Analysis and synthesis of sliding mode control for large scale variable speed wind turbine
Analysis and synthesis of sliding mode control for large scale variable speed wind turbine for power optimization
Jov a n M e rida a,*,Luis T.Aguilar a,Jorge D a vila b
a Instituto Polit e cnico Nacional e CITEDI,Av.del Parque1310,Mesa de Otay,Tijuana,BC22510,Mexico
b Instituto Polit e cnico Nacional,ESIME e Ticom a n,Av.Ticom a n600,Col.San Jos e Ticom a n,Delegaci o n Gustavo A.Madero,M e xico,DF07340,Mexico
a r t i c l e i n f o
Article history:
Received2August2013 Accepted18June2014 Available online15July2014
Keywords:
Sliding mode control
Wind turbines
Maximum power point tracking Renewable energy
Nonlinear control a b s t r a c t
The problem of designing a nonlinear feedback control scheme for variable speed wind turbines,without wind speed measurements,in below rated wind conditions was addressed.The objective is to operate the wind turbines in order to have maximum wind power extraction while also the mechanical loads are reduced.Two control strategies were proposed seeking a better performance.The?rst strategy uses a tracking controller that ensures the optimal angular velocity for the rotor.The second strategy uses a Maximum Power Point Tracking(MPPT)algorithm while a non-homogeneous quasi-continuous high-order sliding mode controller is applied to ensure the power tracking.Two algorithms were devel-oped to solve the tracking control problem for the?rst strategy.The?rst one is a sliding mode output feedback torque controller combined with a wind speed estimator.The second algorithm is a quasi-continuous high-order sliding mode controller to ensure the speed tracking.The proposed controllers are compared with existing control strategies and their performance is validated using a FAST model based on the Controls Advanced Research Turbine(CART).The controllers show a good performance in terms of energy extraction and load reduction.
?2014Elsevier Ltd.All rights reserved.
1.Introduction
1.1.Overview
As a result of population expansion and increased global inte-gration,there has been a high growth in energy consumption.The high rates of electricity consumption supposes a risk for the depletion of natural resources,therefore the demand of renewable energy generation systems has increased[1].Such demand is supported by social and environmental reasons:the debate on climate change,depletion of fossil resources and nuclear damage caused by the use of non-fossil fuels.All these factors have led to the global community,and national governments to set new pol-icies in favor of renewable energy and drive future improvements in related technologies.Wind energy has been proved to be an important source of clean and renewable energy in order to produce electrical energy.Wind energy is currently one of the fastest growing renewable energy technologies in the world[2].
On the other hand,wind turbines present great challenges because they are complex nonlinear systems containing uncertain parameters,unmodeled dynamics,and unknown disturbances. Ongoing research is focused on increasing energy ef?ciency and reducing mechanical stress.One solution is to use advanced control strategies that enhance the performance of the turbine,which allows a better use of resources of the turbine,augmenting the lifetime of mechanical and electrical components,earning higher returns.
There are two primary types of horizontal-axis wind turbines:?xed speed and variable speed[3].In this work,we choose the variable speed because although the?xed speed system is easy to build and operate,it does not have the ability that the variable speed system has in energy extraction,up to a20e30%increase over?xed speed[3].Wind turbine controller objectives depend on the operation area[4e7].Variable speed wind turbine operation can be divided into three operating regions(Fig.1):
Region I:below cut-in wind speed.
Region II:between cut-in wind speed and rated wind speed. Region III:between rated wind speed and cut out wind speed.
*Corresponding author.
E-mail addresses:merida@citedi.mx,jovan21@https://www.sodocs.net/doc/083778658.html,(J.M e rida),laguilarb@ ipn.mx(L.T.Aguilar),jadavila@ipn.mx(J.D a
vila).Contents lists available at ScienceDirect Renewable Energy
journal h omepage:w
https://www.sodocs.net/doc/083778658.html,/locate/renene
https://www.sodocs.net/doc/083778658.html,/10.1016/j.renene.2014.06.030
0960-1481/?2014Elsevier Ltd.All rights reserved.
