Jurnal f Physics: Cnference Series OPEN ACCESS Aerdynamic Separability in Tip Speed Rati and Separability in Wind Speed- a Cmparisn T cite this article: M L Gala Sants et al 14 J. Phys.: Cnf. Ser. 555 139 View the article nline fr updates and enhancements. Related cntent - Wind tunnel experiments t prve a hydraulic passive trque cntrl cncept fr variable speed wind turbines N F B Diepeveen and A Jarquin-Laguna - Mitigating the Lng term Operating Extreme Lad thrugh Active Cntrl Christina Kukura and Anand Natarajan - Extending wind turbine peratinal cnditins; a cmparisn f set pint adaptatin and LQG individual pitch cntrl fr highly turbulent wind W P Engels, S Subhani, H Zafar et al. This cntent was dwnladed frm IP address 148.51.3.83 n 3/9/18 at 1:59
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 Aerdynamic Separability in Tip Speed Rati and Separability in Wind Speed- a Cmparisn M L Gala Sants 1, W E Leithead and P Jamiesn 3 1 CDT Wind Energy Systems, Department f Electrnic and Electrical Engineering, University f Strathclyde. Ryal Cllege Building, rm 3.36, 4 Gerge Street, Glasgw G1 1XW, UK Industrial Cntrl Centre, Department f Electrnic and Electrical Engineering, University f Strathclyde. Ryal Cllege Building, rm.16, 4 Gerge Street, Glasgw G1 1XW, UK 3 CDT Wind Energy Systems, Department f Electrnic and Electrical Engineering., University f Strathclyde. Ryal Cllege Building, rm 3.36, 4 Gerge Street, Glasgw G1 1XW, UK E-mail: 1 maria.gala-sants@strath.ac.uk, w.leithead@strath.ac.uk, 3 peter.jamiesn@strath.ac.uk Abstract. Frm extensive applicatin ver a number f years, it has been established that the nnlinear rtr aerdynamics f typical medium and large wind turbines exhibit an effectively glbal separability prperty, in ther wrds the aerdynamic trque f the machine can be defined by tw independent additive functins. Tw versins f the separability f aerdynamic trque fr variable speed wind turbines are investigated here; the separated functin, related t wind speed, in the first versin is nly dependent n that variable and nt rtr speed and in the secnd versin is nly dependent n tip speed rati. Bth prvide very gd apprximatins t the aerdynamic trque ver extensive neighburhds f T, at least frm t T. 1. Intrductin The aerdynamic trque, T, fr a wind turbine is dependent n the wind speed, V, the blade pitch angle,, and rtr speed, ; that is, T 1 ARV C (, ) ; R / V (1) Q where is the density f air, A is the rtr area, C Q is the trque cefficient and the tip speed rati. The separatin f aerdynamics int tw additive cmpnents, the first independent f wind speed and the secnd independent f pitch angle, is discussed in Jamiesn et al [1], Leithead et al [], [3], [4], [6] and Leith et al [7]. Surprisingly, this separability hlds nt just at the equilibrium perating pints fr variable speed wind turbines in abve rated cnditins, but ver an extensive regin f the perating space. Fr cnstant speed wind turbines, f curse, neither cmpnent depends n rtr speed which des nt vary. Cntent frm this wrk may be used under the terms f the Creative Cmmns Attributin 3. licence. Any further distributin f this wrk must maintain attributin t the authr(s) and the title f the wrk, jurnal citatin and DOI. Published under licence by Ltd 1
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 In this paper, tw different frms f separability f the aerdynamics fr variable speed wind turbines are cmpared. The first frm is purely empiric in derivatin and is a direct extensin f separability fr a cnstant speed wind turbine. The secnd frm is derived frm the nature f the physical relatinship (1). The paper is rganised as fllws. In sectin and sectin 3, separability f the aerdynamics fr cnstant speed wind turbines and fr variable speed wind turbines, respectively, are discussed. In sectin 4, separability f the aerdynamics fr variable speed wind turbines, based n empirical extensin f the cnstant speed case, is presented. In sectin 