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Dynamic Modelling of the Rotating Subsystem of a Wind Turbine for Control Design Purposes
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th World The International Federation of Congress Automatic Control Proceedings of the the 20th9-14, World Congress Available online at www.sciencedirect.com Toulouse, France, July 2017 The International Federation of Toulouse, France,Federation July 9-14, 2017 The International of Automatic Automatic Control Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 9896–9901
Dynamic Modelling of the Rotating Subsystem of a Wind Turbine Dynamic Modelling of the Rotating Subsystem of a Wind Turbine Dynamic Modellingfor ofControl the Rotating Subsystem Design Purposesof a Wind Turbine for Control Design Purposes for Control Design Purposes Adrian Gambier*
Adrian Gambier* Adrian Gambier* *Fraunhofer Institute for Wind Energy and EnergyAdrian System Gambier* Technology Northwest IWES, 27572 Bremerhaven, Germany *Fraunhofer Institute for(Tel: Wind Energy and Energy System [email protected]) Technology Northwest IWES, 27572 Bremerhaven, Germany +49 471 14290-375; e-mail: *Fraunhofer Institute Wind Energy and System Technology *Fraunhofer Institute for for(Tel: Wind Energy and Energy Energye-mail: System [email protected]) Technology Northwest Northwest IWES, IWES, 27572 27572 Bremerhaven, Bremerhaven, Germany Germany +49 471 14290-375; (Tel: (Tel: +49 +49 471 471 14290-375; 14290-375; e-mail: e-mail: [email protected]) [email protected]) Abstract: The present work reviews the dynamic models for the rotating subsystem of a wind turbine, which Abstract: present work reviews purposes the dynamic for the rotating subsystem of a wind turbine, which have beenThe developed for control in models the literature. Modelling approaches are systematically Abstract: The present work work reviews purposes the dynamic dynamic models for the the rotating rotating subsystem of aa wind wind turbine, which which Abstract: The present reviews the models for subsystem of turbine, have been developed for control in the literature. Modelling approaches are systematically organized and classified including advantage,indrawbacks and limitations. Based on theare 5-MW reference have been developed for control purposes the literature. Modelling approaches systematically have been developed for control purposes in the literature. Modelling approaches are systematically organized and classified including advantage, drawbacks andforlimitations. Based on themodels 5-MWare reference turbine developed by NREL, analysis and simulation studies some of the presented carried organized and including advantage, drawbacks and Based on 5-MW reference organized and classified classified including advantage, drawbacks andforlimitations. limitations. Based on the themodels 5-MWare reference turbine developed by NREL, analysis and simulation studies some of the presented carried out. Hence, it is possible to determine models’ properties in for time domain, sampling time, computational turbine developed by NREL, analysis and simulation studies some of the presented models are carried turbine developed by NREL, analysis and simulation studies for some of thesampling presentedtime, models are carried out. Hence, it isdiscrete possible to determine models’ properties infor time computational burden of time models. Finally, recommendations thedomain, selectionsampling of a model according to the out. it possible to models’ properties time domain, time, computational out. Hence, Hence, it is isdiscrete possiblemodels. to determine determine models’ properties in infor time domain, sampling time, computational burden of time Finally, recommendations the selection of a model according to the application are given. burden discrete burden of of time time discrete models. models. Finally, Finally, recommendations recommendations for for the the selection selection of of aa model model according according to to the the application are given. application are given. application are(International given. Keywords: Wind turbine, rotor dynamics, dynamic modelling © 2017, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Wind turbine, rotor dynamics, dynamic modelling Keywords: Keywords: Wind Wind turbine, turbine, rotor rotor dynamics, dynamics, dynamic dynamic modelling modelling
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION Modern multi-MW wind turbines, which are characterized by their Modern multi-MW wind turbines, which are characterized their large dimensions and flexible structures, require advanced by control Modern multi-MW wind turbines, which characterized by their Modern multi-MW wind turbines, which are are characterized by their large dimensions and flexible structures, require advanced control algorithms to reach a satisfactory performance. On the othercontrol hand, large dimensions and flexible structures, require advanced large dimensions and flexible structures, requireOn advanced control algorithms to reach a satisfactory performance. the other hand, advanced control requires suitable dynamic models for control algorithms to aa satisfactory performance. On hand, algorithms control to reach reachrequires satisfactory performance. On the the other other hand, advanced suitable dynamic models fordynamic control purposes, i.e. models that are able to capture the essential advanced control requires suitable dynamic models for control advanced control requires suitable dynamic models for control purposes, i.e. models that are able to capture the essential dynamic behaviour of the machine butable keeping their simplicity such that a purposes, i.e. models that to the dynamic purposes, i.e. models that are are to capture capture the essential essential dynamic behaviour of the machine butable keeping their simplicity such thatina control system design of acceptable complexity can be achieved, behaviour of machine but their such behaviour of the the machine but keeping keeping their simplicity simplicity such that thatinaa control system design of acceptable complexity can be achieved, particular if theydesign have to be implemented in real time. control of acceptable complexity can control system system ofbe acceptable complexity can be be achieved, achieved, in in particular if theydesign have to implemented in real time. particular if be in particular if they theyofhave have towind be implemented implemented in real real time. time. The dynamics largeto turbines generally require a represenThe dynamics large wind turbines generally requirei.e.a equations representation by using of nonlinear partial differential equations, The dynamics of large wind wind turbines generally requirei.e.aa equations represenThe dynamics of large turbines generally require representation by using nonlinear partial differential equations, that are not particularly convenient for control purposes. Mach tation by using nonlinear partial differential equations, i.e. equations tation by using nonlinear partial differential equations, i.e. equations that are effort not particularly convenient for control purposes. Mach research has been done in developing simple models of wind that not convenient for purposes. Mach that are are effort not particularly particularly convenient for control control purposes. Mach research has been done in developing simple models of wind turbines for control. In fact, most works reporting the use ofofmodel research effort has been done in developing simple models wind research for effort has been done in developing simplethe models ofmodel wind turbines control. In fact, most works reporting use of based control algorithms present some reporting kind of model developing turbines for control. In fact, most works the use of model turbines for control. In fact,present most works reporting the use of model based control algorithms some kind of model developing activity. Frequently, the used modelling approach includes a twobased algorithms present some of developing based control control algorithms present some kind kind of model model developing activity. Frequently, theaused modelling approach includes a twomass drivetrain and standard nonlinear expression for the activity. Frequently, the modelling approach includes aa twoactivity. Frequently, theaused used modelling approach includesfor twomass drivetrain and standard nonlinear expression the in the form aerodynamic torque that includes the power coefficient C p mass drivetrain and aaincludes standard nonlinear expression for the mass drivetrain and standard nonlinear expression forform the aerodynamic torque that the power coefficient C p in the of an empirical nonlinear function,the whose coefficients are adjusted by in the form aerodynamic torque that includes power coefficient C p in the form aerodynamic torque that includes the power coefficient C of an empirical nonlinear function, whose coefficients arepadjusted by curve fitting algorithms, e.g. (Boukhezzar & Siguerdidjane, 2011), of an nonlinear whose are by of an empirical empirical nonlinear function, function, whose coefficients coefficients are adjusted adjusted by curve fitting algorithms, e.g. (Boukhezzar & Siguerdidjane, 2011), (Dang,fitting et al., algorithms, 2009) and (Sami & Patton, 2012). Moreover, the2011), forecurve e.g. & curve fitting algorithms, e.g. (Boukhezzar (Boukhezzar & Siguerdidjane, Siguerdidjane, 2011), (Dang, et al., 2009) and (Sami & Patton, 2012). Moreover, the foreaft displacement of the tower &isPatton, added2012). in (Xiao, et al.,the 2014) (Dang, et and Moreover, (Dang, et al., al., 2009) 2009) the and (Sami (Sami 2012). Moreover, the foreforeaft displacement tower & isPatton, added in (Xiao, etofal., and (Schlipf, etof al., 2013). A more complex model the2014) tower aft displacement of the tower is added in (Xiao, et al., 2014) aft displacement ofal., the2013). tower A is more added in (Xiao, etofal., 2014) and (Schlipf, et complex model the tower is used in (Jelavic & Peric, 2009) and a dynamic model for a vertical and (Schlipf, et al., 2013). A more complex model of tower and (Schlipf, et & al., 2013). A and more complex model model for of the the tower is used inand (Jelavic Peric, 2009) a dynamic a vertical inverted non-rotating blade are considered in (Bianchi, et al., is used in (Jelavic & Peric, 2009) and a dynamic model for a vertical is used inand (Jelavic & Peric, blade 2009) are and considered a dynamic model for a vertical inverted non-rotating in (Bianchi, et al., 2007) and (Shan & Shan, 2012). Control applications by using more inverted and non-rotating blade are in et al., inverted and non-rotating blade Control are considered considered in (Bianchi, (Bianchi, et al., 2007) and (Shan & Shan, 2012). applications by using more complex models ofShan, wind2012). turbines can beapplications found, among others, in 2007) and (Shan & Control by using more 2007) and (Shan & Shan, 2012). Control applications by using more complex models of2008), wind (Muyeen, turbines can be found, among others, in (Pournaras, et al., et al., 2007) and (Friis, et al., complex models of wind turbines be among others, in complex models of2008), wind (Muyeen, turbines can can be found, found, among others, in (Pournaras, et al., et al., 2007) and (Friis, et al., 2011). (Pournaras, (Pournaras, et et al., al., 2008), 2008), (Muyeen, (Muyeen, et et al., al., 2007) 2007) and and (Friis, (Friis, et et al., al., 2011). 2011). 2011). It is also possible to find works, which are focused on modelling It is alsoofpossible to find works, which are focused on modelling efforts wind turbines for control design commitments (e.g. It is also alsoofpossible possible to find find works, works, which are focused focused on modelling modelling It is to which are on efforts wind turbines for control design commitments (e.g. (Slootweg, et al., 2003), (Salman & Teo, 2003) and (Jasniewicz & efforts of turbines for design commitments (e.g. efforts of wind wind turbines for control control design commitments (e.g. (Slootweg, et al., 2003), (Salman & Teo, 2003) and (Jasniewicz & Geyler, 2011)). An overview about the2003) modelling aspects of (Slootweg, et al., 2003), (Salman & Teo, and (Jasniewicz & (Slootweg, et al.,An 2003), (Salmanabout & Teo, 2003) and (Jasniewicz & Geyler, 2011)). overview the modelling aspects of wind turbines from the controlabout engineering point ofaspects view can Geyler, 2011)). An overview the modelling of Geyler,turbines 2011)). from An overview about the modelling aspects of wind the control engineering point of view can be found in (Moriarty & Butterfield, 2009). point of view can wind turbines from engineering wind turbines from the the&control control engineering be found in (Moriarty Butterfield, 2009). point of view can be be found found in in (Moriarty (Moriarty & & Butterfield, Butterfield, 2009). 2009).