Renewable Energy71(2014)715e728
In Region I,wind turbines do not move because power available
in the wind is low compared to losses in the turbine system.Region II is an operational mode where it is desirable that the turbine captures as much power as possible from the wind,because wind energy extraction rates are low,and the structural loads are rela-tively small.Generator torque provides the control input to vary the rotor speed while the blade pitch is held constant.Region III occurs when the wind speed are high enough for the turbine,such that it must be limited the fraction of the wind power captured in order to guarantee that the safe electrical and mechanical loads limits are not exceeded.
The problem in Region II is considered in the present work.Classical controllers have been extensively used because linear control theory is a well-developed topic while nonlinear control theory is less developed and dif ?cult to analyze and implement.PI and PID controllers are extensively used [8e 10].PID,gain sched-uling,and LQR controllers are designed in Ref.[11].LQR controllers reduce the pitch activity and power excursions compared to the PID controllers.In Ref.[12],a PI controller is used in conjunction with a gain-scheduled control to accommodate variations in the wind,while the gain-scheduled control allows better power regulation and load reduction.Although some of those classical methods have been successfully applied,they do not consider the nonlinearities on their controllers and do not take into account the dynamical aspect of the wind and the turbine [4,7,13,14].To get a mathematical model that re ?ects accurately the wind dynamics and the me-chanical behavior of the turbine,is a high dif ?cult task because its complexity.That is the main reason to use a simpli ?ed model,mainly if it is pretended to perform a real implementation.One way of circumvent the modeling problem is through a control strategy that compensated the discrepancies between the real plant and the mathematical model.1.2.Literature review
Recently,nonlinear controllers for wind turbines have been of interest to the scienti ?c community;such as variable structure controllers.This approach is robust against parametric un-certainties,external disturbances,and unmodeled dynamics and presents the characteristic of ?nite-time reachability.A ?rst-order sliding mode controller for power regulation is developed in Ref.[15],demonstrating the viability and effectiveness of the con-trol strategy.Beltran et al.[16]extended the control to region II in conjunction with a Maximum Power Point Tracking algorithm,showing that the proposed control strategy is more ef ?cient in
terms of reduction of the drive-train mechanical stresses and output power ?uctuations with respect to standard control.The main disadvantage of the proposed strategy by Beltran et al.[16]is that it uses a monotonic approximation of the signum function in order to avoid the chattering phenomena,losing robustness.An additional drawback of this strategy is that the rotor speed is not limited around its nominal value,allowing high mechanical loads over the drive-train.Four second-order sliding mode controllers are compared in Ref.[17]working in Region II concluding that the super-twisting algorithm is the best option for the studied case.Evangelista et al.[18]synthesized a super-twisting sliding mode control with variable gain,which is compared with a super-twisting algorithm with ?xed gain showing a better performance in terms of chattering,mechanical loads,and power tracking.In Refs.[19],we developed a ?rst-order sliding mode controller to solve the problem of power optimization assuming that all states can be measured.The strategy has a good capture of power,but presents chattering in generator torque.The chattering causes high torque variations increasing mechanical stress.The wind speed measurement is not an easy task,for that reason in Ref.[20]a wind speed estimator,using the wind turbine itself as a measuring de-vice,was designed,the estimator is based on super-twisting observer and Newton e Raphson algorithm.A quasi-continuous second-order sliding mode controller was proposed in Ref.[21]to reduce the effects of chattering in the generated torque.The strategy provides a suitable compromise between conversion ef ?-ciency and mechanical loads,and there is no need for wind speed measured or estimated.The quasi-continuous second-order sliding mode controller was also used to solve the problem of power regulation and load reduction in Region III [22].1.3.Contribution
The objective of this paper is to design controllers that in spite of nonlinear behavior,unmodeled dynamics,and unknown distur-bances,maximize the capture of aerodynamic power and reduce the mechanics loads over the wind turbine.The contribution of this work consists in two proposing strategies using quasi-continuous sliding mode control [23,24]:
The ?rst approach uses a tracking algorithm with a wind speed estimator that ensures the optimal angular velocity for the rotor.The proposed controllers consist of two algorithms.The ?rst one is a sliding mode state output feedback torque controller.The second proposed algorithm uses a quasi-continuous high-order sliding mode controller to ensure the speed tracking.
The second strategy presents a non-homogeneous quasi-continuous high-order sliding mode controller [25]with MPPT algorithm to ensure the power tracking.This strategy only needs to measure the rotor speed and electric power.
