Modelling and Analysis of Variable Speed Wind Turbines with Induction Generator during Grid Fault Bolik, Sigrid Mechthild

Transcription

1 Aalbog Univeitet Modelling and Analyi of Vaiable Speed Wind Tubine with Induction Geneato duing Gid Fault Bolik, Sigid Mechthild Publication date: 4 Document Veion Publihe PDF, alo known a Veion of ecod ink to publication fom Aalbog Univeity Citation fo publihed veion (APA): Bolik, S. M. (4). Modelling and Analyi of Vaiable Speed Wind Tubine with Induction Geneato duing Gid Fault. Aalbog Univeitet: Intitut fo Enegiteknik, Aalbog Univeitet. Geneal ight Copyight and moal ight fo the publication made acceible in the public potal ae etained by the autho and/o othe copyight owne and it i a condition of acceing publication that ue ecognie and abide by the legal equiement aociated with thee ight.? Ue may download and pint one copy of any publication fom the public potal fo the pupoe of pivate tudy o eeach.? You may not futhe ditibute the mateial o ue it fo any pofit-making activity o commecial gain? You may feely ditibute the UR identifying the publication in the public potal? Take down policy If you believe that thi document beache copyight pleae contact u at vbn@aub.aau.dk poviding detail, and we will emove acce to the wok immediately and invetigate you claim. Downloaded fom vbn.aau.dk on: janua, 8

2 Modelling and Analyi of Vaiable Speed Wind Tubine with Induction Geneato duing Gid Fault SIGRID M. BOIK Intitute of Enegy Technology Aalbog Univeity Danih Academy of Technical Science Riø National aboatoy Veta Wind Sytem A/S

3

4 Modelling and Analyi of Vaiable Speed Wind Tubine with Induction Geneato duing Gid Fault by Sigid M. Bolik Intitute of Enegy Technology Aalbog Univeity, Denmak Octobe 4 I

5 II

6 Aalbog Univeity Intitute of Enegy Technology Pontoppidantaede DK 9 Aalbog Eat Copyight Sigid M. Bolik ISBN III

7 Twenty yea fom now you will be moe diappointed by the thing that you didnt do than by the one you did do. So thow off the bowline. Sail away fom the afe habo. Catch the tade wind in you ail. Exploe. Deam. Dicove. -Mak Twain IV

8 V

9 Abtact Duing ecent yea wind tubine technology ha undegone apid development. Gowth in ize and the optimization of wind tubine ha enabled wind enegy to become inceaingly competitive with conventional enegy ouce. A a eult today wind tubine paticipate actively in the powe poduction of eveal countie aound the wold. Thee development aie a numbe of challenge to be dealt with now and in the futue. The penetation of wind enegy in the gid aie quetion about the compatibility of the wind tubine powe poduction with the gid. In paticula, the contibution to gid tability, powe quality and behaviou duing fault ituation play theefoe a impotant a ole a the eliability. The intoduction of the peent wok biefly peent the development of wind tubine technology. Seveal wind tubine type ae dicued and the motivation fo thi poject ae tated. The main motivation ae the challenge elated to the gid connection of wind tubine. The econd chapte elucidate ecent thinking in the aea of gid connection by dicuing eveal gid code o gid equiement. In the dicuion it i tied to peent the view of the tanmiion line opeato a well a the challenge wind tubine manufactue uch a Veta faced. Modelling ha an impotant ole in the eeach and development of ytem change becaue it allow many difficult quetion to be anweed. The vaietie of challenge that mut be addeed in uch modelling ae not met by any ingle modelling oftwae pogam. In addition a huge ange of in-houe pogam fom diffeent companie exit, the mot widely known oftwae fo cuent eeach on the powe gid ae PSS/E, EMTDC/PSCAD and DigSilent. In geneal eeach and epecially fo contol development the oftwae Matlab/ Simulink i ued. In all thee pogam implified geneal model fo diffeent eeach pupoe o pecific detailed model have been developed and ae ued today. Howeve, ince wind tubine technology only ha ecently had to adde gid elated poblem, model do not atify the demand peculia to thi technology. In paticula, the ole of both the wind tubine itelf and the gid with epect to thei influence, inteaction and behaviou unde gid fault o aymmetical opeation need futhe invetigation. The peent thei theefoe deal with the development of an impoved wind tubine model. Fo the modelling the well known and widely ued pogam Matlab/ Simulink ha been choen. The pogam i a poweful tool fo olving and epeenting mathematical diffeential equation and theefoe povide the oppotunity to combine a wide ange of diffeent modelling iue. The impovement of the lage doubly-fed induction geneato model a an inteface between the mechanical and electical chaacteitic of a wind tubine take a cental pat in thi eeach poce. Chapte 3 peent the development and implementation of a detailed analytical thee-phae induction machine model. The model ue only a minimum et of paamete a upplied by the geneato manufactue of Veta. The accuacy of the imulation i impoved by including the atuation of the main and the leakage inductance, which ha been validated by tet bench meauement. The influence of the geneato atuation effect ae dicued with epect to the behaviou of the machine itelf and to the complete wind tubine behaviou. Although the geneato model i of pimay inteet fo impoving the wind tubine model, the implementation of the othe element completing a total wind tubine model have been challenging a well. In chapte 4 the development of a thee-phae tanfome model and epecially a thee-phae autotanfome a ued in a Veta Tubine i hown. Chapte 5 deal with the development of tanmiion line and electical element connecting the tanfome and geneato and giving the poibility to ceate a mall VI

10 electical netwok and theeby give the poibilitie to eeach the dynamic intoduced by a fault in the gid. The wind tubine model ae completed by modelling the mechanical and aeodynamical pat of the wind tubine and include the contol ytem. In thi thei model of the two diffeent wind tubine type, one with puled oto eitance contol (Veta OptiSlip contol) and anothe one with doubly-fed contol (Veta OptiSpeed contol) have been implemented. Finally two fault ituation have been tudied with the two developed wind tubine model. The influence on the geneato a well a the behaviou of the wind tubine duing thee fault ae biefly dicued baed on the imulation eult. It can be concluded, that it ha been poible to build advanced wind tubine model in Matlab/ Simulink fo eeaching fault ituation. Howeve, one concluion of thi thei i that Matlab/ Simulink might not be the ideal tool fo tudying a detailed complete model of a wind tubine including a mall powe ytem. The complexity of the diffeential equation combined with a huge numbe of iteation pocee lead to intabilitie, which limit the ue of the model. Invetigation in optimiation of the implemented model ae advied. Othewie the peented model can be een a bai fo futhe modelling invetigation. The developed model ae open fo futhe extenion fo diffeent pupoe, e.g. eeach of the hamonic. The intoduced atuation effect of the machine definitely have an influence epecially duing the imulation of aymmetic fault ituation and hould be included in model fo invetigation into the fault ituation fo wind tubine, even though the influence i much le than expected at the beginning of thi wok. The educed influence of atuation i elated to the contol of the machine, which baically compenate fo the eo caued by the machine natual behaviou. The atuation effect i deceaed paticula duing the doubly-fed opeation with a vey tong contol. It doe howeve inceae again, if the contol ytem i diabled, a it i initiated fo convete potection duing fault ituation, the effect elated to atuation of the machine inceae again. Futhe wok may be done on the invetigation on the influence of the potection ytem o futhe validation of the whole tubine ytem. Futhemoe it can be aid, that the developed two wind tubine model can be ued fo the eeach of gid fault. The model i amenable to futhe development fo diffeent eeach inteet, fo intance the development of contol tategie o the development of a detailed gea model fo the eeach of tee. VII

11 Acknowledgement Thi thei i initiated fom my daily wok with Veta Wind Sytem A/S and i caied out unde the Indutial PhD fellowhip EF 94. The poject tated on the.. and ended at the 3.9.4, whee the witing of thi thei ha been finihed a well. The poject got financially uppoted by Veta and Oticon though the Danih Academy of Technical Science (ATV). The thei i ubmitted to the Faculty of Engineeing and Science at Aalbog Univeity a patial fulfilment of the equiement fo gaining the PhD degee in Electical Engineeing. Thee have been thee main upevio following the poge of the poject. Pofeo Fede Blaabjeg fom Aalbog Univeity, Poul Soeenen fom Rioe National aboatoy and Toben W. Moelle fom Veta Wind Sytem A/S. I would like to thank all thee fo thei uppot, patient and help in finihing thi ambitiou poject in the given time. I want to thank a Helle, Poul Bandt Chitenen and Thoma Kuege fo poviding pat of wind tubine model to thi tudy, which made it poible to achieve two complete wind tubine model. Futhemoe I like to thank all the people I met on my way fom countie all ove the wold, giving me the poibility to dicu my idea and giving me feedback to my wok and theewith having paticipated in the wok. Epecially I like to highlight the wondeful poibilitie given by the Nodic Netwok fo Multi Diciplinay Optimied Electic Dive (NoMUD; actively uppoted by Pofeo Ewen Ritchie (Aalbog Univeity). I like to give futhe thank to the R&D depatment of ABB in Vaaa and Helinki to help me claifying ome quetion aound machine technology and Veta, who made thi poible. Epecially I want to thank Pofeo Robet E. Betz and the PhD tudent fom Newcatle Univeity, helping me duing my 4 month tay aboad. They paticipated actively to eflect on my accomplihed wok and boaden my view not only in cientific matte. Othe thank i to the Autalian I became fiend with duing my tay aboad and who helped me taying motivated duing the lat phae of my wok. Peonally I want to thank Pia Salminen (appeenanta Univeity of Technology), Ana- Mai Tatau (Aalbog Univeity) and Julija Matevoyan (Royal Intitute of Technology in Stockholm ), who duing my PhD became vey good fiend. Finally I apologie fo all the people they feel fogotten in thi lit. Ringkøbing, Septembe 4 Sigid M. Bolik VIII

12 IX

13 TABE OF CONTENTS ABSTRACT... VI ACKNOWEDGEMENTS... VIII CHAPTER.... INTRODUCTION.... WIND TURBINE TECHNOOGY.... MOTIVATION AND AIM OF THE PROJECT THESIS OUTINE...6 CHAPTER...9. GRID CONNECTION OF WIND TURBINES...9. TRANSMISSION SYSTEM OPERATOR VIEW...9. WIND TURBINE PRODUCERS VIEW....3 CONNECTION REQUIREMENTS Voltage citeia Fequency equiement Active and Reactive powe equiement Dicuing the equiement...5 CHAPTER INDUCTION MACHINE MODES GENERA EQUATIONS IN ABC/ABC REFERENCE FRAME Complex pace Vecto epeentation Uing pace vecto fo the geneal machine equation THE CARK TRANSFORMATION (α, β, EQUIVAENT FRAME) THE PARK TRANSFORMATION (D, Q, EQUIVAENT FRAME) SATURATION EFFECTS Satuation of main eactance Satuation of leakage eactance WINDAGE OSSES AND FRICTION OSSES THE THREE PHASE MODE IN MATAB / SIMUINK Steady tate validation of the diffeent detailed ABC/abc model Dynamical validation of the ABC/abc model Implementation and validation of leakage eactance atuation Influence of the machine coupling...5 CHAPTER MODEING OF TRANSFORMERS THE -WINDINGS 3-PHASE TRANSFORMER MODE THE 3-WINDINGS 3-PHASE TRANSFORMER MODE THE -WINDINGS 3-PHASE AUTOTRANSFORMER THE TRANSFORMER MODES IMPEMENTED IN MATAB/SIMUINK GEOMETRY DEVEOPMENT OF THREE-PHASE TRANSFORMERS...68 CHAPTER POWER SYSTEM MODEING IN MATAB/ SIMUINK DYNAMIC NODE TECHNIQUE POWER SYSTEM EEMENTS Geneal modelling Tanmiion line modelling Model implementation in Matlab/ Simulink...86 X

14 CHAPTER WIND TURBINE MODEING AERODYNAMIC MODE PITCH MODE MECHANICA MODE IBRARY FOR WIND TURBINE MODEING IN MATAB/ SIMUINK...94 CHAPTER CONTRO OF A TURBINE WITH CONTROED ROTOR RESISTANCE THE ROTOR CURRENT CONTRO Modelling of the oto cuent contol THE POWER CONTRO THE SPEED CONTRO THE PITCH CONTRO VAIDATION OF THE OPTISIP WIND TURBINE MODE DISCUSSION... CHAPTER WIND TURBINE WITH A DOUBY-FED INDUCTION GENERATOR CONTRO THE PRINCIPE OF DFIG CONTRO THE DFIG ROTOR CONVERTER CONTRO THE DFIG GRID CONVERTER CONTRO SIMUATIONS WITH THE ROTOR- AND GRID CONTROER Simulation of powe tep fom patial load to full load Simulation of a peed tep Validation of the DFIG contol ytem Dicuion of the imulation eult THE OPTISPEED TM WIND TURBINES OVERA CONTRO STRATEGY...8 CHAPTER FAUT SIMUATIONS WITH TOTA TURBINE MODES WIND TURBINE MODE WITH VARIABE ROTOR RESISTANCE Simulation of a 5% voltage dip Simulation of a phae hot cicuit DISCUSSION WIND TURBINE MODE WITH DFIG Simulation of a 5% voltage dip Simulation of phae hot cicuit DISCUSSION OF THE SIMUATION RESUTS...54 CHAPTER CONCUSION SUMMARY OF THE WORK RESUTS OBTAINED IN THE THESIS FUTURE WORK...59 CHAPTER...6. APPENDIX...6. REFERENCES...6. GOSSARY OF SYMBOS COORDINATE TRANSFORMATION...7 XI

15

16 Chapte. Intoduction The poduced electical powe fom wind ha damatically inceaed in the lat yea. Theefoe today wind tubine, which typically ae centalied in wind pak, have a ignificant influence on the powe poduction. Beide the mege of eveal wind tubine into bigge unit in wind fam fo inceaing the poduction fom wind enegy the wind tubine itelf ha developed in ize and technology apidly in ode to extact moe enegy fom the wind and educe the cot pe poduced kwh. Duing thee development the wind tubine ha to ovecome vaiou kind of poblem and mate new challenge.. Wind tubine technology Thee exit a huge ange of poible wind tubine configuation. Mot commonly wind tubine ae oted into the two majo categoie of fixed peed tubine and vaiable peed tubine. Thee i a wide ange of topologie claified into thee goup. A compaion and dicuion of a few concept can be found in /3/, /34/, /35/, /36/, //. The mot commonly ued concept ae pictued in Figue Fixed peed wind tubine equipped with aynchonou o induction geneato diectly connected to the gid i one of the oldet and implet concept, which wa fit ued in Denmak, and theefoe known a the Danih concept (ee Figue a). Vaiable peed tubine with induction o ynchonou geneato, diectly o indiectly connected to the gid have become moe and moe impotant in ecent yea (ee in Figue concept b) f)). Poibilitie uch a mechanical load eduction and moe efficient enegy poduction, due to intelligent contol tategie, make thee concept a beneficial economic olution. Futhemoe thee type ae bette gid compliant compaed to the fixed peed tubine. The poibility to independently contol the peed decoupled fom the gid, give the poibility to decouple fequencie eulting fom the wind fluctuation fom the gid, which educe flicke contibution. The convete integated in vaiable peed wind tubine give the poibility to actively contol the powe output of the wind tubine, which i inceaingly impotant fo the integation of wind tubine into the gid /3/, //, /4/. The fixed peed wind tubine i a elf contolled concept. With inceaing wind peed the lamina wind cuent flow aound the fixed blade pofile beak and the tubine loe the poibility to obtain enegy fom the wind, the tubine tall. The quiel cage induction geneato which tanfom the obtained mechanical enegy into electical enegy i diectly connected to the gid. The peed of the geneato may diffe fom the gid fequency due to the lip vaiation of the geneato, which i up to %. The vaiable peed wind tubine i uually equipped with a pitch contol, whee the blade can be tuned to inceae o deceae lift foce on the blade pofile and theeby continuouly contol enegy aboption fom the wind. The active pitch contol i deigned to optimize the powe obtained fom the wind by changing the otation peed of the oto and the pitch angle and theewith gain an optimum cuent flow aound the blade. The theewith achieved vaiable peed ange at the tubine haft o geneato axe i diffeent fom the fixed fequency of the powe ytem to 5Hz o 6Hz. A diect coupling of a

17 ynchonou geneato to the gid i theefoe not poible, and a quiel cage induction machine i too mall in peed vaiation poibilitie (<%), which would limit powe poduction only in a ynchonim of the haft peed and gid fequency. To enable an efficient powe poduction at a huge ange of diffeent wind peed the mechanical peed ha to be decoupled fom the gid fequencie. One method to decouple the two ytem i to ue a full cale powe convete between the geneato and the gid. Thi may give a peed vaiation up to %. The ucce of thi concept ha been limited ove many yea due to technical development in the aea of powe electonic and aociated cot. a) diect gid connected aynchonou geneato with hot cicuit oto d) gid connection via convete and ynchonou geneato with excitation ytem geabox n ASG f geabox n SG DC f n (-) f / p...8 (output dependent) inductive powe conume b) gid connection via convete of an aynchonou geneato with hot cicuit oto n.5... f / p (contollable) ) with thyito convete - inductive powe conume ) with pule invete - contollable eactive powe output e) gid connection via convete and ynchonou geneato with excitation ytem geabox n ASG f n SG DC f n f / p (contollable) inductive powe conume c) dynamic lip contol with lip ing geneato n.5... f / p (contollable) ) with thyito convete - inductive powe conume ) with pule invete - contollable eactive powe output f) doubly-fed lip ing aynchonou geneato geabox n ASG f geabox n ASG f n (-) f / p...(.3) (output dependent, dynamic) inductive powe conume n f / p (output dependent, dynamic) inductive powe conume Figue Example of geneato concept fo wind tubine /35/ (note the ued figue fom the book define poitive lip value fo geneato opeation) Anothe way to connect a vaiable peed wind tubine to the gid i to ue a doubly fed induction geneato (DFIG). Wind tubine equipped with DFIG have become moe and moe common duing the lat yea. It combine the advantage of pitch contol with an efficient tanmiion of the powe to the gid and the poibility of dynamic contol of active and eactive powe. Wind tubine technology benefit fom the development in the aea of dive contol. Eential poge in the dynamical contol of machine bought the intoduction of field oiented contol by Blachke /7/. In the following yea until today eeach to impove the pinciple ha been done /6/, /33/, /36/ - /38/, /59/, /64/, /7/, /8/.

