Identification of Wind Turbine Model for Controller Design

Transcription

1 Identfcaton of Wnd Turbne Model for Controller Desgn M. Jelavć *, N. Perć *, I. Petrovć * * Unversty of Zagreb / Faculty of Electrcal Engneerng and Computng, Zagreb, Croata Abstract Wnd power ncreases rapdly wth ncrease of wnd speed. In order to keep wnd turbne runnng even at strong wnds wnd power capture has to be constraned. Effcent way for constranng wnd power capture s the use of ptchable blades. For ptch controller desgn a sutable wnd turbne model s needed. Fndng mathematcal model that would be a good representaton of wnd turbne system and at the same tme sutable for controller desgn s very dffcult task snce wnd turbne system s strongly nonlnear. Descrpton of wnd turbne system usng many lnear models dentfed at partcular operatng ponts s explored n ths paper. Effects occurrng when swtchng between models are examned and ther repercussons on controller desgn are dscussed. Use of fuzzy logc s proposed as possble soluton for avodng negatve effects caused by swtchng between models. I. INTRODUCTION Wnd turbne operaton dependng on wnd speed can be dvded n two very dfferent operaton regons. The frst regon descrbes wnd turbne operaton durng weak wnds. In ths regon power contaned n the wnd s lower than the rated power output of wnd turbne generator. Therefore, the man task of control system n ths regon s to maxmze wnd turbne power output by maxmzng wnd power capture. The second operaton regon descrbes wnd turbne operaton durng strong wnds. In ths regon wnd power s greater than the rated power output of wnd turbne generator and t ncreases wth the thrd power of wnd speed [1]. The man task of control system n ths regon s to constran wnd power capture. A method for constranng wnd power capture that s analyzed n ths paper s ptchng of rotor blades. Ptchng of the blades changes the angle between blade chord and local wnd speed. Ths angle s termed "angle of attack". Increase n angle of attack results n reducton of wnd energy capture [1], what makes blade ptchng an effcent way for wnd power capture constranng. The border between two operaton regons s the lowest wnd speed at whch turbne generator reaches ts rated power output. Ths wnd speed s termed "rated wnd speed". Hence the wnd turbne operaton regons are known as below and above rated regons. In ths paper we shall focus on above rated operaton regon. Ths paper s organzed as follows. Secton II gves the descrpton of wnd turbne power control system n both operaton regons. In secton III wnd turbne rotor speed behavor n above rated operaton regon s modeled usng set of lnear models dentfed at partcular operatng ponts. Behavor of dentfed models and effects that occur when swtchng between them are analyzed n ths secton. The use of fuzzy logc as an alternatve approach to wnd turbne modelng s proposed n secton IV. II. WIND TURBINE CONTROL SYSTEM The man task of wnd turbne control system s enablng contnuous power producton under all operatng condtons determned by varous wnd speeds. As output power s drectly proportonal to generator speed, power control can be done by controllng generator speed. In ths paper we focus on wnd turbnes that use generators drectly connected to ther rotor,.e. wthout a gearbox. For those turbnes, whch are known as drect drve systems, generator speed equals turbne rotor speed. The prncple scheme of wnd turbne rotor speed control system s shown n Fg. 1. In ths paper, as seen n Fg. 1., converson of wnd energy nto rotor knetc energy s consdered as a black box. Detals on ths converson physcal bass can be found n e.g. [1]. We must pont out that ths converson s hghly nonlnear and s dependent on current wnd speed v w, turbne rotor speed ω and ptch angle β as shown n Fg. 1. Generator electromagnetc torque M g opposes drvng mechancal torque M t whch s caused by wnd and ther dfference accelerates/decelerates turbne rotor. Frcton torque M fr s mostly very small and n our consderaton t wll be neglected. As s shown n Fg. 1., turbne rotor speed can be nfluenced and thus controlled by two means by generator torque whch opposes drvng torque and by ptch angle whch alters the wnd energy converson. For ths reason turbne rotor speed control system conssts of two control loops. Both control loops operate all the tme but dependng on operaton regon one of them s domnant. In the below rated operaton regon the torque control loop s used to control turbne rotor speed to values that wll result n maxmal wnd power capture. Ths can be done owng to the fact that wnd turbnes n scope of ths paper are not drectly connected to the electrcal grd but a frequency converter s used as an nterface between turbne generator and the grd. Ths confguraton enables turbne generator to operate at varable speed. Fg. 1. Prncple scheme of wnd turbne control system