Effective improvements are brought regarding a previously proposed control strategies,such a better energy extraction and load alleviation without wind speed measurement.
The proposed controllers are validated using the high-order nonlinear aeroelastic model FAST (fatigue,aerodynamics,struc-tures,and turbulence)of the CART [26e 28]https://www.sodocs.net/doc/083778658.html,anization of the paper
The paper is organized as follows.In Section 2,the wind turbine model and problem formulation are presented.The design of the ?rst strategy of control is covered in Section 3.In Section 4,the second strategy of control is given.Simulation results are provided and discussed in Section 5.Section 6presents some
conclusions.
Fig.1.Power curve for the CART.
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2.Wind turbine model and problem statement
This section presents the dynamic model of a variable speed
wind turbine and the problem statement.The wind turbine model incorporates aerodynamic characteristics,turbine mechanics,and generator dynamics.Detailed descriptions of the different dynamic sub-models are given in the following subsections.2.1.Aerodynamics
The aerodynamic power captured by the rotor is given by the nonlinear expression [29]
P a ?
1
2
r p R 2C p el ;b Tv et T3(1)
where v (t )is the wind speed,r is the air density,and R is the rotor
radius.The ef ?ciency of the rotor blades is denoted as C p ,which depends on the blade pitch angle b ,or the angle of attack of the rotor blades,and the tip speed ratio l ,the ratio of the blade tip linear speed to the wind speed.The parameters b and l affect the ef ?ciency of the system.The coef ?cient C p is speci ?c for each wind turbine.The relationship of tip speed ratio is given by
l ?R
u r
v et T
(2)
where u r is the rotor speed.The turbine estimated C p surface as a function of tip speed ratio and blade pitch based on the CART is illustrated in Fig.2.This surface was created using modeling soft-ware WTPerf which uses blade-element-momentum theory to predict the performance of wind turbines [29,30].
Fig.2indicates that there is a unique l at which the turbine is most ef ?cient.From (1)and (2),notice that the rotor ef ?ciency is highly nonlinear and makes the entire system a nonlinear system.The ef ?ciency of power capture is a function of the tip speed ratio and the blade pitch.The power captured from the wind follows the relationship
P a ?T a u r
(3)
where
T a ?
1
r p R 3C p el ;b Tv et T2(4)
is the aerodynamic torque which depends nonlinearly upon the tip speed ratio.The main components of a variable speed wind turbine are:an aeroturbine,a gearbox,and a generator.The energy in the wind turns two or three propellers around a rotor.Because the rotor is connected to the main shaft which in turn is connected to the high speed shaft through a gearbox,the aerodynamic power is transferred to generator.Finally,the electrical power generated is transferred to the grid.2.2.Turbine mechanics
The mechanical model of the two-mass wind turbine (Fig.3)can be described as follows [7,31]:
_u r ?T a eu r ;b ;v TàK ls eq r àq ls TàD ls eu r àu ls TàD r u r
J r
_u
g ?àT e n g tK ls eq r àq ls TtD ls eu r àu ls TàD g n g u g
g g
_d
?u r àu g g
(5)
where T e is the generator (electromagnetic)torque,u ls is the low shaft speed,q r is the rotor side angular deviation,q ls is the gearbox side angular deviation,J r >0is the rotor inertia,J g >0is the generator inertia,u g is the generator speed,D r is the rotor external damping,D g is the generator external damping,D ls is the low speed shaft damping,K ls is the low speed shaft stiffness,and d described the de ?ection of the drive-shaft.Assuming an ideal gearbox with transmission n g
n g ?
u g u ls ?T
ls T hs
(6)
where T ls ?K ls eq r àq ls TtD ls eu r àu ls Tis the low speed shaft tor-que.A simple rigid body model (single-mass)of a wind turbine can be considered if u r ?u ls .Therefore,upon using (6)and (5),one gets [16,32]:
J t _u
r ?T a eu r ;b ;v TàD t u r àT g (7)
where J t ?J r tn 2g J g ,D t ?D r tn 2g D g ,and T g ?n g T e are the turbine
total inertia,turbine total external damping,and generator torque in the rotor side,respectively.The controllers are synthesized using a single-mass model of a variable speed wind turbine (7).