18 In a ytem with DFIG the convete i placed to feed into the oto of the machine while the tato i diect connected to the gid. Though the convete it i poible to contol the upply, o extact the enegy to the oto of the induction machine. Theeby the machine can be contolled to un between ub ynchonou peed and ove ynchonou peed (peed highe than ynchonou peed). Uually a vaiation fom -4% to +3% of ynchonou peed i choen. The total peed vaiation i between 6% and 7%. Unde thee condition the powe convete ha only the ize of a thity, maximum a foty of the ated powe, which i beneficial both economically and technically. In the wold leading company Veta the wind tubine concept with DFIG contol i known a the OptiSpeed tubine (Figue ). P, Q DC P Pwind geabox n DFIG P, Q P, Q pitch Figue Veta OptiSpeed wind tubine with Doubly Fed Induction Geneato Beide thi concept Veta alo ue anothe concept, which technically i a vaiant between the fixed peed tubine and a vaiable peed tubine. The o called OptiSlip i baed on the concept of peed o toque contol while uing vaiable impedance at the oto of the induction machine (imila a Figue c). Veta ue thi concept to achieve a peed vaiation of % (ub ynchonou). The benefit ae the implicity of the ytem, a in a fixed peed tubine but at the ame time having the poibility of flicke egulation and load minimiation epecially at high wind peed.. Motivation and Aim of the Poject A wide ange of eeach and development in the field of wind tubine technology to find the optimal concept fo wind tubine opeation ha been done in the pat. The main focu theeby wa the maximiation of enegy poduction fom the wind and cot eduction. The inceaing numbe of wind tubine connected to the gid hifted the focu of the eeach towad gid tabiliation. With the pefeed boad ue of the Danih concept the awaene of new powe quality iue ha ien. The fixed peed wind tubine couple the dynamic of the wind diectly to the gid though the induction geneato, which caue a poo quality powe output of the tubine. An inceaed numbe of thi wind tubine type integated into the gid affect the powe quality of the whole ytem. Even the vaiable peed wind tubine ae geneally bette gid compatible than fixed peed wind tubine, Netwok opeato, who have to enue well function powe tanmiion and upply to thei cutome have given the wind tubine moe and moe attention. Induction geneato fo enegy poduction ae not known, epecially not of the 3

19 G G G... G G G G G G ize today ued in today wind tubine (NEG Micon 4.MW, Veta 3.MW). Hitoically powe poduction i made by ynchonou geneato. Theefoe thee ae no expeience with the integation of a lage numbe of induction geneato into the gid. Fo anweing thee quetion, meauement have been caied out and imulation model developed, mainly fo the fixed peed wind tubine with quiel cage induction geneato. Reult of the eeach fom impact on the voltage quality and powe fluctuation ae hown in /5/, /5/, /9/, /45/, /73/, /89/, //. Impotant tak ae the gid connection and theeby connection and diconnection of the wind tubine, gid diappeaance, hot cicuit powe contibution, voltage dip and powe quality duing the poduction //, /44/, /56/, /58/, /76/, /77/, /85/. Netwok opeato tated to intoduce gid connection equiement fo the connection of wind tubine all ove the wold //, /6/, /7/, /6/, /67/, /9/. Howeve, the equiement ae fo all tubine type and the development of wind tubine model fo the tudy of vaiable peed wind tubine with DFIG i till being developed. Wind tubine with DFIG have the advantage of fat electical contol. Thi give the oppotunity to ue a wind tubine like a powe plant. With pecial contol tategie the wind tubine can even uppot the gid e.g. the powe facto /84/, /9/, /93/. Howeve exiting model ae mainly made with the pupoe to develop contol tategie and theefoe do not expe the dynamic behaviou of the whole tubine on the gid, e.g. influence of dive tain ocillation and tanfome aymmety i miing //, /75/, /84/. WT... WT WT Contol & Potection Contol & Potection load WT WT WT Wind N N IG U gid WT WT WT Figue 3 Powe Sytem Stability eeach with a wind fam connected to a lage gid Figue 4 Wind Tubine te eeach AC AC P Whee on one ide the netwok opeato need wind tubine model to enue well functioned powe ytem (Figue 3) on the othe ide wind tubine manufactue have to deign a high quality poduct fo a eaonable pice. A the wind tubine affect the gid, the gid affect the function of the wind tubine itelf. Detemining the load caued by voltage dip, aymmetie o hot cicuit on the electical a well a mechanical ytem equie detailed model (Figue 4). Thee model contibute with a detailed knowledge about e.g. the toque on the wind tubine haft caued by fault in the gid o tanient cuent ie which i impotant fo the etimation of the load on the wind tubine. An accuate a poible knowledge enue the availability of the tubine fo the etimated life time and low cot a well a help fo the deign of new wind tubine. The typical model tudie equiement ae ummaized in Table -. 4

20 Table - Typical tudy deciption fo wind tubine model Netwok opeato model Wind tubine manufactue Type of analyi Powe ytem tability /55/ Wind tubine tability - Steady tate (load flow) - Tanient dynamic - Tanient dynamic (lage ditubance) - oad analyi - Small ignal tability (mall ditubance) - Stability tudie including hot and long tem dynamic (angula, fequency, voltage tability) Time cale m min u m Model type - tubine model a wind tubine unit -> eveal - detailed tubine model with e.g. unit in wind fam detailed tanfome model, geneato model Wind tubine modelling i theefoe an impotant pat in the tudy of contol, deign, poduction and gid integation of wind tubine. One impotant iue while developing model i the claification of the pupoe of the model. Any aumption made duing the model development, which lead to implification of the model itelf, ae impotant in ode to validate the coectne of the eult. Ceating a complete wind tubine model with pedefined element in the available pofeional oftwae like EMTDC/ PSCAD, DigSilent o PSS/E can intoduce a difficult tak to find the aumption duing the development of the pogam. Futhemoe ome element ae not epeented in the oftwae package and foce eithe modification of the model with an equivalent deign while uing available element o the ceation of own model. Validation of the coect behaviou of the total wind tubine model unde thee cicumtance can be quite a difficult tak. The mot common way fo claifying the coectne of the model i the ue of meauement, although, in ome opeation it might be difficult to obtain ueable meauement e.g. hot-cicuit of a 3MW wind tubine. But epecially thee abnomal ituation have ecently come in the focu. Although netwok opeato and wind tubine manufactue have a lot of imila inteet and equiement, wind tubine poduce have an additional focu on the poduct itelf, which equie the ue of additional oftwae tool. The effectivene of the model in tem of imulation peed i theeby not that a impotant a the quality of the imulation eult. With conideing the diffeent apect towad wind tubine imulation fo a wind tubine manufactue uch a Veta the following equiement to a wind tubine model can be tated a. Development of a total wind tubine model which i appopiate fo fault imulation into one pogamme. Development of an open wind tubine model with an ability to add featue fo futue wind tubine development. Development of a detailed geneato model able to handle aymmetie, fault opeation, a huge ange of diffeent geneato and a minimum data input. Poible eeach on diffeent geneato model to gain knowledge about geneato effect e.g. atuation influence. The main goal of the poject can be concluded in the following point:. Claification about the equiement due to wind tubine opeation with focu on non-nomal opeation mode.. Achievement of a geneato model in Matlab/ Simulink handling the above mentioned equiement. 3. Implementation of two total pecific wind tubine model in Matlab/ Simulink appopiate fo gid fault imulation tudie. 4. Reeach of the geneato, gid fault influence to the wind tubine. 5

21 .3 Thei outline Thi thei deal with the development and eeach of wind tubine model fo the imulation of wind tubine opeation in nomal and epecially abnomal opeation. The model ae developed with the pecial focu on imulation of the tubine in gid fault ituation and othe tough opeation mode (gid code). A elf developed model ha the advantage that aumption and limitation ae known. Thi model can be ued additionally to veify model in othe pogam and eveal potential diffeence, which would othewie be ovelooked. Anothe advantage i the feedom to develop the model expeing the wind tubine a it exit in eality, while uing the available data. Fo the peent wok the known tool Matlab/ Simulink ha been choen. Matlab i a wold known and ecognied pogam. The gaphical ue inteface Simulink i widely known fom the eeach in contol tategie and give the advantage to olve time dependent mathematical equation without deepe pogamming kill and eem theefoe ueable oftwae fo the olving the tak. The main pat of the thei theefoe deal with the modelling and implementation of diffeent component of the wind tubine in the fom of a libay, which give a vaiety of imulation poibilitie. Special focu ha been given to the modelling of the geneato. Detailed modelling of the geneato i cucial in ode to gain ealitic behaviou of the wind tubine duing fault. Simplification of the model i theefoe only limited poible. The main limitation to the model i the available et of data and the oftwae itelf. A thee phae geneato model implementation i a futhe focu of the wok, but othe known implified model will be dicued a well. The tanfome model and the netwok model ae implemented in diffeent gade of detail and can be abitaily exchanged in the total wind tubine model. The two common Veta tubine type OptiSlip and OptiSpeed have been implemented a example fo the wind tubine modelling tudy. In Figue 5 modelling focu i viualied. 6

22 imple model detailed model imple model Contol & Potection Wind OptiSlip with Veta Roto Cuent Contol N N IG gid OptiSpeed with Veta Convete Sytem IG AC AC Figue 5 Schematic of modelling the Veta OptiSlip and OptiSpeed ytem With thee two wind tubine model example of the tee occuing duing a tandad fault will be hown. The type of the fault i a typical ituation a dicued in gid connection equiement, which will be dicued in the beginning of the thei. Following one main focu of thi wok, which i the development of a geneato model fo a wind tubine unde gid fault, the thei tat with the claification of the equiement with looking at the ecent gid code dicuion. To point out the main eeach aea thi tudy i followed diectly by the development of the detailed model; the electical component (ee Figue 5). To continue the detailed modelling of the electical component of the wind tubine and finih the electical development iue the thei will decibe the gid, befoe hifting to the le dicued topic of mechanical component modelling of the wind tubine. The mechanical and aeodynamical model of wind tubine ae the focu in many othe invetigation and theewith will not be dicued a much, although it i impotant fo the completion of a total wind tubine model and can theefoe not be left out of the deciption. The imila matte i the contol tategie. The both peented contol tategie ae not the focu of the eeach, although have an impotant influence on the behaviou of the wind tubine, and theewith ae biefly dicued in the end pat of the thei befoe etting the focu towad the fault imulation with the wind tubine model. 7

23 The content of the thei in chapte afte thi intoduction i: Chapte Gid connection of Wind Tubine give an oveview ove the main point in the ecent gid connection dicuion. Chapte 3 Induction Machine Model how the development of induction machine model, including atuation effect. In thi chapte pace vecto theoy i intoduced and applied fo machine modelling. Chapte 4 Modelling of Tanfome explain the development of thee-phae tanfome model. Chapte 5 Powe ytem modelling in Matlab/ Simulink decibe a olution to connect electical element in Simulink. Additionally tanmiion line theoy and modelling ae intoduced. Chapte 6 Wind Tubine modelling how the modelling of the non-electical component of a wind tubine excluive the contol ytem. Chapte 7 Contol of a wind tubine with contolled oto eitance explain on the ued example of the Veta OptiSlip tubine contol ytem and theewith conclude one complete wind tubine model. Chapte 8 Wind tubine with a doubly-fed induction geneato contol complete the econd wind tubine model with uing the aangement of the Veta OptiSpeed contol ytem. An oveview of the mot impotant contol featue of thi tubine i given. Chapte 9 Fault imulation with total wind tubine model peent two fault imulation with both the achieved Veta model. Chapte Concluion ummaize and conclude the mot impotant eult obtained in thi wok and ketche the pepective and futue wok egading the wind tubine model. Appendix include all efeence ued in thi thee a well a ymbol explanation and a well a copie of two publihed pape duing the Ph.D. wok. 8

24 Chapte. Gid connection of Wind Tubine In the intoduction the impotance of gid connection iue wee biefly intoduced. The following chapte will highlight two majo viewpoint to thi iue. The fit point i the view fom the ytem opeato o the eponible intitution fo powe tanmiion and poduction in a county. A an example five diffeent countie ae tudied. The econd view i fom the wind tubine manufactue, who i poducing the powe plant and theefoe the poduct to the enegy upplie companie. Finally a cloing dicuion tate ome of the point, which hould be conideed fo the gid connection of wind tubine.. Tanmiion ytem opeato view Even though powe poduction fom wind tubine ha been known fo eveal yea the dicuion fo connection of wind tubine ha aien in ecent yea a the poduced powe by wind tubine inceae. Futhemoe, it became moe and moe common to intall eveal wind tubine connected in wind fam o wind pak, which change the view fom wind powe a a upplementay powe ouce to the iue of poible ubtitution of conventional powe plant with wind powe. Connection equiement fo electical device to the gid have a long hitoy in ode to enue table opeation of the gid. Thee equiement ae beide juitic tem concening fequency, voltage, powe poduction and many othe chaacteitic of the ytem. Becaue common powe plant ae uually equipped with lage ynchonou machine, gid connection equiement ae fitted to meet the opeation fom electical netwok with ynchonou machine. Mot commonly wind tubine ae equipped with induction machine. Hitoically induction machine have been teated a conume of the electical gid and not a upplie. Expeience with induction machine connected in lage numbe of wind tubine a enegy upplie on the gid have jut ecently been tudied /3/, /4/, /84/ and thee ae till many quetion to anwe. Anothe impotant diffeence fom conventional powe plant i the enegy eouce fo poduction. Unlike powe poduction fom nuclea powe plant o coal powe plant, whee the poible powe poduction i dependent on the ma of aw mateial and the conumed powe, wind tubine ae dependent on wind foecating to guaantee a pedicted powe poduction. Thi foecating of powe poduction fom wind tubine i not a imple tak. Howeve it i quite impotant fo ytem opeato, who need to have eliable enegy ouce. They have to enue the poduced powe offe i fitting to the powe demand, in ode to keep the powe ytem table. Vaying powe demand can caue voltage and fequency change in the ytem. Thee change can be nomalied with contolling the active and eactive powe upply and theewith i anothe equiement the tanmiion ytem opeato have towad powe plant. 9

25 . Wind tubine poduce view A wind tubine i a tate of the at poduct, which all the time hould meet the maket demand. In addition, the cot pe kilowatt hould be low. Theefoe it i impotant to know the load in the tubine and the behaviou of the tubine exactly. Only with pecie knowledge i it poible to avoid ove- o unde izing of the poduct, which will lead to eithe athe expenive olution o of not holding the life time expectancie of yea fo the tubine. It take typically thee to five yea to develop a new wind tubine type fitting to the new demand fom deign to ale. A an example, the Veta poduct ove the pat yea ae hown in Figue 6. Hub height, m Capacity 55 kw 75 kw 9 kw kw kw 5 kw 5 kw 6 kw 66 kw 85 kw 75 kw kw Fit intall Roto 5/6 m 7 m 9 m m 5 m 7/9 m 39 m 44 m 47 m 5 m 66 m 8 m Contol Stall Stall Stall Stall Stall Pitch Pitch/OptiSlip Pitch/OptiSlip Pitch/OptiSlip Pitch/OptiSpeed Pitch/OptiSpeed Pitch/OptiSpeed Tow e m m m 3 m 3 m 39 m 5 m 5 m 55 m m 6-78 m 6- m MWh/yea kw 9 m Pitch/OptiSpeed 8 m Figue 6 Veta Type Development duing the lat 3 yea While the development demand in ecent yea have been diven by mainly extacting mot powe out of the wind, the ecent development moved the focu to gid compliancy. Unfotunately the ecent equiement to wind tubine o wind fam acting like conventional powe plant ae not neceaily eay to combine with the idea and challenge that wind tubine have been deigned in line with in yea. A balance between the wind tubine handling the fluctuating wind enegy eouce and on the othe hand being a eliable enegy ouce uppoting the electical netwok ha to be found. In thi poce the geneal equiement to poduce a cheap poduct with low maintenance cot and high eliability have to be met.

