-wise load of blades. In authors’ previous
works (Zhang et al., 2019a,b), both the first control objectives (power,
platform pitch motion) are achieved by collective blade pitch control.
Here, the blade load (especially the blade flap-wise load) alleviation is
also considered and can be ensured by separately adjusting the pitch angle
of each blade, namely, by using the individual blade pitch control (Bossanyi,
2003; Selvam et al., 2009; Van Engelen, 2006).
The overall control scheme is shown in Fig. 3. The IBP angle
adjustment is added to the CBP control input but has a limited effect
on the global behavior of the power and platform pitch motion; in
other words, there is a very reduced coupling between the CBP and IBP
control (Bossanyi, 2003; Jelavić et al., 2010). Hence, these latter can be
separately designed as two independent control loops while achieving their
own control objectives.
Fig. 2. Rotor azimuth angle 𝜓 (left) and blade#1 flap-wise bending deflection
(right) (Cheon et al., 2019; Liu et al., 2017).
3.1. Formalization of the collective blade pitch control problem
2.2. Reduced IBP control model
The task of CBP control loop is to regulate power at rated 𝑃0
meanwhile reducing the platform pitch motion. Usually,1 the generator
torque is supposed to be fixed at its rated 𝑇𝑔0 in the above rated region,
then, the power regulation turns into regulate the rotor speed at its
rated value 𝛺𝑟
This model is focused on the blade behavior. In this case, 3 DOFs
are enabled, i.e. the first flap-wise bending mode of each blade. Hence,
the state vector includes the flap-wise bending deflection of each blade
𝑞1,2,3 (see 𝑞1 for blade#1 — Fig. 2-right) and its corresponding velocity
𝑞̇ 1,2,3 , i.e. 𝑥𝐼 = [𝑞1 𝑞2 𝑞3 𝑞̇ 1 𝑞̇ 2 𝑞̇ 3 ]𝑇 . The control input vector 𝑢𝐼 =
[𝛽̃1 𝛽̃2 𝛽̃3 ]𝑇 is such that each blade pitch angle has its own value. As
previously, the system model can be written as (4)
𝑥̇ 𝐼
=
𝛺𝑟 =
with 𝑛𝑔 the gear box ratio between high speed shaft and low speed
shaft.
(6)
𝑓𝐼 (𝑥𝐼 , 𝑡) + 𝑔𝐼 (𝑥𝐼 , 𝑡)𝑢𝐼
However, the dynamics of each blade depends on the azimuth angle
𝜓, i.e. the angle between a vertical axis and the current position of
the blade symmetrical axis (see Fig. 2-left), which induces a periodic
system. Therefore, analysis and control design could be not straightforward. In order to avoid the periodic dynamics, the most conventional
method in IBP control is the application of the multi-blade coordinate
(MBC) transformation (Bir, 2008), also known as Coleman transformation. MBC transformation allows to write the dynamics into a fixed
non-rotating frame; by this way, the controller is designed without
considering the periodic property. Such coordinates transformation also
allows to decouple the IBP control that is focused on load reduction,
from the CBP control (Stol et al., 2009). Notice that the control based
on MBC transformation has almost same results as the directly periodic
control (Stol et al., 2009), but without complexity.