30
β
C p (λ,β)
Fig.2.Power coef ?cient curve.Fig.3.Two-mass model.
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2.3.Generator dynamics
The generator model is described by a?rst-order linear model[7]:
_T
e?à1
t T T et
1
t T
T e;r(8)
where T e;r is the requested generator torque and T e is the actual generator torque with the time constant t T?0:01s.
2.4.Problem statement
The main objective in Region II is to maximize the power extracted from the wind.While energy is captured from the wind, the aerodynamic power should be maximized below rated wind speed.This operating strategy uses system speed control to track the maximum ef?ciency curve developed in the torque speed plane in order to maximize the ef?ciency of the extraction of energy[33]. This makes the ef?ciency crucial.In(2),the tip speed ratio can be altered to include the optimized points
l opt?R u ropt
v(9)
leading to a unique maximum point given by
C p à
l opt;b opt
á
?C p
max
(10)
that corresponds to the maximum power production.
To maximize the extracted energy,the maximum rotor ef?-ciency must be maintained during operation.For this purpose,b is ?xed to b opt and u r
opt
should change depending on the wind speed variations
u r
opt ?l opt
v
R
(11)
Then,the control objective is to?nd a control law T e to maximize power extraction by adjusting the rotational speed of the wind changes,such that u r follows u r
opt
and the aerodynamic power remains at its maximum value,while also reducing transient loads presented in the turbine components.
In order to make a measurement of the proposed controller performance,a comparison will be done with some existing control laws.A brief description of the controllers to be compared is given below.
The following control law tries to keep the turbine operating at the peak of its curve C p
max
T e?K opt u2ràD t u r
n g
;with K opt?1
2
p r R5C p max
l3opt
(12)
The optimal tip speed ratio l opt of the CART is equal to8.5and C p
max
?0:4291.This method is known as Indirect Speed Control (ISC)[32].Boukhezzar and Siguerdidjane[32]proposed two state feedback controllers.The controllers show a good performance in terms of ef?ciency with acceptable efforts on the rotor speed and control torque.The nonlinear static state feedback control (NSSFC)is:
T e?T aàD t u ràJ t_u r optàJ t a0e u
n g
(13)
where e u?u r optàu r is the rotor speed error and a0>0.The nonlinear dynamic state feedback control(NDSFC)is:
_T
e?
_T
aàD t_u ràJ t€u r optàJ t b1_e uàJ t b0e u
n g
(14)
The coef?cients b0and b1are found by using the pole-placement criteria,that is
s2tb1stb0?s2t2x u0stu20(15) leading to b0?u20and b1?2x u0.Further,the gains a0,b0,and b1 are chosen such the control action keeps a good tradeoff between power generation and load reduction.These existing control techniques are not robust enough against unmodeled dynamics, parametric uncertainties,and external disturbances.The proposed control strategies,therefore,shall overcome this problem in order to improve the performance.
3.Sliding mode control design
In this section,a combination of?rst and second-order sliding mode controllers with a wind estimator is presented(Fig.4).
3.1.Wind estimator
For a wind turbine,it is usual to measure the wind speed by an anemometer installed on the top of the nacelle.The measured wind speed is called point wind speed.The point wind speed does not represent the rotor effective wind speed since it is impossible to represent the wind speed v by a unique measure.The wind speed varies spatially on the swept rotor area,consequently it is dif?cult to obtain an accurate value of the rotor effective wind speed.From Eq.(9),the estimation of the wind speed b v is related to the T a by the following equation:
T aà
1
r p R3C q
b l
b v2?0(16)
where
b l?b u r R
b v
and C qeb lT?C qeb l;b optTis a tabulated function of b l.b v is calculated using the Newton e Raphson algorithm[6].The obtained value of b v allows to deduce the optimal rotor speed b u r
opt
?l opt b v=R.The closed-loop system,under the proposed controllers,will track the wind speed in order to achieve b u r
opt
.
3.2.Quasi-continuous sliding mode control
The quasi-continuous of an arbitrary order sliding mode controller was suggested in Refs.[24]and[34].For i?1;…;nà1, let us
denote
Fig.4.First proposed control scheme.