Thus, much research has been done in order to propose simplified Thus, research has been done in the order to propose simplified modelsmuch for control purposes. However, proposed models are not Thus, research has done order to simplified Thus, much much research has been been done in in the order to propose propose simplified models for control purposes. However, proposed models not comparatively analysed in the same context. Therefore, theare main models for control purposes. However, the proposed models are not models for control purposes. However, the proposed models aremain not comparatively analysed in the same context. Therefore, the contribution of analysed the present work is to context. review and analyse the different comparatively in the same Therefore, main comparatively analysed in work the same context. Therefore, the main contribution of the present is to review and analyse different dynamic models forpresent the rotating subsystem of and the wind turbine, i.e. contribution of work is analyse different contribution of the the work subsystem is to to review review analyse different dynamic models forpresent the rotating of and the wind turbine, i.e. the rotor, drivetrain and generator. Simulation studies are done by dynamic models for the subsystem of wind turbine, i.e. dynamic models for and the rotating rotating subsystem of the the windare turbine, i.e. the rotor, drivetrain generator. Simulation studies done by using the 5-MW reference wind turbine from NREL (Jonkman, et al., the rotor, drivetrain and Simulation studies are by the rotor, drivetrain and generator. generator. Simulation studies are done done by using the 5-MW reference wind turbine from NREL (Jonkman, et al., 2009). The paper is organized as follows: in Section 2, general using the 5-MW reference wind turbine from NREL (Jonkman, et al., using the 5-MW reference wind turbine from NREL (Jonkman, et al., 2009). Theconsiderations paper is organized as follows: modelling are given, followedin bySection Sections2, 3general and 4, 2009). paper as in 2, 2009). The Theconsiderations paper is is organized organized as follows: follows: inbySection Section 2,3general general modelling are given, followed Sections and 4, where the different modelsare aregiven, described and formulated. Sections 5 modelling considerations followed by Sections 3 modelling considerations are given, followed by SectionsSections 3 and and 4, 4, where the different models are described and formulated. and 6 the are different devoted models to the description of and the formulated. reference turbine, the55 where are described Sections where models are described Sectionsthe5 and 6 the are different devoted to theand description of and theofformulated. reference turbine, simulation environment the analysis the results. Finally, and to description of turbine, the and 6 6 are are devoted devoted to the theand description of the theofreference reference turbine, the simulation environment the analysis the results. Finally, conclusions are drawn in Section 7. simulation environment and the analysis of the results. Finally, simulation environment and the7.analysis of the results. Finally, conclusions are drawn in Section conclusions conclusions are are drawn drawn in in Section Section 7. 7. 2. GENERAL MODELLING ASPECTS 2. GENERAL MODELLING ASPECTS 2. MODELLING 2. GENERAL GENERAL MODELLING ASPECTS Wind turbines can be modelled by usingASPECTS approaches of largeWind turbines since can bethey modelled by using approaches of largescale systems, are complex systems that brings into Wind turbines can modelled by approaches of Windsystems, turbines since can be bethey modelled by using using approaches of largelargescale are complex systems that brings into play many state variables. These approaches allow the application scale systems, since they are systems that brings into scalemany systems, since theyThese are complex complex systems that brings into play state variables. approaches allow the application of typical techniques for large-scale systems,allow such as decomposiplay many state variables. These approaches the application play many state variables. These approaches allow the application of typical techniques for large-scale systems, such as decomposition and coordination (Chen & Zhang, 2016),such (Mahmoud, et al., of typical techniques large-scale systems, as of typical techniques for for large-scale systems, such as decomposidecomposition and coordination (Chen & Zhang, 2016), (Mahmoud, et al., 1985) and (Leithead, et al., 1991). These techniques can also be tion and coordination (Chen & 2016), (Mahmoud, et tion and coordination (Chen & Zhang, Zhang, 2016), (Mahmoud, et al., al., 1985) and (Leithead, et al., 1991). These techniques can also be found in the modelling of wind turbines (e.g. (Mirzaei, et al., 1985) (Leithead, et 1991). These can be 1985) and and (Leithead, et al., al., 1991). These techniques techniques can also also be found in the modelling of wind turbines (e.g. (Mirzaei, et al., 2012), in (van der Tempel of & wind Molenaar, 2002)). found the modelling turbines (e.g. found in theder modelling turbines (e.g. (Mirzaei, (Mirzaei, et et al., al., 2012), (van Tempel of & wind Molenaar, 2002)). 2012), 2012), (van (van der der Tempel Tempel & & Molenaar, Molenaar, 2002)). 2002)). 2.1 Plant Decomposition 2.1 Plant Decomposition 2.1 Plant Decomposition 2.1the Plant Decomposition At present time, there is no suggestion in the literature on how to At the present time, there isHowever, no suggestion the literature on how to decompose wind turbines. there in is some tacit agreement to At the present time, there no in the on At the present time, there is isHowever, no suggestion suggestion in the literature literature on how how to to decompose wind turbines. there is some tacit agreement use a spatio-temporal decomposition. Thisis technique consists oftoa decompose wind turbines. However, there some tacit agreement decompose wind turbines. However, there is some tacit agreement use a spatio-temporal decomposition. This oftoa combination of physical decomposition, i.e.technique physical consists components, use decomposition. This technique consists of use aa spatio-temporal spatio-temporal decomposition. This technique consists of aa combination of physical decomposition, i.e. physical components, Thus, a physical decomposition is and functional decomposition. combination of physical decomposition, i.e. physical components, combination of decomposition. physical decomposition, i.e. physical components, Thus, a physical decomposition is and functional used for the main machine and functional decomposition for the obThus, decomposition is and decomposition. Thus, aa physical physical decomposition is and functional functional decomposition. used for the main machine and functional decomposition for the obsubsystems. Fig. 1and shows an example including the tained used for the main machine functional decomposition for the obused for the main machine and functional decomposition for the obsubsystems. Fig. 1 shows an example including the tained involved coordinating variables. This decomposition is inspired subsystems. Fig. 1 an example the tained subsystems. Fig.variables. 1 shows showsThis an decomposition example including including the tained involved coordinating is inspired by (Leithead, et al., 1991) and it will support the modelling involved coordinating variables. This decomposition is inspired involved coordinating variables. This decomposition is inspired by (Leithead, et al.,in1991) and itThe willcoordinating support thevariables modelling activities described this work. are by et and will support modelling by (Leithead, (Leithead, et al., al.,in1991) 1991) and it itThe willcoordinating support the thevariables modelling activities described this work. are described in the next section. activities described in this work. The coordinating variables activities in described in this work. The coordinating variables are are described the next section. described described in in the the next next section. section.
Copyright © 2017, 2017 IFAC 10310 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 responsibility IFAC 10310 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 10310 Copyright © 2017 IFAC 10310 10.1016/j.ifacol.2017.08.1621
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Adrian Gambier et al. / IFAC PapersOnLine 50-1 (2017) 9896–9901
Pitch Tpitch subsystem
vw
Tb r Ta
Aerodynamics
given by the generator. The three mass model and its utilization can be found e.g. in (Gonzalez-Longatt, et al., 2012), (Xu, et al., 2013) and (Morisse, et al., 2016). Fig. 1 shows this concept.
Rotating Subsystem
r Drive Rotor xl Train
g
P Tg Generator
Ft
Br Klss
Tower
t
Ta Jr
Fig. 1. System decomposition of a wind turbine.
r
2.2 The Rotating Subsystem The rotating subsystem consists of the wind turbine rotor, the drive train and the generator. In turn, the wind turbine rotor includes the three rotor blades and the hub. The drive train contains in this work the low-speed shaft, the gearbox and the high-speed shaft. In case of direct drive systems, only a low-speed shaft is present and no gearbox. A scheme for the rotating subsystem is presented in Fig. 2.
Tb1 Blade 1 1 r Tb2 Blade 2 2 Hub xl Tb3 Blade 2 3
g Drive Train
Tg
9897
P Generator
Rotor
Dlss
Bxl
nx Jxl Bxh Khss Bg
xl Jxh
Jg
Tg
xh Dhss g
Gearbox
Generator
Fig. 4. Three-mass rotating subsystem with one-mass rotor. The equations of motion for this approach are
Jr r (Br Dlss )r (Dlss / nx )xh (Klss / nx )(nxr xh ) Ta , (1) Jexh Bexh nxDlssr nx2Dhssg Klss (nxr xh ) nx2Khss (xh g ) 0 , (2) J g g ( Dhss Bg )g Dhssxh Khss ( xh g ) Tg ,
(3)
Je Jxl nx2 Jxh and Be Bxl Dlss nx2 (Bxh Dhss ) .
(4)
where
Fig. 2. Scheme for the rotating subsystem. The drive train and the generator are represented here by either a twomass system (one for the gearbox and one for the generator) or by a single mass system assuming a massless gearbox. The mass of the slow-speed shaft is considered to be incorporated to the hub and the mass of the high-speed shaft to the generator rotor. The different approaches that are analysed in this study are distinguished by the different representations of the rotor and are divided into two categories. The first one represents the whole rotor as a single mass. Hence, the rotating subsystem can be modelled as a three-mass system (rotor, gearbox, generator), a two-mass system (rotor, generator) or a single mass system (rotor + generator). In the second category, the rotor is represented as a multi-mass system, i.e. the rotor can have two, three or four masses, which leads to a rotating system either with four, five and six masses or with three, four and five masses, respectively depending if the mass of the gearbox is included or not. This description is illustrated in Fig. 2. Rotating Subsystem
Parameters J, B, D, K and n represent mass second moments of inertia, viscous friction coefficients, damping coefficients, stiffness coefficients and the gearbox ratio, respectively. Variables , and T correspond to rotation speed, rotation angle and torque, respectively. Subscripts r, g, a, e, x, xl, xh, lss and hss denote rotor, generator, aerodynamic, equivalent, gearbox, low-speed side of the gearbox, high-speed side of the gearbox, low-speed shaft and high-speed shaft, respectively. The aerodynamic torque is computed by (5) Ta R2CQ ( , )ver2 , 2 where , R, ver are the air density, the rotor ratio, and the effective and relative wind speed. is the tip-speed-ratio defined as Rr/vw and is the pitch angle. CQ is the torque coefficient defined as Cp(,)/ and Cp is the power coefficient, which is usually given as a two-dimensional table for and calculated by a precise aerodynamic simulation based on the blade element theory for a given blade design. Defining the state-space variables as x1 = r, x2 = xh, x3 = nxr – xh, x4=g and x5=xh –g and rearranging (1)-(3), a fifth order state-space representation is obtained
Rotor as single mass system
Rotor as multi mass system
One-mass Rotor
Two-mass Three-mass Four-mass Rotor Rotor Rotor
x1
One-mass Two-mass Three-mass System System System
Three/ Four/five- Five/sixfour-mass mass mass System System System
x 2
( Br Dlss ) D K 1 x1 lss x2 lss x3 Ta Jr nx J r nx J r Jr
n x Dlss B K n2 D n2 K x1 e x2 lss x3 x lss x4 x lss x5 Je Je Je Je Je
x3 n x x1 x2
Fig. 3. Classification of the rotating subsystem according to the number of masses used for the representation. 3. MODELLING THE ROTOR AS SINGLE-MASS SYSTEM The simplest and most frequently case of modelling the rotating subsystem is representing the rotor is by only a single mass. Thus, three cases can be found, three-mass, two-mass and one-mass rotating subsystems (see Fig. 3). They are formulated in the next subsections. 3.1 Three-mass rotating subsystem
. (6)
B g Dhss D K 1 x 4 hss x2 x4 hss x5 Tg Jg Jg Jg Jg x5 x 2 x4
3.2 Two-mass rotating subsystem A two-mass model is used for example in (Ashwini & Archana, 2016) and in (Boukhezzar & Siguerdidjane, 2011). The model is obtained by considering a massless gearbox. In this case, (2) becomes
The rotating subsystem is modelled by three masses: the rotor as a single mass, the gearbox as a second mass and the third one is 10311
Bexh nx Blssr nx2 Bhssg Klss (nxr xh ) nx2Khss (xh g ) 0 (7)
Proceedings of the 20th IFAC World Congress 9898 Adrian Gambier et al. / IFAC PapersOnLine 50-1 (2017) 9896–9901 Toulouse, France, July 9-14, 2017
and this leads to
xh nx1 r nx22g (Klss / Be )(nxr xh ) nx2 (Khss / Be )(xh g ) (8) where, l Blss / Be and h Bhss / Be . Introducing (8) in (1) and (3), the equations for the two-mass system are obtained,
Jr r [Br (1l )Blss ]r nxl Bhssg (Klss / nx )(1l )(nxr xh ) nxl Khss (xh g ) Ta
,
h Klss (nxr xh ) (1 nx2h )Khss ( xh g ) Tg
.