Consider the following state coordinates transformation, called MBC
transformation (Bir, 2008)
However, there are two control objectives with a single control
input that is the collective blade pitch angle 𝛽𝑐𝑜𝑙 . A solution to such
problem regarding to the FWT control is to modify the rotor speed
reference by including the platform pitch rate 𝜑̇ that gives a new rotor
speed reference 𝛺∗ defined as
𝛺∗ = 𝛺𝑟 − 𝐾 ⋅ 𝜑̇
with
⎡
1 ⎢2 cos(𝜓)
𝑇 = ⎢
3 ⎢ 2 sin(𝜓)
⎣
2𝜋
)
3
2𝜋
2 sin(𝜓 +
)
3
2 cos(𝜓 +
As conclusion, the controlled output associated to the system (5) is
defined as
⎤
2 cos(𝜓 + 4𝜋
)
3 ⎥
⎥
2 sin(𝜓 + 4𝜋
)⎥
3 ⎦
(8)
𝑦𝐶
Then, applying this transformation to (5) gives the following system
(that will be called in the sequel, MBC system)
=
𝛺 − 𝛺∗
(11)
Remark 2. From Remark 1 and (Zhang et al., 2019b; Zhang and
Plestan, 2020), it can be verified that, in the considered operating
domain, the relative degree (Isidori, 1999) of system (5) with the
previous output 𝑦𝑐 is equal to 1, i.e. the first time derivative of 𝑦𝑐
explicitly depends on the control input 𝑢𝑐 . ■
(9)
𝑥̇ 𝐼𝑛𝑟 = 𝑓𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡) + 𝑔𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡)𝑢𝐼𝑛𝑟
]𝑇
(10)
with 𝐾 a positive constant. Such solution takes advantage of the
physical features of the rotor rotation and platform pitching in response
to aerodynamic torque and thrust; consider that forward pitching is
appearing and suppose that the control is efficient. In this case, 𝜑̇ <
0: in order to fulfill (10), thank to the action on the blade pitch
angle that increases the aerodynamic torque of blades, the rotor speed
increases and is greater than 𝛺𝑟 . At the same time, the aerodynamic
thrust increases preventing the platform forward pitching. Therefore,
the platform pitch velocity 𝜑̇ will be reduced, and thereby the rotor
speed will converge to the rated (Lackner, 2009; Cunha et al., 2014).
(7)
𝑥𝐼𝑛𝑟 = 𝑇 𝑥𝐼
𝑃0
𝑛𝑔 𝑇𝑔0
]𝑇
with 𝑥𝐼𝑛𝑟 = [𝑞𝑡𝑖𝑙𝑡 𝑞𝑦𝑎𝑤 𝑞̇ 𝑡𝑖𝑙𝑡 𝑞̇ 𝑦𝑎𝑤 and 𝑢𝐼𝑛𝑟 = [𝛽𝑡𝑖𝑙𝑡 𝛽𝑦𝑎𝑤 the state and
input vectors in the non-rotating frame. 𝑞𝑡𝑖𝑙𝑡 and 𝑞𝑦𝑎𝑤 the fictitious tilt
and yaw component of blade flap-wise deflections respectively; 𝛽𝑦𝑎𝑤
and 𝛽𝑡𝑖𝑙𝑡 the yaw and tilt component of blade pitch angles.
1
In the paper, for a sake of simplicity, the generator torque is supposed to
be constant, the objective being to focus the attention on the control of the
hydrodynamic part of the wind turbine. However, the authors are fully aware
that it is necessary to also consider the control by generator point-of-view.
That will be the object of future works.
3. Problem statement
Recall that the control objectives of the current study are to ensure
the power output at rated meanwhile reducing the platform pitch
3
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
Fig. 3. Control scheme of the whole closed-loop system.
Fig. 4. Coordinate system of the rotational blade root (left) and the fixed hub center (right), 𝑖 = 1, 2, 3 refers to the 𝑖th blade (Jelavić et al., 2010).
3.2. Formalization of the individual blade pitch control problem
moment (see Fig. 4). As shown in (Wang et al., 2016; Xiao et al., 2013),
this output can be written as
The rotor of wind turbine transforms the wind power into aerodynamic torque that drives the generator; at the same time, partial wind
energy is transformed into thrust on the rotor that induces load. Due
to the wind shear, tower shadow and turbulence, the wind speed and
direction are varying across the rotor plane; these factors cause additional loads on the blades. These loads are related with the frequency
of the rotor speed and can be decomposed along different modes, the
main one being at the rotor speed frequency — this mode is denoted
the 1p-mode (once-per-revolution-see Fig. 9). Other modes are existing at
multiples of rotor speed and are denoted 2p, 3p ... (Bossanyi, 2003). The
reduction of the 1p-mode for each blade appears being a main objective
of IBP control.