J.M e rida et al./Renewable Energy71(2014)715e728 718
40;n?s;N0;n?j s j;J0;n?signesT(17)
4i;n?seiTtb i Nenà1T=erànt1T
ià1;n
J ià1;n(18)
N i;n?
seiT
tb i Nenà1T=erànt1T
ià1;n
(19)
J i;n?4i;n
N i;n(20)
where b i;…;b nà1are positives numbers.The controller takes the form
u?àaFexTJ nà1;n
s;_s;…;senà1T
(21)
FexT?k1k x ktk2(22) where a,k1,and k2are constants suf?ciently large,and FexTis the so-called gain function[25].It was shown in Refs.[24,25,34]that, provided the suf?ciently large tuning parameters b i;…;b nà1and a and considering(22),then the control law de?ned by(17)e(22) ensures the convergence of s?_s?/?senT?0in?nite-time. Moreover,the control is globally boundedeu aFexTTand continuous everywhere except in the origin of the n-dimensional error space.Remark1.If FexTis equal to1,then we have the case of homo-geneous control with constant gain and other case we have the non-homogeneous control.
It has been proved that the quasi-continuous high-order sliding mode controllers present robustness to parametric uncertainties, robustness with respect to external disturbances,robustness to unmodeled dynamics,exactness,?nite-time convergence,and asymptotic accuracies.On the other hand,quasi-continuous high-order sliding mode controllers provide better transient features than other high-order sliding mode controllers.The non-homogeneous high-order sliding mode controller preserves the characteristic of homogeneous sliding mode controller,but allows to adjust the amount of energy supplied into the system in order to maintain the desired performance reducing at the same time the effects of chattering.
3.2.1.First-order sliding mode controller
Here,a?rst-order sliding mode controller(FOSMC)is synthe-sized.For this purpose,let us consider the next sliding variable
s1?b u r optàu r(23) where s1is the rotor speed https://www.sodocs.net/doc/083778658.html,puting the time derivative of (23)and using Eq.(7),we get
_s1?J t_b u r
opt
tD t u rtn g T eàT a(24) The following controller
T e?
T aàD t u ràJ t_b u r optàa1s1àa1signes1T
g
;with a1;a1>0
(25) drives s1to the origin in?nite-time.
3.2.2.Stability analysis
Consider the Lyapunov function candidate
V?
1
2
s21(26) The time derivative of(26)
is:
Fig.5.Second proposed control
scheme.
Fig.6.(a)Wind speed and(b)wind speed estimation.
J.M e rida et al./Renewable Energy71(2014)715e728719
_V ?s 1_s 1?s 1?J t _b u r opt tD t u r tn g T e àT a ?
?às 1?a 1s 1ta 1sign es 1T ?àa 1s 21àa 1k s 1k
(27)
Notice that _V is negative de ?nite if and only if a 1and a 1are positives.
3.2.3.Second-order sliding mode controller
Now,a second-order sliding mode controller (SOSMC)is syn-thesized to maximize the ef ?ciency of aerodynamic power extraction.We impose a second-order dynamics to (23)
€s 1td 1_s 1td 0s 1?0(28)
where d 0>0,d 1>0.Substituting the second time derivative of s 1
into (28),we obtain
J t €b u r opt tJ t d 1_s
1tJ t d 0s 1tD t _u r tn g _T e à_T a ?0(29)
The following controller is thus derived for (29)based on [24]
_T e ?
_T a àJ t b u €r opt àD t _u r àJ t d 1_s 1àJ t d 0s 1g
àa 2_s 1t s 1 1=2sign s 1eT n g _s
1 t s 1 1=2
;with d 0;d 1;a 2>0(30)
Note that controller (30)requires the time derivative of s 1,there-fore,we use the sliding mode differentiator from Levant [23,35]to
estimate _s
1which consist of a ?rst-order real-time differentiator of the
form
Fig.7.Rotor
speed.
Fig.8.Generator speed.
J.M e rida et al./Renewable Energy 71(2014)715e 728
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_z0?z1àl2j L j1=2j z0às1j1=2signez0às1T
_z1?àl1L signez1à_z0T(31)
where z0and z1are the real-time estimations of s1and_s1,respec-tively.The time derivatives that appear in the generator torque expression(13),(14),(25),and(30)are obtained by a?ltered de-rivative s=eqt1T.