In plane motion
Effective blade stiffness & damping
(10) Blade tips
B (1l )Dlss K (1l )Dlss n B nK 1 x1 r x1 lss x2 x l hss x3 x l hss x4 Ta Jr nx Jr Jr Jr Jr
Bg (1n )Dhss (1n )Khss nxhDlss hKlss 1 x1 x2 x3 x4 Tg Jg Je Jr Jr Jr 2 x h
. (11)
Br
r
r
b Db 3 Jb_tip 3 Blade tips
Hub and blade roots
r
Jh + 3Jb_root
Fig. 5. Abstraction process to represent the rotor as a twomass rotating system (Ramtharan, et al., 2007).
Jb b Dbb Dbr Kb (b r ) Ta ,
(13)
Jr r (Db Br Dlss )r Dbb (Dlss / nx )xh Kb (b r ) (Klss / nx )(nxr xh ) 0
2 x hss
Klss nK x2 (1nx2h)x3 x4 Be Be
, (14)
and (2) and (3) stay unchanged. The final state-space model is of order seven. Notice that the model presented in (Li & Chen, 2007) leads to the presented above after the proposed simplifications.
3.3 One-mass rotating subsystem The one-mass representation is not often used but is described in several works. For example, it is analysed in comparison with the two-mass system in (Fortmann, 2014). In order to derive the onemass system, it is assumed that the shafts are rigid, then r = xl = nx xh = nx g and in this case, (1)-(3) are reduced to
Jdt g Bdtg nx (Ta nxTg ) ,
Three-mass rotor representation The concept presented above can be extended by introducing a second breaking point in the blades in order to introduce an additional frequency of vibration. In this case, the abstraction process is shown in Fig. 4. The resulting rotor has three masses.
(12)
b
2 x
where Jdt Jr Jxl n (Jxh Jg ) and Bdt Br Bxl n (Bxh Bg ) .
b
In plane motion
0
2 x
Kb
b
The model formulation under this approach can be carried out changing (1) by the following two equations
K nB n2K x2 nx (1l )x1 lss x2 x hss x3 x hss x4 Be Be Be
x4 nxhx1
By doing the process of abstraction shown in Fig. 3, the four-mass representation is simplified in (Ramtharan, et al., 2007) to a two-mass representation.
(9)
Defining the state-space variables as x1 = r, x2 = r –xh, x3 = g and x4 = xh–g and rearranging, the state-space representation is obtained as
x3
Two-mass rotor representation
b
J g g [Bg (1 nx2h )Bhss ]g nxh Blssr
2 x h
drive train and therefore it is not necessary to include them in the model representation of the drive train dynamics.
0
b
b
4. MODELLING THE ROTOR AS MULTI-MASS SYSTEM Modelling the rotor as a multi-mass system allows the introduction of the dynamics of flexible blades in the equations of motion. This idea was proposed in (Wasynczuk, et al., 1981) for the MOD-2 wind turbine. The multi-mass model approach consists in dividing the blades in two sections connected by spring-damper devices. The point, at which the blades are divided, is named hinge point or breaking point. The blade roots are rigidly linked to the hub. The spring-dampers permit edge and flap bending of the blade tips. The bending modes of the blades can be described in two orthogonal planes. The first one is called in-plane and describes the motion of the blade in the rotor plane, and the second one is the out-of-plane describing the motion of the blade perpendicularly to the rotor plane. Thus, the rotor is represented in general by four masses. Three masses represent the blade tips and the fourth mass corresponds to the hub and the blades roots.
Kb0
Kb
b0 Db0
b Db
3 Jb_tip
3 Jb
Br
r Jh + 3Jb_root
Fig. 6. Abstraction process to represent the rotor by three masses. The mathematical formulation for this extension is reached by changing (1) with the following three equations
Jb0 b0 Db0b0 Dbb Kb (b0 b ) Ta ,
(15)
J r r Dbb ( Db Br Dlss )r ( Dlss / nx )xh , Kb (b r ) ( Klss / nx )(nxr xh ) 0
(17)
Jbb (Db0 Db)b Db0b0 Dbr Kb0(b0 b) Kb(b r ) 0, (16)
4.1 Modelling including only the edge blade vibration Due to the fact that the motion of the out-of-plane modes is normal to the direction of motion of the rotor, they do not directly couple to the
r
r
The final state-space representation is, in this case, of order nine.
10312
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Adrian Gambier et al. / IFAC PapersOnLine 50-1 (2017) 9896–9901
5. REFERENCE WIND TURBINE
Four-mass rotor representation A rotor with a representation of four-masses and two additional masses (for the gearbox and the generator), i.e. six masses in total, is presented in (Papathanassiou & Papadopoulos, 2001) and also mentioned but not formulated in (Muyeen, et al., 2007). This is an extension of the model given in (Wasynczuk, et al., 1981) for two blades. Blades and hub are connected by using spring-dampers, as illustrated in Fig 7. Jb1
Tb1
Kb1, Db1, Bb1
Klss
Br
Jh Tb3
r
Kb3, Jb2 Kb2, Jb3 Db3, D Bb3 b2, Tb2 B
Dlss
xl Jxh
Jxl Bxh Khss
xh
Bg
Jg
Table 1. Properties of the 5 MW-NREL Wind Turbine
Te
Parameter
Dhss g
b2
Gearbox
Rotor
In order to study the dynamic properties of the proposed models, a reference wind turbine was selected. There are several options in the literature, including the 20 MW reference turbine proposed in (Ashuri, et al., 2016), the 10 MW reference turbine from DTU (Bak, et al., 2013) and the 5-MW machine specified in (Jonkman, et al., 2009). However, the 5-MW NREL turbine has been the most frequently reported in the literature and hence more information is available. For this reason, the 5-MW NREL turbine is chosen for the present study. 5.1 Parameters according to the specifications The parameters for the wind turbine are summarized in Table 1.
nx
Bxl
Generator
Fig. 7. Rotating subsystem with a three-mass rotor.
Jb1b1 (Bb1 Db1)b1 Db1r Kb1(b1 r ) Tb1 ,
(18)
Jb2 b2 (Bb2 Db2)b2 Db2r Kb2(b2 r ) Tb2 ,
(19)
Jb3b3 (Bb3 Db3)b3 Db3r Kb3(b3 r ) Tb3 ,
(20)
Jh r Db1b1 Db2b2 Db3b3 (Br Dlss Db1 Db2 Db3)r (Dlss / nx )xh Kb1(b1 r ) Kb2(b2 r )
, (21)
Kb3(b3 r ) (Klss / nx )(nxr xh) 0
J g g Dhss xh ( Bg Dhss ) g K hss ( xh g ) Tg , (23)
where Je and Be are already given in (4). Moment Tbi is the moment acting on the i-th blade. The aerodynamics of each blade is computed at one point on each blade In the simplest case, this moment is composed by the aerodynamic torque acting on the blade and the weight, i.e.
6
2 R3CT (ri , i )vwri mbi g rg cos(r i ) ,
(24)
where g, i and mbi are the gravitational acceleration, pitch angle and the mass of blade i, respectively. rb is the distance from the axes of rotation to the blade centre of mass.
Values
Units
Jr 38759228 Kg m2 115926 Kg m2 Jh Jb 11776047 Kg m2 534.116 Kg m2 Jg Ke 867637000 N m/rad De 6215000 N m s/rad 97 nx
rated 1.267109
rad/s
Tgrated 43093.55
Nm
5.2 Assumed Parameters Many parameters required for the models are not available in the specification and therefore they were collected from the literature or calculated indirectly. These parameters are summarized in Table 2. Table 2. Added parameters for the Wind Turbine Parameter
Name
Mass second moment of inertia of Gearbox Viscose bearing coeff, of the rotor Viscose bearing coeff, of the ls-side of gearbox Viscose bearing coeff, of the rotor Viscose bearing coeff, of the generator Stiffness coeff. of slow-speed shaft Damping coeff. of slow-speed shaft Stiffness coeff. of high-speed shaft Damping coeff. of high-speed shaft
Values
Units
Kg m2 409.04 Jge 818.231 N m/rad Br Bxl 945.874 N m/rad Bxh 710.655 N m s/rad 781.721 N m s/rad Bg N m/rad 9.0867e+08 Klss N m s/rad 6.6822e+06 Dlss N m/rad 2042120 Klss N m s/rad 9.0867e+08 Dhss
5.3 Selection of the Breaking points The breaking points for the blades were chosen by using the stiffness distribution given in (Jonkman, et al., 2009) and presented in Fig. 8. 10 Blade 2 10
4.2 Modelling including edge and flap blade vibration
Name
Rotor mass second moment of inertia Hub mass second moment of inertia Blade mass second moment of inertia Generator mass second moment of inertia Equivalent drive train stiffness coeff. Equivalent drive train damping coeff. Gearbox ratio Rated rotor speed Rated generator torque
Jexh Bexh nxDlssr nx2Dhssg Klss (nxr xh) nx2Khss (xh g ) 0 , (22)
Tbi
9899
stiffness distribution
1.8
All models presented in this section include the vibration of the blades only in plane. The out-of-plane deflection of the blades is not included. However, these approaches are not sufficient for a modelbased individual pitch control, where it is necessary that the model captures the root bending moments in the flap-wise direction. Modelling approaches that include the out-of-plane dynamic are, for example, (Eggleston & Stoddard, 1987) and (Gentile & Trudnowski, 2002). The model of the last reference is a generalization of the model given in the previous one, where constant spring stiffnesses in the flap motion are included. Fig. 7 can also be used as a schematic representation of this modelling approach, but assuming that the blade tips move not only in the rotation plane but also orthogonally to it. This model is nonlinear and not included in this work.