26 .3 Connection Requiement Howeve, the lack of expeience enfoce the ue of exiting gid code of common geneating unit o powe plant, which can be quite challenging fo wind tubine. Special connection condition fo wind tubine have to be dicued. A few countie ae aleady woking on gid code pecifically fo wind tubine o wind fam, o a modification of the exiting code, which alo meet wind tubine chaacteitic /3/. Figue 7 i viualie the gid code development compaed to the tubine development and appoximate tubine development time. The Hon Reef poject i illutated a an example. 5 yea Denmak (Elta) Wind elated gid code Denmak (Defu. ed.) Denmak (Elta) Gemany (E.ON) Gemany (E.ON) Iland ESB National Gid Gemany (E.ON) Scotland Englih Ieland Autalia... Denmak (Elta) Veta Tubine V47 V5, (pitch V66 V9 (pototype) V8(pototype) OptiSlip) (pitch (pitch (pitch OptiSpeed) OptiSpeed) OptiSpeed) 4 3 yea Tende Hon Reef Figue 7 Wind Tubine development time and gid code demand Unfotunately not all countie whee wind tubine ae intalled have pecific o modified gid code. Thi make the development of futue wind tubine vey difficult. Thee i a big ik involved in developing new technique to fulfil unpecified demand. Mot wind tubine poduce today delive wind tubine to an intenational maket and ae faced with vaiou numbe of gid equiement. ooking at a few of the exiting gid code it appea that they ae vey diffeent in the mot impotant chaacteitic. To viualie the tate of the at technique, the diffeent gid code will be compaed to the tandad Veta V8 MW OptiSpeed tubine. The ame tubine type i choen fo the modelling example in thi Thei..3. Voltage citeia The voltage in powe ytem can vay due to e.g. load change o fault occuence. To avoid lo of powe geneation in epecially citical ituation a voltage opeation ange i defined. The voltage toleance can be een fom two apect. Fitly, how much voltage vaiation hould the wind tubine ytem be able to withtand, without tipping out and econdly, when and how hould the diconnection fom the gid occu. Figue 8 peent the limit in tem of connection of the wind tubine to the gid fo diffeent countie and the pefomance of the Veta V8 tubine.

27 voltage [%] time [] Autalia Ieland Scotland Gemany Denmak Veta V8 MW Figue 8 Voltage toleance equiement in diffeent countie and Veta V8 capability A hown in Figue 8 the demand between the countie ae vey diffeent. In Autalia the gid connected unit ha to utain % voltage fo 75 m, while in Denmak it i only 5% voltage fo m without diconnecting fom the gid. The Veta tandad V8 MW can atify thee equiement uing additional option, which expand the uual voltage toleance of the wind tubine. In pactice thee i a majo diffeence to develop a olution fo no et voltage o ome et voltage duing a voltage dip. The wot cae i zeo voltage, what i equivalent to a teminal hot cicuit. The induction geneato will looe all magnetiation and i not able to opeate futhe on. A poible way of handling uch a cae i decibed in //. The DFIG i not able to uppot the voltage and build up the gid again a a ynchonou machine would be able to do. Anothe cae i a majo voltage dip e.g. 5% voltage demanded in Gemany o 5% voltage in Denmak. Duing a voltage dop the cuent in the wind tubine inceae. To avoid a diconnection of the tubine due to high cuent, olution fo limiting thee cuent have to be developed. Howeve, handling a voltage to a cetain et voltage i le complicated than the handling of no voltage. Seveal olution fo fault ide though ae aleady peented /5/, /6/. Though thee ae good olution fo the gid, all the method intoduce high mechanical te in the wind tubine..3. Fequency equiement Hitoically powe plant ue big ynchonou geneato. Diect gid connected ynchonou machine have tability poblem due to load change and fequency change. In the cae of a machine oveload, the ynchonou machine will dop in peed, which mean eithe they deceae the gid fequency a well o become untable and dop out. The fequency change in a gid can be caued by lo of tanmiion, udden demand change o ome poblem of bigge powe plant (lo of geneation), which would nomally tabilie the gid fequency. A change in gid fequency will affect all othe

28 machine connected to it. Theefoe an impotant iue i to opeate in a wide ange of fequencie (ee Figue 9). Wind tubine with induction geneato have no poblem with aynchonou opeation. The fequency opeation ange i moe an iue of contol tategy. A high fequency opeation ange fo wind tubine and fequency contol i theefoe only inteeting, if the powe ytem opeato ue the wind fam fo tabiliing the gid fequency. Although wind tubine ae able to toleate majo fequency change they have difficultie to uppot the fequency contol opeation in a wide ange, hence the inetia in the ytem i vey mall. It i poible to initiate a udden upply o eduction of powe, which will actively help to tabilie the gid again. Howeve, it ha to be mentioned, that a upply of powe into the gid i only poible, if the wind tubine i uing an advanced opeation at educed powe and theeby ha enough powe eeve o uing anothe new contol tategy. An example i given in the opeation of the Hon Reef wind fam /4/ f/ f ated 6% Autalia (NECA) daft Scottih (SP Tanmiion & Ditibution, Scottih Hydo Electic) daft Ieland (CER & ESBN) Denmak (Elta&Elkaft) diconnection afte m 6 % Gemany (ENE -.8.3) Veta diconnection 5% 4% 3% % % % 99% diconnection afte.4 Powe deceae % pe. Hz (initial powe > 85%). % continuou % min 99 % diconnection within max 4% Powe deceae % pe. Hz.8 % no mentioning 4% 6 min 4 % % continuou 99 % min % continuou fat automatic diconnection Powe eduction 4% pe. Hz 3 % % continuou 99 % Powe eduction % continuou 99 % 98% 97% 96% 95% 94% 93% 9% 9% Powe inceae % pe. Hz (initial powe < 85%) diconnection afte.4 94 % continuou 95 % diconnection within max 6 min 94 % 95 % no mentioning 5 min 5 min 94 % 95 % 96 % 98 % diconnection afte m time limited Powe eduction % pe. Hz 3 min min min 95 % fat automatic diconnection time limited, powe inceaing 94 % diconnection 9% 89% 88% Figue 9 Fequency equiement in diffeent countie compaed to the pefomance of a Veta V8 MW wind tubine 3

29 .3.3 Active and Reactive powe equiement The eactive powe upply and aboption i impotant to contol the voltage of the powe ytem. No aboption o upply of eactive powe fom the wind tubine to the gid would mean only negligible influence of the wind tubine on the ytem voltage. Howeve, a contol to a powe facto of one (only active powe) i natually compenate malle eactive powe change in the ytem and theewith uppot the gid. In thi way wind tubine with DFIG have a poitive effect on the gid. Uually the powe facto capabilitie ae decibed in one point, a the tatic eactive powe exchange with the gid duing % powe poduction fom the tubine. Theefoe the focu i on eactive powe exchange with the powe ytem, without lo of powe poduction. But doe thi equiement make any ene fo wind tubine? No, thi one opeation point conideation doe validate the wind tubine without eeing the poibilitie a wind tubine offe in it othe opeation point. The connection demand, which ae alo the bae fo daft veion of wind tubine connection equiement, ae pimaily made fo integation of geneating unit into the gid. Thee geneating unit uually poduce full powe afte connection on to the gid and ae low in deceaing o inceaing the powe. The powe poduction of a wind fam can vay between and %, depending on the wind eeve and type. Becaue of the natue of wind powe poduction fom fluctuating wind, wind tubine have a fat powe contol. Wind fam ae theefoe capable in a hot time to adjut powe poduction. On the othe hand thi give the poibility to ue the wind tubine eactive powe capability fo compenation o uppot of the gid in a ituation, whee no active powe i needed. Figue how the poibility of the V8 MW to upply eactive powe hown, and in Table - the pecific numbe ae collected. Hence Veta V8 i changing to ta connection at mall powe ange the eactive powe capability i mall a well. But it i poible to tay in delta connection with a lo of ome efficiency. Even though the wind tubine in malle powe i capable of an excellent powe facto contol it will cot exta money to expand the eactive powe capability at ated powe. Figue Requiement egading eactive powe in diffeent countie and the technical capability of a Veta V8 MW tubine (pe unit of ated active powe) 4

30 Table - Specific value fo active- and eactive powe equiement County P-Q demand Scotland.95 lead to.85 lag Denmak ± %P Ieland.93 lead to.85 lag (% powe).7 lead to.4 lag (35% powe) Gemany (E.ON).95 lead.95 lag (<MW).95 unit geneating eactive powe (> MW dependant on voltage ange) E.ON additional WT - ule.975 lead to,975 lag (P geneating) lead to.95 lag (P conumption) Q tep: <.5%, > 5 kva Autalia (NECA).93 lead to.9 lag ynchonou geneato unity to.95 lagging aynchonou geneato Veta (V8 MW) unity to.98 cap (high voltage ide tanfome) + at low voltage ide tanfome.98 cap.96 ind. (% powe) limited contant eactive powe ( - % active powe). cap.. ind. (powe -> ) + contol powe facto adjutment due to voltage, powe contol, powe facto contol, cutome demand.3.4 Dicuing the equiement The pecification given in the above chapte to voltage, fequency and active-eactive powe equiement how clealy the dicepancy and the vaiety in the gid code. It alo ha to be mentioned that all equiement ae in a apid development, and thee ae changing at thi moment. One of the tendencie i to follow the E.ON in Gemany /3/ etablihed equiement fo gid connection of wind tubine. Howeve, evey gid i diffeent and it might not be a good idea jut copying the equiement. Although it i a good example fo the Geman gid, a moe pactical olution could be a epaation into two goup wind tubine connected to tong gid, wind tubine connected to weak gid and the intoduction of pecial equiement depend on the cae. Gid code hould not have the pupoe to avoid the connection of wind tubine to the gid. They hould be open enough fo dicuion poject pecific iue. The following point hould be teated. Tubine hould tat automatically with a pe-defined time delay and powe amp Tubine hould top automatically with a pe-defined diconnection pocedue (high wind o fault) and powe amp The continuou fequency ange hould be defined. A ditinction between ynchonou and aynchonou geneato hould be made. A dynamical deciption duing ated opeation and fault ituation egading voltage, active powe and eactive powe hould be made. Conideing the poible powe poduction dependent on the wind input 5

31 The poblem howeve tay. Wind tubine poduce have a numbeed poduct ange, which hould be able to fulfil a many gid equiement a poible. With the lack of confomability wind tubine will be deigned fo the code fo the bigget maket and the oute limit. Thi conequently limit the pice eduction of wind tubine technology. Theefoe a continuou poitive dicuion between all gid opeato, cutome and wind tubine poduce i equied to find acceptable demand and olution in the futue. One tep fowad i to gain knowledge of the behaviou of wind tubine with induction machine duing ituation uch a voltage dip. In a tet bench it i had to poduce the neceay opeation mode. Theefoe the development of wind tubine model a peented in thi thei i a neceay tep on the way to define gid code and deign wind tubine. 6

32 Chapte 3 3. Induction Machine Model A mentioned in the chapte above, one main inteet of wind tubine manufactue i the pecie knowledge about the influence of cetain gid condition to the wind tubine itelf. Detailed knowledge about e.g. the toque on the wind tubine haft caued by fault in the gid o tanient cuent ie i impotant fo the etimation of the load on the wind tubine. Accuate knowledge of thee effect enue low cot and the availability fo the etimated life time of the wind tubine. Theewith the geneato a connection point between the electical gid and the mechanical wind tubine play a cental ole. Though fo undetanding and calculation of the dynamic behaviou in a wind tubine, the ealitic epeentation of the geneato i impotant. Howeve wind tubine poduce face eveal poblem which ae limiting the infomation gained fom a machine model. Becaue mot wind tubine manufactue don t poduce thei own geneato, thee i only a minimum acce to data. Repeenting a ange of geneato in a model, facing the additional vaiation in each geneato dimenion and mateial, detailed FEM tudie of the geneato ae not ufficient. Theefoe the pupoe of the peented wok i to find an analytical geneato model fo the imulation of fault condition on vaiable peed Wind Tubine e.g. aymmetical voltage upply o -,-, 3-phae hot cicuit uing the pogam Matlab/ Simulink. 3. Geneal equation in ABC/abc Refeence Fame In ode to make a mathematical deciption of the double fed induction machine it will be aumed that a unifom ai gap machine can be modelled initially a two concentic cylinde with an ai gap of contant adial length /68/. The tato and oto have wounded thee phae winding. Futhemoe ion loe, and changing of paamete with tempeatue, ae not conideed. A balanced thee-phae machine i expeed in Figue. The conceptual winding ae placed along the magnetic axe of the winding. The ubcipt indicate the tato winding with effective numbe of tun N ξ and the oto winding with effective numbe of tun N ξ. The coupling between the phae ae depending fom the electical angle between the phae θ. If the thee winding ubcipted with a, b, c ae diplaced by ± the angle θ i only dependent fom the oto diplacement angle ρ. The baic equation fo developing a thee phae model can be found in vaiou book about electical machine and dive /5/, /5/. The vaiable of the machine ae defined in vecto, whee each element i epeented pe phae: The equation fo tato and oto voltage can be witten a: u u R R i i d Ψ + dt d Ψ + dt () () 7

33 Whee the voltageu, the cuent i and Flux Ψ ae vecto exemplified fo tato value defined by (3). A imila definition can be witten fo oto value. ua u ub, uc ia i ib, ic Ψ Ψ Ψ Ψ a b c (3) b - axi b - axi ρ θ i b u b a - axi i b u a ρ u b θ i a u a i a a - axi i c u c i c u c c - axi ρ c - axi Figue Magnetic axe, concentic tato and oto winding, cuent and voltage and angle dependencie of a thee phae induction machine /67/ The flux in tato of the machine can be expeed with the flux ceated by the tato phae itelf and the flux pat influencing the tato oiginated fom the oto phae. In the ame way the oto flux can be epaated into the flux belonging to the oto and a pat penetating the oto winding oiginating fom the tato phae. Ψ Ψ Ψ Ψ + Ψ + Ψ ( ) ( ) ( ) ( ) The flux can be expeed a a poduct of an inductance matix and the cuent vecto i. The flux ceated fom the tato winding i expeed in equation (6). (4) (5) 8