2 Ths control loop s not n the scope of ths paper. Detals on ts specfcs can be found n e.g. []. In the above rated regon ths loop holds generator torque at ts rated value but temporally t can allow generator torque to exceed ts rated value n order to produce addtonal brakng torque durng short wnd gusts. The ptch control loop s used for fndng adequate ptch angle that wll keep turbne rotor speed at ts reference value under all operatng condtons determned by varous wnds. Below rated wnd speed ths loop sets ptch angle to value that assures maxmal wnd power capture whch s usually around o. In ths paper we assume that all blades have the same ptch angle what s known as "collectve ptch". Controller n ths loop, although used to control turbne rotor speed, s commonly termed ptch controller. From Fg 1. we could conclude that wnd turbne rotor speed ω can be expressed as: f ( vw, β, M g) ω =, (1) where f s general nonlnear functon. To be able to desgn ptch controller sutable mathematc descrpton of general model (1) s needed. III. WIND TURBINE MODELING In ths paper we consder a generc drect drve ptch controlled wnd turbne wth rated power output of 1 MW and rated turbne rotor speed of 9 rpm. Instead of real wnd turbne system we consder ts model created and smulated usng professonal wnd turbne smulaton software GH []. uses very complex aerodynamc and structural model of wnd turbnes what makes ts results very relable and n accordance wth behavor of real wnd turbnes. Ths has been recognzed by all major standardzaton and certfcaton nsttutons (e.g. Germanscher Llyod) so we have reason to consder results as they were taken on the real wnd turbne. On the other hand expermentng usng allows us greater freedom than expermentng on real turbnes whch s often restrcted due to safety reasons. The wnd turbne system s hghly nonlnear as prevously stated. To prove ths fact on selected generc turbne we conducted followng experment. Wnd turbne model n was excted wth step change of wnd speed. The postve step of 1 m/s occurred at t = 5 s whle at t = 15 s wnd speed stepped down to ts steady state value. Three steady state wnd speeds were consdered: rated wnd speed of 11 m/s, 15 m/s and m/s whch was the hghest speed at whch selected turbne could produce power (cut-out wnd speed). At each of three wnd speeds ptch angle was adjusted to keep turbne rotor speed at ts reference value durng wnd steady state. Durng wnd step change ptch angle wasn't changed nether was generator torque. In other words system was smulated n open loop to determne ts dynamcs. Fg.. shows turbne rotor speed responses to appled wnd speed change at three selected steady state wnd speeds. It can be observed that turbne rotor speed response to the same wnd speed step change s almost 1 tmes faster at wnd speed of m/s than t s at 11 m/s. The magntude of turbne rotor speed change was around 7 tmes greater at rated wnd speed than at cut-out wnd speed for the same Turbne rotor speed (rad/s) v w = 11 m/s v w = 15 m/s v w = m/s Tme (s) Fg.. Response of turbne rotor speed to wnd speed step change magntude of wnd speed change. Equvalent open loop experments were done for ptch angle and generator torque and showed that system dynamcs change more than 1 tmes through above rated operaton regon. The observed stuaton makes t mpossble to model wnd turbne system usng one lnear model no matter how many smplfcatons we allow for. A possble soluton s usng set of lnear models where one model descrbes wnd turbne behavor around one operatng pont. Wnd turbne behavor or operatng pont s determned by wnd speed so t seems straghtforward to use wnd speed as an operatng pont ndcator. Wnd speed can not be measured fast and relably enough to be used as operatng pont ndcator n wnd turbne control system due to many facts that can be found n [7]. Instead of wnd speed, nstantaneous blade angle wll be used as operatng pont ndcator. It can be measured fast and relably enough and, under assumpton that the control system s workng properly, there s unambguous relatonshp between wnd speed and blade ptch angle. We dvded above rated operaton regon n even spaced operatng ponts based on ptch angles from o to o. Each operatng pont has dfferent steady state wnd speed whch wth operatng pont ptch angle keeps the turbne rotor speed at ts reference value. The lnear model that descrbes wnd turbne behavor n -th operatng pont can be wrtten as: ω = + +, () ( k) G ( q ) v ( k) G ( q ) β ( k) G ( q ) M ( k) w w β ref Mg gref where β ref and M gref are ptch angle and generator torque reference values. These reference values are used snce ths form s more sutable for controller desgn. Because of that transfer functon G β wll nclude any dynamcs between ptch angle reference and obtaned value, what s n fact the