5, separability f / (, ) / T CQ () is discussed and, in sectin 6, separability f the aerdynamics fr variable speed wind turbines, based n the nature f this physical relatinship, is presented. In sectin 7, the tw frms f separability are discussed and, in sectin 8, sme cncluding remarks are made.. Cnstant speed wind turbines Fr a cnstant speed wind turbine, the rtr speed is and rated trque is T. In abve rated cnditins, there is a pitch angle at which rated trque is attained fr each wind speed; that is, there is a lcus f equilibrium perating pints n which the trque is T. Prvided ( ) is mntnic, where ( ) is the relatinship f t alng this lcus f equilibrium pints, and ( ) and ( ) are chsen such that ( ) ( ( )) (3) ( ) ( ( ) ) (4) then, n the lcus, ( ) ( ) ( ) ( ( ) ( )) (5) Furthermre, the partial derivatives by cnstructin are, als, crrect n the lcus. The relatinship f ( ) t ( ) is ( ) ( ) ( ( )) (6) Hence, n the lcus, ( ( ) ( )). ( ) ( ( )) (7) Lcally t the lcus f equilibrium perating pints, ( ) is a gd representatin f ( ), that is, within a sufficiently small neighburhd f the lcus the difference can be made arbitrarily small. Mre generally, near the lcus, fr any functin ( ) ( ) ( ( ( ) ( ))) (8) such that, ( ) (9) i ( ) (1) By judicius chice f ( ), the size f the neighburhd can be maximised. In effect, in a regin that includes the whle set f equilibrium perating pints, the dependence f aerdynamic i On the lcus f equilibrium pints ( ) ( ) and
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 trque n pitch angle is thrugh ( ) and n wind speed thrugh ( ) whilst the dependence n displacement f the perating pint frm the lcus is thrugh [5]. Smewhat surprisingly, ( ) ( ( ( ) ( ))) applies fr a large neighburhd fr all rtrs, catering satisfactrily fr a very brad range f values f pitch angle and wind speed that includes all the nrmal perating pints. Furthermre, ( ), is generally a very weak nnlinear functin. Typically, the neighburhd includes values f aerdynamic trque frm t 3T. Indeed, it nly starts t break dwn n the rtr entering stall. The significance f the abve, particularly in the cntext f wind turbine cntrl, is that the wind turbine can be viewed as a nnlinear dynamic system dependant nly n the pitch angle (the nnlinearity ( )) with an additive external disturbance dependent nly n wind speed ( ( )) ii [5]. In Figure 1, the aerdynamic trque fr a 3MW HAWT is pltted against its separated representatin, ( ). The range f pitch angles is frm 3 t 19 and wind speed frm 11.41m/s t 4.1 m/s. Rated trque is 1 6. Clearly, ( ) is a gd apprximatin t ( ) even fr perating pints far frm rated trque. A best fit t the data wuld be an apprpriate chice fr ( ). 8 x 16 x 1 6 Figure 1. Separability fr a cnstant speed pitch regulated wind turbine. 3. Variable speed wind turbines Fr variable speed wind turbines, separatin functins, ( ), ( ) and ( ), can be determined fr different values f rtr speeds,. Hence, fr the lcus f equilibrium perating pints, i.e all perating pints, (, V, ), fr which T (, V, ) is T, where ( ) and ( ) are defined by (,V) 8 ( ) ( ( ) ( )) (11) ( ) ( ) (1) ( ) ( ) (13) ii Functins ( ) and ( ) are unique fr each and rtr 3
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 Nte, has a range f pssible values, that includes rated rtr speed,, and the lcus is nw a -dimensinal surface in a 4-dimensinal space. In the neighburhd f the lcus f equilibrium perating pints ( ) ( ( ( ) ( ))) (14) where ( ) ( ). Cnsider the set f pints, ( ), n the lcus, fr which the aerdynamic trque has cnstant value, such that Since ( ). These pints can be cntinuusly parameterised by ( ) ( ( ) ( ) ( )) (15) ( ) the partial derivative with respect t is simply related t the ther tw partial derivatives. Since the partial derivatives with respect t and V f the separated frm, ( ( ) ( )), n the lcus f equilibrium pints are crrect by cnstructin, the partial derivative with respect t is als crrect. ( ) ( ) (16) 8 x 16 6 4 V 8% V 1% V 1% 1 13 16 19 5 wind velcity [m/s] Figure. Functin ( ) fr 8%, 1% and 1% f ω. 5 x 16 1 x 16-3 -7 8% 1% 1% 5 1 15 5 [degree] Figure 3. Functin ( ) fr 8%, 1% and 1% f ω. 8 x 16 8% 1% 1% 5 8 T + [h(, )-g(v, )] x 1 6 T + [h(, )-g(v, )] x 1 6 Figure 4. Functin fr 8%, 1% and 1% f Figure 5. Separability fr case studi 8%, 1% ω. and 1% f ω. The separated representatin f the aerdynamics is determined fr a variable speed HAWT with the same rtr characteristics as the cnstant speed HAWT in sectin 1. The rtr speed range is 8% 1% 1% 4