34 9 m m m m m m m m m i Ψ σ σ σ ) ( (6) In the ame way the flux ceated fom the oto winding can be expeed in equation (7). m m m m m m m m m i N N N N N N N N N N N N N N N N N N Ψ ) ( ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ σ σ σ (7) The coupling between the oto and tato i depending fom the oto diplacement angle. The flux penetating the tato initiated in the oto i hown in equation (8) and the tato flux penetating the oto i hown in equation (9) m i N N Ψ ρ π ρ π ρ π ρ ρ π ρ π ρ π ρ ρ ξ ξ co /3) co( /3) co( /3) co( co /3) co( /3) co( /3) co( co ) ( (8) m i N N Ψ ρ π ρ π ρ π ρ ρ π ρ π ρ π ρ ρ ξ ξ co 3) / ( co 3) / ( co 3) / ( co co 3) / ( co 3) / ( co 3) / ( co co ) ( (9) The winding tun in tato and oto ae in mot cae diffeent. Fo expeion of the tato and oto ytem in an equivalent coupled ytem the atio ue i intoduced in (). Thi atio i aleady implied in the flux equation. N N ue ξ ξ () u u ue () ue Ψ Ψ () i ue i (3) m m m ue ue with: m m m (4) R R ue (5) ue σ σ (6)

35 When uing the atio ue the flux equation ae implified to: Ψ Ψ σ + m m m σ + m m m σ m + σ m m + m m m σ m m + m i + m m i + + σ m m m coρ co( ρ π /3) co( ρ+ π /3) coρ co( ρ + π /3) co( ρ π /3) co( ρ+ π /3) coρ co( ρ π /3) co( ρ π /3) coρ co( ρ + π /3) co( ρ π /3) co( ρ+ π /3) i coρ co( ρ + π /3) co( ρ π /3) i coρ (7) (8) Often the expeion in one matix i ued //, /9/ expeing the angle dependency in a function: f co ρ (9) f co ( ρ + / 3) () π f co ( ρ / 3) () 3 π Ψ Ψ Ψ Ψ Ψ Ψ a b c a b c σ + m m m m f m f m f 3 σ + m m m m m f 3 f f m 3 σ m m m m m + f f 3 f m σ m m m + m m f f 3 f m m m m f f f σ m + m 3 m m m m f f f σ 3 m m + m ia ib ic ia i b ic () The inductance matix i expeing the coupling of the flux in the tato and oto winding. It take a cental poition in the machine modelling. Depending on the fom of the inductance matix, the machine model i moe o le detailed. In poge of thi chapte, conideation of atuation effect will change the contant inductance in thi matix to time dependent function. 3.. Complex pace Vecto epeentation A lot of autho /5/, /55/, /67/ intoduce the method of complex pace vecto fo a moe convenient mathematical deciption of the machine vaiable. It i widely ued in the field of machine modelling and machine contol. The phae cuent of the thee machine winding diplaced by ± a aleady hown in Figue can be decibed a complex vecto. With the help of the complex veo a (3) thi complex vecto can be expeed by (4). Fo convenience the eal pat of the

36 complex pace vecto i aligned to phae a of the 3 phae tato ytem o 3 phae oto ytem. 3 j π o 3 + j e a (3) i a i, i a i, i a i (4) a b b c c Whee i a, i, b i c ae the abolute value of the cuent time vecto. A eulting cuent vecto can be achieved though an addition of the thee phae cuent vecto (5). i i a + i b + i c i + a i a b + a i c (5) Some autho /3/, /56/ intepet thi equation a a definition fo the complex pace vecto. In intepetation a complex pace vecto the time vecto itelf can be complex vecto. Intead of the definition (5) a moe common definition i achieved though multiplying (5) by the facto /3 (6). i 3 ( i + a i + a i ) a b c (6) I j β a b I b a j 9 o o + 3 j i(t) / ic(t) ia(t) ib(t) t o I 3 ( I a + ai b + a I c ) -/ t 4 { I a } Ia I Re α I b a I c a - I c a I c 3 j Figue Space vecto contucted fom a thee phae ytem Figue 3 Phae cuent in a time baed ytem The advantage of uing definition (6) ae that the pojection fom the pace vecto i to the magnetic axe achieve the abolute time value of the exiting axe-cuent (7). The vecto change it length and poition with the phae cuent (Figue, Figue 3). Futhemoe the pace vecto can be ued fo voltage and fluxe in the ame way a fo the cuent (8), (9). The calculated powe uing the vecto defined a in (6), (8) i invaiant. The calculation of powe and toque (ee equation (3), (33), (34) ) give the ame expeion like the geneal equation of ymmetical thee phae ytem /55/. Theefoe thi definition i often called the eal o coect powe definition.

37 i u a Ψ { i }, i b Re{ a i }, i c { a i } ( u + a u + a u ) Re Re 3 3 a b ( Ψ + a Ψ + a Ψ ) a b c c (7) (8) (9) The powe calculated uing the method of complex pace vecto i hown in equation (3) { } ( ) ( ) 3 3 S u i ua + a ub + a uc ia + a ib + a ic ( ) ( ( ) ( ) ( )) 3 ua ia + ub ib + uc ic ua ib + ic + ub ia + ic + uc ia + ib j ( ia ( ub uc) + ib ( uc ua ) + ic ( ua ub ) ) 9 (3) Auming the phae cuent can be added to zeo, a hown in equation (3). The active tato powe in equation (3) and eactive tato powe equation (33) can be achieved a eal and imaginay pat of the total powe hown in equation (3). The aumption of equation (3) mean thee i no zeo component exiting, o the zeo component i not conideed in the calculation. It can be eaily conideed epaately on a late (ee chapte 3. ). i i + i (3) a + b c 3 Re { } P u i ua ia + ub ib + uc ic (3) 3 * Q Im { u i } ( ia ( uc ub) + ib ( ua uc ) + ic ( ub ua )) (33) 3 and the toque in an analogou manne (damping due to fiction and windage i not included): T e 3 p pp 3 p Im { Ψ i} ( i ( Ψ Ψ ) + i ( Ψ Ψ ) + i ( Ψ Ψ )) a c b b a c c b a (34) 3.. Uing pace vecto fo the geneal machine equation Uing the above intoduced complex pace vecto, the geneal machine equation can be expeed moe conveniently compaed to the vecto peentation fo the pe phae peentation. The voltage equation (35), (36) uing the complex pace vecto epeentation fo the tato and oto can be expeed though the tato flux oto flux Ψ tanient and winding loe: Ψ and

38 3 dt d i R u Ψ + (35) dt d i R u Ψ + (36) The flux i defined with the following equation /67/ ( ) ( ) Ψ Ψ + Ψ (37) ( ) ( ) Ψ Ψ + Ψ (38) Whee the flux component ae expeed a: m i + Ψ ) 3 ( ) ( σ (39) m i + Ψ ) 3 ( ) ( σ (4) ρ j m e i Ψ 3 ) ( (4) ρ j m e i Ψ 3 ) ( (4) If the mutual inductance and couple inductance ae equal a aumed in a one phae equivalent diagam, then the magnetizing inductance can be witten a: m m m m (43) The flux equation become: ρ σ j m m e i i + + Ψ ) ( (44) ρ σ j m m e i i + + Ψ ) ( (45) The oto voltage equation elated to the tato become: dt d i R u Ψ + (46) The vecto expeing the oto cuent, voltage and flux ae till elated to thei own efeence fame, maked by the uppe index, while the tato vecto ae elated to the tato efeence fame, maked by the uppe index. Thi diffeence of the two coodinate ytem i expeed with the oto angle ρ (Figue ). With uing vecto tanfomation (Chapte Appendix) and thi angle, the oto voltage equation can be witten a: j dt d i R u Ψ Ψ + ρ (47) Finally all geneal equation fo modelling an induction machine uing the epeentation of complex pace vecto ae:

39 u u Ψ Ψ T e d Ψ R i + dt (48) d Ψ R i + j ρ Ψ dt (49) ( + ) i + i j ρ e (5) σ m m j ρ ( σ + m ) i + m i e (5) 3 p p Im { Ψ } i dω Te Tm J + Dω (/98/ page 56) (53) dt dω dω p e p pp ρ ( Te Tm Dω ) (54) dt dt J The achieved equation and paamete can be viualized in the dynamical one phae equivalent diagam een in Figue 4. (5) i R j ω Ψ σ σ j ( ω ω ) Ψ R i i + i i µ u dψ dt m dψ dt u Figue 4 Dynamical pe phae equivalent diagam fo induction machine 3. The Clak Tanfomation (α, β, equivalent fame) The complex pace vecto epeentation i the bae fo eveal diffeent deciption of an electical machine. One of the poible intepetation of the with the complex pace vecto deduced machine equation ae the o called α, β - component. The α, β - component, alo known a Clak Tanfomation ae deduced by epaating the complex pace vecto of the machine into thei eal and imaginay pat and efeing them to a tationay efeence ytem. The o achieved component can be phyically imagined a two field in nomative diection to each othe, meaning the thee-phae machine ha been deduced to an equivalent two-phae machine ee Figue 5. Equivalent to the ued x, y component in ome liteatue 4

40 j β i β i i α Figue 5 Equivalent two-phae machine howing Clak-Tanfomation The eal component i calculated a hown in equation (55) and the imaginay pat equation (56). ib + ic iα Re i Re ia + a ib + a ic ia 3 3 iβ Im{ i} Im ( ia + a ib + a ic ) ( ib ic ) 3 3 { } ( ) (55) (56) The complex pace vecto can be witten a: i i + j i (57) α β If the addition of the thee phae doe not equate to zeo an additional component the o called zeo component ha to be ued a well. i ia + ib + ic (58) 3 Vey poitive i the fact, that in cae of no exiting zeo component the eal pat i identical to the phae a of the 3-phae ytem. ib + ic ia + ic ic iα Re{ i} i i i 3 3 a a a ia + ib + ic (59) 5

41 In the following matice the complete tanfomation (including zeo component) i hown. iα i β i ia i b i c ia 3 3 i b 3 i c iα 3 i β 3 i (6) (6) Whee equation (6) i ued fo the tanfomation of the thee-phae ytem into the α, β - component the equation (6) i ued fo the back tanfomation. The equivalent diagam fo the machine epeentation can be een in Figue 6 and Figue 7. i αβ R σ σ j ( ω ) Ψ αβ R i αβ αβ + αβ µ i i i u αβ Ψ αβ d dt m Ψ αβ d dt u αβ Figue 6 Equivalent cicuit diagam fo α and β - component i R σ σ R i u d dt Ψ d dt Ψ u Figue 7 Equivalent cicuit diagam fo the zeo component The equation fo deciption of the machine ae concluded in equation (6) to (7). The zeo component ae aumed to be only linked via the leakage field. 6

42 d Ψ u R i + dt (6) dψ u R i + dt (63) d Ψ u R i + j ρ Ψ dt (64) dψ u R i + dt (65) Ψ ( + ) i + i j ρ e (66) σ σ m m Ψ i (67) Ψ j ρ ( σ + m ) i + m i e (68) Ψ σ i (69) 3 3 Te pp Im{ Ψ i } pp ( Ψ α iβ Ψβ i α ) (7) dω Te Tm J + Dω (/98/ page 56) dt (7) ρ dω dω e pp pp ( Te Tm Dω ) dt dt J (7) The powe i calculated a hown in equation (73) and equation (75). 3 { u i } Re ( u + ju ) ( i j i ) 3 Re * 3 { } ( u i u i ) P α β α β α α + β β (73) P 3 u i (74) Q 3 { u i } Im ( u + ju ) ( i j i ) 3 Im * 3 { } ( u i u i ) α β α β β α α β (75) 3.3 The Pak Tanfomation (d, q, equivalent fame) In the ame way a in the α, β - component, hown in the chapte 3., the d, q component ae a epaation of the complex pace vecto in eal and imaginay pat. j i id + j iq ( ia + aib + a ic ) e γ 3 (76) In oppoite to efeing the pace vecto to a tationay efeence fame a done in the Clak component, in the o called Pak tanfomation the complex pace vecto i expeed in the otating othogonal ytem linked with the oto of the machine. The method i howeve ued to expe the complex pace vecto in any otating efeence fame. The advantage by doing o otating pace vecto change to appea tationay in a otating efeence fame, which mean altenating tace of the pace vecto ae teady and the change in abolute value of the vecto i depicted clealy. Thi i beneficially Be awae - Definition i vaying in the liteatue 7

43 ued fo fat contol of electical machine. j q i β i i q ϕ IR ϕis i d d ω γ i α Figue 8 Pak - tanfomation In Figue 8 the Pak tanfomation in elation to the α, β - component ae hown. The tanfomation between thee component and an abitay otating fame i: d q j ( α β ) ϕ i + j i i + j i e (77) and fo etun to the α, β - component j ( d q ) i + j i i + j i e ϕ (78) α β The zeo component i equal fo both tanfomation, hence it fomed duing expeing the 3-phae machine into a two-phae equivalent machine. i dq i αβ (79) Expeing the tanfomation in Matice, the following Matice fo fowad and back tanfomation can be witten. ( ϕ ) ( ϕ ) ( ϕ ) ( ϕ ) iα co in id iβ in co iq i αβ i dq i d iq i ( ϕ ) ( ϕ ) ( ϕ ) ( ϕ ) co in i α in co iβ i dq αβ (8) (8) 8

44 Tanfomation fom a 3-phae ytem diect to an abitay otating ytem can be done in the following way: id i q i ia i b i c co γ coγ π coγ + π 3 3 ia in γ in γ π in γ π i b i c co γ in γ id co γ π in γ π i q 3 3 i coγ + π inγ + π 3 3 (8) (83) The equivalent diagam fo a machine in Pak tanfomation i hown in Figue 9. The equivalent fo the zeo component i equal a hown in Figue 7 in chapte 3.. Since all machine equation ae vey imila to the deciption of the machine in Clak component it i not neceay to how them again. Jut the toque and powe equation ae hown in equation (84), (85) and (86). 3 3 Te pp Im{ Ψ i } p p iq Ψd Ψq i d (84) 3 * 3 P Re{ u i } ( udid + uqiq ) (85) 3 * 3 Q Im{ u i } ( uqid udiq ) (86) The equivalent diagam fo the d,q component i hown in Figue 9. The equivalent diagam fo the zeo component i identical to the diagam hown in Figue 7. i R j ω Ψ σ σ j ( ω ω ) Ψ R i i + i i µ u dψ dt m dψ dt u Figue 9 Equivalent cicuit diagam fo d, q - component 9

45 3.4 Satuation effect In the ideal machine it i aumed, that the leakage eactance and magnetizing eactance ae independent of the cuent and the voltage. In a eal machine the magnetic voltage though the ion i not diectly popotional to the cuent and theefoe thi aumption i not valid. Fo taking a cloe look at the phenomena, the influence of atuation of the leakage eactance and the magnetizing eactance hould be taken into account. /8/ 3.4. Satuation of main eactance The main eactance o magnetizing eactance X m i mainly defined though the tato induced electomotive foce (EMF/ EMK). The eactance would not be changing, if the E would be a taight line. In eality the cuve i bending in the uppe chaacteitic of ( ) I µ pat a hown in Figue..4. E U [p.u.] I µ [ p. u.] Figue Example of a meaued no load (magnetizing) cuve dahed line and the theoetical not atuating cuve olid line of an induction machine When inceaing the induced voltage (E ) the magnetizing eactance lightly deceae. The conequence i a fate deceae of the magnetizing cuent a the deceae of the induced oto EMF. The effect of main eactance atuation on the tato cuent i mot noticeable duing no load opeation. In Figue thi influence of magnetizing eactance i viualized. 3

46 4 3 eal pat tato cuent [p.u.] - I Ik I imaginay pat tato cuent [p.u.] Figue Cuent cicle diagam howing the effect of main inductance atuation; ated-olid line, halve Xm (atuated) dahed line, I k -hot cicuit cuent at lip and I - cuent at infinite lip To find the dependency it i theefoe bet to meaue the no load cuve. Thee ae two poibilitie of meauing the cuve. If a lip-ing machine i ued it i poible to meaue diectly the oto cuent and tato cuent. Thi implifie the machine complexity, the machine i at tand till and the oto winding ae open and voltage and cuent on both teminal can be meaued (diect meauement of magnetizing cuve). Howeve moe common i the meauement at nealy ynchonou peed (ued fo quiel cage moto), the o called no load cuve. In no load opeation at ynchonou peed o at tandtill the oto cuent can be aumed to zeo. Theefoe it can be aumed that the cuent of the tato winding and the magnetizing cuent ae equal. The voltage dop i now only ove the tato eitance, tato leakage eactance, ion lo eitance and the magnetizing eactance. The magnetizing inductance dependent at the cuent can be achieved with the following equation. It i aumed that the eitance and the tato leakage inductance ae not changing with the cuent. The o deduced function of the main inductance fom the magnetizing cuent i a common way to include atuation effect into a machine model (87). U X I I R R m ( µ ) m ( µ ) ω ( ) + m σ ω 3 Iµ (87) In Figue the main inductance dependency fom the magnetizing cuent, which wa achieved though the no load cuve meauement and then calculated with equation (87) i hown. Vey often ated inductance i aumed at low cuent. In eality thee i not enough flux in the machine becaue of the low voltage and the inductance i deceaing again towad low cuent. 3