In this regard, the flap-wise bending moment of each blade are
considered as the outputs of IBP control loop. Consider the MBC system
(9) and denote the control output 𝑦𝐼𝑛𝑟 = [𝑀𝑡𝑖𝑙𝑡 𝑀𝑦𝑎𝑤 ]𝑇 with 𝑀𝑡𝑖𝑙𝑡 and
𝑀𝑦𝑎𝑤 respectively the tilt and yaw component of blade root flap-wise
𝑦𝐼𝑛𝑟 = ℎ𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡) + 𝑙𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡)𝑢𝐼𝑛𝑟
(12)
Remark 3. Notice that the output 𝑦𝐼𝑛𝑟 depends on the control input
vector 𝑢𝐼𝑛𝑟 ; in this case, the relative degree of system (9) with output
𝑦𝐼𝑛𝑟 , is equal to 0. ■
The main idea of IBP control is to force the magnitudes of 𝑀𝑦𝑎𝑤
and 𝑀𝑡𝑖𝑙𝑡 close to zero that reduces the blade flap-wise load. MBC
approach allows the decoupling between the IBP control that is responsible for load reduction, and the CBP control. Furthermore, it
has been shown (Bossanyi, 2003) that 𝑀𝑦𝑎𝑤 and 𝑀𝑡𝑖𝑙𝑡 can be treated
independently by 𝛽𝑦𝑎𝑤 and 𝛽𝑡𝑖𝑙𝑡 respectively, i.e. it is possible to use two
single input single output controllers for the 𝑀𝑦𝑎𝑤 and 𝑀𝑡𝑖𝑙𝑡 alleviation.
Hence, the control objectives of IBP control are to ensure the 𝑀𝑡𝑖𝑙𝑡 and
𝑀𝑦𝑎𝑤 close to zero by the control 𝛽𝑡𝑖𝑙𝑡 and 𝛽𝑦𝑎𝑤 respectively.
4
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
Fig. 5. Control scheme of IBP control loop.
Notice that the following inverse MBC transformation should be
applied after the controllers design in order to generate IBP control
command 𝛽̃1 , 𝛽̃2 and 𝛽̃3 (see Fig. 5)
[
]𝑇
[
]𝑇
𝛽̃1 𝛽̃2 𝛽̃3 = 𝑇 −1 𝛽𝑦𝑎𝑤 𝛽𝑡𝑖𝑙𝑡 ,
⎡ 𝑐𝑜𝑠(𝜓)
⎢
2𝜋
𝑇 −1 = ⎢𝑐𝑜𝑠(𝜓 + 3 )
⎢𝑐𝑜𝑠(𝜓 + 4𝜋 )
⎣
3
𝑠𝑖𝑛(𝜓) ⎤
⎥
𝑠𝑖𝑛(𝜓 + 2𝜋
)
3 ⎥
4𝜋 ⎥
𝑠𝑖𝑛(𝜓 + 3 )⎦
fact gives a ‘‘high gain’’ control that is not good for chattering reduction. Therefore, adaptive version of STW (Shtessel et al., 2012) is
well-adapted: it allows to dynamically adapt the gain versus uncertainties and perturbations while keeping high level of performances, even
in case of very reduced knowledge of the system.
(13)
4.1. Recalls
Consider the nonlinear system
3.3. Overall control scheme
𝑧̇ = 𝑓 (𝑧, 𝑡) + 𝑔(𝑧, 𝑡)𝜐
(14)
𝑦 = ℎ(𝑧, 𝑡)
By a structural point-of-view, the overall control scheme is the
combination of CBP control and IBP control. Then, the overall control
system design process can be summarized as follows
with 𝑧 ∈ 𝑍 ⊂ R𝑛 the state vector and 𝜐 ∈ 𝑈 ⊂ R the control
input, 𝑓 (𝑧, 𝑡) and 𝑔(𝑧, 𝑡) the bounded unknown uncertain functions, 𝑦
the control output, i.e. the control objective is to force 𝑦 to 0. Define the
sliding variable 𝜎 = 𝜎(𝑧, 𝑡) such that control objective is achieved when
𝜎 = 0. The objective of the control design is to define a control input 𝜐
that drives the sliding variable 𝜎 to the sliding surface 𝜎(𝑧, 𝑡) = 0 in a
finite time despite the uncertainties and perturbations. Note that sliding
variable is defined according to control objective 𝑦 and its relative
degree (Isidori, 1999).
• design the CBP control 𝛽𝑐𝑜𝑙 for regulation of the power and
reduction of the platform pitch motion;
• transform the three flap-wise blade flap-wise bending moments
𝑀𝑦1 , 𝑀𝑦2 and 𝑀𝑦3 into the fictitious ones 𝑀𝑦𝑎𝑤 and 𝑀𝑡𝑖𝑙𝑡 , and
design the control loop that provides 𝛽𝑦𝑎𝑤 and 𝛽𝑡𝑖𝑙𝑡 respectively,
and obtain the components 𝛽̃1 , 𝛽̃2 and 𝛽̃3 thanks to the inverse
MBC transformation;
• the real blade pitch angles 𝛽1 , 𝛽2 and 𝛽3 , that are the control
inputs, equal to the sum of 𝛽𝑐𝑜𝑙 with 𝛽̃1 , 𝛽̃2 and 𝛽̃3 .