4.Sliding mode control design with MPPT algorithm
In this section,a second-order quasi-continuous control in combination with a maximum power point tracking(MPPT)algo-rithm is presented.In most of the wind energy conversion systems, MPPT algorithm is implemented using wind speed data obtained from wind speed sensors.However,accurate measurement of wind speed is not an easy task mainly in large size wind turbines. Therefore,a lot of researches are underway in order to develop controllers which do not require measurements of wind speed.In Refs.[36,37],a reviewing state of the art of MPPT algorithms for wind energy systems is done,making a comparison between the different strategies.We choose the power output as the controlled variable,such that,a simple structure controller can be utilized.We are using power signal feedback(PSF)control to generate the po-wer reference,no wind velocity measurement is required in this method.Taking the data supplied by WTPerf[30]for the CART and based on tests and design calculations we obtain the electrical power output P e
opt
against shaft speed u r characteristic of the system.
4.1.Non-homogeneous quasi-continuous control design
Here,the non-homogeneous quasi-continuous sliding mode controller is developed to achieve robust power tracking.The following sliding variable is
proposed
Fig.9.Generator
torque.
Fig.10.Low speed shaft torque.
J.M e rida et al./Renewable Energy71(2014)715e728721
s 2?P e opt àP e
(32)
where s 2is the power tracking error and P e ?n g u r T e is the elec-trical power.Then,we obtain the ?rst time derivative of s 2
_s 2?_P e opt àn g _u
r T e àn g u r _T e tx
(33)
where x et ;u r T2?represents the uncertainties and disturbances.It is assumed that x et ;u r Tis Lebesgue measurable and it is matched and uniformly bounded,that is,j x et ;u r Tj
_T
e ?a 3F es 2T
_s
2tb 1j s 2j 1=2sign es 2T .
j _s 2j tb 1j s 2j
1=2 n g r
;
with a 3>0
(34)
F es 2T?k 1jj s 2jj tk 2;
k 1>0;
with k 2> _P e opt t n g _u
r T e t n g u r x t
(35)
The time derivative _s
2is computed using (31).Remark 2.If F es 2Tis equal to 1,then we have the case of homo-geneous control with constant
gain.
Fig.11.Electric
power.
Fig.12.(a)Rotor speed;(b)generator speed.
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The proposed generator power control strategy is as shown in
Fig.5.The non-homogeneous controller with variable gain (NHVG)(34)and (35)will be compared with the homogeneous controller with constant gain (HCG).We assume for these controllers that only the rotor speed and electric power are available from mea-surements on the wind turbine.
5.Simulation results under turbulent wind
The dynamic model of a horizontal-axis turbine CART is simu-lated in Matlab-Simulink and validated with FAST simulator [26]which is interfaced with Simulink.Based on [6],the simulations are carried out under the following operating conditions:
Choosing dynamics that track the mean tendency wind speed,along a short-time interval,while avoiding to track the wind speed local high-turbulence ?uctuations.
Filtering the generator torque T e using a low-pass ?lter in order to smooth the control action.
Filtering the reference speed u r opt and its time derivatives to obtain a less turbulent signal.
Filtering the rotor speed signal by a low-pass ?lter for the con-trol laws NHVG and HCG to reduce the effects of drive-train oscillations and provide a smoother reference signal.
In presence of an additive measurement noise on u r with an SNR around 7dB.
In presence of a constant additive control input disturbance 115.8346Nm in the generator torque T e .
The wind speed is described as a slowly varying average wind speed superimposed by a rapidly varying turbulent wind speed.The model of the wind speed v at the measured point is
v ?v m tv t
(36)
where v m is the mean value and v t is the turbulent component.The hub-referenced wind ?eld was generated using Turbsim [38].The wind data consist of 600s [29]in v m ?7.5m/s with 18%turbulence intensity,via Kaimal turbulence model.
In Fig.6a,we show the wind speed v and Fig.6b is the corre-sponding wind speed estimation b v .Fig.6b shows that the wind speed estimation is very closely tracked.The discrepancies are caused by the effects of unmodeled turbine dynamics,errors in the C p curve,changes in the power function due to dynamic effects,and speed of convergence of Newton e Raphson algorithm.The estimator provides a good estimated of wind speed,this allows to get a better rotor
speed
Fig.13.(a)Generator torque;(b)low speed shaft
torque.