47 . m [p.u.] I µ [ p. u.] Figue Main inductance dependency of the magnetizing cuent ( ) I µ m 3.4. Satuation of leakage eactance Fo the eeach in fault ituation like hot cicuit the opeation with high cuent i fa moe impotant than no load opeation. Theefoe a lage impotance i the conideation of the dependency of leakage eactance X σ and X σ on the cuent. With inceaing cuent the leakage eactance will deceae a well a the magnetizing eactance. In Figue and Figue 3 the cuent cicle diagam i calculated fo a machine with half of the ated magnetizing eactance o leakage eactance and compaed to a machine with ated value. While in ated opeation ange, which i aound p.u., the influence of the magnetizing inductance i dominating, the change duing fault opeation with high cuent ae clealy dominated by the leakage eactance. The cuent conideing a change of leakage eactance i eentially lage than without thi conideation eal pat tato cuent [p.u.] I Ik I imaginay pat tato cuent [p.u.] Figue 3 Cuent cicle diagam howing the effect of leakage eactance atuation; ated value-olid line, halve X σ and X σ (atuated) dahed line 3

48 The leakage eactance can be achieved fom the blocked oto tet. In thi tet with hot cicuit oto coil the magnetizing cuent i vey mall and can be neglected. With meauement of the voltage and cuent and the powe facto co ϕ the total leakage eactance can be calculated a: X σ ϕ k U in (88) I To epaate the leakage eactance X σ into the tato leakage eactance X σ and the oto leakage eactance X σ the no load meauement can be ued again. In the no load meauement the eactance X m + X σ ae meaued and theewith the pat of the oto leakage eactance can be etimated. Anothe facto helping the moe exact epaation between the tato and oto leakage eactance i the meauement of the leakage facto σ. m X σ (89) X X whee X m X + X and X σ X m + X σ Howeve, the method ae detailed dicued in book /8/ and othe machine book and theefoe will not futhe deepened in thi thei. 3.5 Windage loe and Fiction loe Windage and fictional loe of a machine can be divided into two pat, which have a diffeent popotion to the otational peed ω 3 Ventilation loe ae popotional to ω. Thi include all ventilation loe (end winding ventilation lo, oto cooling duct ventilation lo etc.) 3 7 / 3 Fictional loe ae popotional to le than ω ( P ω ). Thee loe include fictional loe in beaing. The coect popotional atio i quite had to etimate, ince it may fluctuate. e.g. tempeatue of the geae ha ome influence into fictional loe. Thi mean that in diffeent evice condition the popotional atio may be diffeent. When combining the above component total windage and fictional loe ae achieved..5.7 Total loe can be appoximated to a popotion between ω... ω. Unfotunately it i vey had to include uch non intege potential dependency into the model. Since the diffeence between the diffeent potential dependencie i vey mall in the opeating aea it i choen to ue a dependency in the econd ode. A econd ode dependency due to powe loe lead to a imple dependency of the toque, which can be found in vaiou book. 33

49 The damping facto can be fom the no load loe: P D (9) ω fw The windage and fiction loe depending on the peed will be calculated uing the contant damping facto achieved above. Pfw ( ω) D ω (9) Though the damping facto i only an equivalent thee ae a ange of poible method conideing the leakage and windage loe. In Figue 4 and Figue 5 the influence of the diffeent exponential dependencie i hown. The damping facto D i detemined by meauing the otational loe at ynchonou peed and dividing it by the exponent choen. P fw D (9) ω...,5...,7...3 The windage and fiction loe in the model would accodingly be calculated in the following way. Pfw...,5...,7...3 ( ω) D ω (93) 8 6 ω ω.5 ω.7 ω 3 4 ω ω.5 ω.7 ω 3 4 P fw [kw] 8 P fw [kw] ω [pm] ω [pm] Figue 4 Diffeent exponential dependencie of D between 5 pm Figue 5 Diffeent exponential dependencie of D at ove peed In the opeation ange fom appox. 5 pm up to 7 pm the diffeence in the calculation of the potential dependencie i vey mall. If ued fo the pupoe of modelling at ove peed ituation, a change fom the choen dependency of econd ode might be econideed, and intead the ue of a model with a dependency of thid ode i moe appopiate. 34

50 3.6 The Thee Phae Model in Matlab / Simulink The machine model i baed on the equation (), (), (), (34), (54) and implemented in Matlab/ Simulink a hown in Figue 5. Figue 6 The unmaked Machine model in Matlab/ Simulink The input of the machine model ae the thee phae voltage of tato and oto and the mechanical load toque T m of the geneato axi/haft. Output ae the thee phae tato and oto cuent, the peed of the machine expeed a electical angula peed of the oto o mechanical peed and the electic magnetic ai gap toque T e of the machine. The powe of the machine i calculated fom the voltage and cuent of tato and oto a decibed in the chapte above. The model i independent of the uage a geneato o moto. The mechanical load toque i defined negative fo geneato opeation. The powe i negative fo powe poduction and poitive fo powe conumption. Fo oto powe calculation a aleady fo the tato deduced equation (3), (33) ae equivalently ued. P Q ua ia + ub ib + uc ic (94) ( ia ( uc ub ) + ib ( ua uc ) + ic ( ub ua )) 3 (95) The block called angle fiction expee the equation of motion. With thi equation the geneato peed and the neceay oto diplacement angle ρ i achieved. Futhemoe the fiction loe ae implemented with uing the damping facto D a hown in Figue 7. 35

51 Figue 7 Block angle - fiction The machine block called flux cuent i expeing the flux equation to calculate the cuent with help of the inductance matix. The atuation effect i taken into account a vaiable inductance in the inductance matix f f f σ ma ma ma maa mab mac f f f mb σ mb mb mba 3 mbb mbc f f f mc mc σ mc mca mcb 3 mcc f f f maa mab 3 mac f mba mbb mbc 3 mb σ mb mb f f σ ma ma ma mca 3 mcb mcc mc mc σ mc f f f (96) In chapte 3.4 the influence of the atuation on the machine i decibed. Fo the vaiable main inductance the magnetizing cuent i detemined in the following way. iµ i + i (97) Uing (97) in the tato flux equation (98): Ψ ( σ + m ) i + m i (98) Ψ + i + i i (99) ( σ ) ( µ ) m m the magnetizing cuent can be calculated with the tato flux and tato cuent to: Ψ σ i iµ () m Becaue the magnetizing cuve i baed on m value, the m value of the magnetizing cuent i calculated a well. Thi magnetizing cuent i ued with magnetizing cuve to achieve the I ). To ecue a table functioning of the model the cuent and theewith m ( µ the main inductance ae limited to the known value. Figue 8 how the implementation in Matlab/ Simulink. 36

52 Figue 8 Block flux cuent ued in the machine model (Figue 6) 3.6. Steady tate validation of the diffeent detailed ABC/abc model In thi chapte a compaion between the machine model including diffeent featue ae made. At fit thee diffeent model duing ated condition ae choen. One model how the ideal machine, without fiction loe o atuation effect of the eactance; the econd model i a model conideing fiction loe and the thid model i a model including fiction loe and atuation of main magnetizing eactance. The input paamete fo the thee-phae model ae deduced fom the one phae equivalent upplied by the machine manufactue, which ae exemplay data of a.8 MW machine (Figue 4). Duing the imulation the oto of the induction machine i hot cicuit, which i modelled with etting the input oto voltage to zeo. The input tato voltage ae ated. Theewith a tat of the machine, thee load change and a thee phae hot cicuit ae imulated. The thee diffeent opeation point duing the load change ae then compaed by the manufactue upplied meaued opeation point of the machine. The machine tat up in moto mode and then ettle into geneato mode poducing electical powe accoding to the load toque on the axi of the machine. In Figue 9 and Figue 3 the toque and the active powe ove the whole imulation peiod fom the model with atuation ae hown. 37

53 6 Electomagnetic Toque and oad Toque/ ated toque Time [] Figue 9 Simulated geneato electomagnetic eaction toque and input haft toque in opeation machine tat; tepping fom ated load toque deceaing to no load toque to 3- phae hot cicuit in [p.u.] 3 Active Stato powe Powe/ ated powe Time [] Figue 3 Simulated active tato powe [p.u.] coeponding to the input toque In Figue 3 and Figue 3 the tato cuent fo all thee model ae hown in compaion to the meaued m value taken fom manufactue upplied data. The peak of the time 38

54 value hould theoetically touch the line of the tato peak value ( m ). Figue 3 compae the ideal model and the model with implemented damping. The model with damping (fiction and windage loe) how a malle tato cuent output with the ame load, which i eay to explain. Since the fiction loe ae depending on the peed and the peed i inceaing with the load, thi conequently mean highe loe and theewith a malle tato cuent. At no load opeation the fiction loe ae malle than at ated load and theewith had to notice in the imulation..5 ated load halve half of ated load no load meaued m(i)*qt() I/I ated Time [] Time [] Time [] Figue 3 Stato cuent elated to ated tato cuent; olid line ideal model, dahed line model with fiction Figue 3 compae the model with atuation and damping to the model without atuation though with included damping. Duing the opeation with ated load it i had to ee a diffeence in the imulated tato cuent. A it i explained in chapte 3.4, the atuation of main inductance ha only an inignificant influence in the aea of ated opeation up to the maximal opeation point. With including leakage atuation a highe cuent would appea. The influence of main flux atuation i clealy een in half of ated load opeation and no load. A expected due to the theoy lead the conideation of main magnetizing atuation to an impovement of the model and the imulated eult. 39

55 . ated load halve half of ated load no load meaued m(i)*qt() I/I ated Time [] Time [] Time [] Figue 3 Stato cuent elated to ated tato cuent; olid line model with atuation, dahed line model with fiction and windage loe A futhe citeion fo impovement of the model i hown in the evaluation of the powe facto o the eactive powe. The powe i geneally moe peciely meauable than the cuent. Hee the model with included atuation and damping how again the impovement compaed to the ideal model. Calculation of the powe facto alo appove thi eult. 4

56 Reactive Stato powe meaued value model with atuation.9.8 Q/Q ated ].7.6 ideal model and model with fiction loe Time [] Figue 33 Stato eactive powe elated to ated eactive tato powe [p.u.]; model with atuation uppe line, on top of each othe model with fiction loe and ideal model lowe line The imulation eult made with the.8 MW machine how only mall diffeence between the diffeent machine model. Thi i due to the fact that the data ued a input fo the model i given to meet the expected opeation aea equiement. Satuation ha in thi aea only minimal influence. Machine with tonge atuation effect will how a moe ignificant diffeence. Fo uing the model fo hot cicuit imulation, the leakage field atuation hould be included a well. 4

57 3.6. Dynamical validation of the ABC/abc model The aimed ue of the developed model i fo imulation of dynamical ituation including fault ituation. The above teady tate validation though ae not ufficient enough to veify the ue fo thi kind of imulation. Theefoe a Diect On ine (DO) tat of a 85/ 8 kw machine i initiated in a laboatoy tet et-up. The DO tat ha been choen to poduce high cuent and achieve atuation effect in the machine, ince eal hot cicuit fo validation of the machine model ae not feaible in the eeached machine ize of 85kW to 3MW. Duing the tet peed, thee-phae tato cuent, thee phae oto cuent and thee phae tato voltage have been meaued. The meaued voltage i late on ued a input fo the machine model. At fit the model conideing main magnetizing eactance atuation and fiction loe will be compaed to the meauement. The machine i connected ta/ ta and the oto winding ae hot cicuited. Due to the impedance in the gid the tato voltage dop with connection the machine diect to the gid. The tato voltage keep the deceaed voltage amplitude until the machine ha tated up. With finihing the tat up the voltage ie to the ated value again (Figue 34). The expeienced long tating equence i athe unuual fo induction machine. The machine i deigned fo geneato opeation and not fo moto opeation and theefoe develop vey little tating toque. The long tating peiod give the poibility to ue it fo validation. Thee exit an aymmetical tanient and a quai tationay pat duing the DO tat of the machine. quaitationaypeiod.8.6 U/ U ated [p.u.] cut in 5 5 time [] Figue 34 Meaued Gid voltage/stato voltage duing the DO of a 85/8 kw geneato In Figue 35 the peed duing the DO tat i hown. Thee ae only mall diffeence between the imulated peed and the meaued peed. In the lat phae of the tat, befoe eaching the final teady peed the model how a mall ovehoot. Thee ae 4

58 many diffeent poibilitie caing thi ovehoot /95/. 6 4 peed [pm] t [] Figue 35 Speed duing DO tat of a 85/8 kw geneato; model with main atuation and fiction loe olid line, meauement peed dahed line Figue 36 and Figue 37 how a window of the imulated and meaued tato cuent. The meaued tato cuent ae bigge than the imulated value. Thi i not upiing, ince diffeent effect ae not included in the model e.g. the atuation of leakage eactance. The bigget diffeence though can be een in the tanient ituation a in Figue 36, duing the fit econd afte connecting the machine to the gid. The bigget diffeence between the imulated value and the meauement ae at the highet peak value, which i 38% malle than the meaued value. The mallet diffeence can be een at the fit peak, which i the mallet of all, with % diffeence efeed to the meaued peak value. Duing the quai tationay peiod while unning up of the machine, the cuent i quai tationay a well. The aveage diffeence between the peak value of the tato cuent i now % (Figue 37). 43

59 8 6 4 I/I ated t [] Figue 36 Zoom into the tato cuent at tat of the DO-tat (olid- model including main atuation and fiction loe, dahed - meaued) I/I ated t [] Figue 37 Window of the tato cuent in the middle of the DO-tat (olid- model including main atuation and fiction loe, dahed - meaued) Afte the tat up the machine it i unning in no load opeation and conume only the powe fo coveing the own loe. Figue 38 how the cuent duing thi no load 44

60 condition. The imulated cuent i inignificantly bigge than the meaued, which veifie the coect implementation of the main magnetizing eactance atuation. The noticeable aymmetic between the phae cuent i due to the lightly aymmetical eal voltage input I/ I ated t [] Figue 38 Window of tato cuent at no load opeation afte DO- tat 45

61 3.6.3 Implementation and validation of leakage eactance atuation It i highly deiable to minimie the diffeence between the imulated and expeimental value in Figue 36 and Figue 38. The exact knowledge of the cuent peak enable a bette deign of potection ytem o the component withtanding the dynamical load. A aleady implied in the chapte about atuation effect an implementation of leakage eactance atuation could impove the model epecially duing ituation with high cuent e.g. hot cicuit. Unfotunately, thee i a lack of data to have a detailed implementation of the nonlineaity of the leakage eactance. Howeve the atuated leakage eactance of tato and oto ae etimated fom the blocked oto meauement with malle voltage and available in the data heet upplied by the manufactue. Alo known ae the atuated tato cuent o blocked oto tato cuent. Uing the two decibed point of ated and atuated opeation a linea appoximation ha been choen to expe change in the leakage inductance. Stato cuent eakage inductance ated ated value ated Blocked oto tato atuated value at lip cuent x ated cuent value atuated value at lip Table 3- eakage inductance atuation efeence oto and tato leakage inductance [p.u.] tato cuent [p.u.] Figue 39 Example fo tato (olid line) and oto (dahed line) leakage inductance Changing the leakage inductance in the peented manne might eem odd, ince commonly the change in inductance ae made depending on the lip of the machine /86/. Dependency fom the lip though ha the diadvantage that all 3-phae value ae changed imultaneouly. The dependency on the cuent give the poibility of changing the phae inductance epaately due to the phae intantaneou cuent. 46