Assumption 1. The relative degree of (14) is equal to 1.
■
Define 𝜎 = 𝑦; its dynamic reads as
𝜎̇ =
4. Control design
As previously detailed, floating wind turbine is a class of nonlinear
system with model uncertainties and perturbations that introduced
from the flexible structures, wind and waves. Furthermore, the traditional controllers of FWT based on the linearized model such as LQR,
MPC and GSPI need great effort of tuning (due to the large set of operating points) in order to keep high performances all over the operating
domain. Then, there is a real interest to design a robust controller with
a reduced tuning effort and a single set of parameter tuning meanwhile
keeping the control efficiency among the large operating range despite
of the uncertainties and perturbations.
The robust nonlinear strategy selected in this work is based on
sliding mode control (SMC) theory (Utkin, 1977; Utkin et al., 2009),
a well-known nonlinear control strategy with properties of robustness,
accuracy and finite time convergence. In fact, the standard first order
SMC can be easily implemented; however, the control of standard SMC
is discontinuous. Due to the discontinuous term of the control input,
chattering is introduced and can damage the physical components
such as blade pitch actuator. In order to reduce chattering and keep
robustness, super-twisting (STW) (Levant, 1993) control is introduced.
Furthermore, given the unknown terms of system dynamics, the uncertainties on turbine and the perturbations..., the gains of the controller
must be chosen sufficiently large to accommodate those effects. This
𝜕𝜎 𝜕𝜎
𝜕𝜎
+
𝑓 (𝑧, 𝑡) +
𝑔(𝑧, 𝑡) 𝜐
𝜕𝑡
𝜕𝑧
𝜕𝑧
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟
𝑎(𝑧, 𝑡)
𝑏(𝑧, 𝑡)
(15)
Assumption 2. 𝑎(𝑧, 𝑡) and 𝑏(𝑧, 𝑡) are unknown but bounded functions,
such that |𝑎| ≤ 𝑎𝑀 , 0 ≤ 𝑏𝑚 ≤ 𝑏 ≤ 𝑏𝑀 for 𝑧 ∈ 𝑍 and 𝑡 > 0, 𝑎𝑀 , 𝑏𝑚 and
𝑏𝑀 being the positive constant. ■
As mentioned above, standard SMC with sufficient large controller
gains can establish a sliding mode, namely, drive 𝜎 to zero in a
finite time, but with important chattering. The super-twisting algorithm
(STW) (Levant, 1993) given by
1
𝜐 = −𝑘1 |𝜎| 2 ⋅ sign(𝜎) + 𝑤
(16)
𝑤̇ = −𝑘2 ⋅ sign(𝜎)
with 𝑘1 and 𝑘2 the controller gains satisfying
𝑘1 >
𝑎𝑀
,
𝑏𝑀
𝑘22 ≥
4𝑎𝑀
𝑏2𝑚
⋅
𝑏 𝑀 𝑘 1 + 𝑎𝑀
⋅
𝑏 𝑚 𝑘 1 − 𝑎𝑀
(17)
ensures the establishment of a second order sliding mode, i.e. 𝜎 = 𝜎̇ = 0,
in a finite time. In practice, especially due to sampling period, only real
second order sliding mode is established, that is defined as (Levant,
1993)
|𝜎| < 𝜇1 𝑇𝑒2 , |𝜎|
̇ < 𝜇2 𝑇𝑒
5
(18)
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
with 𝑇𝑒 the sampling time of controller, 𝜇1 and 𝜇2 positive constants.
Thanks to the continuous nature of STW algorithm, the chattering
is greatly reduced while the robustness is kept. However, due to the
uncertainties and perturbations of the real systems, sufficiently large
gains are necessary; such gains are always overestimated which limits
the interest of this control strategy.