dynamcs of controlled servo drve used for the blades postonng. Smlarly G Mg ncludes dynamcs of generator and frequency converter. G w models only wnd energy converson and turbne rotor dynamcs so t doesn't nclude any of the control system elements. Although they are completely dfferent n nature, n terms of system modelng all three quanttes from the rght sde of () can be consdered as nput sgnals and so wll be referred to from now on. The lnear model () s gven n dscrete form snce wnd turbne controllers are mostly mplemented dgtally. Relable descrpton of the way n whch rotor blades convert wnd energy to rotatonal torque s gven by combned blade element and momentum theory whch s explaned n [1] and []. Ths methodology s used by

3 and yelds very realstc results that correlate well wth real wnd turbne behavor. The problem wth ths theory s that t uses mplct equatons to descrbe wnd energy converson. Those equatons can only be solved teratvely and can not be used drectly to derve needed explct relatons gven n (). Instead of tryng to fnd analytcal models we tred to estmate expermental models usng system dentfcaton technques. Usng nstead of real wnd turbne allowed us to use artfcal test sgnals to excte wnd turbne system. The frst step was frequency characterstcs dentfcaton of lnear transfer functons G w, G β and G Mg gven n (). Ths has been done to determne frequency band that s relevant for system descrpton and to choose approprate samplng tme for dscrete models. Although wnd turbne rotor dynamcs are relatvely slow t s shown that some hgher frequences are relevant for system behavor as well. The system was excted wth chrp sgnal whose frequency ncreased from to 5 Hz. Only one nput was varyng at the tme whle other two were kept on ther steady state values determned by operatng pont. Fg.. shows part of the chrp sgnal used as reference ptch angle and correspondng turbne rotor speed response at the operatng pont determned wth ptch angle of 5 o. Experment was repeated for each operatng pont. From recorded nput and output sgnals we were able to estmate frequency characterstcs usng spectral analyss descrbed n [4]. Fg. 4. shows estmated frequency characterstc of G β for one operatng pont. It can be seen that frequency characterstc has n general expected low pass character determned by large moment of nerta of the turbne rotor. The phase s postve at low frequency regon as a consequence of negatve gan n transfer functon - ncrease n ptch angle results n decrease of turbne rotor speed. It can be clearly seen that frequency characterstc has rses and dps at certan frequences. Careful analyss of estmated frequency characterstc shows that observed rses and dps occur at frequences that correspond to wnd turbne tower and blades modal frequences. Tower and blades modal frequences can be calculated from structural propertes of wnd turbne components such as mass and stffness dstrbuton []. Analyss shows that these frequences play crucal role n wnd turbne behavor [] and that wnd turbnes of ths type can never be consdered as absolutely stff structures. Smlar nfluence of wnd turbne tower and blades modal frequences was present n G w and G β as well. Reference ptch angle (deg) Turbne rotor speed (rad/s) Tme (s) Fg.. Chrp sgnal used as reference ptch angle and correspondng turbne rotor speed response Magntude Phase (deg) Estmated usng spectral analyss Lnear model Frequency (rad/s) Fg. 4. Estmated frequency characterstcs of G β It was observed that hghest structural modal frequency that sgnfcantly contrbutes to system behavor was somewhat lower than 5 Hz, so we chose samplng tme of.1 s. The next step was estmaton of lnear parametrc models for G w, G β and G Mg. We chose Output Error (OE) structure for two reasons. Lke frst t s very smple and sutable for controller desgn. The second reason was that we dd our experments usng so we could turn off measurements nose and all effects that couldn't be modeled wth lnear models. Those effects were wnd shear and tower shadow. Wnd shear s phenomenon that wnd speed decreases as wnd approaches the ground and tower shadow s the degeneraton of ar flow around tower []. Both effects ntroduce perodcal dsturbances n turbne rotor speed behavor. The dentfcaton experments were done usng Pseudo Random Bnary Sgnal (PRBS) as an exctaton. Its repeatng perod was chosen to be about 1.5 tmes greater than turbne rotor speed open loop rse tme. As wnd turbne dynamcs change wth operatng ponts so does the PRBS duraton and shape. Magntude of PRBS sgnal was 1 o for reference ptch angle,.5-1 m/s for wnd speed dependng on operatng pont and knm for generator reference torque. Model dentfcaton was done n Matlab usng Ident toolbox. For each transfer functon from () we dentfed several models wth dfferent orders at each operatng pont and