1.6 1.4 1.2 1.0
Breaking points
0.8 0.6 0.4 0.2 00
Blade length 4.7
9.7
Jbroot = 337086.84
14.7
21.7
31.7
41.7
2
Jb = 1.2544e+07 Kg m Kb = 2.552e+08 N m/rad Db = 9.3581e+05 N m s/rad
51.7
61.2
Jbtip = 1.18783e+06 Kg m2 Kbtip = 8.7546e+08 N m/rad Dbtip = 9.1065e+05 N m s/rad
Fig. 8. Breaking points and parameters.
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6. ANALYSIS AND SIMULATION RESULTS All presented models were implemented in Matlab/Simulink by using the parameters described in Section 5. The electromagnetic torque is calculated by using a steady-state simple model of an induction generator. The models are named as follows: 3M-R1 (3masses in total, 1 mass in rotor), 4M-R2 (4-masses in total, 2 in rotor), 5M-R3 (5-masses in total, 3 in rotor), 2M-R1 (2-masses in total, 1 in rotor), 3M-R2 (3-masses in total, 2 in rotor), 4M-R3 (4-masses in total, 3 in rotor) and 1M-R1 (only one mass). The most important properties are summarized in Table 3. Because of the careful selection of parameters, it was possible to obtain the same poles for identical components. Only additional poles that occur as a result of add complexity are different. Discrete-time models of order nine requires about 2.8 ms to perform one time-step evaluation. Thus, these models are able to perform in real time. Models with more than one mass in the rotor are able to reflect blade vibrations in the drive train. This is not the case for two or three mass models but for modelling the whole rotor as only one mass. Rotors with three masses react slower than the others. The simulation results show that the mass of the gearbox can be neglected because the differences between models with and without gearbox are not significant (overlapped curves in Fig. 9, where the rated generator speed has been scaled to one). 7. CONCLUSIONS In the present work, different models for the rotating subsystem of a wind turbine are presented. Simulation results show that simple models like one-mass or two mass models can be used in a first stage. However, the dynamics of the blades are coupled with the drive train, and in the case of using model-based control, more accurate models are necessary. This work is a first study. The next step is to include in the models the flap-wise dynamic of the blades as well as the dynamics of the tower. REFERENCES Ashuri, T., Martins, J. R. R. A. & Zaaijer, G., 2016. Aeroservoelastic design definition of a 20 MW common research wind turbine model. Wind Energy. Ashwini, P. & Archana, T., 2016. Mathematical modeling of wind energy system using two mass model including generator. 3M-R1
International Journal of Emerging Trends in Electrical and Electronics, 12(2), pp. 18-23. Bak, C. et al., 2013. Description of the DTU 10 MW Reference Wind, Roskilde: DTU Wind Energy. Bianchi, F. D., de Battista, H. & Mantz, R. J., 2007. Wind Turbine Control Systems. London: Springer-Verlag. Boukhezzar, B. & Siguerdidjane, H., 2011. Nonlinear control of a variable-speed wind turbine using a two-mass model. IEEE Transactions on Energy Conversion, 26(1), pp. 149-162. Chen, H. & Zhang, Y., 2016. Power System Optimization: Largescale Complex Systems Approaches. Singapore: John Wiley & Sons. Dang, D. Q., Wang, Y. & Cai, W., 2009. A multi-objective optimal nonlinear control of variable speed wind turbine. Christchurch, IEEE, pp. 17-22. Eggleston, D. M. & Stoddard, F. S., 1987. Wind Turbine Engineering Design. New York: Van Nostrand Reinhold. Fortmann, J., 2014. Modeling of wind turbines with doubly fed generator system. Wiesbaden: Springer Vieweg. Friis, J. et al., 2011. Repetitive model predictive approach to individual pitch control of wind turbines. Orlando, IEEE, pp. 3664 - 3670. Gentile, A. & Trudnowski, D., 2002. A low-order wind-turbine model for power system transient stability studies. Portland, AWEA, pp. 1-11. Gonzalez-Longatt, F., Regulski, P., Novanda, H. & Terzija, V., 2012. Impact of shaft stiffness on inertial response of fixed speed wind turbines. Automation of Electric Power Systems, 36(8). Jasniewicz, B. & Geyler, M., 2011. Wind turbine modelling and identification for control system applications. Brussels, EWEA, 280-284. Jelavic, M. & Peric, N., 2009. Wind turbine control for highly turbulent winds. Automatika, 50(3-4), p. 35–151. Jonkman, J., Butterfield, S., Musial, W. & Scot, G., 2009. Definition of a 5-MW Reference Wind Turbine for Offshore System Development, Golden, Colorado: NREL. Leithead, W. E., de la Salle, S. A., Reardon, D. & Grimble, M. J., 1991. Wind turbine modelling and control. Edinburgh, IET. Li, H. & Chen, Z., 2007. Transient stability analysis of wind turbines with induction generators considering blades and shaft flexibility. Taipei, s.n., pp. 1604-1609. Mahmoud, M. S., Hassan, M. F. & Darwish, M. G., 1985. LargeScale Systems. s.l.:Marcel Dekker, Inc..
Table 3. Most important properties of the models 2M-R1 4M-R2 3M-R2 5M-R3
4M-R3
1M-R1
0.172195196155469 6.31613143199687 33.7685374724714 319.587511087030
0.09476529316288 5.67703654144849 7.43023555206874 33.7678465647983 328.204202710417 12217.2433220824
0.0947713406472 5.6789117730058 7.4307353119597 33.768892180226 319.58751108703
1.436896961944
Natural frequencies
0.17151048175339 13.9531121097397 328.429748918498 12216.6102572282
0.17153020734761 13.9586693958588 319.793562256054
0.17217516717445 6.31378691250415 33.7674917547955 328.204202710416 12217.2433220823
Smallest time constant
0.09672192281243
0.0967680722065
0.102956772557
0.10301442267
0.025166626833
0.025225167051
1.025225167051
Sampling time
0.01
0.01
0.01
0.01
0.002
0.002
0.1
-1.2217e+04 -3.2843e+02 -1.34967 ± i 1.3887 -0.1715
-3.1979e+02 -1.3507 ± i 1.3893 -0.1715
-1.2217e+04 -3.2820e+02 -3.8514 ± i 3.355 -0.6500 ± i 6.2802 -0.17217
-3.1959e+02 -3.8517 ± i 3.3548 -0.6506 ± i 6.2825 -0.17219
-1.2217e+04 -3.2820e+02 -3.8513 ± i 3.355 -0.1870 ± i 7.4279 -0.5267 ± i 5.6525 -9.4765e-02
-3.8516 ± i 3.3548 -0.094771 -0.1874 ± i 7.4284 -0.5269 ± i 5.6544 -3.1959e+02
-1.4369 0.0000
2.9059e-03
2.7860e-03
2.8721e-03
2.8090e-03
2.8842e-03
2.8146e-03
1.6132e-03
Poles
Time execution (mean value in sec)
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Rotational speed of generator in rad/s
1.0 0.8 Black: 3M-R1 Red: 2M-R1 (overlapped with black) Magenta: 5M-R3 Blue: 4M-R3 (overlapped with magenta) Green: 4M-R2 Light blue: 3M-R2 (overlapped with green)
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 0
10
20
30
40
50
t in sec
60
Fig. 9. Simulation results. Mirzaei, M. et al., 2012. Individual pitch control using LIDAR measurements. Dubrovnik, IEEE, pp. 1646-1651. Moriarty, P. J. & Butterfield, S. B., 2009. Wind turbine modeling overview for control engineers. Saint Louis, s.n., pp. 2090-2095. Morisse, M., Bartschat, A. & Wenske, J., 2016. Dependency of the lifetime estimation of power modules in fully rated wind turbine converters on the modelling depth of the overall system. Karlsruhe, IEEE. Muyeen, S. M. et al., 2007. Comparative study on transient stability analysis of wind turbine generator system using different drive train models. IET Renewable Power Generation, 1(2), pp. 131-141. Muyeen, S. M. et al., 2007. Comparative study on transient stability analysis of wind turbine generator system using different drive train models. IET Renewable Power Generation, 1(2), pp. 131-141. Papathanassiou, S. . A. & Papadopoulos, M. P., 2001. Mechanical stresses in fixed-speed wind turbines due to network disturbances. IEEE Transactions on Energy Conversion, 16(4), pp. 361-367. Pournaras, C., Riziotis, V. & Kladas, A., 2008. Wind turbine control strategy enabling mechanical stress reduction based on dynamic model including blade oscillation effects. Vilamoura, IEEE, pp. 1-6. Ramtharan, G., Anaya-Lara, O., Bossanyi, E. & Jenkins, N., 2007. Influence of structural dynamic representations of FSIG wind turbines on electrical transients. Wind Energy, March, 10(4), pp. 293-301. Salman, S. K. & Teo, A. L. J., 2003. Windmill modeling consideration and factors influencing the stability of a grid-connected wind power-based embedded generator. IEEE Transactions on Power Systems, 18(2), pp. 793-802. Sami, M. & Patton, R. J., 2012. An FTC approach to wind turbine power maximisation via T-S fuzzy modelling and control. Mexico City, IFAC, pp. 349-354.
Schlipf, D., Schlipf, D. J. & Kühn, M., 2013. Nonlinear model predictive control of wind turbines using LIDAR. Wind Energy, Volume 16, p. 1107–1129. Shan, W. & Shan, M., 2012. Gain scheduling pitch control design for active tower damping and 3P harmonic reduction. Copenhagen, EWEA. Slootweg, J. G., de Haan, S. W. H., Polinder, H. & Kling, W. L., 2003. General model for representing variable speed wind turbines in power system dynamics simulations. IEEE Transactions on Power Systems, 18(1), pp. 144-151. Sørensen, P., Hansen, A. D. & Carvalho Rosas, P. A., 2002. Wind models for simulation of power fluctuations from wind farms. Journal of Wind Engineering and Industrial Aerodynamics, 90(1215), p. 1381–1402. van der Tempel, J. & Molenaar, D.-P., 2002. Wind turbine structural dynamics - A review of the principles for modern power generation, onshore and offshore. Wind Engineering, 26(4), pp. 211-220. Wasynczuk, 0., Man, D. T. & Sullivan, J. P., 1981. Dynamic behavior of a class of wind turbine generators during random wind fluctuations. IEEE Transactions on Power Apparatus and Systems, PAS-100(6), pp. 2837-2845. Xiao, S., Yang, G. & Geng, H., 2014. Nonlinear pitch control design for load reduction on wind turbines. Hiroshima, IEEE, pp. 543-547. Xu, H., Xu, H., Chen, L. & Wenske, J., 2013. Active damping control of DFIG wind turbines during fault ride through. Istambul, IEEE. ACKNOWLEDGEMENTS This work is financed by the Federal Ministry of Economic Affairs and Energy (BMWi).