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 between 8% and 1% f rated rtr speed,. The functins, ( ) and ( ), are depicted in Figures and 3, respectively, with values f 8%, 1% and 1%. The similarity f the functins, ( ), is striking. In Figure 4, the aerdynamic trque ( ), is pltted against its separated representatin, ( ( ) ( )). The separated representatin is again a gd apprximatin. 4. Separability in wind speed The bservatin, that the plts in Figure are very similar, suggests strngly that separability culd have a simplified frm with ( ) nly dependent n V; that is, n the lcus f equilibrium perating pints ( ) ( ) ( ( ) ( )) (17) Fr any chice f cnstant rtr speed,, n the lcus f equilibrium perating pints, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) ( ( )) ( ) ( ( )) ( ) Hence, chsing ( ) t be ( ), the functin at, an apprpriate chice fr ( ) is (18) ( ) ( ( )) (19) In Figure 6, ( ) and ( ( )) are cmpared with values f 8%, 1% and 1%. The pairs f functins match clsely. 1 x 16-3 -5 h ( ), 8% h ( ), 1% h ( ), 1% g (f ( ), 8% g (f ( ), 1% g (f ( ), 1% -7 5 1 15 5 [degree] Figure 6. Validatin f simplified wind speed separability 5. Separability in tip speed rati The aerdynamic trque is defined as ( ) ( ) ( ) () ( ) where is a cnstant. Separability is applied t ( ) alng the lcus f perating pints n which ( ) ( ) with such that ( ) ( ( ) ( )) (1) 5
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 n the lcus. It can be seen frm Figure 7 that the separated frm, ( ( ) ( )), is a gd representatin fr ( ) fr a very large neighburhd enclsing the lcus f equilibrium pints. It includes values f ( ) that vary frm t. 5 x 1-3 Cq/.6694 Figure 7. Separabity fr cnstant This suggests strngly that separability culd have the frm fr variatins f 8%, 1% and 1% f ω and ( ) ( ( ( ) ( ))) () n the lcus f equilibrium pints with ( ). Nte, the lcus, n which ( ), is different frm that, n which ( ). Mre generally, ( ) ( ( ( ) ( ))) (3) fr sme ( ) such that ( ) and ( ). The functins, ( ) and ( ), are determined fr the separated frm f ( ) using data sets fr 8%, 9%, 1%, 11%, 1% and 14% fr which ( ) is clse t. Of curse, these n lnger remain lcal t as varies. It can be seen frm Figure 8 that the separated representatin, ( ( ( ) ( ))), is a gd representatin fr ( ). The functin, ( ) is als determined independently using data sets fr 8%, 1% and 1%. In Figure 9, the ( ) scaled by are shwn. T/ 8 x 15 7 6 5 4 3 1 Figure 8. Equilibrium pints after separatin & T, 8% & T, 1% & T, 1% - -4.6694 C + [H( )-G( )] 5 x 1-3 Equilibrium pints linear fit 4 6 8 T / x 1 5 + [H( )+G( )] 7 x 16 8% 1% 1% 7 T x 1 6 + [H( )+G( )] Figure 9. Functin ( ) scaled by 6. Cmparisn f the tw versins f separability In Sectins 4 and 5, tw different versins f separability are discussed, specifically, a separatin using wind speed, (17), and a separatin using tip speed rati, (3); that is, 6