compared ther behavor on valdaton data that wasn't used for model dentfcaton. The models wth hgher order generally showed better agreement wth valdaton data but after certan order mprovement of model behavor wth order ncrease became very small. So we choose the model wth order after whch no sgnfcant mprovement n model behavor wth order ncrease was ganed to be representatve model for partcular operatng pont. In order to determne the expected model order we performed followng analyss. Rotaton of turbne rotor could be descrbed as 1 st order system f the turbne structure was completely stff. Modal analyss n shows that wnd turbne structural propertes can be well descrbed wth 7 modal frequences 5 for blades and for tower. In addton to that servo drve was modeled as the second order system. Wnd energy converson model ddn't ntroduce addtonal dynamcs. So t turns out that model order for G β should be around 1 th. For G w the same analyss can be done except there s no need to nclude servo drve, whle for G Mg servo drve s replaced wth generator and frequency converter model. Electrcal transents n generator and frequency converter have much faster dynamcs than turbne rotor so for ths analyss they

4 need not to be modeled n detals. In ths paper we used the frst order system to model generator and frequency converter. So the expected orders for G w was 8 th and for G Mg 9 th. Models that were chosen as representatve accordng to prevously explaned crteron were mostly of 11 th order for G β and 8 th for G w and G Mg what confrmed our expectatons. Fg 5. shows lnear model output for one operatng pont when one nput at the tme vares, whle other hold ther steady state values. The turbne rotor speed response obtaned n usng same nput sgnals s gven for comparson. It can be seen that obtaned lnear models can model turbne rotor speed behavor around operatng pont very successfully. The frequency characterstc of lnear model shown n Fg. 4. shows very good agreement wth frequency characterstcs that was estmated usng spectral analyss. Ths leads to concluson that lnear models can relably descrbe nfluence of wnd turbne structural propertes on turbne rotor dynamcs. Ths s very mportant because lnear models wll be used for controller desgn and controller must take nto account structural propertes of the system n order not to drve t nto resonant behavor. The next step was to test how lnear model behaves when all three nput sgnals vary smultaneously. We could vary all three nputs ndependently but t mght lead us to abnormal stuatons that can not happen durng normal turbne operaton (e.g. small wnd speed and large ptch angle). These stuatons can only happen f there s a falure n control system but ths s out of the scope of ths paper. For that reason we used to smulate operaton of wnd turbne system when ts rotor speed s controlled n closed loop lke shown n Fg.1. In that case control system wll react on wnd speed changes by adjustng ptch angle and generator torque. Wnd speed, ptch and torque reference values are consdered as nputs for lnear model () regardless the fact that wnd speed s the only ndependent exogenous nput whle ptch angle and generator torque reference values are produced by control system as reacton to mposed wnd speed. So we ddn't use lnear model () n closed loop but we recorded wnd speed, blade ptch angle and generator torque reference sgnals n and used them as nputs for dentfed lnear models. The nput sgnals recorded n can be seen n Fg. 6. The transent behavor of turbne rotor speed n closed loop s not of our nterest at ths moment. Turbne rotor speed (rad/s).4. Reference ptch angle varatons Reference generator torque varatons Wnd speed varatons Lnear model Wnd speed (m/s) Ref. Ptch angle (deg) Ref. Generator torque (knm) Tme (s) Fg. 6. Input sgnals used to test model behavor around operatng pont Here we are explorng the ablty of dentfed lnear models to approxmate turbne rotor speed behavor when they are excted wth same nput sgnals as model n. The closed loop control s used only to produce reasonable combnatons of nput sgnals that can be expected n real turbne operaton. Let's just menton here that ptch control was done usng PI controller whle torque control was done usng look-up table on measured turbne rotor speed. Fg. 7. shows turbne rotor speed response recorded n and ts approxmaton wth two lnear models that were dentfed at two "nearest" operatng ponts. We could say that overall behavor of models s satsfyng. The steady state error s present for each model as nput sgnals go further from operatng pont at whch model was dentfed. To model wnd turbne behavor through entre above rated operaton regon we used followng method. All models were smultaneously excted wth the same nput sgnals. As an actve output we took the output of model whch was