Tel (31) Fax (31)

Size: px

Start display at page:

Download "Tel (31) Fax (31)"

Daisy Clark
6 years ago
Views:

1 Publed n Wnd engneerng 23 Iue Page 57 Te energy yeld of roof mounted wnd turbne S. Merten Secton Wnd Energy, Faculty Cvl Engneerng and Geocence Delft Unverty of Tecnology, Stevnweg 1, 2628 CN Delft, Te Neterland Tel (31) Fax (31) e-mal:.merten@ctg.tudelft.nl ABSTRACT Te roof a poble te for mall wnd turbne. T a new but more and more accepted dea. Several Dutc compane n cloe cooperaton wt te Secton Wnd Energy of Delft Unverty of Tecnology are buy wt te degn of mall roof wnd turbne at te moment. Te mportant flow feature for a wnd turbne on te roof owever are unknown. Te common wnd engneerng knowledge only focued on extreme wnd peed n order to ave a afe buldng. So, model to calculate te flow feature on te roof, mportant for a wnd turbne, ave to be developed. T paper gve a frt attempt to decrbe te flow feature and gudelne neceary for tng of mall wnd turbne on te roof. Tey concern: te derable egt above te roof, te cange of te undturbed wnd to te wnd peed above te roof and te probablty dtrbuton of te wnd peed above te roof. Wt ue of te gudelne tree example calculaton are gven to at te reader n te calculaton procedure. Keyword: wnd energy, bult envronment, roof 1 THEORY Predcton of te wnd peed n te bult envronment dffcult. Due to te large rougne n te bult envronment, te wnd peed cloe to te ground a local parameter and one can not gve a local parameter on ba of ome average caractertc of te rougne n te bult envronment. At roof of buldng te local flow feature on te flow depend on an even larger number of local parameter. So, a calculaton baed on ome average caractertc of te bult envronment can only gve an average and general decrpton of te flow feature on te roof. Local parameter wll be able to gnfcantly cange t average flow feature. However, n order to predct te average energy yeld of a wnd turbne at a roof, a model to predct te average wnd peed at te roof eental and terefore derved. 1.1 WIND IN THE BUILT ENVIRONMENT Let u frt look at te wnd peed n te buld envronment. Suppoe we ave te tuaton depcted n Fgure 1 and we want to know te wnd peed n te Internal Boundary Layer (IBL) n order to predct te wnd peed at te roof of a certan buldng n te IBL. External boundary layer z Internal boundary layer z d+z k z,ref x Rougne 1 Rougne 2

2 Fgure 1 Cange n wnd peed due to a tep n rougne. Te wnd come from te left de, aumed to be a rural area wt rougne lengt z,ref and experence a tep n rougne from z,ref to te rougne of te bult envronment z. Downwnd of te tep a new boundary layer wll develop: te Internal Boundary Layer. Above te IBL te wnd peed approxmately undturbed and equal to te wnd peed upwnd of te rougne cange. For te calculaton of te egt of te IBL Garret [ 1] ow tat a varety of model avalable. Some are baed on te vertcal wnd peed varance tat aumed to determne te growt rate of te IBL but we all ue a mple relaton baed on te boundary layer growt over a moot plate a uggeted by Wood [ 2] for a neutral boundary layer (wnd peed >6 [m/] ee Wernga and Rjkoort [ 5]): x k ( x) =.28z, ( 1 ).8,max ( ) z,max were (x) denote te IBL egt, x denote te dtance from te tep n rougne and k z,max te larger of te uptream and downtream rougne, wc z for t cae. Relaton ( 1 ) only vald for te wall regon o tat k ( x) <.2δ were δ denote te boundary layer egt (Wood [ 2]) wc approxmately 1 m for neutral flow over a cty. For frequently cangng rougne relaton ( 1 ) not convenent. Wernga and Rjkoort [ 5] owed tat t approprate to ue ( x) = 6 m n tat cae. k Far downwnd of te tep n rougne te IBL n equlbrum wt te ear force of te external boundary layer and te new rougne. Ten te profle of te IBL wll be logartmc. We tu arrve at a wnd peed profle wtn te IBL of k x ( ) z d ln z ln ref z u z, ( ) = uref, ( 2 ) ref k ( x) d ln ln z z, ref were d denote te dplacement lengt (ee Fgure 1 and te defnton of d n equaton ( 4 )) and te ndex ref refer to te parameter of te undturbed wnd tat can be found n a wnd atla o tat: ref =1 m,z,ref =.3 m and u ref from te wnd atla. Te logartmc profle n te IBL vald for 5 m < x < 5 km (Smu and Scanlan [ 3]) and a Wernga and Rjkoort [ 5] tate for egt z 2 z + d. ( 3 ) For maller egt te wnd peed nfluenced by te local rougne. Relaton ( 3 ) tate a very mportant ue for wnd energy converon n te bult envronment. Below te egt gven by relaton ( 3 ) t not relable to calculate te local wnd peed wt relaton ( 2) nce t can only gve an average value a t baed on average value of te rougne. Nevertele te average value from relaton ( 2) can be ued to calculate te average wnd peed n te IBL. Te dplacement lengt n relaton ( 3 ) rougly equal to 7% 8% of te average buldng egt of te upwnd area (Panofky and Dutton [ 4]). Were te border of te upwnd area a mnmum of 1 buldng egt from te pont were te rougne a to be etmated nce te lope of te IBL growt about 1/1. We tu approxmate te dplacement lengt wt d =. 75 b ( 4 )