To this end, gain adaptation is adopted in order to further increase
the control performance; the adaptation law dynamically adapts the
controller that avoids the gain overestimation and strongly reduces
the effort of uncertainties/perturbations evaluation. Adaptive supertwisting (ASTW) controller proposed in (Shtessel et al., 2012) is used
here, the adaptive law being defined as
√
⎧
𝜒
𝜔
sign(|𝜎| − 𝜇) if 𝑘1 > 𝑘1𝑚
⎪
2
𝑘̇ 1 = ⎨
(19)
⎪𝑘
if 𝑘1 ≤ 𝑘1𝑚
⎩
with
𝑦𝐼𝑛𝑟
𝑥̄ 𝐼𝑛𝑟
]
= ℎ𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡) + 𝑙𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡)𝑢𝐼𝑛𝑟
(22)
=
(23)
𝑢𝐼𝑛𝑟
and 𝑣𝐼𝑛𝑟 = 𝑢̇ 𝐼𝑛𝑟 the new control input, system (9) can be reformulated
as
𝑥̇ 𝐼𝑛𝑟
𝑥̄̇ 𝐼𝑛𝑟
𝑦𝐼𝑛𝑟
=
=
=
𝑓𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡) + 𝑔𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡)𝑥̄ 𝐼𝑛𝑟
𝑣𝐼𝑛𝑟
ℎ𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡) + 𝑙𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡)𝑥̄ 𝐼𝑛𝑟
(24)
with 𝑣𝐼𝑛𝑟 = [𝛽̇𝑡𝑖𝑙𝑡 𝛽̇𝑦𝑎𝑤 ]𝑇 the new control input.
Then, the relative degree of (24) with respect to [𝛽̇𝑡𝑖𝑙𝑡 𝛽̇𝑦𝑎𝑤 ]𝑇 is equal
to 1. Therefore, ASTW algorithm can be applied: the sliding variables
of IBP loop are defined as [𝜎2 𝜎3 ]𝑇 = [𝑀𝑡𝑖𝑙𝑡 𝑀𝑦𝑎𝑤 ]𝑇 . Fig. 5 depicts the
IBP control scheme.
Then, one has the sliding variable vector
with 𝑘1𝑚 , 𝜖, 𝜔, 𝜒, 𝜇, 𝑘 positive constants, and 𝑘1 (0) > 𝑘1𝑚 . From (19),
one can find that
• |𝜎| < 𝜇: the control accuracy is high. So, the controller gains can
be reduced because they are certainly sufficient: 𝑘̇ 1 is negative;
• |𝜎| > 𝜇: the control accuracy is low. The gains could be too
small. It is necessary to increase the gains in order to improve
the accuracy: 𝑘̇ 1 is positive;
• the parameter 𝑘1𝑚 is a very small positive constant that ensures
the positiveness of controller gains.
⎡ 𝜎1 ⎤ ⎡ 𝛺 − 𝛺𝑟 + 𝐾 𝜑̇ ⎤
⎥
𝜎 = ⎢ 𝜎2 ⎥ = ⎢
𝑀𝑡𝑖𝑙𝑡
⎢
⎥ ⎢
⎥
𝑀𝑦𝑎𝑤
⎣ 𝜎3 ⎦ ⎣
⎦
(25)
and its dynamics reads as
(26)
𝜎̇ = 𝑎(⋅) + 𝑏(⋅)𝑣
with 𝑎(⋅) and 𝑏(⋅) unknown but bounded functions obtained from (5)–
(24). The control input 𝑣 is defined as
Remark that
• 𝜔 and 𝜒 determine 𝑘1 -dynamics (and also 𝑘2 -dynamics given
that 𝑘1 and 𝑘2 are proportional). If they are stated large, 𝑘1
will strongly increases (resp. decreases) when real second order
sliding mode is lost (resp. established). Large dynamics of 𝑘1 and
𝑘2 could induce large variations of the control input; it could be
damageable for the actuator ;
• 𝜖 defines the ratio between 𝑘1 and 𝑘2 that depends on the system;
• 𝜇 acts on the accuracy of the close-loop system (see the items
following (19)). A smaller 𝜇 means a higher accuracy but induces
a more intensive control given that it gives a larger gain 𝑘1 . On
the other hand, a larger value for 𝜇 means a lower accuracy; in
this case, the control gain is smaller that gives a less intensive
control.