62 Figue 4 how the implementation of main and leakage atuation in the machine model in Matlab/ Simulink. The et of the machine model i the ame a aleady decibed above. Figue 4 Modified Matlab/ Simulink block flux cuent including leakage inductance atuation The ame imulation ha been pefomed a aleady done with the model conideing only the main magnetizing inductance atuation. Thee i not a noticeable change in the peed chaacteitic compaed to the model including only main atuation. Figue 4 how the model with the implementation of the leakage inductance atuation compaed to the meauement of the tet bench. 47

63 6 4 peed [pm] t [] Figue 4 Speed duing DO tat (olid line model including main and leakage inductance atuation, dahed - meauement) In Figue 4 and Figue 43 the tato cuent window fom the cut in of the machine and fom the quai tationay ituation ae hown. Implementation how a lage impovement in the tato cuent imulation compaed to the imulation with the model not conideing leakage atuation. The diffeence in the maximum peak value duing the DO tat have been deceaed fom 38% to %. In the quai tationay tate afte the tat up the aveage diffeence between the peak value of the tato cuent i deceaed fom % to % (Figue 43) I [A] Time [] Figue 4 Zoom into tato cuent imulation at tat of the DO tat (olid- model including main and leakage atuation, dahed - meaued) 48

64 5 4 3 I [A] Time [] Figue 43 Window of Stato cuent in the quai tationay tate of the DO-tat (olidmodel including main and leakage atuation, dahed - meaued) Hence the influence of the atuation of main and leakage inductance could be clealy hown, the influence duing no load opeation i inignificant. Figue 44 how the cuent duing the no load tate afte the DO tat. The miing behaviou duing no load opeation i eaily explained with the ued implementation method. While the main inductance i moe coectly changing accodingly the available point fom the no-load cuve, the leakage inductance ae aumed contant to thei ated value and do not change dependant fom the cuent between zeo and ated cuent. The Figue 44 give alo only infomation about the functionality of the model even with vey mall cuent I [A] Time [] Figue 44 Zoom into tato cuent at no load opeation afte DO- tat; olid line model including main and leakage inductance atuation, dahed line - meaued 49

65 3.6.4 Influence of the machine coupling In the above chapte the influence of atuation effect and fiction loe have been dicued. The dicued dependencie ae in pinciple the ame independence of the ue of a thee phae model o made with Clak o Pak tanfomation. In the following the influence of the machine coupling will be dicued. Unfotunately it i not only econday dicued, though found impotant duing the conideation of chooing the method of machine modelling fo the pupoe of imulating non ated condition. Theefoe ome peumption done duing developing the analytical model will be paid attention. The geneal ABC/abc geneato modelling ha been decibed. Gaining the input fo thi model i vey challenging unde the given cicumtance. Though it i woth the effot, ince a thee-phae ABC/abc geneato model give the poibility to look at each phae epaately a well a the coupling between the phae /4/ Theeby the ABC/abc model i not limited by condition of ymmety of eithe upply voltage o phae impedance /68/. The thee-phae analytical model can be eaily impoved with acceibility of moe input paamete, without complete model change. All thee advantage ae impoving the imulation eult. Anothe poible peentation of the ABC/abc pe phae machine model, point out the field coupling between tato and oto and the coelation to the voltage equation i given in Figue 45. i A R ψ A ψ a R i a u A i B R ψ B ψ b R i b u a u B i C R ψ C ψ c R i c u b u C u c u d d R i + Ψ R i u dt dt Ψ + ABC ABC ABC abc abc abc Figue 45 Thee phae equivalent ABC/abc cicuit diagam of a doubly fed induction machine The coupling i deived to be expeed a a poduct of the inductance matix /79/ and the cuent vecto i. The inductance matix of the machine model take a cental poition. Duing the computing of the machine model the invee of the inductance matix ha to be calculated. Thi intoduce a athe difficult tak and i lowing down the imulation poce. Including atuation effect impove the ealitic peentation of the machine but alo complicate the model even moe, ince the invee of the matix ha to be computed evey time tep. Taking the electical coupling in the induction matix into account give a poibility to elax the complexity of the model it elf and theewith implify the computing of the invee matix a popoed by Pillay /79/ and Goldembeg /3/. The inductance matix of the induction machine including atuation of the inductance ha been peviouly detemined to: 5

66 +.5.5 f f f f f f f f f σ A ma ma ma maa mab mac 3 mb σ B mb mb mba 3 mbb mbc mc mc σ C mc mca mcb 3 mcc maa f mab f3 mac f σ a+ ma.5 ma.5 ma mba f mbb f mbc f3.5 mb σ b + mb.5 mb mca f3 mcb f mcc f.5 mc.5 mc σ + mc () Auming a ta/ta (YY) connected machine with no extenal acceible ta point, the addition of the oto cuent pe phae and tato cuent pe phae i zeo in the ta point a hown in equation (). ia + ib + ic, ia + ib + ic () Taking thi into account, the induction matix take the following fom (3). +.5 f f f +.5 f f f +.5 f f f σ A ma maa mab mac 3 σ B mb mba 3 mbb mbc σ C mc mca mcb 3 mcc maa f mab f3 mac f σ a +.5 ma mba f mbb f mbc f3 σ b +.5 mb mca f3 mcb f m cc f σ +.5 mc (3) Notice, the tato and oto mutual inductance m, m ae of /3 m the one phae equivalent diagam. Fo model epeented in Clak o Pak tanfomation the YY coupling of the machine i aumed in the mot cae and the expeience made with the thee phae model ae exemplay fo the model made with a tanfomation a well. In cae of a Y connected machine the coupling of the oto phae can be implified a hown above in the YY example, while leaving the tato phae untouched. The inductance matix fo a Y connected machine take the fom hown in (4) f f f f f f f f f σ A ma ma ma maa mab mac 3 mb σ B mb mb mba 3 mbb mbc mc mc σ C mc mca mcb 3 mcc maa f mab f3 mac f σ a+.5 ma mba f mbb f mbc f3 σ b +.5 mb mca f3 mcb f mcc f σ +.5 mc (4) eaving the magnetic coupling of the tato phae in the oiginal allow the tato cuent to flow in a cicle. Thi give a moe ealitic eult than the uually ued method of auming a magnetically YY connected machine and aftewad calculation of the electical delta coupling of the tato winding, which i hown in the following example. A delta coupled geneato expeience a -phae line to line hot cicuit at the teminal. 5

67 voltage [p.u.] time [p.u.] Figue 46 Voltage at geneato tato teminal duing a two-phae hot cicuit While thee i no diffeence between the imulated tato cuent unde ymmetical condition befoe the hot cicuit, the diffeence between the model can be een in the aymmetical condition Figue 47. The cuent modelled with the ta connected machine model and a calculated delta coupling how diffeence in phae and amplitude to the cuent imulated with the delta connected machine model. tato cuent [p.u.] time [] Figue 47 Stato cuent of the geneato dahed line ta modelled machine with calculated delta coupling and olid line delta modelled machine They ae many diffeent way of machine modelling, but a the machine ae deigned fo thei application, the model ha to be developed fo the expected ue and the eality. The modelling in thi thei ue a thee-phae model becaue it ha the pupoe of accuately fault imulation in Veta vaiable peed wind tubine. Theeby it ha been hown, that diffeent analytical deciption of the e.g. connection of the machine phae have an inceaed impotance, while looking at aymmetical condition. eakage atuation ha an impotant influence, while looking at hot cicuit contibution to the gid, which play an impotant ole fo gid connection of the tubine. Not all influence have been tudied in thi peented wok. The influence of ion loe and kin effect a well a the change due to tempeatue change ae open fo futue eeach activitie. The ABC/abc machine model i beneficial to ue, becaue it doe not only implify the incluion of connection 5

68 condition influence and pe phae vaiability of atuation. The thee-phae model can be eaily expanded fo othe imulation pupoe e.g. aymmety in the machine paamete o hamonic. 53

69 Chapte 4 4. Modelling of Tanfome The Tanfome i an impotant pat to connect the wind tubine to the gid. Howeve, it i vey common to neglect the dynamic behaviou of the tanfome and only conide a vey imple ideal model o the ue of equivalent impedance fo calculation. In thi thei it i vey impotant to conide the dynamic behaviou of the wind tubine tanfome. The content of thi chapte ae the development of the -winding 3-phae tanfome model and the 3-winding 3-phae tanfome model. In the end the -winding 3-phae Autotanfome model i deived. 4. The -winding 3-phae tanfome model The -winding 3-phae tanfome (w 3ph tanfome) connect a thee phae ytem with two diffeent voltage level. The tanfome conit of winding, which ae connected though a magnetic cicuit in the coe. The two diffeent voltage level on the tanfome ae called high- and low-voltage ide o commonly alo known a pimay and econday ide, becaue the high voltage ide i uual connected to the t winding and the low voltage ide i connected to the nd winding. Auming thee i no ai gap in the coe, no voltage dop in the magnetic cicle (infinite coe pemeability), and no hyteei loe, the eitance of the winding ae negligible. The eult of all aumption togethe i a lo-le (ideal) tanfome, which i only epeenting a imple elationhip between the two voltage and cuent. u i atio a w w i u The voltage equation fo the two winding of uch an ideal tanfome i: dψ dψ u e u e dt dt Hence the whole flux i cloing inide the coe the flux in winding and winding ae identical and theewith the flux linkage can be expeed a: Ψ w Φ Ψ w Φ ideal tanfome In thi lo-le tanfome the ummation of the cuent conideing the numbe of winding i w i + i w. Uing the cuent coelation in combination with the voltage equation the elationhip between pimay and econday voltage can be eaily expeed with the winding atio a. u u w w a w w u a u u The econday voltage tanfomed with the winding atio i equivalent to the pimay voltage. Similaly the cuent can be expeed with the winding atio to: 54

70 i i i a Thee with the ideal model achieved imple coelation ae the mot baic equation fo tanfome conideation. In a ealitic tanfome a pat of the magnetic field will cloe via the ai and theefoe only a pat of the magnetic field i linked to both winding. The pat of the flux, which i not linked though the coe, i called the leakage flux, which only link to eithe pimay o econday winding. The loe in the winding of the tanfome ae expeed with an electical eitance. i R Φσ Φ σ u u w Φ w m + Φm R i Figue 48 -winding -phae Tanfome The definition of the flux i vey imila a in an electical machine, divided into main flux in the coe and the leakage flux. Ψ w Φ + w Φ + w Φ (5) σ m m Ψ w Φ + w Φ + w Φ (6) σ m m A in the electical machine theoy the tanfome i often expeed in a one phae equivalent. The inductance ae achieved a: w ( Φ σ + Φm ) σ + m i w ( Φ σ + Φm) σ + m i wφ m i wφ m i w Φ m m (7) i w Φ m m (8) i w (9) m w w () m w w m m w () and decibe the mutual inductance between the two winding and, the elf-induction of the winding. The flux coupling between winding and can be witten a: 55

71 Ψ i i Ψ () The voltage equation theewith become: d dt [ u] [ R] [ i] + ( [ ] [ i] ) u i d i [ R R ] u i + dt i (3) Thi one phae equivalent ytem can be depicted in the well-known equivalent T cicuit of two magnetically coupled coil. Pefeed i to elate the paamete and vaiable to one ide of the tanfome and uing the ideal tanfome to expe the tanfomation. The paamete ae tanfomed uing the intoduced atio a. The tanfomation fom all value to the pimay ide can be made in the following way: u a u i i a Ψ a Ψ a a a R a R (4) The calculation of the value to the econday ide can be made in the ame way. The paamete itelf can be detemined with a no load and a hot cicuit tet /5/, /6/, /67/, /75/, /8/. i R σ σ R i atio a u m u u ideal tanfome Figue 49 Equivalent cicuit of a tanfome efeed to pimay ide atio a R i σ R σ i u u m u ideal tanfome Figue 5 Equivalent cicuit of a tanfome efeed to econday ide 56

72 Fo aymmetical dynamic imulation of the w3ph tanfome the equivalent one phae equation developed above ae not ufficient enough. The coupling between the phae ha to be conideed a well. The geneal voltage equation fo a two winding w3ph tanfome conideing all phae can now be witten a: d dt [ u] [ R] [ i] + ( [ ] [ i] ) ua i a u b i b u c i c R a R b R c Ra Rb Rc u + a ia u b i b uc ic a ab ac a f ab f ac f3 i a ba b bc ba f3 b f bc f i b d ca cb c ca f cb f3 c f i c + dt a f ab f3 ac f a ab ac ia ba f b f bc f3 ba b bc i b ca f3 cb f c f ca cb c ic (5) f co( ϑ) f co( ϑ + π ) f 3 co( ϑ π ) 3 3 In the inductance matix equation (5) new facto f, f, f 3 ae intoduced. The facto conide the extenal connection of the winding and the elated tun in the phae. The facto i eay explained with an example: A vey common coupling of the winding of a thee-phae tanfome i DYN5. The expected phae diffeence between the voltage in phae a of the pimay winding compaed to the voltage in phae a of the econday winding i 5 3gad 5gad. To achieve thi phae diffeence with the model the angle theta hould be et to ϑ.68ad. 57

73 4. The 3-winding 3-phae tanfome model The thee-winding thee-phae (3w3ph) tanfome i modelled in the ame way a the w3ph tanfome. The geneal equation imply get expanded with a thid ytem epeenting the additional thee phae in the thid winding: d dt [ u] [ R] [ i] + ( [ ] [ i] ) u a a Ψ a u b i b Ψ b u c i c Ψ c ua ia Ψ a u d b R i a R b R c R a Rb Rc R3a R3 b R 3c b + Ψ b dt u c i c Ψc u3 a i3a Ψ 3a u 3b i Ψ 3b 3b u 3c 3c i Ψ 3c i (6) Ψ a a ab ac a f ab f ac f3 3 a f 3 ab f 3 ac f3 i a Ψ b ba b bc ba f3 b f bc f 3 ba f3 3 b f 3 bc f i b Ψ c ca cb c ca f cb f3 c f 3 ca f 3 cb f3 3 c f i c Ψ a a f ab f3 ac f a ab ac 3 a f4 3 ab f5 3 ac f6 i a Ψ b ba f b f bc f3 ba b bc 3 ba f6 3 b f4 3 bc f 5 i b Ψ c ca f3 cb f c f ca cb c 3 ca f5 3 cb f6 3 c f4 i c Ψ 3a 3 a f 3 ab f3 3 ac f 3 a f4 3 ab f6 3 ac f5 33 a 3 ab 3ac i 3a Ψ3 b 3 ba f 3 b f 3 bc f3 3 ba f5 3 b f4 3 bc f6 3 ba 33 b 3 bc i3 b Ψ 3c 3 ca f3 3 cb f 3 c f 3 ca f6 3 cb f5 3 c f4 3 ca 3 cb 33 c i 3c f co( ϑ) f co( ϑ + π ) f 3 co( ϑ π ) 3 3 f 4 co( ϑ) f 5 co( ϑ + π ) f 6 co( ϑ π ) 3 3 In the ame way multiphae and multi-winding ytem can be modelled. 58

74 One poblem fo modelling uch a tanfome i the detemination of the input paamete fo the model. A poible way i to detemine the paamete of the 3w3ph tanfome fom the data heet of the tanfome, which ae uually given in the following fom. Typical data Voltage Un, Un, Un3 Appaent powe S n, S, S 3, S 3 Vecto goup e.g. DY5 Shot cicuit voltage u k, u k3, u k3 Shot cicuit voltage (eitive) u, u 3, u 3 No load excitation loe I [%] Ion loe P [W] S I I U n U n S 3 I I 3 Figue 5 Simplification of 3w3ph tanfome U n U n3 The coupling between the 3 winding of the 3w3ph tanfome hown in Figue 5 i epaated into thee ingle equivalent diagam Figue 5 in ode to undetand the intepetation of the typical data et. The typical data decibe uually the coupling between the diffeent winding and theefoe the deived paamete ae alway a mixtue of both winding. Fo the 3w3ph dynamical tanfome model the individual paamete pe winding have to be achieved. Howeve, it i not exactly poible, but they can be etimated with auming a ta connection of the winding. The paamete though can be ued late on in the geneal model a decibed above. U n I S 3 I 3 U n3 Figue 5 Simple equivalent cicuit of a 3w3ph tanfome 59