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Dynamic Modelling of the Rotating Subsystem of a Wind Turbine Dynamic Modelling of the Rotating Subsystem of a Wind Turbine Dynamic Modellingfor ofControl the Rotating Subsystem Design Purposesof a Wind Turbine for Control Design Purposes for Control Design Purposes Adrian Gambier*
Adrian Gambier* Adrian Gambier* *Fraunhofer Institute for Wind Energy and EnergyAdrian System Gambier* Technology Northwest IWES, 27572 Bremerhaven, Germany *Fraunhofer Institute for(Tel: Wind Energy and Energy System [email protected]) Technology Northwest IWES, 27572 Bremerhaven, Germany +49 471 14290-375; e-mail: *Fraunhofer Institute Wind Energy and System Technology *Fraunhofer Institute for for(Tel: Wind Energy and Energy Energye-mail: System [email protected]) Technology Northwest Northwest IWES, IWES, 27572 27572 Bremerhaven, Bremerhaven, Germany Germany +49 471 14290-375; (Tel: (Tel: +49 +49 471 471 14290-375; 14290-375; e-mail: e-mail: [email protected]) [email protected]) Abstract: The present work reviews the dynamic models for the rotating subsystem of a wind turbine, which Abstract: present work reviews purposes the dynamic for the rotating subsystem of a wind turbine, which have beenThe developed for control in models the literature. Modelling approaches are systematically Abstract: The present work work reviews purposes the dynamic dynamic models for the the rotating rotating subsystem of aa wind wind turbine, which which Abstract: The present reviews the models for subsystem of turbine, have been developed for control in the literature. Modelling approaches are systematically organized and classified including advantage,indrawbacks and limitations. Based on theare 5-MW reference have been developed for control purposes the literature. Modelling approaches systematically have been developed for control purposes in the literature. Modelling approaches are systematically organized and classified including advantage, drawbacks andforlimitations. Based on themodels 5-MWare reference turbine developed by NREL, analysis and simulation studies some of the presented carried organized and including advantage, drawbacks and Based on 5-MW reference organized and classified classified including advantage, drawbacks andforlimitations. limitations. Based on the themodels 5-MWare reference turbine developed by NREL, analysis and simulation studies some of the presented carried out. Hence, it is possible to determine models’ properties in for time domain, sampling time, computational turbine developed by NREL, analysis and simulation studies some of the presented models are carried turbine developed by NREL, analysis and simulation studies for some of thesampling presentedtime, models are carried out. Hence, it isdiscrete possible to determine models’ properties infor time computational burden of time models. Finally, recommendations thedomain, selectionsampling of a model according to the out. it possible to models’ properties time domain, time, computational out. Hence, Hence, it is isdiscrete possiblemodels. to determine determine models’ properties in infor time domain, sampling time, computational burden of time Finally, recommendations the selection of a model according to the application are given. burden discrete burden of of time time discrete models. models. Finally, Finally, recommendations recommendations for for the the selection selection of of aa model model according according to to the the application are given. application are given. application are(International given. Keywords: Wind turbine, rotor dynamics, dynamic modelling © 2017, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Wind turbine, rotor dynamics, dynamic modelling Keywords: Keywords: Wind Wind turbine, turbine, rotor rotor dynamics, dynamics, dynamic dynamic modelling modelling
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION Modern multi-MW wind turbines, which are characterized by their Modern multi-MW wind turbines, which are characterized their large dimensions and flexible structures, require advanced by control Modern multi-MW wind turbines, which characterized by their Modern multi-MW wind turbines, which are are characterized by their large dimensions and flexible structures, require advanced control algorithms to reach a satisfactory performance. On the othercontrol hand, large dimensions and flexible structures, require advanced large dimensions and flexible structures, requireOn advanced control algorithms to reach a satisfactory performance. the other hand, advanced control requires suitable dynamic models for control algorithms to aa satisfactory performance. On hand, algorithms control to reach reachrequires satisfactory performance. On the the other other hand, advanced suitable dynamic models fordynamic control purposes, i.e. models that are able to capture the essential advanced control requires suitable dynamic models for control advanced control requires suitable dynamic models for control purposes, i.e. models that are able to capture the essential dynamic behaviour of the machine butable keeping their simplicity such that a purposes, i.e. models that to the dynamic purposes, i.e. models that are are to capture capture the essential essential dynamic behaviour of the machine butable keeping their simplicity such thatina control system design of acceptable complexity can be achieved, behaviour of machine but their such behaviour of the the machine but keeping keeping their simplicity simplicity such that thatinaa control system design of acceptable complexity can be achieved, particular if theydesign have to be implemented in real time. control of acceptable complexity can control system system ofbe acceptable complexity can be be achieved, achieved, in in particular if theydesign have to implemented in real time. particular if be in particular if they theyofhave have towind be implemented implemented in real real time. time. The dynamics largeto turbines generally require a represenThe dynamics large wind turbines generally requirei.e.a equations representation by using of nonlinear partial differential equations, The dynamics of large wind wind turbines generally requirei.e.aa equations represenThe dynamics of large turbines generally require representation by using nonlinear partial differential equations, that are not particularly convenient for control purposes. Mach tation by using nonlinear partial differential equations, i.e. equations tation by using nonlinear partial differential equations, i.e. equations that are effort not particularly convenient for control purposes. Mach research has been done in developing simple models of wind that not convenient for purposes. Mach that are are effort not particularly particularly convenient for control control purposes. Mach research has been done in developing simple models of wind turbines for control. In fact, most works reporting the use ofofmodel research effort has been done in developing simple models wind research for effort has been done in developing simplethe models ofmodel wind turbines control. In fact, most works reporting use of based control algorithms present some reporting kind of model developing turbines for control. In fact, most works the use of model turbines for control. In fact,present most works reporting the use of model based control algorithms some kind of model developing activity. Frequently, the used modelling approach includes a twobased algorithms present some of developing based control control algorithms present some kind kind of model model developing activity. Frequently, theaused modelling approach includes a twomass drivetrain and standard nonlinear expression for the activity. Frequently, the modelling approach includes aa twoactivity. Frequently, theaused used modelling approach includesfor twomass drivetrain and standard nonlinear expression the in the form aerodynamic torque that includes the power coefficient C p mass drivetrain and aaincludes standard nonlinear expression for the mass drivetrain and standard nonlinear expression forform the aerodynamic torque that the power coefficient C p in the of an empirical nonlinear function,the whose coefficients are adjusted by in the form aerodynamic torque that includes power coefficient C p in the form aerodynamic torque that includes the power coefficient C of an empirical nonlinear function, whose coefficients arepadjusted by curve fitting algorithms, e.g. (Boukhezzar & Siguerdidjane, 2011), of an nonlinear whose are by of an empirical empirical nonlinear function, function, whose coefficients coefficients are adjusted adjusted by curve fitting algorithms, e.g. (Boukhezzar & Siguerdidjane, 2011), (Dang,fitting et al., algorithms, 2009) and (Sami & Patton, 2012). Moreover, the2011), forecurve e.g. & curve fitting algorithms, e.g. (Boukhezzar (Boukhezzar & Siguerdidjane, Siguerdidjane, 2011), (Dang, et al., 2009) and (Sami & Patton, 2012). Moreover, the foreaft displacement of the tower &isPatton, added2012). in (Xiao, et al.,the 2014) (Dang, et and Moreover, (Dang, et al., al., 2009) 2009) the and (Sami (Sami 2012). Moreover, the foreforeaft displacement tower & isPatton, added in (Xiao, etofal., and (Schlipf, etof al., 2013). A more complex model the2014) tower aft displacement of the tower is added in (Xiao, et al., 2014) aft displacement ofal., the2013). tower A is more added in (Xiao, etofal., 2014) and (Schlipf, et complex model the tower is used in (Jelavic & Peric, 2009) and a dynamic model for a vertical and (Schlipf, et al., 2013). A more complex model of tower and (Schlipf, et & al., 2013). A and more complex model model for of the the tower is used inand (Jelavic Peric, 2009) a dynamic a vertical inverted non-rotating blade are considered in (Bianchi, et al., is used in (Jelavic & Peric, 2009) and a dynamic model for a vertical is used inand (Jelavic & Peric, blade 2009) are and considered a dynamic model for a vertical inverted non-rotating in (Bianchi, et al., 2007) and (Shan & Shan, 2012). Control applications by using more inverted and non-rotating blade are in et al., inverted and non-rotating blade Control are considered considered in (Bianchi, (Bianchi, et al., 2007) and (Shan & Shan, 2012). applications by using more complex models ofShan, wind2012). turbines can beapplications found, among others, in 2007) and (Shan & Control by using more 2007) and (Shan & Shan, 2012). Control applications by using more complex models of2008), wind (Muyeen, turbines can be found, among others, in (Pournaras, et al., et al., 2007) and (Friis, et al., complex models of wind turbines be among others, in complex models of2008), wind (Muyeen, turbines can can be found, found, among others, in (Pournaras, et al., et al., 2007) and (Friis, et al., 2011). (Pournaras, (Pournaras, et et al., al., 2008), 2008), (Muyeen, (Muyeen, et et al., al., 2007) 2007) and and (Friis, (Friis, et et al., al., 2011). 2011). 2011). It is also possible to find works, which are focused on modelling It is alsoofpossible to find works, which are focused on modelling efforts wind turbines for control design commitments (e.g. It is also alsoofpossible possible to find find works, works, which are focused focused on modelling modelling It is to which are on efforts wind turbines for control design commitments (e.g. (Slootweg, et al., 2003), (Salman & Teo, 2003) and (Jasniewicz & efforts of turbines for design commitments (e.g. efforts of wind wind turbines for control control design commitments (e.g. (Slootweg, et al., 2003), (Salman & Teo, 2003) and (Jasniewicz & Geyler, 2011)). An overview about the2003) modelling aspects of (Slootweg, et al., 2003), (Salman & Teo, and (Jasniewicz & (Slootweg, et al.,An 2003), (Salmanabout & Teo, 2003) and (Jasniewicz & Geyler, 2011)). overview the modelling aspects of wind turbines from the controlabout engineering point ofaspects view can Geyler, 2011)). An overview the modelling of Geyler,turbines 2011)). from An overview about the modelling aspects of wind the control engineering point of view can be found in (Moriarty & Butterfield, 2009). point of view can wind turbines from engineering wind turbines from the the&control control engineering be found in (Moriarty Butterfield, 2009). point of view can be be found found in in (Moriarty (Moriarty & & Butterfield, Butterfield, 2009). 2009).