𝜐 = [𝛽𝑐𝑜𝑙 𝛽̇𝑡𝑖𝑙𝑡 𝛽̇𝑦𝑎𝑤 ]𝑇 = [𝜐1 𝜐2 𝜐3 ]𝑇
1
𝑡
⎡ 𝜐1 ⎤ ⎡ −𝑘11 |𝜎1 | 2 sign(𝜎1 ) − ∫0 𝑘12 sign(𝜎1 )𝑑𝜏 ⎤
⎢
1
⎢ 𝜐 ⎥ = ⎢ −𝑘 |𝜎 | 2 sign(𝜎 ) − ∫ 𝑡 𝑘 sign(𝜎 )𝑑𝜏 ⎥⎥
21 2
2
2
0 22
⎢ 2 ⎥ ⎢
1
⎥
𝑡
⎣ 𝜐3 ⎦ ⎣
−𝑘31 |𝜎3 | 2 sign(𝜎3 ) − ∫0 𝑘32 sign(𝜎3 )𝑑𝜏 ⎦
5. Simulation results
The nonlinear OC3-Hywind 5MW floating wind turbine model from
NREL, especially controlled by the previously detailed controller, is
simulated in this section. The used model is built in the FAST software
and is regarded as a benchmark in many wind turbines studies; the
parameters of this wind turbine are shown in Table 1.
The control is developed in the SIMULINK environment and link
with the FAST model by an s-function. Finally, the co-simulations
between FAST and SIMULINK are made on the full DOFs FAST nonlinear model while the control has been designed based on the reduced
DOFs model as detailed previously. Three controllers are used in the
following simulations
• the first one is the CBP control loop focusing on the control of
rotor speed and platform pitch motion;
• the second one is the IBP control loop producing an additional
term to each blade pitch angle in order to reduce the variation of
blade root flap-wise bending moments.
As previously recalled, these two control loops can be independently
designed (Bossanyi, 2003; Jelavić et al., 2010).
CBP loop. As claimed in Remark 2, the relative degree (5) with 𝑦𝐶
(11) is equal to 1. Therefore, according to Assumption 1, the sliding
variable of CBP control can be defined as
• GSPI-CBP: the baseline GSPI controller with collective blade pitch
control (Jonkman, 2008);
• ASTW-CBP: the adaptive super-twisting controller with collective
blade pitch control; in this case, only 𝜐1 (28) is considered (Zhang
et al., 2019b).
• ASTW-CIBP: the adaptive super-twisting controller (28) that combines collective with individual blade pitch control.
(20)
IBP loop. As recalled in Remark 3, the relative degree of system
(9) with output 𝑦𝐼𝑛𝑟 , is equal to 0. Given that ASTW algorithm must be
applied to system with relative degree equal to 1, consider again the
system (9)
𝑓𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡) + 𝑔𝐼𝑛𝑟 (𝑥𝐼𝑛𝑟 , 𝑡)𝑢𝐼𝑛𝑟
(28)
The gains 𝑘∗1 and 𝑘∗2 (∗= 1, 2, 3) are evolving according to adaptation
law (19).
As detailed in Section 3, the control scheme includes two control
loops
𝜎1 = 𝑦𝐶 = 𝛺 − 𝛺∗ = 𝛺 − (𝛺𝑟 − 𝑘𝜑)
̇
(27)
with
4.2. Application to the floating wind turbine system
=
𝑀𝑡𝑖𝑙𝑡
𝑀𝑦𝑎𝑤
A solution consists in defining a dynamic control input that increases
the relative degree of the system. Defining
𝑘2 = 𝜖𝑘1
𝑥̇ 𝐼𝑛𝑟
[
=
The use of these controllers has several objectives: comparison
between standard (GSPI) and advanced controllers (STW), and comparison between CBP and CBP/IBP control structures. In addition, two
cases of wind and wave conditions are simulated in the sequel
(21)
6
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
Table 1
Wind turbine parameters Jonkman et al. (2009).
Table 2
ASTW-CIBP controller parameters.