Fig.14.Electric power.
Table 1
Comparison of control strategies using FAST simulator.
ISC
NSSFC FOSMC NDSFC SOSMC HCG NHVG h aero (%)80.6881.9782.0182.3382.3682.7682.83h elect (%)
81.2482.2782.3082.4282.4683.0883.14std(T ls )(kN m)
13.186
13.072
13.113
12.437
12.325
12.269
12.902
max(T ls )
(kN m)
77.95478.07578.04772.58772.38175.14076.714std(T e )
(kN m)
0.30140.29210.29380.29060.28790.28410.2905max(T e )
(kN m)
1.791 1.645 1.640 1.589 1.588 1.668 1.721J.M e rida et al./Renewable Energy 71(2014)715e 728
723
reference.We chose the following gains for the controllers (12)e (14),
(25),(30),(34)and (35):K opt ?5:3813?103,a 0?0:115,b 0?0:0005,b 1?0:0402,a 1?4:746?104,b 1?1,a 1?389:0947,d 0?0:0005,d 1?0:0402,a 2?19:4547,a 3?50?103,k 1?0:00006,and k 2?0:00001,under the following initial conditions:u r e0T?33:7042rpm,u g e0T?1454:8417rpm,
T e e0T?1408:5919Nm,and b e0T?à1 .We used q ?100for the
?ltered derivative.
https://www.sodocs.net/doc/083778658.html,ing the mathematical model
First,we describe the results obtained using the mathematic model.The controllers are applied to optimize the extracted aerodynamic power and reduce strong torque variations in the generator that could lead to increase mechanical stresses.Follow the optimal rotor speed (see Fig.7)means following the short-time changes fast in wind speed.Due to the dynamics of the rotor is impossible to achieve this optimal value;thus,an inter-mediate tracking dynamics should be chosen to establish a compromise between energy capture improvement and dynamic loads reduction.Looking at Figs.7e 14we see the commitment made between power extraction and loads reduction of each controller.In general,ISC shows the lowest performance.Rotor and generator speed are shown in Figs.7,8and 12.SOSMC and NDSFC follow more closely the optimal value of the average trend of the optimal rotor speed u r opt avoiding following the short-time turbulent component,but HCG and NHVG have a smoother tracking.Because ISC and NSSFC are unable to reject the additive disturbance input,tracking the optimal value is slower than that achieved by the other controllers,which results in a loss of power extraction.The power obtained by FOSMC is a little better than NSSFC,but smaller than NDSFC (see Fig.11).Examining Fig.9,
we
Fig.15.Rotor
speed.
Fig.16.Generator speed.
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see that the torque with FOSMC is longer,causing strong efforts on the drive-train(Fig.10),even so,the generator torque remains below the limit value of3.753kN m.SOSMC has better perfor-mance than NDSFC.NHVG increases the electric power and me-chanical loads compared to HCG(see Figs.13and14).SOSMC produces the lower mechanical stresses while NHVG has the best capture and generation power avoiding long?uctuations(Figs.9 and14).
5.2.Validation using FAST simulator
The performance of the proposed control strategies have been veri?ed using a FAST model of the CART interfaced with Simulink. The FAST model has three DOFs active including the generator speed,drive-train?exibility,and blade teeter DOFs.The criteria for assessing the performance of the controllers are[6]:the aerodynamic h aero and electrical h elec ef?ciency,the standard deviation and maximum value of the generator and low speed shaft torque.The aerodynamic and electrical ef?ciency are de?ned as follows:
h aeroe%T?
Z t
fin
t ini
P aetTd t
Z t
fin
t ini
P a
opt
etTd t
;h elece%T?
Z t
fin
t ini
P eetTd t
Z t
fin
t ini
P a
opt
etTd t
(37)
where P a
opt
?0:5r p R2C p
max
v3is the optimal aerodynamic power corresponding to the wind speed and P e?T e u g.The controllers performance are summarized in Table1.
The rotor and generator speed of all controllers is depicted in Figs.15,16and20.Based on the results shown in Table1,ISC
has Fig.17.Generator
torque.
Fig.18.Low speed shaft torque.