Thus, much research has been done in order to propose simplified Thus, research has been done in the order to propose simplified modelsmuch for control purposes. However, proposed models are not Thus, research has done order to simplified Thus, much much research has been been done in in the order to propose propose simplified models for control purposes. However, proposed models not comparatively analysed in the same context. Therefore, theare main models for control purposes. However, the proposed models are not models for control purposes. However, the proposed models aremain not comparatively analysed in the same context. Therefore, the contribution of analysed the present work is to context. review and analyse the different comparatively in the same Therefore, main comparatively analysed in work the same context. Therefore, the main contribution of the present is to review and analyse different dynamic models forpresent the rotating subsystem of and the wind turbine, i.e. contribution of work is analyse different contribution of the the work subsystem is to to review review analyse different dynamic models forpresent the rotating of and the wind turbine, i.e. the rotor, drivetrain and generator. Simulation studies are done by dynamic models for the subsystem of wind turbine, i.e. dynamic models for and the rotating rotating subsystem of the the windare turbine, i.e. the rotor, drivetrain generator. Simulation studies done by using the 5-MW reference wind turbine from NREL (Jonkman, et al., the rotor, drivetrain and Simulation studies are by the rotor, drivetrain and generator. generator. Simulation studies are done done by using the 5-MW reference wind turbine from NREL (Jonkman, et al., 2009). The paper is organized as follows: in Section 2, general using the 5-MW reference wind turbine from NREL (Jonkman, et al., using the 5-MW reference wind turbine from NREL (Jonkman, et al., 2009). Theconsiderations paper is organized as follows: modelling are given, followedin bySection Sections2, 3general and 4, 2009). paper as in 2, 2009). The Theconsiderations paper is is organized organized as follows: follows: inbySection Section 2,3general general modelling are given, followed Sections and 4, where the different modelsare aregiven, described and formulated. Sections 5 modelling considerations followed by Sections 3 modelling considerations are given, followed by SectionsSections 3 and and 4, 4, where the different models are described and formulated. and 6 the are different devoted models to the description of and the formulated. reference turbine, the55 where are described Sections where models are described Sectionsthe5 and 6 the are different devoted to theand description of and theofformulated. reference turbine, simulation environment the analysis the results. Finally, and to description of turbine, the and 6 6 are are devoted devoted to the theand description of the theofreference reference turbine, the simulation environment the analysis the results. Finally, conclusions are drawn in Section 7. simulation environment and the analysis of the results. Finally, simulation environment and the7.analysis of the results. Finally, conclusions are drawn in Section conclusions conclusions are are drawn drawn in in Section Section 7. 7. 2. GENERAL MODELLING ASPECTS 2. GENERAL MODELLING ASPECTS 2. MODELLING 2. GENERAL GENERAL MODELLING ASPECTS Wind turbines can be modelled by usingASPECTS approaches of largeWind turbines since can bethey modelled by using approaches of largescale systems, are complex systems that brings into Wind turbines can modelled by approaches of Windsystems, turbines since can be bethey modelled by using using approaches of largelargescale are complex systems that brings into play many state variables. These approaches allow the application scale systems, since they are systems that brings into scalemany systems, since theyThese are complex complex systems that brings into play state variables. approaches allow the application of typical techniques for large-scale systems,allow such as decomposiplay many state variables. These approaches the application play many state variables. These approaches allow the application of typical techniques for large-scale systems, such as decomposition and coordination (Chen & Zhang, 2016),such (Mahmoud, et al., of typical techniques large-scale systems, as of typical techniques for for large-scale systems, such as decomposidecomposition and coordination (Chen & Zhang, 2016), (Mahmoud, et al., 1985) and (Leithead, et al., 1991). These techniques can also be tion and coordination (Chen & 2016), (Mahmoud, et tion and coordination (Chen & Zhang, Zhang, 2016), (Mahmoud, et al., al., 1985) and (Leithead, et al., 1991). These techniques can also be found in the modelling of wind turbines (e.g. (Mirzaei, et al., 1985) (Leithead, et 1991). These can be 1985) and and (Leithead, et al., al., 1991). These techniques techniques can also also be found in the modelling of wind turbines (e.g. (Mirzaei, et al., 2012), in (van der Tempel of & wind Molenaar, 2002)). found the modelling turbines (e.g. found in theder modelling turbines (e.g. (Mirzaei, (Mirzaei, et et al., al., 2012), (van Tempel of & wind Molenaar, 2002)). 2012), 2012), (van (van der der Tempel Tempel & & Molenaar, Molenaar, 2002)). 2002)). 2.1 Plant Decomposition 2.1 Plant Decomposition 2.1 Plant Decomposition 2.1the Plant Decomposition At present time, there is no suggestion in the literature on how to At the present time, there isHowever, no suggestion the literature on how to decompose wind turbines. there in is some tacit agreement to At the present time, there no in the on At the present time, there is isHowever, no suggestion suggestion in the literature literature on how how to to decompose wind turbines. there is some tacit agreement use a spatio-temporal decomposition. Thisis technique consists oftoa decompose wind turbines. However, there some tacit agreement decompose wind turbines. However, there is some tacit agreement use a spatio-temporal decomposition. This oftoa combination of physical decomposition, i.e.technique physical consists components, use decomposition. This technique consists of use aa spatio-temporal spatio-temporal decomposition. This technique consists of aa combination of physical decomposition, i.e. physical components, Thus, a physical decomposition is and functional decomposition. combination of physical decomposition, i.e. physical components, combination of decomposition. physical decomposition, i.e. physical components, Thus, a physical decomposition is and functional used for the main machine and functional decomposition for the obThus, decomposition is and decomposition. Thus, aa physical physical decomposition is and functional functional decomposition. used for the main machine and functional decomposition for the obsubsystems. Fig. 1and shows an example including the tained used for the main machine functional decomposition for the obused for the main machine and functional decomposition for the obsubsystems. Fig. 1 shows an example including the tained involved coordinating variables. This decomposition is inspired subsystems. Fig. 1 an example the tained subsystems. Fig.variables. 1 shows showsThis an decomposition example including including the tained involved coordinating is inspired by (Leithead, et al., 1991) and it will support the modelling involved coordinating variables. This decomposition is inspired involved coordinating variables. This decomposition is inspired by (Leithead, et al.,in1991) and itThe willcoordinating support thevariables modelling activities described this work. are by et and will support modelling by (Leithead, (Leithead, et al., al.,in1991) 1991) and it itThe willcoordinating support the thevariables modelling activities described this work. are described in the next section. activities described in this work. The coordinating variables activities in described in this work. The coordinating variables are are described the next section. described described in in the the next next section. section.
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Pitch Tpitch subsystem
vw
Tb r Ta
Aerodynamics
given by the generator. The three mass model and its utilization can be found e.g. in (Gonzalez-Longatt, et al., 2012), (Xu, et al., 2013) and (Morisse, et al., 2016). Fig. 1 shows this concept.
Rotating Subsystem
r Drive Rotor xl Train
g
P Tg Generator
Ft
Br Klss
Tower
t
Ta Jr
Fig. 1. System decomposition of a wind turbine.
r
2.2 The Rotating Subsystem The rotating subsystem consists of the wind turbine rotor, the drive train and the generator. In turn, the wind turbine rotor includes the three rotor blades and the hub. The drive train contains in this work the low-speed shaft, the gearbox and the high-speed shaft. In case of direct drive systems, only a low-speed shaft is present and no gearbox. A scheme for the rotating subsystem is presented in Fig. 2.
Tb1 Blade 1 1 r Tb2 Blade 2 2 Hub xl Tb3 Blade 2 3
g Drive Train
Tg
9897
P Generator
Rotor
Dlss
Bxl
nx Jxl Bxh Khss Bg
xl Jxh
Jg
Tg
xh Dhss g
Gearbox
Generator
Fig. 4. Three-mass rotating subsystem with one-mass rotor. The equations of motion for this approach are
Jr r (Br Dlss )r (Dlss / nx )xh (Klss / nx )(nxr xh ) Ta , (1) Jexh Bexh nxDlssr nx2Dhssg Klss (nxr xh ) nx2Khss (xh g ) 0 , (2) J g g ( Dhss Bg )g Dhssxh Khss ( xh g ) Tg ,
(3)
Je Jxl nx2 Jxh and Be Bxl Dlss nx2 (Bxh Dhss ) .
(4)
where
Fig. 2. Scheme for the rotating subsystem. The drive train and the generator are represented here by either a twomass system (one for the gearbox and one for the generator) or by a single mass system assuming a massless gearbox. The mass of the slow-speed shaft is considered to be incorporated to the hub and the mass of the high-speed shaft to the generator rotor. The different approaches that are analysed in this study are distinguished by the different representations of the rotor and are divided into two categories. The first one represents the whole rotor as a single mass. Hence, the rotating subsystem can be modelled as a three-mass system (rotor, gearbox, generator), a two-mass system (rotor, generator) or a single mass system (rotor + generator). In the second category, the rotor is represented as a multi-mass system, i.e. the rotor can have two, three or four masses, which leads to a rotating system either with four, five and six masses or with three, four and five masses, respectively depending if the mass of the gearbox is included or not. This description is illustrated in Fig. 2. Rotating Subsystem
Parameters J, B, D, K and n represent mass second moments of inertia, viscous friction coefficients, damping coefficients, stiffness coefficients and the gearbox ratio, respectively. Variables , and T correspond to rotation speed, rotation angle and torque, respectively. Subscripts r, g, a, e, x, xl, xh, lss and hss denote rotor, generator, aerodynamic, equivalent, gearbox, low-speed side of the gearbox, high-speed side of the gearbox, low-speed shaft and high-speed shaft, respectively. The aerodynamic torque is computed by (5) Ta R2CQ ( , )ver2 , 2 where , R, ver are the air density, the rotor ratio, and the effective and relative wind speed. is the tip-speed-ratio defined as Rr/vw and is the pitch angle. CQ is the torque coefficient defined as Cp(,)/ and Cp is the power coefficient, which is usually given as a two-dimensional table for and calculated by a precise aerodynamic simulation based on the blade element theory for a given blade design. Defining the state-space variables as x1 = r, x2 = xh, x3 = nxr – xh, x4=g and x5=xh –g and rearranging (1)-(3), a fifth order state-space representation is obtained
Rotor as single mass system
Rotor as multi mass system
One-mass Rotor
Two-mass Three-mass Four-mass Rotor Rotor Rotor
x1
One-mass Two-mass Three-mass System System System
Three/ Four/five- Five/sixfour-mass mass mass System System System
x 2
( Br Dlss ) D K 1 x1 lss x2 lss x3 Ta Jr nx J r nx J r Jr
n x Dlss B K n2 D n2 K x1 e x2 lss x3 x lss x4 x lss x5 Je Je Je Je Je
x3 n x x1 x2
Fig. 3. Classification of the rotating subsystem according to the number of masses used for the representation. 3. MODELLING THE ROTOR AS SINGLE-MASS SYSTEM The simplest and most frequently case of modelling the rotating subsystem is representing the rotor is by only a single mass. Thus, three cases can be found, three-mass, two-mass and one-mass rotating subsystems (see Fig. 3). They are formulated in the next subsections. 3.1 Three-mass rotating subsystem
. (6)
B g Dhss D K 1 x 4 hss x2 x4 hss x5 Tg Jg Jg Jg Jg x5 x 2 x4
3.2 Two-mass rotating subsystem A two-mass model is used for example in (Ashwini & Archana, 2016) and in (Boukhezzar & Siguerdidjane, 2011). The model is obtained by considering a massless gearbox. In this case, (2) becomes
The rotating subsystem is modelled by three masses: the rotor as a single mass, the gearbox as a second mass and the third one is 10311
Bexh nx Blssr nx2 Bhssg Klss (nxr xh ) nx2Khss (xh g ) 0 (7)
Proceedings of the 20th IFAC World Congress 9898 Adrian Gambier et al. / IFAC PapersOnLine 50-1 (2017) 9896–9901 Toulouse, France, July 9-14, 2017
and this leads to
xh nx1 r nx22g (Klss / Be )(nxr xh ) nx2 (Khss / Be )(xh g ) (8) where, l Blss / Be and h Bhss / Be . Introducing (8) in (1) and (3), the equations for the two-mass system are obtained,
Jr r [Br (1l )Blss ]r nxl Bhssg (Klss / nx )(1l )(nxr xh ) nxl Khss (xh g ) Ta
,
h Klss (nxr xh ) (1 nx2h )Khss ( xh g ) Tg
.