Parameters
Value
Gains
Parameters
Cut-in, rated, cut-out wind speed
Rotor, hub diameter
Hub height
Rated rotor speed 𝛺0
Minimum, maximum blade pitch angle
Maximum blade pitch rate
3 m/s, 11.4 m/s, 25 m/s
126 m, 3 m
90 m
12.1 rpm
0 deg, 90 deg
±8 deg/s
𝑘11 , 𝑘12
𝑘21 , 𝑘22
𝑘31 , 𝑘32
𝑘1𝑚 = 10−4 , 𝜖 = 0.03, 𝜔 = 1, 𝜒 = 0.001, 𝜇 = 0.05, 𝑘 = 10−4
𝑘1𝑚 = 10−6 , 𝜖 = 0.05, 𝜔 = 1, 𝜒 = 0.003, 𝜇 = 0.4, 𝑘 = 0.01
𝑘1𝑚 = 10−6 , 𝜖 = 0.05, 𝜔 = 1, 𝜒 = 0.003, 𝜇 = 0.2, 𝑘 = 0.01
As previously mentioned, there is no coupling between CBP and
IBP control loops; hence, ASTW-CBP and ASTW-CIBP have similar
performances on rotor speed and platform pitch rate. As shown by
Fig. 12, the time series of ASTW-CBP and ASTW-CIBP in terms of rotor
speed (power) and platform pitch angle are almost identical.
Furthermore, ASTW-CBP controller has also reduced the platform
roll and yaw rate; on the contrary, ASTW-CIBP controller induces more
important platform roll and yaw rate due to the greatly increased
blade pitch actuation (Namik and Stol, 2014). However, given that the
magnitude of platform roll and yaw are relatively small (see Fig. 12),
they have a very limited influence on the stability of the whole system.
Fig. 13 shows the DEL results: it is clear that ASTW-CBP control law
reduces the platform base loads while increasing the blade root flapwise load. For the ASTW-CIBP, the tower base side-to-side and fore–aft
loads have similar reductions than ASTW-CBP, while the torsional
load increases by 3%. Nonetheless, Fig. 12 shows the torsional load is
very reduced comparing to the side-to-side and fore–aft loads of tower
base: then, an increasing of 3% is meaningless for the load of tower.
Furthermore, ASTW-CIBP reduces the blade root flap-wise load (1p load
— see Fig. 14).
Generally, ASTW-CIBP control strategy has not only better performance on the rotor speed (power) regulation and platform pitch motion
reduction than GSPI-CBP as ASTW-CBP, but also can reduce the fatigue
load of blade, all of which being crucial problems of the floating wind
turbine control. Moreover, this controller requires very few knowledge
of system model and the controller gains can be dynamically adapted
with the uncertainties and perturbations (see Fig. 15) that largely
reduced the parameters tuning effort.
However, such improvement has a cost that is a more aggressive
actuator use, as shown by Fig. 11: the variation of ASTW-CBP increases
by 82% versus CBP-GSPI whereas it is worst with ASTW-CIBP controller. Notice that, given that the dynamics of blade pitch actuators
is taken into account in the simulations, such a use of these actuators
is practically acceptable (see Fig. 16).
• Case 1. 18 m/s constant wind velocity with still water (i.e. no
wave).
• Case 2. 18 m/s wind velocity with 15% turbulence intensity;
irregular wave with significant height of 3.25 𝑚 and peak spectral
period of 9.7 𝑠 (see Fig. 6).
Note that the wind speed of both cases are in above rated region.
All the simulations are made in 10 min and Euler integration is used
with sample time fixed at 0.0125 s. Moreover, the blade pitch angle
and its rated are saturated as shown in Table 1.
5.1. Case 1: constant wind velocity with still water
In this case, the ASTW-CBP and ASTW-CIBP control strategies are
compared, the objective being to check the interest to include a IBP
control loop. Firstly, Fig. 7 displays that both the CBP and IBP controllers ensure the rotor speed around its rated value 12.1 rpm and
reduced the platform pitch motion (i.e. reduced the platform pitch angle
variation. Furthermore, Fig. 8 shows that the tilt and yaw moment are
forced around zero with the CIBP controller, as a consequence, the
blade root flap-wise moment are strongly reduced comparing with the
CBP controller (see Fig. 8-right).
Specifically, from the power spectral density (PSD) of blade#1 root
flap-wise moment displayed by Fig. 9, one can find that the load
reduction of IBP control is acting on the 1p component of the blade
load. Meantime, the rotor speed and platform pitch motion are not
affected as shown in Fig. 7 (the trajectories of CBP and CIBP control are
highly coincidence), as mentioned in previous section, the collective
pitch control and the individual pitch control are decoupled. Fig. 10
shows the blade#1 pitch angle obtained with the two controllers.