3 were b te average egt of te buldng at te upwnd area. Wernga and Rjkoort [ 5] gve te rougne a a functon of te average buldng egt were te area A z =. 5A b ( 5 ) a percentage of te total area occuped by obtacle of average egt b. It recalled tat te border of tat area ould at leat be 1b from te pont were te rougne etmated. 1.2 SEPARATION OF THE BOUNDARY LAYER AT THE UPWIND ROOF EDGE At te upwnd roof edge te boundary layer eparate and a eparaton bubble formed on te roof. A a conequence te wnd velocty vector at te roof not parallel wt te orzontal roof but make an angle wt t: te kew angle (ee Fgure 2). Merten [ 7] owed tat t kew angle depend on te poton at te roof, te ape of te buldng and te urroundng rougne. A a conequence of te up flow on te upwnd de of te buldng, te kew angle at te upwnd edge of te buldng 9 o and decreae downwnd. Furtermore t own tat for a buldng wt dmenon l, w, {= 3,1,2 m} (ee Fgure 4 for te dmenon defnton), te kew angle at te centre of te roof can vary between approxmately o and 2 o. Te low kew angle aceved at a cty rougne (z =1 m) and te g kew angle aceved at te rougne of rural area (z =.3 m). Te dependence of te performance of a HAWT on te kew angle dcued n everal andbook (performance n yawed condton, ee for ntance Burton et al. [ 9]). Te Glauert momentum teory for yawed flow ow good agreement wt meaurement. It ow omewat le decreae n power compared to axal momentum teory. T gve a power decreae by a malgnment factor m wt m = co 3 γ ( 6 ) were γ te yaw angle. Skewed flow gve te ame power decreae of te HAWT nce te yaw angle can be ntercanged wt te kew angle becaue of te ymmetry of te rotor. Becaue of t mplcty and te many uncertante n wnd peed n te bult envronment 3 te power decreae n kewed flow found wt axal momentum teory ( m = co γ ) wll be ued ere. Skew angle Wnd velocty vector Yaw angle z y x Fgure 2 Defnton of te kew angle and yaw angle.