75 The paamete elated to the pimay ide can be calculated a followed: Z Z Z R k k3 k 3 3 uk U [ Ω ] Zk Zk + Zk % S uk3 U [ Ω ] Zk3 Zk + Zk 3 % S 3 uk 3 U [ Ω ] Zk 3 Zk + Zk 3 % S 3 3 uk3 U [ Ω ] R3 R + R3 % S R uk U [ Ω ] R R + R % S (7) R 3 uk3 U [ Ω ] R3 R + R3 % S 3 k k [ ] X Z R Ω k3 k3 3 [ ] X Z R Ω k 3 k 3 3 [ ] X Z R Ω X X + X k σ σ X X + X k3 σ σ 3 X X + X k 3 σ σ 3 Z Z + Z Z [ ] k3 k k 3 k Ω Z Z + Z Z [ ] k k 3 k3 k Ω (8) Z Z + Z Z [ ] k 3 k3 k k3 Ω The eitance and the leakage eactance can be found equivalent to the impedance calculation. Notice though that eitance and impedance, which ae calculated in thi way, ae only etimated atificial value and ae not diectly elated to the eal paamete. In ome cae the calculated value become negative, epecially if uing the technique fo an autotanfome /39/. The main flux inductance can be calculated fom the no load meauement. X m I % U S [ Ω ] R [ ] fe U Ω (9) P 6

76 The Matix paamete can now be detemined fo e.g. ymmetical geomety to: R R R R a R R 3 3 a3 σ Xσ ω X X σ σ ω a m m m m ω ω a X X σ 3 σ 3 ω a3 X m m3 ω a3 () X m ω a 3 3 X m ω a 3 a 3 3 m The -winding 3-phae autotanfome Uing the ame appoach a decibed in the chapte 4., the thee-phae Autotanfome can be een a a modification of the 3w3ph tanfome; even it belong to the goup of w3ph tanfome. While a 3w3ph tanfome ha thee epaated theephae winding, the Auto tanfome ha a galvanic connection between the econd and thid winding. Thi can be expeed in the equivalent diagam a hown in Figue 53. In S In S 3 Un Un In R Un In3 R 3 In S 3 In3 Un3 ( ) In 3 Un Un3 Un S 3 Figue 54 Coupling of the econd and thid winding in an autotanfome In In3 Un Un3 Figue 53 Simple equivalent diagam of the theephae thee-winding autotanfome 6

77 The imple equivalent diagam of the 3w3ph tanfome hown in Figue 5 ha only changed the coupling between the econd and thid winding. In Figue 54 i the coupling between the econd and thid winding i hown epaately to viualie the detemination of the eitance paamete. R3 R + R3 R R + R R3 R + R3 () If the definition of the eitance between the winding i till valid a hown in equation () and cuent between gound and the haed winding i: I U () 3 n3 ( ) I n3 U The coupling eitance between the pimay and tetiay winding can be calculated to: U R R + R (3) 3 3 ( ) U While the eitance of the tetiay winding i: R U R 3 3 U (4) Following thi path the ame equation et a hown fo the 3w3ph tanfome i deduced. The model fo the 3w3ph tanfome can alo be ued fo decibing the autotanfome. The only diffeence i the calculation of the paamete input, ince the paamete of winding thee i dependent on the paamete fom winding two. 6

78 4.4 The tanfome model implemented in Matlab/ Simulink The w3ph tanfome model a developed with the equation (5) i implemented in Matlab/ Simulink a depicted in Figue 55. In the block Flux - cuent i the 6x6 inductance matix i pogammed. The input value ae the thee-phae voltage of winding and winding. The output ae the thee-phae cuent of both winding. Figue 55 The w3ph tanfome model in Matlab/ Simulink Included in the model i the phae diffeence due to the coupling. In Figue 56 a imulation of a DY5 (delta-ta) tanfome coupling i demontated. The pimay and econday voltage efe to thei ated value at a no load imulation i hown. The 5-degee phae diffeence fo the 5 Hz ignal i clealy viible Upim Uec [p.u.] t [] Figue 56 Phae diffeence pimay (thick line) and econday (thin line) voltage imulating a DY5 w3ph tanfome 63

79 In Figue 57 and Figue 58 the tanfome i hown at no load and loaded opeation. At no load thee the no load loe can be meaued. The voltage i not ideal and a no load cuent i exiting. Duing load the tanfome voltage i deceaing at the ame time the cuent ie Uec [p.u] Ipim, Iec [p.u] t [] t [] Figue 57 Ideal (thin line) and eal econday voltage (thick line) at no load and full loaded Figue 58 Pimay (thick line) and econday cuent (thin line) at no load and loaded Typically the tanfome i gounded at the ta coupling of the econday ide. Thi gounding can be implemented in the tanfome model. The gound cuent can be calculated a a diffeence between the load cuent and the cuent output of the tanfome and initiate a voltage dop to gound. Thi potential diffeence i conideed at the econday voltage of the tanfome. The w3ph tanfome model incluive with a eitive gounding i hown in Figue 59. The input and output of the model change due to the implementation. The input i now the 3-phae pimay voltage and the econday ide 3-phae cuent. The output of the model i now the econday 3-phae voltage and the 3-phae pimay cuent. Figue 59 The w3ph tanfome incluive gounding ytem 64

80 The 3w3ph tanfome model o the w3ph autotanfome model i equivalent to the wph model. The only change i the additional thid input vecto of the voltage and the thid output epeenting the tetiay cuent. The block Flux cuent now include a 9x9 inductance matix. Figue 6 The 3w3ph tanfome model in Matlab/ Simulink including ta point gounding In the ame way a applied in the induction machine model the main atuation can be implemented a a vaiable induction dependent upon fom the magnetizing cuent. The change in the main inductance due to atuation i moe than in an induction machine, elated to the diffeent flux path in tanfome and induction machine. The beginning of the magnetizing cuve chaacteize the inductance of the flux though the ai, while the econd pat of the cuve chaacteize the inductance influenced fom the magnetizing mateial. Becaue in the tanfome a huge pat of the flux i coupled ove the ai, whee contay to the induction machine which ha only a minimal ai gap, the cuve i iing vey fat in the beginning (Figue 6). 65

81 Figue 6 Magnetizing cuve of a. MVA 3w3ph tanfome elated to the ated cuent The magnetizing impedance can be calculated fom the no load meauement. The value of the magnetizing impedance conit of an ion loe eitance in paallel to the main inductance. The magnetizing inductance can be calculated with equation (5). Z m U I Z 3 Rm ωm Zm I Rm + ω m m Rm Zm ω R Z m m (5) Figue 6 Main inductance atuation cuve of a. MVA 3w3ph tanfome 66

82 Figue 63 Block flux cuent of a 3w3ph tanfome with main atuation 67

83 4.5 Geomety Development of thee-phae Tanfome The thee phae model ha beide the non-lineaity caued by the atuation anothe natual aymmety caued by the geometical contuction of the thee-phae Tanfome. Thi fact i uually neglected, though play an impotant pat if the pupoe i thee phae imulation of ymmetical and aymmetical condition. Geneally a thee phae tanfome can be achieved by connecting thee one-phae tanfome unit. a a b b c c Figue 64 Thee one-phae E-coe tanfome in ta connection Figue 65 Thee one-phae U-coe tanfome in ta connection The diffeence between the two diffeent tanfome coe hape i the ditibution of the flux. In compaion to the aembly in Figue 65, the flux in Figue 64 i epaating and cloing ove the oute coe, which caue a bigge ize of the tanfome unit. The connection of 3 one-phae tanfome i not the mot ued olution fo a thee-phae ytem. Rathe than thi aembly a combination fom the 3 tanfome to one theephae tanfome i ued. In Figue 67 one tep fo the development of uch an aangement i hown. 68

84 u 3 i 3 Pimay and Secinday winding Φ 3 Φ m u i Φ i Figue 66 Magnetic connection of 3 one phae tanfome to a thee phae tanfome aangement u Figue 67 Slice of the thee-phae tanfome aangement The flux in the middle coe m i the addition of the flux fom the thee phae. Φ Φ + Φ + Φ m 3 In the cae of ymmetical contuction and excitation the flux in the common coe add by upepoed to zeo, that mean no flux i peent in the common leg. Φ Φ + Φ + Φ m 3 Theefoe the common leg can be cut out of the contuction. Pimay and Secinday winding Figue 68 Symmetical thee-phae tanfome Figue 69 Symmetical thee-phae tanfome Howeve, thi ymmetical aangement ue too much pace and theefoe it i widely ued to aange the thee leg of the tanfome in one plan. Howeve thi i changing the flux path and a pat of the now aymmetical flux ha to cloe ove the ai outide of the tanfome coe Φ + Φ + Φ 3 Φ ai. 69

85 Figue 7 Unymmetical thee-phae tanfome Figue 7 Unymmetical thee-phae tanfome If it i aumed the flux in the ai i minimied, the flux coupling fom the middle leg i hote that fom the oute leg and caue an unymmetical flux contibution. Thi caue a malle magnetiing cuent in the middle leg and alo a malle phae cuent /6/, /8/. Thi unbalance can alo be obeved duing opeation on a ymmetical gid and in no load opeation. 7

86 Chapte 5 5. Powe ytem modelling in Matlab/ Simulink The achieved geneato and machine model ae pat of the electical ytem in wind tubine imulation. Beide thee two hown component, othe element ae neceay in ode to tudy the wind tubine behaviou duing hot cicuit. Any mall netwok conit of a ouce, line and a load. In the peent cae the wind tubine i the load of the netwok, connected to a voltage ouce and a tanmiion line o cable, which epeent netwok dynamic. The wind tubine itelf conit of diffeent electical element. The in Chapte 3 developed machine model and the in Chapte 4 developed tanfome model ae two main pat of the wind tubine model. In thi chapte the othe electical element will be modelled. In the beginning the dynamic node technique will be intoduced, followed by the ouce modelling and finally a tanmiion line model i deduced. 5. Dynamic node technique In the cae of connecting the diffeent electical element like geneato tanfome, cable, load, ouce etc. to a malle electical netwok in Matlab/ Simulink a modelling method ha to be found. A uable method i the Dynamic node technique fom F. Flinde and W. Oghanna in /5/, /7/ which i developed fo powe convete imulation. Thi method can be ued fo any electical cicuit imulation in Matlab/ Simulink. The mot difficultie with modelling electical netwok in Simulink ae given due to imulation tability. Complicated mathematical tuctue with diffeential equation and nonlinea behaviou often lead to untable imulation and algebaic loop. The mot efficient way i to model electical machine and tanfome due to the cuent, which mean with voltage a input vaiable and cuent a output vaiable. If it i deied to connect a tanfome and machine the voltage need to be a function of the cuent. The poblem i dafted in Figue 7. Ua Ia Za Ua Node Ub Ib Zb U b U ouce U N Figue 7 Electical netwok daft fo imulation of the powe ytem To Next Element Figue 73 Block diagam expeing the theoy in Matlab/ Simulink The poblem can be olved by conideing the paaitic capacitance between the element, which ae deied to connect. The capacitance can be aumed a lumped at the cicuit node. The voltage ove the capacito can be calculated with the ummation of the cuent connected tough thi node. 7

87 u node indt C (6) node Duing teady tate the node voltage i contant and ha no influence on the imulation. Kichhoff cuent law would be atified fo evey node within the cicuit. i n ( teady tate) (7) The pope function of thi method duing dynamical imulation, without influencing the imulation eult, i enued by the choice of a vey mall nodal capacitance (le than nf). The integal natue of the node decouple the mathematical loop ytem and theewith how the additional advantage of counteact algebaic loop. With thi technique diffeent device can eaily be modelled. In Table 5- a few example ae given. Table 5- Diffeent electical element modelled with the dynamic node technique in Matlab/ Simulink Device Cicuit Block (cuent -> voltage) node i i i 3 u n C node u ef Reito R i R u R Paallel inductance/ eitance R i R // eial inductance/ eitance u R // R ir u R 7

88 paallel capacito/ eitance R C i R // C eial capacito/ eitance u R // C R C i RC u RC 5. Powe ytem element 5.. Geneal modelling The voltage ouce ued fo the imulation i an ideal voltage ouce, with diffeent imulation option. The model implemented in Matlab/ Simulink can be deived fom the equation (8) ( π ϕ ) ( π ϕ ) ( π ϕ ) u Uˆ in f t + a a a u Uˆ in f t + b b b u Uˆ in f t + c c c (8) Theeby the baic fequency f i aumed a equal in all thee phae and ha the 5 Hz fo a 5 Hz ytem. The phae angle ϕ and voltage amplitude U ˆ can be changed independently fo each phae. In cae of a ymmetical thee phae ytem the phae angle between the phae can be aumed contant a a phae diffeence of o /3. Figue 74 how the implementation in Matlab/ Simulink fo uch a ytem. Figue 74 Thee-phae ideal voltage ouce in Matlab/ Simulink Thi baic pinciple wa expanded with eveal function to imulate hot cicuit, voltage dop and fequency change in the ytem voltage. The input maked fo the Matlab/ Simulink block i hown in Figue 75. The model i in an advanced model futhe 73

89 expanded with the upepoition of hamonic to the decibed fundamental function. ( π ϕ ) ˆ ( π ϕ ) u Uˆ in f t + + U f t + υ υ (9) The index υ theeby decibe the numbe of the hamonic. Figue 75 Input mak fo the imple Matlab/ Simulink block 3 phae voltage ouce To achieve a moe ealitic behaviou a eial eitance and inductance can be added to the ouce. Uing the dynamic node technique thi i implemented a hown in Figue 76 and Figue

90 R R tay i R R // i iload u abc C node u node node C node u u load u ef Figue 76 poible R equivalent netwok Figue 77 Poible R equivalent netwok in Matlab/ Simulink A pobably noticed thee i a paallel eitance added to the inductance. Thi eitance intoduce a damping of the inductance. The uual value fo thi eitance i vey lage a e.g. R M Ω. tay In Figue 78 the entie ytem ouce with a eial R tanmiion i hown. Figue 78 Souce and R tanmiion pat in Matlab/ Simulink The ame tategy can be ued fo connecting abitay two device. 75

91 5.. Tanmiion line modelling The common model of tanmiion line i eaily modelled and implemented in the ame way a decibed fo the connection of two element. Fo the pupoe of dynamical imulation a highe ode tanmiion line model including the coupling between the phae i deived. In ode to claify the input paamete fo thi dynamical model baic coelation ae decibed again. The oveall model i hown in Figue 79. a R b c R R C g G g Figue 79 Dynamical tanmiion line model Howeve, the tak of modelling a highe ode tanmiion line o cable i not a eay a tak a it eem. Patly thi i becaue thee ae many diffeent type of line, which complicate the tak of finding a geneal model. In ode to give a pope intoduction in thi field, the backgound of tanmiion line modelling i explained in moe depth than the peviou developed modelled component. ine Impedance It i aumed that an ovehead line hould be modelled, which i implified to an infinite long, paallel and taight line. Thee aumption give the poibility to deive the neceay equation, which decibe ufficiently the behaviou of a ealitic ytem a well /6/, /94/. The magnetic field tength outide and inide a line, which i caying a cuent, can be calculated with the help of Maxwell equation to: I H ( ) π I H ( ) P π P P P > (outide line) (3) < (inide line) (3) 76

92 I P H Figue 8 Magnetic field tength of a line The inductive coupling between the two line can be deived fom the upepoition fom the field of the line. If two paallel line, ae tudied which have the ditance D and a adiu (Figue 8), the electomagnetic field i defined though the magnetic field foce B and electical field foce H. B µ H µ µ µ (3) Φ B da (33) B + (34) B B Φ H + H ) da µ ( µ 4π 7 µ ( ai) H m (35) Φ l µ I D π ln (36) A D Figue 8 Inductance of two paallel line The inductance between the two line i dependent on the length and calculated fom the field: Φ n µ D l I π ln with n (37) The intenal inductance of the line itelf can be calculated like in equation (38). It i noticed that the intenal inductance and field i not dependent on the adiu. Thi i caued due to the aumption of homogeneou cuent denity of the conducto. 77