In plane motion
Effective blade stiffness & damping
(10) Blade tips
B (1l )Dlss K (1l )Dlss n B nK 1 x1 r x1 lss x2 x l hss x3 x l hss x4 Ta Jr nx Jr Jr Jr Jr
Bg (1n )Dhss (1n )Khss nxhDlss hKlss 1 x1 x2 x3 x4 Tg Jg Je Jr Jr Jr 2 x h
. (11)
Br
r
r
b Db 3 Jb_tip 3 Blade tips
Hub and blade roots
r
Jh + 3Jb_root
Fig. 5. Abstraction process to represent the rotor as a twomass rotating system (Ramtharan, et al., 2007).
Jb b Dbb Dbr Kb (b r ) Ta ,
(13)
Jr r (Db Br Dlss )r Dbb (Dlss / nx )xh Kb (b r ) (Klss / nx )(nxr xh ) 0
2 x hss
Klss nK x2 (1nx2h)x3 x4 Be Be
, (14)
and (2) and (3) stay unchanged. The final state-space model is of order seven. Notice that the model presented in (Li & Chen, 2007) leads to the presented above after the proposed simplifications.
3.3 One-mass rotating subsystem The one-mass representation is not often used but is described in several works. For example, it is analysed in comparison with the two-mass system in (Fortmann, 2014). In order to derive the onemass system, it is assumed that the shafts are rigid, then r = xl = nx xh = nx g and in this case, (1)-(3) are reduced to
Jdt g Bdtg nx (Ta nxTg ) ,
Three-mass rotor representation The concept presented above can be extended by introducing a second breaking point in the blades in order to introduce an additional frequency of vibration. In this case, the abstraction process is shown in Fig. 4. The resulting rotor has three masses.
(12)
b
2 x
where Jdt Jr Jxl n (Jxh Jg ) and Bdt Br Bxl n (Bxh Bg ) .
b
In plane motion
0
2 x
Kb
b
The model formulation under this approach can be carried out changing (1) by the following two equations
K nB n2K x2 nx (1l )x1 lss x2 x hss x3 x hss x4 Be Be Be
x4 nxhx1
By doing the process of abstraction shown in Fig. 3, the four-mass representation is simplified in (Ramtharan, et al., 2007) to a two-mass representation.
(9)
Defining the state-space variables as x1 = r, x2 = r –xh, x3 = g and x4 = xh–g and rearranging, the state-space representation is obtained as
x3
Two-mass rotor representation
b
J g g [Bg (1 nx2h )Bhss ]g nxh Blssr
2 x h
drive train and therefore it is not necessary to include them in the model representation of the drive train dynamics.
0
b
b
4. MODELLING THE ROTOR AS MULTI-MASS SYSTEM Modelling the rotor as a multi-mass system allows the introduction of the dynamics of flexible blades in the equations of motion. This idea was proposed in (Wasynczuk, et al., 1981) for the MOD-2 wind turbine. The multi-mass model approach consists in dividing the blades in two sections connected by spring-damper devices. The point, at which the blades are divided, is named hinge point or breaking point. The blade roots are rigidly linked to the hub. The spring-dampers permit edge and flap bending of the blade tips. The bending modes of the blades can be described in two orthogonal planes. The first one is called in-plane and describes the motion of the blade in the rotor plane, and the second one is the out-of-plane describing the motion of the blade perpendicularly to the rotor plane. Thus, the rotor is represented in general by four masses. Three masses represent the blade tips and the fourth mass corresponds to the hub and the blades roots.
Kb0
Kb
b0 Db0
b Db
3 Jb_tip
3 Jb
Br
r Jh + 3Jb_root
Fig. 6. Abstraction process to represent the rotor by three masses. The mathematical formulation for this extension is reached by changing (1) with the following three equations
Jb0 b0 Db0b0 Dbb Kb (b0 b ) Ta ,
(15)
J r r Dbb ( Db Br Dlss )r ( Dlss / nx )xh , Kb (b r ) ( Klss / nx )(nxr xh ) 0
(17)
Jbb (Db0 Db)b Db0b0 Dbr Kb0(b0 b) Kb(b r ) 0, (16)
4.1 Modelling including only the edge blade vibration Due to the fact that the motion of the out-of-plane modes is normal to the direction of motion of the rotor, they do not directly couple to the
r
r
The final state-space representation is, in this case, of order nine.
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5. REFERENCE WIND TURBINE
Four-mass rotor representation A rotor with a representation of four-masses and two additional masses (for the gearbox and the generator), i.e. six masses in total, is presented in (Papathanassiou & Papadopoulos, 2001) and also mentioned but not formulated in (Muyeen, et al., 2007). This is an extension of the model given in (Wasynczuk, et al., 1981) for two blades. Blades and hub are connected by using spring-dampers, as illustrated in Fig 7. Jb1
Tb1
Kb1, Db1, Bb1
Klss
Br
Jh Tb3
r
Kb3, Jb2 Kb2, Jb3 Db3, D Bb3 b2, Tb2 B
Dlss
xl Jxh
Jxl Bxh Khss
xh
Bg
Jg
Table 1. Properties of the 5 MW-NREL Wind Turbine
Te
Parameter
Dhss g
b2
Gearbox
Rotor
In order to study the dynamic properties of the proposed models, a reference wind turbine was selected. There are several options in the literature, including the 20 MW reference turbine proposed in (Ashuri, et al., 2016), the 10 MW reference turbine from DTU (Bak, et al., 2013) and the 5-MW machine specified in (Jonkman, et al., 2009). However, the 5-MW NREL turbine has been the most frequently reported in the literature and hence more information is available. For this reason, the 5-MW NREL turbine is chosen for the present study. 5.1 Parameters according to the specifications The parameters for the wind turbine are summarized in Table 1.
nx
Bxl
Generator
Fig. 7. Rotating subsystem with a three-mass rotor.
Jb1b1 (Bb1 Db1)b1 Db1r Kb1(b1 r ) Tb1 ,
(18)
Jb2 b2 (Bb2 Db2)b2 Db2r Kb2(b2 r ) Tb2 ,
(19)
Jb3b3 (Bb3 Db3)b3 Db3r Kb3(b3 r ) Tb3 ,
(20)
Jh r Db1b1 Db2b2 Db3b3 (Br Dlss Db1 Db2 Db3)r (Dlss / nx )xh Kb1(b1 r ) Kb2(b2 r )
, (21)
Kb3(b3 r ) (Klss / nx )(nxr xh) 0
J g g Dhss xh ( Bg Dhss ) g K hss ( xh g ) Tg , (23)
where Je and Be are already given in (4). Moment Tbi is the moment acting on the i-th blade. The aerodynamics of each blade is computed at one point on each blade In the simplest case, this moment is composed by the aerodynamic torque acting on the blade and the weight, i.e.
6
2 R3CT (ri , i )vwri mbi g rg cos(r i ) ,
(24)
where g, i and mbi are the gravitational acceleration, pitch angle and the mass of blade i, respectively. rb is the distance from the axes of rotation to the blade centre of mass.
Values
Units
Jr 38759228 Kg m2 115926 Kg m2 Jh Jb 11776047 Kg m2 534.116 Kg m2 Jg Ke 867637000 N m/rad De 6215000 N m s/rad 97 nx
rated 1.267109
rad/s
Tgrated 43093.55
Nm
5.2 Assumed Parameters Many parameters required for the models are not available in the specification and therefore they were collected from the literature or calculated indirectly. These parameters are summarized in Table 2. Table 2. Added parameters for the Wind Turbine Parameter
Name
Mass second moment of inertia of Gearbox Viscose bearing coeff, of the rotor Viscose bearing coeff, of the ls-side of gearbox Viscose bearing coeff, of the rotor Viscose bearing coeff, of the generator Stiffness coeff. of slow-speed shaft Damping coeff. of slow-speed shaft Stiffness coeff. of high-speed shaft Damping coeff. of high-speed shaft
Values
Units
Kg m2 409.04 Jge 818.231 N m/rad Br Bxl 945.874 N m/rad Bxh 710.655 N m s/rad 781.721 N m s/rad Bg N m/rad 9.0867e+08 Klss N m s/rad 6.6822e+06 Dlss N m/rad 2042120 Klss N m s/rad 9.0867e+08 Dhss
5.3 Selection of the Breaking points The breaking points for the blades were chosen by using the stiffness distribution given in (Jonkman, et al., 2009) and presented in Fig. 8. 10 Blade 2 10
4.2 Modelling including edge and flap blade vibration
Name
Rotor mass second moment of inertia Hub mass second moment of inertia Blade mass second moment of inertia Generator mass second moment of inertia Equivalent drive train stiffness coeff. Equivalent drive train damping coeff. Gearbox ratio Rated rotor speed Rated generator torque
Jexh Bexh nxDlssr nx2Dhssg Klss (nxr xh) nx2Khss (xh g ) 0 , (22)
Tbi
9899
stiffness distribution
1.8
All models presented in this section include the vibration of the blades only in plane. The out-of-plane deflection of the blades is not included. However, these approaches are not sufficient for a modelbased individual pitch control, where it is necessary that the model captures the root bending moments in the flap-wise direction. Modelling approaches that include the out-of-plane dynamic are, for example, (Eggleston & Stoddard, 1987) and (Gentile & Trudnowski, 2002). The model of the last reference is a generalization of the model given in the previous one, where constant spring stiffnesses in the flap motion are included. Fig. 7 can also be used as a schematic representation of this modelling approach, but assuming that the blade tips move not only in the rotation plane but also orthogonally to it. This model is nonlinear and not included in this work.