4 A frt attempt to decrbe te performance of a lft drven VAWT n kewed flow found n Merten [ 6]. It own by mplfed Blade Element Momentum teory tat, n contrat to te HAWT, te VAWT ould ave a power ncreae n kewed flow. Te reaon found n te ncreang rotor area tat experence undturbed flow (t aumed tat bot te HAWT and VAWT ave vertcal upport and are not tlted). In Merten [ 7] a more elaborated model of te performance of a lft drven VAWT n kewed flow explaned and compared wt meaurement on an H-Darreu of.74 m dameter, egt.5 m wt Tp Speed Rato around 3 and cord lengt of te blade.8 m. A kew angle of 1 o owed a power ncreae of 14% wle a kew angle of 2 o owed a power ncreae of 2%. We tu ave a malgnment factor m v for te VAWT wt te above mentoned dmenon (oter dmenon reult n oter malgnment factor) of m =1.2 at γ = 2 m v v =1.14 at γ =1 ( 7 ) Furtermore Merten [ 8] owed meaurement of te performance of a larger H-Darreu wt twted blade (prototype of Turby, dameter 1.5 m, egt 1.12 m, cord lengt blade.8 m (ee [ 1]) n kewed flow wc demontrate te poblty of a large ncreae of te power output of an H-Darreu n moderate kewed flow. Note tat t aumed tat all HAWT and VAWT for tee purpoe are lft drven macne, not drag drven. Wle kewed flow can be benefcal for te power producton, te g turbulence n and cloe to te eparaton bubble on te roof certanly not! It damage te wnd turbne by caung fatgue and t reult n a power decreae. Te g turbulence regon ould tu be avoded by tng te wnd turbne well above te eparaton bubble on te roof (ee Fgure 3 and furtermore te reult n ecton 2). Fgure 3 Computatonal Flud Dynamc Calculaton of te treamlne around te roof of a buldng wt flow at 45 o to te wndward de of te buldng. Te roof wnd turbne (n t cae Turby ee [ 1]) ould be ted well above te eparaton bubble on te roof (ee reult n ecton 2). 1.3 PROBABILITY DISTRIBUTION OF THE WIND SPEED AT THE ROOF Te eparaton wnd peed u ( +, ) at a egt above te roof wt egt, for wnd from wnd roe ector, can be defned n term of te wnd peed n te IBL ( ) found wt relaton ( 2 ) u ( ) C u ( ), = r,, u, + ( 8 )

5 were te parameter at egt C r, above te roof. gve te cange of te undturbed wnd peed to te wnd peed Te Webull dtrbuton read f u, ) of te undturbed wnd peed u, from wnd roe ector (, f k k 1 u, k u, a u = e a a (,, ) ( 9 ) were te cale parameter a of te undturbed wnd peed can be found from u, ( ) u, a = Γ k ( 1 ) were Γ te gamma functon and k te ape parameter wc taken a 2 for all wnd drecton and egt. T an approxmaton nce k vare wt egt and rougne. However for a egt around 2 m Wernga and Rjkoort [ 5] ow tat k=2 a good approxmaton. For k=2 te Webull dtrbuton called te Rayleg dtrbuton and te gamma functon found to be 1 π Γ 1 + =. ( 11 ) 2 4 Te Rayleg dtrbuton and te accompanyng value for te gamma functon wll be ued n te calculaton n ecton 2.1. Te probablty of a wnd peed at te roof, for undturbed wnd from ector wt u u, C c,, ), undturbed wnd peed at roof egt f ( u, du. We tu ave f u, ) du Cc, f ( u,, ) du, ( = ( 12 ) were C c, te probablty of undturbed wnd from ector wt N = 1 C 1 ( 13 ) c, = were N te number of ector of te wnd roe. Subttuton of ( 8 ) and ( 9 ) n ( 12 ) gve u te probablty of a wnd peed at te roof u caued by undturbed wnd wt wnd peed u, from wnd roe ector f k k 1 u k u a C r, ( u, ) du = C c, e du. ( 14 ) ac r ac, r, Te probablty dtrbuton of te wnd peed on te roof for undturbed wnd from ector tu read f k k 1 u k u ac r, ( u, ) = C c, e. ( 15 ) ac r ac, r,