93 i µ (38) 8π The total inductance pe length of line fom the two-line ytem i: µ D + i ln + π 4 (39) The equation (39) can alo be ued to achieve an aveage inductance pe conducto fo a thee-phae ytem, a e.g. hown in Figue 8. D ab D bc D ac Figue 8 Inductance of a thee phae ovehead line The inductance of the thee conducto can be diffeent, becaue of it dependency on the ditance between them and the cuent in the line. In mot tanmiion line the method of tanpoition i ued. The phae-line change theeby thei geometical place. In thi way the geometical aymmety get adjuted to a nealy ymmetical ytem. If it i futhemoe aumed that the cuent in the thee phae ae Ia + Ib + Ic an equivalent aveage ditance (4) and can be ued to calculate the aveage pe phae inductance (4): D 3 nm DabDbcDac (4) Dnm abc + i abc ln + π B 4 µ (4) A imila method fo the calculation of the aveage inductance can be ued fo the n- bundle conducto figuation. B n n n T n n 3 n 4 T T T a 8 in n o a a a B 3 a a B B 4 a Figue 83 n-bundle conducto line 3 78

94 Whee n define the numbe of conducto, the adiu of the conducto, T the equivalent cicle adiu, a i the ditance between the conducto and B the equivalent adiu of the n-bundle conducto. The equivalent adiu eplace the adiu of one conducto in equation (4). The eitance of the line can be calculated fom the co ection aea A of the conducto and the electical conductivity ρ. R i ρ l l A κ A (4) m S m S The electical conductivity fo coppe i κ 56 and fo aluminium κ mm mm Although it i moe appopiate to ue efeence book fo the detemination of eitance. Calculation can caue not accuate enough eult, becaue the additional effect uch a kin and poximity effect ae vey difficult to etimate in the co ection aea. The above decibed inductance ae appopiate fo the ue in an equivalent model uch a the common -line model. In ode to achieve a dynamical tanmiion-line model the influence of the gound ha to be conideed a well. If an ideal gound i aumed which ha an infinite conductivity, the influence can be calculated with uing the method of eflected conducto. With thi method a econd conducto can be imagined in the gound, which i eflected on the gound uface and caie the ame amount of cuent, ee Figue 84. The extenal inductance of a ingle conducto with gound i calculated to: hj (43) π µ j ln In eality the gound doe not pefectly conduct. Theefoe the equation (43) give no ealitic eult. In /44/ two main method fo calculation of the gound impedance ae dicued. If it i aumed that the conducto adiu i mall compaed to the ditance between conducto and gound, an expeion deived fom Calon /44/, /94/ can be ued. The othe inductance can now be witten a: j µ δ g ln (44) π The value δ g i the equivalent depth fo the cuent penetation in the gound. ρg δ g 66 (45) f ρ g i the gound conductivity and it value lie between and Ω m and f i the fequency. Typical value fo the equivalent depth and the gound conductivity ae given in //. If the gound i not known a conductivity of Ωm can be ued, which lead to an equivalent depth of 93 m at 5 Hz. 79

95 j D jk k h j j h j h k µ gound D j µ h j h j j h k Figue 84 Single ine-gound coupling Figue 85 Two line ytem with gound coupling The eitance of the gound i R Ω π f (46) km 4 g It i only dependent of the fequency and independent fom the cuent penetation. Thi i due to the abitay big aea fo the cuent flow in the gound. The impedance of one conducto to gound finally complete to: jg j g ω j i Z R + R + j ( + ) (47) A moe appopiate equation to include in a modelling pogam i the equation developed by Wedepohl /44/. By intoducing a complex depth of penetation δ gw in the gound: δ ρ g gw (48) jωµ The gound impedance can be achieved though: jωµ h + δ gw Z g Rg + jω g ln π (49) If a econd conducto i intoduced a hown in Figue 85 an additional mutual coupling between the two cicuit ha to be intoduced. The inductance can be calculated to /94/. 8

96 jk µ D j kj ln π D jk (5) The equation fo the mutual inductance found by Wedepohl /44/ i: jωµ Z jk Z kj Rkj + jω kj ln π ( hj + hk + δ gw ) + D jk ( hk hj ) D jk (5) If it i futhemoe aumed that the equivalent depth fo the cuent penetation in the gound i much bigge than the ditance fom the conducto to gound fo the mutual inductance fom a two-line ytem to gound which can be implified to: jk µ δ g kj ln π D jk (5) Expeing the inductance of the two-line o two conducto ytem a the length elated inductance in matice we can wite: (53) Whee the elf-inductance, ae the addition of extenal inductance and intenal inductance and, ae the mutual inductance. + + i i j k (54) jk Equivalent the eitance matix can be found. R R R R R (55) Whee eitance R and R, R ae the total elf-eitance including the etun path and the mutual R i the value of the etun path /44/. R R + R i g R R + R j g R R R (56) g ine admittance Thee i an analogy between the electical field calculation method and the magnetical field calculation method. Theefoe the eult fo the capacitance calculation of a line i quite imila to the inductance calculation. A cylindical line i aumed, which ha chaged paticle on the uface. The chage 8

97 denity of the line i q and Q define the chage of the line and can be calculated with the flux denity D fom Maxwell equation and the Gau law: D da q dl Q D da (57) Becaue the vecto D i paallel to the uface aea vecto, the whole value i in thi diection and a cala deciption can be ued. Q dl e z e e α P(, α, z) D, da Figue 86 Flux denity and electical field intenity of a line The electical field intenity calculate fom the flux denity to: D ε E ε ε ε (58) ε i the mateial contant decibing the pemittivity. The pemittivity of the fee pace ε 8.85 F / m. Solving Poion diffeential equation the capacity of the line i [ ] calculate fom the potential diffeent voltage. U ϕ E d D d ε (59) Q C U (6) It i common to ue the potential coefficient P a the ecipocal value fom the capacitance C. The equation (6) i eaanged to: U P Q (6) Uing the hown equation the field denity of the line i q D (6) π Theewith the potential diffeence between the line and a point outide line i: U P q q P d ln (63) ε π πε The capacity i theefoe: 8

98 Q C q P P ln ln πε πε l (64) C C l πε P ln F m (65) A hown fo the inductance the poibility fo a conducto bundle can be ued a well. The adiu ha to be eplaced with the adiu of the bundle. +q -q D Figue 87 Capacitance of two paallel line The mutual capacitance of two-line ytem can be achieved with upepoition of the two field. U U U C _ q D ln πε _ ln q πε D q ln q ln πε πε πε D ln (66) (67) (68) (69) The Capacitance of one line to gound can be detemined with the eflecting method decibed in Figue 84. The gound i a potential aea fo zeo potential. It i vey obviou, that the gound capacitance become: C C πε h ln g jj (7) The potential coefficient to gound become with ineting the pemittivity of the fee pace to: 83

99 P g 9 h m Pjj 8 ln F (7) If thee i an aangement of a two-line ytem like in Figue 85 the mutual capacitance coupling i. C C πε D j ln D jk jk kj (7) P jk D m 9 j 8 ln D jk F (73) It can be concluded that the potential coefficient o the capacitance pe line i a upepoition fom the potential coefficient/ capacitance of the elf-chage and the chage fom anothe chage. U j C jk k U k j C jj Cii µ, ε Figue 88 The patial capacity between two line Often thee i an inteet in the individual capacitance value. The coelation between the voltage and chage i given to /6/: ( ) ( ) Q C U + C U U i jj j jk j k Q C U U + C U k jk k j kk k ( ) Q C + C U C U i jj jk j jk k ( ) Q C U + C + C U k jk j jj jk k (74) (75) Now it i poible to define the coefficient dimenion of a capacitance. K jj, K kk, K jk and K jk, which have the ( ) K C + C K C jj jj jk jk jk ( ) K C K C + C kj kj jj jj jk (76) 84

100 In geneal fo a ytem of n line the following i valid: K C + C K jj jj jk jk C jk (77) The capacitance matix notation can be witten: K K K K K (78) Accoding to the equation (77), (78) the capacitance matix fo a thee-phae ytem can be deduced to: a C ac c C ab b C bc Caa C bb Ccc Figue 89 Model capacitance in a thee phae ovehead line Caa + Cab + Cac Cab Cac K Cba Cbb + Cba + Cbc Cbc C C C + C + C ca cb cc ca cb (79) Fom the Maxwell equation we can deduce a coelation between the chage and the cuent can be deduced to: I jω Q (8) Including the loe of the capacitance the Admittance coefficient to: A G + jω K (8) And the complete geneal deciption a: i Au (8) 85

101 5..3 Model implementation in Matlab/ Simulink The baic equation fo a dynamic 6 th ode model of the thee phae line hown in Figue 79 ae: d abcr abc abc + abc abc dt (83) d iabc Gabc uabc + Kabc uabc dt (84) [ u ] [ R ] [ i ] [ ] [ i ] [ ] [ ] [ ] [ ] [ ] In the model the admittance loe G ae not neglected. To olve the diffeential equation (83), (84) in Matlab/ Simulink equation have to be ewitten. d [ i abc ] [ abc ] ( [ u abcr ] [ R abc ] [ i abc ]) dt (85) d [ u abc ] [ P abc ][ i abc i load ] dt (86) [ ] abc [ ] aa ab ac ba bb bc ca cb cc R R R R abc R R R R R R aa ab ac ba bb bc ca bc cc aa i ag + ab abg Raa Ri + Rg Rab Rg (87) Caa + Cab + Cac Cab Cac P Cba Cbb + Cba + Cbc Cbc C C C + C + C [ ] ca cb cc ca cb The model paamete decibed a matice ae contant, which take the coupling between the 3 phae and gound into account. The line can be modelled with line element a hown in Figue 9. x i abc x R line x line i abc _ load u abc x u abc x P line u abc x u abc Figue 9 -phae equivalent fo 3-phae model of the line 86

102 Dynamical input i the load cuent at the end of the line and the voltage at the beginning of the line. Output i the cuent in the line egment and the voltage at the end of the line. Figue 9 Example fo tanmiion line model implemented in Matlab/ Simulink The deduced equation can be ued fo a cable model a well. A cable model diffe fom the tanmiion line model in the point, that the ingle phae ae placed cloe togethe and theewith kin and poximity effect play an impotant ole. Futhemoe thee i uually an exta cable fo the gound, which o ha to be conideed in the coupling matice. 87

103 Chapte 6 6. Wind Tubine modelling In the peviou chapte the impoved model of the electical component ae decibed and hown. Thee advanced electical model combined with the mechanical model, the aeodynamical model and contol ytem model conclude a complete wind tubine model. In thi chapte the aeodynamic and the mechanic model of the wind tubine ae decibed. Thee model ae implemented in Matlab/ Simulink in the block called mechanical ytem a hown in Figue 9. Figue 9 Block Mechanical Sytem in Matlab/ Simulink of the Wind Tubine 6. Aeodynamic Model The kinetic enegy obtained by the blade fom the wind i tanfomed to mechanical toque on the oto haft of the wind tubine. The blade ae attached to the oto haft and otate with the tip peedω ot, whee i the length of the blade. The blade pofile expeience a elative wind velocity geneated by the upepoition of the tip peed and the wind velocity v w. While wind i paing the pofile it intoduce lift () and dag (D) foce on the blade, which eult in movement of the blade. Fom thee foce the powe obtained fom the wind can be calculated a hown in equation (88). 88

104 Roto plane α v el ω ot v w D ω ot Roto axi ϑ Figue 93 Co ection of a wind tubine blade (-angle of attack). 3 Pw ρai A cp ( λ, ϑ ) vw (88) Theeby ρ ai i the ai denity, v w the fee wind peed expeienced by the oto, A the wept oto aea and c p the powe coefficient. The powe coefficient depend upon the pitch angle ϑ (often denoted a β ) and the tip-peed-atio λ. λ ω v ot (89) w The tip-peed atio i defined by the atio of the tip peed ot ω and the wind peed w v. The powe coefficient i typically given in a fom of a table a hown in Figue

105 .5.4 c p λ Figue 94 The powe coefficient tip peed atio cuve a a function of the pitch angle The toque on the oto haft, which i impotant fo the haft model late on (Figue 97), theewith can be calculated fom the powe with help of the otational peed. T P w A (9) ω In Figue 95 the implementation of the aeodynamical model i hown. Notice that the toque calculated with equation (9) i called Tot in the pictue. It i the toque, which i obtained by the oto haft though the blade elated to the low peed ide of the tubine haft. Figue 95 The aeodynamical model implemented in Matlab/ Simulink 9

106 6. Pitch model The mechanical aangement fo pitching the blade of the tubine i late found in the Matlab/ Simulink block mechanical ytem/pitch ytem and hown in Figue 96. The pitch model conit of the pitch actuato. The pitch actuato i expeed a a non-linea function between the contol voltage ignal and the eal pitch angle. The chaacteitic i modelled a a non-lineaity with a dead-zone. Figue 96 Model of the pitch ytem implemented in Matlab/ Simulink 6.3 Mechanical Model In thi chapte the implified mechanical model of the wind tubine will be decibed. Thi dynamical model, which can be found in a boad ange of liteatue /56/ - /75/ i widely accepted a expeing the dynamical behaviou of the dive tain fo the aea of eeach. The implified model of the dive tain i hown in Figue 97. In thi model all mae ae lumped at the low and high peed haft. The inetia of the low peed haft come mainly fom the otating blade and the inetia of the high peed haft fom the geneato. Thu it i moe impotant to include all mall mae of the high peed haft, becaue they have an impotant influence on the dynamic ytem due to the gea atio tanfomation. The ma of the geabox itelf i inignificant and neglected. Stiffne and damping of the haft ae combined in one equivalent tiffne and damping placed at the low peed ide. 9

107 low peed haft high peed haft Roto geabox geneato T A TS ϕ K n J D ϕ T S T ϕ m ϕ g Tm Te T fic J g Figue 97 Dive tain chematic fo modelling of a wind tubine Input to the model fo a two ma ytem i the toque T A obtained by the aeodynamic ytem and the geneato eaction toquet e. Output i the change in oto peed geneato peedω g. ω and The high peed geneato dynamic can be expeed a aleady hown in the deciption of the machine model. Pleae notice the ign adjutment due to geneato opeation. The diffeence of the mechanical diving toque T m, the geneato eaction toque T e and the toque due to fiction loe T fic eult in the change of the geneato angula peed ω g. Tm Te Tfic J g ωg (9) ω ϕ (9) g g The change of the angula oto peed ω ae caued by the diffeence of the aeodynamic toque T A and the haft toque T S at the low peed ide. TA TS J ω (93) ω ϕ (94) The mechanical diving toque T m and the haft toque T S ae connected by the ue of the gea atio in the following way: T m T n S (95) The dynamic of the haft can be decibed with the equation (96) TS KS ϕ + DS ϕ (96) ϕ g ω g ϕ ϕ ω n n (97) 9

108 Ineting the equation (96) and (97) into (9) and (93) and uing the equation (9), (94) and (95) the equation (98) and (99) will be obtained. The equation ae ued to decibe the dive tain dynamic. D ω S g ω TA DS ω ω g KS ω dt J + n n (98) DS DS K ω S g ω g Te Dg ω g ω ω dt J g n n n n (99) To obtain the tiffne contant K S the eigenfequency of the dive tain ha to be known. Thinking of a two ma fee winging ytem the eigenfequency i defined a: ω K () S S π fs J ge The total inetia of the fee winging ytem on the low peed ide can be calculated in the following way: J ge J J n g g J + J n () The tiffne contant of the low peed haft i: K J ( π f ) S ge S () The damping contant of the haft D S can be obtained with help of the logaithmic decementξ S. A tandad appoximation of the logaithmic decement i %. The damping contant can be obtained accoding to: D S ξ K ξ J S ge + 4π (3) Figue 98 how the dive tain a decibed implemented in Matlab/ Simulink. Figue 98 Dive tain model implemented in Matlab/ Simulink 93

109 6.4 ibay fo Wind Tubine modelling in Matlab/ Simulink The electical and mechanical component modelled in Chapte 3 to Chapte 6 ae geneally ueful fo the development of wind tubine model. In ode to make thee model acceible fo futhe development a libay in Matlab/ Simulink collect the diffeent model. The libay i acceible though the Simulink bowe a hown in Figue 99. Figue 99 Simulink ibay Bowe with opened Wind Tubine Toolbox Though the libay the diffeent component model ae acceible. In the Aynchonou Machine ibay the diffeent advanced machine type can be found (Figue ). A in thi libay, vaiou model can be found in the othe ubgoup. A detailed deciption of the entie libay option extent thi wok. Theewith thi how only an example about the libay. 94