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 g(v) h(, ) x 1 6 8 8% 1% 6 1% g(v) OPTIMISED 4 1 x 1 6 15 V [m/s] 5 - -4 ( ( ( ) ( ))) ( ( ( ) ( ))) (4) Fr wind speed separability, ( ) and ( ) tgether with ptimised plynmial fits, ( ) and g( V ) av bv c, are depicted in Figure 1. Fr tip speed rati separability, ( ) and ( ), tgether with ptimised fits ( ) and ( ) ( ), are depicted in Figure 11. g( ) h( ) 4 x 16 3 1 8% 1% 1% 3 4 5 6 7 8 9 1 x 16 -.5.5 - -6 5 1 15 5 [degree] Figure 1. Wind speed separability equatins -.5 5 1 15 5 [degree] Figure 11. Tip speed rati separability equatins In Figures 1, Figure 13 and Figure 14 the tw versins f separability are directly cmpared fr rtr speed f 8%, 1% and 1%, respectively. Bth are gd apprximatins t the aerdynamic trque fr large neighburhds enclsing the lcus f equilibrium pints. As wuld be expected, the 1% case exhibits the clsest match. The least accurate is the 8% case but that is nt unexpected since its perating pints are the clsest t stall. Fr the tip speed rati separability, the neighburhd, fr which the separated representatin is a clse apprximatin, extends in every case at least frm t, expanding as the set f perating pints fr the different rtr speeds mve away frm the stall regin. It is particularly gd apprximatin when. Fr wind speed separability, the neighburhd, fr which the separated representatin is a clse apprximatin, als extends in every case at least frm t, It is particularly gd when Althugh there is n first principle justificatin fr the wind speed versin f separability, i.e. fr a ( ) being independent f, the latter even lcally t T, it is at least as gd as tip speed rati separability. Each figure is further accmpanied by a blw up f the data clse t, i.e. the data lcal t the equilibrium perating pints. It can be seen that there is a small inaccuracy, particularly clear in Figure 1, in the tip speed rati versin f separability. The underlying reasn fr this inaccuracy is the implicit assumptin in Sectin 5 that the data sets fr 8%, 9%, 1%, 11%, 1% and 14% shuld all lie n a straight line; that is ( ) n the perating regin defined by these data sets. That this assumptin is nt apprpriate can be seen frm Figure 7 r frm Figure 9. 7
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 8 x 16.1 x 16 8 x 16 1.811 x 16 1.59 1.65.1 x 1 6 1.451 1.451 1.831 x 1 6 T +h( )-g(v) T +h( )-g(v) [T +[h( )+g( )] ]x [T +[h( )+g( )]] x 8 8 x 1 6 x 1 6 Figure 1. Separability cmparisn ω 8% Figure 13. Separability cmparisn ω 1% 9 x 16 1.781 x 16 1.438 1.415 1.783 x 1 6 T +h( )-g(v) [T +[h( )+g( )]] x 8 x 1 6 Figure 14. Separability cmparisn ω 1% 7. Cnclusin Tw versins f separability f aerdynamic trque fr variable speed wind turbines are investigated; the separated functin, related t wind speed, in the first versin is nly dependent n that variable and nt n rtr speed and in the secnd versin is nly dependent n tip speed rati. Bth prvide very gd apprximatins t the aerdynamic trque ver extensive neighburhds f, at least frm t. If anything, the wind speed separability is the mre accurate but has the least analytic supprt being purely empirically mtivated. The tw versins f separability are illustrated by applicatin t a 3MW HAWT. Acknwledgments Authrs wish t thank EPSRC fr their supprt t the prject thrugh the CDT Wind Energy Systems based at the University f Strathclyde, Glasgw, UK. References [1] Jamiesn, P., Leithead, W.E., Gala Sants, M.L., The Aerdynamic Basis f a Trque Separability Prperty. Prceedings f EWEA 11, Brussels, 11. [] Leithead, W.E., Rgers, M.C.M., Agius, RR.D., Dynamic analysis f the cmpliant tip, University f Strathclyde, Reprt prepared fr A.E.A. Technlgy, May 199. [3] Leithead, W.E., Rgers, M.C.M., Leith, D.J., Cnnr, B., Design f Wind Turbine Cntrllers. Prceedings f EURACO Wrkshp Recent Results in Rbust & Adaptive Cntrl, Flrence, 1995. [4] Leithead, W., Leith, D., Harden, F., Marku, H., Glbal gain-scheduling cntrl fr variable speed wind turbines, (Nice, France) EWEC, 1999 [5] Leithead W.E., D.J. Leith, On the Separability f Wind Turbine Rtr Aerdynamics. Internal Reprt, University f Strathclyde, 8
The Science f Making Trque frm Wind 1 Jurnal f Physics: Cnference Series 555 (14) 139 di:1.188/174-6596/555/1/139 [6] Leithead, W., Harden, F., Leith, D., Identificatin f aerdynamics and drive-train dynamics fr variable speed wind turbines, (Madrid, Spain), EWEC, July 3. [7] Leith, D.J., Leithead, W.E., Implementatin f Wind Turbine Cntrllers, Int. J. Cntrl, 66, pp. 349-38, 1997. 9