5.2. Case 2: turbulent wind and irregular wave
6. Conclusion
Previous case shows that both the ASTW controllers allow to achieve
all the control objectives in ideal conditions and proves the necessity
to introduce a IBP control loop. Case 2 now allows to evaluate the
controllers (GSPI-CBP, ASTW-CBP, ASTW-CIBP) in a more realistic
situation.
The performance of the 3 controllers are compared by using the
following indicators
Super-twisting algorithms with gain adaptation algorithm based on
collective/individual blade pitch control are applied to the floating
wind turbines control problem in above rated region. Such control
algorithms strongly reduce the workload of parameters tuning: only
few knowledge of system model is required that makes such control
strategy well adapted to the floating wind turbine systems.
The control goals are the regulation of the rotor speed, the reduction
of the platform pitch motion and the reduction of the fatigue load
of the blades. The simulations made on FAST software show that the
collective control loop and individual blade pitch control loop are well
decoupled by the MBC transformation. Then, the CIBP based ASTW
algorithm gives not only better performances on the power regulation
and platform pitch motion reduction than CBP controllers, but also
provides better performances on the blade load reduction.
Future works will be focused on the application of the proposed
control solution on experimental set-up and on the use of simpler
adaptive super-twisting algorithm (Zhang et al., 2021).
• root mean square (RMS) of rotor speed error, power error, platform angular rates — these values evaluate the accuracy of the
closed-loop system for the 3 controllers;
• variation (VAR) (Wang et al., 2014) of the blade pitch angle —
this value evaluates the activity of the actuator;
• damage equivalent load (DEL) (Hayman, 2012) of the tower base
fore–aft, side-to-side and torsional moment, and DEL of averaged
blade root flap-wise and edge-wise bending moment of the three
blades. - these values evaluate the level of mechanical constraints
acting on the blades.
Fig. 11 shows that, for two of the main control objectives (rotor
speed/ power regulation and platform pitch motion reduction), ASTWCBP and ASTW-CIBP controllers have similar performances allowing
reduction of rotor speed error (by 8%–9%) and platform pitch rate (by
22%–23%), versus GSPI-CBP.
CRediT authorship contribution statement
Cheng Zhang: Conceptualization, Simulation, Draft writing. Franck
Plestan: Supervision, Conceptualization, Reading-writing-reviewing.
7
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
Fig. 6. Wind (top) and wave (bottom) conditions of Case 2.
Fig. 7. Case 1. Rotor speed and platform motion versus time (sec).
Fig. 8. Case 1. Yaw moment 𝑀𝑦𝑎𝑤 (left-top), tilt moment 𝑀𝑡𝑖𝑙𝑡 (left-bottom) and blade#1 root flap-wise moment (right) versus time (sec).
Declaration of competing interest
this work has been supported by WEAMEC program from Région Pays
de la Loire, France, thanks to O2GRACE grant.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Appendix
Acknowledgments
The controller gains of GSPI-CBP controller can be found in
(Jonkman et al., 2009; Jonkman, 2008) whereas the gains of the ASTWCBP controller can be found in authors’ previous work (Zhang et al.,
2019b) (see Table 2).
This work is a part of the Ph.D. thesis of Cheng Zhang who has
been supported by Chinese Scholarship Council (CSC). Furthermore,
8
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
Fig. 9. Case 1. Power spectral density of blade#1 root flap-wise moment.
Fig. 10. Case 1. Blade pitch angles versus time (sec).
Fig. 11. Case 2. Normalized RMS and VAR values obtained with the 3 controllers. The reference (red horizontal line) is the result obtained by GSPI-CBP.
Fig. 12. Case 2. System variables of versus time (sec).
9
Ocean Engineering 228 (2021) 108897
C. Zhang and F. Plestan
Fig. 13. Case 2. Normalized tower base and blade root DEL obtained with the 3 controllers. The reference (red horizontal line) is the result obtained by GSPI-CBP.
Fig. 14. Case 2. PSD of blade#1 root flap-wise moment.
Fig. 15. Case 2. Gains of ASTW-CIBP algorithm (28) versus time (sec).
Fig. 16. Case 2. Blade pitch angle of ASTW-CBP and ASTW-CIBP versus time (sec).
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