6 Te probablty dtrbuton of te wnd on te roof averaged over all drecton gven by k 1 k u acr, N k u f = ( u ) Cc, e. ( 16 ) = 1 acr, acr, Equaton ( 16) ow tat te probablty dtrbuton of te wnd peed on te roof te um of a wegted Webull dtrbuton wt dfferent cale parameter compared to te undturbed cale parameter. a a C c, Due to te dfferent cale parameter te bandwdt of te probablty dtrbuton on te roof can be larger: te probablty of low and g wnd peed can ncreae. Te bandwdt depend on te locaton at te roof nce te cale parameter a functon of te locaton. A well degned roof wnd turbne a to take t nto account. A a functon of t locaton on te roof, t ould pobly operate n a wder range of wnd peed n order to arvet all te avalable wnd energy at te roof. On te oter and t quetonable weter tee locaton ave to be condered. Tey provde extreme wnd beavor wc mot certanly very demandng for te wnd turbne. Wt equaton ( 15 ) te average wnd peed on te roof can be found to be u = N = 1 A meaure for te energy denty at a roof locaton found wt u 3 = N = 1 u f ( u, ) du. ( 17 ) 3 u f ( u, ) du. ( 18 ) Wt t te dfferent roof locaton can be compared concernng avalable energy. However t a to be kept n mnd tat te kew angle of te flow a well a te bandwdt of te probablty dtrbuton on te roof can be very demandng for te wnd turbne o tat t dffcult to arvet all te avalable energy at ome locaton. If t aumed tat te wnd turbne can operate equally at all kew angle at te rotor or f te kew angle mall and approxmated wt zero, relaton ( 16) gve u te followng dfferental of te energy yeld of a wnd turbne at te roof de ( u ) P( u ) T f ( u ) du =. ( 19 ) wt power of te wnd turbne P(u ) and operatng tme T. For all velocte operatng range we tu ave u = uco u = uc ( ) f ( u ) u wtn te E = T P u du, ( 2 ) were u c te cut n wnd peed and u co te cut out wnd peed of te wnd turbne. Unle te wnd turbne equpped wt an expenve mecanm tat andle te kew angle, te wnd turbne wll cange power a a functon of te kew angle. So, n contrat to te aumpton made for relaton ( 19) n mot cae te power output wll be a functon of te wnd drecton and te ummaton preented n relaton ( 16 ) ould be evaluated a follow. Wt relaton ( 15 ) we ave

7 ( u ) P( u, ) T f ( u ) du de, =,, ( 21 ) o tat u = uco ( ) = T P( u ) f ( u ) de,, du ( 22 ) u = uc and E = T = N u = uco = 1 u = uc ( ) f ( u, ) P u, du. ( 23 ) 2 RESULTS In t ecton te nput parameter for te model wll be dcued togeter wt example of calculaton wt te dcued model. 2.1 CHANGE OF THE UNDISTURBED WINDSPEED TO THE ROOF WINDSPEED A CFD calculaton of te flow around a buldng, wt dmenon l,w, {= 3,1,2 m}, n an atmoperc boundary layer owed te cange of wnd peed from undturbed wnd peed at roof egt to te wnd peed at a certan egt above te roof (ee Merten [ 7] for detal on te CFD calculaton). Untl date tere wa no approprate data avalable tat could be compared wt te CFD calculaton. T off coure a conequence of te fact tat t paper gve te frt attempt to develop teory wt neceary nput data to predct te performance of a roof wnd turbne. On te roof, tree poton were coen for te reference wnd peed above te roof. Tey are depcted n Fgure 4. / =.5 ϕ corner edge centre u / =.25 / =.5 u o l w Fgure 4 Poton of te coen wnd peed evaluaton. Te egt above te roof for te centre locaton coen n order to avod te low velocte and g turbulence cloe to te eparaton bubble. Accordng to te reult of te CFD calculaton (Merten [ 7]) depcted n Fgure 5 a egt of 5 m or / =. 25 above te roof at te centre poton enoug to avod te g turbulence and low velocte wtn te eparaton bubble. At te corner and edge locaton on te roof te egt above te roof coen at te poton of maxmum wnd peed for ϕ = (ee Fgure 4 and Fgure 5 ). Accordng to te wnd peed profle t condton wa aceved at a egt above te roof of approxmately 1 m o / =.5 wa coen for te corner and edge poton.