1.6 1.4 1.2 1.0
Breaking points
0.8 0.6 0.4 0.2 00
Blade length 4.7
9.7
Jbroot = 337086.84
14.7
21.7
31.7
41.7
2
Jb = 1.2544e+07 Kg m Kb = 2.552e+08 N m/rad Db = 9.3581e+05 N m s/rad
51.7
61.2
Jbtip = 1.18783e+06 Kg m2 Kbtip = 8.7546e+08 N m/rad Dbtip = 9.1065e+05 N m s/rad
Fig. 8. Breaking points and parameters.
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Proceedings of the 20th IFAC World Congress 9900 Adrian Gambier et al. / IFAC PapersOnLine 50-1 (2017) 9896–9901 Toulouse, France, July 9-14, 2017
6. ANALYSIS AND SIMULATION RESULTS All presented models were implemented in Matlab/Simulink by using the parameters described in Section 5. The electromagnetic torque is calculated by using a steady-state simple model of an induction generator. The models are named as follows: 3M-R1 (3masses in total, 1 mass in rotor), 4M-R2 (4-masses in total, 2 in rotor), 5M-R3 (5-masses in total, 3 in rotor), 2M-R1 (2-masses in total, 1 in rotor), 3M-R2 (3-masses in total, 2 in rotor), 4M-R3 (4-masses in total, 3 in rotor) and 1M-R1 (only one mass). The most important properties are summarized in Table 3. Because of the careful selection of parameters, it was possible to obtain the same poles for identical components. Only additional poles that occur as a result of add complexity are different. Discrete-time models of order nine requires about 2.8 ms to perform one time-step evaluation. Thus, these models are able to perform in real time. Models with more than one mass in the rotor are able to reflect blade vibrations in the drive train. This is not the case for two or three mass models but for modelling the whole rotor as only one mass. Rotors with three masses react slower than the others. The simulation results show that the mass of the gearbox can be neglected because the differences between models with and without gearbox are not significant (overlapped curves in Fig. 9, where the rated generator speed has been scaled to one). 7. CONCLUSIONS In the present work, different models for the rotating subsystem of a wind turbine are presented. Simulation results show that simple models like one-mass or two mass models can be used in a first stage. However, the dynamics of the blades are coupled with the drive train, and in the case of using model-based control, more accurate models are necessary. This work is a first study. The next step is to include in the models the flap-wise dynamic of the blades as well as the dynamics of the tower. REFERENCES Ashuri, T., Martins, J. R. R. A. & Zaaijer, G., 2016. Aeroservoelastic design definition of a 20 MW common research wind turbine model. Wind Energy. Ashwini, P. & Archana, T., 2016. Mathematical modeling of wind energy system using two mass model including generator. 3M-R1
International Journal of Emerging Trends in Electrical and Electronics, 12(2), pp. 18-23. Bak, C. et al., 2013. Description of the DTU 10 MW Reference Wind, Roskilde: DTU Wind Energy. Bianchi, F. D., de Battista, H. & Mantz, R. J., 2007. Wind Turbine Control Systems. London: Springer-Verlag. Boukhezzar, B. & Siguerdidjane, H., 2011. Nonlinear control of a variable-speed wind turbine using a two-mass model. IEEE Transactions on Energy Conversion, 26(1), pp. 149-162. Chen, H. & Zhang, Y., 2016. Power System Optimization: Largescale Complex Systems Approaches. Singapore: John Wiley & Sons. Dang, D. Q., Wang, Y. & Cai, W., 2009. A multi-objective optimal nonlinear control of variable speed wind turbine. Christchurch, IEEE, pp. 17-22. Eggleston, D. M. & Stoddard, F. S., 1987. Wind Turbine Engineering Design. New York: Van Nostrand Reinhold. Fortmann, J., 2014. Modeling of wind turbines with doubly fed generator system. Wiesbaden: Springer Vieweg. Friis, J. et al., 2011. Repetitive model predictive approach to individual pitch control of wind turbines. Orlando, IEEE, pp. 3664 - 3670. Gentile, A. & Trudnowski, D., 2002. A low-order wind-turbine model for power system transient stability studies. Portland, AWEA, pp. 1-11. Gonzalez-Longatt, F., Regulski, P., Novanda, H. & Terzija, V., 2012. Impact of shaft stiffness on inertial response of fixed speed wind turbines. Automation of Electric Power Systems, 36(8). Jasniewicz, B. & Geyler, M., 2011. Wind turbine modelling and identification for control system applications. Brussels, EWEA, 280-284. Jelavic, M. & Peric, N., 2009. Wind turbine control for highly turbulent winds. Automatika, 50(3-4), p. 35–151. Jonkman, J., Butterfield, S., Musial, W. & Scot, G., 2009. Definition of a 5-MW Reference Wind Turbine for Offshore System Development, Golden, Colorado: NREL. Leithead, W. E., de la Salle, S. A., Reardon, D. & Grimble, M. J., 1991. Wind turbine modelling and control. Edinburgh, IET. Li, H. & Chen, Z., 2007. Transient stability analysis of wind turbines with induction generators considering blades and shaft flexibility. Taipei, s.n., pp. 1604-1609. Mahmoud, M. S., Hassan, M. F. & Darwish, M. G., 1985. LargeScale Systems. s.l.:Marcel Dekker, Inc..
Table 3. Most important properties of the models 2M-R1 4M-R2 3M-R2 5M-R3
4M-R3
1M-R1
0.172195196155469 6.31613143199687 33.7685374724714 319.587511087030
0.09476529316288 5.67703654144849 7.43023555206874 33.7678465647983 328.204202710417 12217.2433220824
0.0947713406472 5.6789117730058 7.4307353119597 33.768892180226 319.58751108703
1.436896961944
Natural frequencies
0.17151048175339 13.9531121097397 328.429748918498 12216.6102572282
0.17153020734761 13.9586693958588 319.793562256054
0.17217516717445 6.31378691250415 33.7674917547955 328.204202710416 12217.2433220823
Smallest time constant
0.09672192281243
0.0967680722065
0.102956772557
0.10301442267
0.025166626833
0.025225167051
1.025225167051
Sampling time
0.01
0.01
0.01
0.01
0.002
0.002
0.1
-1.2217e+04 -3.2843e+02 -1.34967 ± i 1.3887 -0.1715
-3.1979e+02 -1.3507 ± i 1.3893 -0.1715
-1.2217e+04 -3.2820e+02 -3.8514 ± i 3.355 -0.6500 ± i 6.2802 -0.17217
-3.1959e+02 -3.8517 ± i 3.3548 -0.6506 ± i 6.2825 -0.17219
-1.2217e+04 -3.2820e+02 -3.8513 ± i 3.355 -0.1870 ± i 7.4279 -0.5267 ± i 5.6525 -9.4765e-02
-3.8516 ± i 3.3548 -0.094771 -0.1874 ± i 7.4284 -0.5269 ± i 5.6544 -3.1959e+02
-1.4369 0.0000
2.9059e-03
2.7860e-03
2.8721e-03
2.8090e-03
2.8842e-03
2.8146e-03
1.6132e-03
Poles
Time execution (mean value in sec)
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9901
Rotational speed of generator in rad/s
1.0 0.8 Black: 3M-R1 Red: 2M-R1 (overlapped with black) Magenta: 5M-R3 Blue: 4M-R3 (overlapped with magenta) Green: 4M-R2 Light blue: 3M-R2 (overlapped with green)
0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 0
10
20
30
40
50
t in sec
60
Fig. 9. Simulation results. Mirzaei, M. et al., 2012. Individual pitch control using LIDAR measurements. Dubrovnik, IEEE, pp. 1646-1651. Moriarty, P. J. & Butterfield, S. B., 2009. Wind turbine modeling overview for control engineers. Saint Louis, s.n., pp. 2090-2095. Morisse, M., Bartschat, A. & Wenske, J., 2016. Dependency of the lifetime estimation of power modules in fully rated wind turbine converters on the modelling depth of the overall system. Karlsruhe, IEEE. Muyeen, S. M. et al., 2007. Comparative study on transient stability analysis of wind turbine generator system using different drive train models. IET Renewable Power Generation, 1(2), pp. 131-141. Muyeen, S. M. et al., 2007. Comparative study on transient stability analysis of wind turbine generator system using different drive train models. IET Renewable Power Generation, 1(2), pp. 131-141. Papathanassiou, S. . A. & Papadopoulos, M. P., 2001. Mechanical stresses in fixed-speed wind turbines due to network disturbances. IEEE Transactions on Energy Conversion, 16(4), pp. 361-367. Pournaras, C., Riziotis, V. & Kladas, A., 2008. Wind turbine control strategy enabling mechanical stress reduction based on dynamic model including blade oscillation effects. Vilamoura, IEEE, pp. 1-6. Ramtharan, G., Anaya-Lara, O., Bossanyi, E. & Jenkins, N., 2007. Influence of structural dynamic representations of FSIG wind turbines on electrical transients. Wind Energy, March, 10(4), pp. 293-301. Salman, S. K. & Teo, A. L. J., 2003. Windmill modeling consideration and factors influencing the stability of a grid-connected wind power-based embedded generator. IEEE Transactions on Power Systems, 18(2), pp. 793-802. Sami, M. & Patton, R. J., 2012. An FTC approach to wind turbine power maximisation via T-S fuzzy modelling and control. Mexico City, IFAC, pp. 349-354.
Schlipf, D., Schlipf, D. J. & Kühn, M., 2013. Nonlinear model predictive control of wind turbines using LIDAR. Wind Energy, Volume 16, p. 1107–1129. Shan, W. & Shan, M., 2012. Gain scheduling pitch control design for active tower damping and 3P harmonic reduction. Copenhagen, EWEA. Slootweg, J. G., de Haan, S. W. H., Polinder, H. & Kling, W. L., 2003. General model for representing variable speed wind turbines in power system dynamics simulations. IEEE Transactions on Power Systems, 18(1), pp. 144-151. Sørensen, P., Hansen, A. D. & Carvalho Rosas, P. A., 2002. Wind models for simulation of power fluctuations from wind farms. Journal of Wind Engineering and Industrial Aerodynamics, 90(1215), p. 1381–1402. van der Tempel, J. & Molenaar, D.-P., 2002. Wind turbine structural dynamics - A review of the principles for modern power generation, onshore and offshore. Wind Engineering, 26(4), pp. 211-220. Wasynczuk, 0., Man, D. T. & Sullivan, J. P., 1981. Dynamic behavior of a class of wind turbine generators during random wind fluctuations. IEEE Transactions on Power Apparatus and Systems, PAS-100(6), pp. 2837-2845. Xiao, S., Yang, G. & Geng, H., 2014. Nonlinear pitch control design for load reduction on wind turbines. Hiroshima, IEEE, pp. 543-547. Xu, H., Xu, H., Chen, L. & Wenske, J., 2013. Active damping control of DFIG wind turbines during fault ride through. Istambul, IEEE. ACKNOWLEDGEMENTS This work is financed by the Federal Ministry of Economic Affairs and Energy (BMWi).
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