dentfed for the operatng pont that s closest to the current ptch angle. The lnear model () descrbes, of course, devaton of turbne rotor speed from ts steady state value. The nput sgnals at the rght sde of () are also devatons of partcular sgnal from ts steady state value. It should be ponted out that same wnd speed and same ptch angle wll produce dfferent devaton dependng on the partcular operatng pont snce steady state values for wnd speed and ptch angle are dfferent for each operatng pont. On the other hand steady state values for turbne rotor speed and generator torque are always the same and equal to reference values. Ths fact s not mportant when models are used for controller desgn but snce we are now usng models n open loop t should be taken nto account. Turbne rotor speed (rad/s).1.5 Lnear model for 11 o o Lnear model for Tme (s) Fg. 5. Outputs from lnear models when only one nput sgnal s vared Tme (s) Fg. 7. Turbne rotor speed response from and output of two "nearest" models

5 To test the possblty of modelng wnd turbne rotor speed behavor by swtchng between many lnear models we used closed loop system excted wth wnd speed that ncreased from 11 m/s to m/s n steps of 1.5 m/s. The closed loop control was done usng PI controller wth fxed parameters wthout any adaptaton to current operatng pont. That produced greater oscllatons n turbne rotor speed than t should be allowed n practce, but at ths pont we are not concerned wth that as prevously explaned. Part of used nput sgnals can be seen n Fg. 8., whle actve model output n comparson wth turbne rotor speed response from can be seen n Fg. 9. From Fg. 9. we could say that overall performance of ths modelng method s rather good except for the steady state error. Another test usng nput sgnals shown n Fg. 1. was performed when wnd speed, nstead of n steps, ncreased lnearly. As system operated n closed loop, ptch angle ncreased as well keepng turbne rotor speed around ts reference value. The actve model output for ths test n comparson wth turbne rotor speed response from can be seen n Fg. 11. It s notceable that swtchng between lnear models gave poor agreement wth response from. As the operatng pont changed actve model output experenced sudden jumps caused by swtchng between models. Ths leads to concluson that desgnng lnear controllers for each operatng pont and swtchng between them based on current ptch angle, what s very common n practce, can produce sudden jumps n controller output placng undesred stress on ptch drve and entre wnd turbne structure. One possble soluton to ths problem s to use many controllers that operate n parallel and to assure bumpless transton between them as proposed n [5]. Here we shall explore dfferent approach to solvng ths problem by modfyng wnd turbne model. Wnd turbne system can be modeled usng nonlnear models such as ones based on neural networks. We ddn't use ths methodology for the followng reasons. Lke frst, as t can be seen n Fg. 5., and 7. lnear models show good capablty of descrbng turbne rotor dynamcs around operatng pont for whch they were dentfed. They are also capable of modelng nfluence of wnd turbne structural propertes on turbne rotor dynamcs, as can be seen from frequency characterstcs shown n Fg. 4. Another reason why lnear models should be awarded attenton s the fact that they are very sutable for controller desgn. So we decded to keep usng dentfed lnear models for certan operatng pont and to seek the soluton for above dscussed problems usng dfferent swtchng logc. Wnd speed (m/s) Ref. ptch angle (deg) Ref. generator torque (knm) Tme (s) Fg. 8. Input sgnals used for model testng n entre above rated operaton regon Turbne rotor speed (rad/s) Lnear models Takag-Sugeno model Tme (s) Fg. 9. Turbne rotor speed response when wnd speed ncreases n steps IV. FUZZY MODEL OF WIND TURBINE Instead of "sharp" swtchng between models based on current ptch angle we used fuzzy logc for choosng approprate model. Very smple logc was used. For dentfed models (.e. for each operatng pont) we used a fuzzy set that can be named "Around n" where n stands for partcular ptch angle. Usng those sets we could form smple fuzzy rules: If β s "Around n" then ω = ω n, () where ω n s the output of lnear model () calculated at operatng pont for whch ptch angle equals n. Lnear model output n one tme nstant s sharp numerc value that can be regarded as sngleton fuzzy set. Usng ths swtchng logc we use n fact Takag-Sugeno fuzzy approach whch s descrbed n detal n [6]. Output from Takag-Sugeno model n each tme nstant k can be calculated as: ω ( k ) = nr µ β ω = 1 nr = 1 ( ( k) ) ( k) ( ( k )) µ β, (4) where nr s the number of fuzzy rules whch n ths case corresponds to number of chosen operatng ponts and µ s the value of -th membershp functon (MF). The expresson (4) shows how turbne rotor speed s calculated by usng Takag-Sugeno model as a combnaton of many local lnear models. Snce local lnear models have already been dentfed, Takag-Sugeno model output wll depend upon the shape of MFs used n fuzzy rules. Wnd speed (m/s) Ref. Ptch angle (deg) Ref. Generator torque (knm) Tme (s) Fg 1. Input sgnals when wnd speed ncreases lnearly