J.M e rida et al./Renewable Energy71(2014)715e728725
the lowest performance as established in the previous analysis of
the controllers.Only ISC and NSSFC are unable to reject the input disturbance,in consequence these controllers do not follow the wind speed adequately.SOSMC and NDSFC track more closely the optimal rotor speed u r opt ,but NHVG and HCG track u r opt without tracking the large variations of wind speed allowing to have a smooth power reference.Observe that SOSMC has better power generation (see Fig.19)and load reduction than NDSFC.In Table 1,we can see that SOSMC has the best dynamic characteristics (has the lowest maximum and minimum standard deviation in T e value)with slower mechanical stresses on low speed shaft (the standard deviation of T ls is minimal,as well as,its maximum value is the lowest)than the other controllers as it is corroborated in Figs.17,18and 21.HCG increases power capture with a very close performance in loads reduction to SOSMC.The NHVG has the best generation power (see Fig.22),but the dynamic characteristics,in comparison with the homogeneous controller,are slightly lower generating high mechanical stresses as illustrated in Fig.21.Table 1shows that NHVG and HCG have a better power generation avoiding long electrical power ?uctuations (see Fig.22).NHVG and HCG only need measurements of the rotor speed and electrical power,while SOSM,FOSM,NDSFC,and NSSF need to measure the rotor speed and aerodynamic torque,estimate the wind speed,
estimate the ?rst derivative of aerodynamic torque _T
a ,estimate the ?rst derivative of rotor speed error and the second derivative
of optimal rotor speed €u
r opt .Another advantages of NHVG and HCG are the simplicity in terms of adjustment of their coef ?cients and the design procedure is more https://www.sodocs.net/doc/083778658.html,pared to SOSM and NDSFC,NHVG and HCG require less number of real-time
differentiators.
Fig.19.Electric
power.
Fig.20.(a)Rotor speed;(b)generator speed.
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726
6.Conclusions
This paper addresses the problem of power generation control in variable speed wind turbines without wind speed measurement.The objective was to synthesize robust controllers to maximize the energy extracted from the wind while ensuring reduction of me-chanical loads.Two strategies of sliding mode control were pro-posed.The ?rst strategy uses a wind speed estimator.The second strategy uses an MPPT algorithm that does not need wind speed measurements.The resulting controllers provide better power extraction and dynamic characteristics with respect to the compared control strategies.The proposed strategies com-plemented with high-order sliding modes controllers ensure better performance thanks to their features:robustness against unmod-eled dynamics,parametric uncertainties,external disturbances,and chattering attenuation;providing a suitable compromise among conversion ef ?ciency,mechanical stresses,and perturbation rejection.The validation results,developed with FAST model,have shown the feasibility of the proposed strategies.
Acknowledgments
L.Aguilar and J.D a
vila gratefully acknowledge the ?nancial support from CONACYT (Consejo Nacional de Ciencia y Tecnología)under Grants 127575and 151855.
Appendix A
The CART is variable speed one,in which the rotor speed in-creases and decreases with changing wind speed producing electricity with a variable frequency.The parameters of the model are given in Table A.1.Those parameters are based on the CART which is a two-bladed,teetered,active-yaw,upwind,variable pitch,and horizontal-axis wind turbine which is located at the National Wind Technology Center in Colorado [27,28].The nom-inal power is 600kW;the startup wind speed is 5m/s,the rated wind speed of 12m/s,and a cut out wind speed of 26m/s [39].The rated rotor speed is 41.7rpm.The pitch system can pitch the blades up to 18 /s with pitch accelerations up to 150 /s 2[11].The required constraints for torque and rotor speed are 162kN m and 58rpm,respectively [28].The gearbox is connected to an induc-tion generator via the high speed shaft,and the generator is connected to the grid via power electronics.In this work,we ignore the power electronics control and an ideal performance will be assumed [7,31]
.
Fig.21.(a)Generator torque;(b)low speed shaft
torque.
Fig.22.Electric power.
Table A.1
Two-mass model parameters.Notation Numerical value Units R
21.65m r
1.308
kg/m 3J r 3.25?105kg m 2J g 34.4kg m 2
D r 27.36N m/rad/s D g 0.2N m/rad/s K ls 9500
N m/rad D ls 2.691?105N m/rad/s P enom 600?103W T emax 3.753?105N m
n g
43.165
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727
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