6 Turbne rotor speed (rad/s) Lnear models Takag-Sugeno model Tme (s) Fg. 11. Turbne rotor speed response when wnd speed ncreases lnearly We choose trangular MFs snce they are qute smple and gve good results, as t wll be shown. Trangular MF for -th operatng pont can be descrbed as: for β < β ( β β ) for β < β < β (5) β β µ ( β ) = ( β β ) + 1 for β < β < β β β for β β, 1 where β s the top of trangular MF, whle β and β are ts base lower and upper ends. The top of trangular 1 MF β was set equal to the ptch angle for partcular operatng pont. Snce t would be very demandng to manually tune MF parameters we used an adaptaton algorthm to tune MF parameter n order to mprove the model behavor. The top of trangular MF wasn't tuned snce t corresponds to ptch angle for operatng pont at whch partcular lnear model was dentfed. In each tme nstant k β and β were adapted n the followng way: 1. Nether of those sgnals was used for MFs adaptaton. Fg. 9. shows output of Takag-Sugeno model when t s excted wth nput sgnals shown n Fg. 8. whle Fg. 11. shows model output when t s excted wth nput sgnals shown n Fg. 1. Output from swtchng lnear models and output are shown as well. From Fg. 9. t can be seen that Takag-Sugeno model elmnates steady state error present for swtchng lnear models. Fg. 11. shows that the output s stll not perfectly matchng output but the accuracy s drastcally mproved. More mportant s the fact that sudden jumps n model output are elmnated as model gradually shfts from one local model to another. It leads to concluson that smlar fuzzy logc n schedulng controller parameters should be used to avod sudden jumps n controller output. CONCLUSION Wnd turbne operaton n above rated operaton regon s consdered. Turbne rotor speed changes are modeled usng lnear models where wnd speed, ptch angle and generator torque are consdered to be model nputs. Lnear models are dentfed at number of operatng ponts usng system dentfcaton methods. Lnear models demonstrated good descrpton of wnd turbne behavor around partcular operatng pont and the ablty to model nfluences of wnd turbne structural propertes on turbne rotor dynamcs. Wnd turbne operaton n entre above rated regon was modeled usng swtchng between lnear models based on current ptch angle. Ths methodology ntroduces sudden jumps n model output n some stuatons. Ths leads to concluson that f turbne rotor speed control s done usng separate controller for each operatng pont and swtchng between them based on measured ptch angle, can ntroduce undesred stress on ptch drve and entre wnd turbne structure. The use of fuzzy logc s proposed as an alternatve method for choosng operatng pont and swtchng between local lnear models. Ths logc leads to Takag-Sugeno fuzzy approach n modelng nonlnear systems. It showed better accuracy and removed unwanted jumps from model output. Ths leads to concluson that same swtchng logc should be used for controller parameter schedulng. ( ) ( k ) ( ) ( k ) J k β ( k+ 1) = β ( k) K β J k β ( k+ 1) = β ( k) K β, where J s crteron functon whch was chosen to be: 1 ( ) = e( k) J k In (7) e(k) s the dfference between model output and tranng sgnal whch was taken to be output from. The gan K n (6) was used to control adaptaton speed. At one tme nstant k only one MF was modfed usng (6). Ths was the MF whose modfcaton lead to the greatest reducton of crteron functon J. We wll show model behavor, after adaptaton of MFs was completed, usng nput sgnals shown n Fg. 8. and. (6) (7) REFERENCES [1] G. S. Heder, "Grd ntegraton of wnd energy converson system", John Wely & Sons, 1998 [] P.Novak, T.Ekelund, I. Jovk and B. Schmdtbauer, "Modelng and Control of Varable-Speed Wnd-Turbne Drve-System Dynamcs", Control system magazne, vol. 15, num. 4, August [] Theory manual, GH report, 8/BR/9, December. [4] L. Ljung, "System dentfcaton: Theory for the User", Prentce- Hall, New Jersey, [5] I. Kraan and P. M. M. Bongers, "Control of a wnd turbne usng several lnear robust controllers", Proceedngs of the nd Conference on Decson and Control, San Antono, Texas, December 199. [6] T. Takag, M. Sugeno, Fuzzy Identfcaton of Systems and ts Applcaton to Modelng and Control", IEEE Transactons on Systems, Man and Cybernatcs, Vol 15, No. 1. pp , January/February [7] M. Jelavć, N. Perć and S. Car, "Estmaton of Wnd Turbulence Model Parameters", Proceedngs of the 5 Internatonal Conference on Control and Automaton, Budapest, Hungary, June 5.