8 z [m] roof center 3 roof center roof edge roof edge roof corner roof corner z [m] u u (z)/u (z=2 [m]) [.] (z)/u (z=2 [m]) [.] Fgure 5 Non-dmenonle velocte above te roof wt wnd perpendcular to te wndward de of te buldng ( ϕ = ) (left grap z =1 m, rgt grap z =.3 m ). Te tree-dmenonal CFD calculaton owed te followng cange from undturbed wnd peed at roof egt to te wnd peed above te eparaton bubble at te roof at a certan egt above te roof. C r, Centre: / =. 25 Edge: / =. 5 Corner: / =. 5 ϕ z = 1m z =. 3 m z = 1m z =. 3 m z = 1m z =. 3 m Table 1 Calculated wnd peed cange for dfferent poton at te roof C r, Te ometme low value of C r, n Table 1 pont toward preence n te eparaton bubble. Obvouly te eparaton bubble maller for g rougne ( z 1m) nce mall value for C, are only aceved for z =. 3 m. T ndeed te cae nce for low rougne tere r a lot more up flow at te upwnd de of te buldng, reultng n a bgger eparaton bubble. Te econd tng tat eem clear from Table 1 tat te acceleraton from undturbed to roof wnd peed bgger for ger rougne. T a bt mleadng. Te reaon tat for ger rougne te meaurement poton a a larger dtance to te eparaton bubble on te roof (te eparaton bubble maller). A a conequence of te ncreang wnd peed above te eparaton bubble te wnd peed muc more ncreaed compared to te ncreae for g rougne were te meaurement poton cloe to te eparaton bubble. 2.2 ENERGY DENSITY A meaure for te energy denty at te dfferent roof locaton for a unform wnd roe calculated wt equaton ( 17 ), ( 18 ) and Table 1. Te reult for rougne z =1 m and z =.3 m are mentoned n Table 2. Roof 3 u / u u 3 / u / locaton Centre z =1 m Edge Corner =

9 Centre z =.3 m Edge Corner Table 2 Energy denty and average wnd peed cange from undturbed wnd to wnd at a locaton on te roof. Obvouly te centre locaton a te get energy denty for bot z =1 m and z =.3 m. But of equal mportance te value of te kew angle of te wnd velocty vector at te corner and edge wc g compared to te kew angle at te centre locaton becaue of te up flow at te de of te buldng. For t reaon and becaue of te g energy denty te centre locaton certanly preferred for operaton of a HAWT, moreover becaue te g wnd peed regon at te corner and edge locaton a a mall egt (ee Fgure 5). Tere only room for a mall wnd turbne. In contrat to te HAWT, te VAWT ow an ncreaed power output for ger kew angle. For te VAWT, te edge poton can tu gve g energy yeld. However te vertcal egt of te g wnd peed regon mall and te effect tu not very mportant. 2.3 EXAMPLE 1 Suppoe we ave a buldng wt dmenon l,w, {=3,1,2 m} n a cty centre 3 km downwnd of te cty border wt a rural area were te potental wnd peed (we cooe a value from te Dutc wnd atla cloe to Utrect n te mddle of te Neterland) 4.5 m/. At te centre of te roof of t buldng we lke to ntall a mall HAWT wt a power curve depcted n Fgure 6. 3 Power [Watt] wnd peed [m/] Fgure 6 Power curve of te example wnd turbne Furtermore uppoe tat te average egt of te buldng around our buldng about 1 m and rougly 2% of te area around our buldng unformly occuped wt buldng of t egt. Accordng to ( 5 ) te rougne of tat area wll be z =1 m. Relaton ( 4 ) gve an accompanyng dplacement lengt of 7.5 m. Relaton ( 1 ) gve u a egt of te nternal boundary layer at a poton 3 km downwnd of te cty border of 17 m. Wt relaton ( 2 ) we are now able to calculate te undturbed wnd peed at roof egt. We fnd a wnd peed of 3.3 m/. From Fgure 5 we ee tat a egt of 5 m above te roof enoug to avod te g turbulence level and low velocte n te eparaton bubble. Furtermore Merten [ 7] (ee ecton 1.2) ow tat, for a rougne of z =1 m, te kew angle of te wnd vector on te roof approxmately equal to zero. Te nfluence of te kew angle on te wnd turbne performance can tu be neglected. We coe to te te wnd turbne at 5 m egt above te roof o te cange n wnd peed from undturbed to roof wnd peed found wt relaton ( 17 ) and denoted n Table 1 applcable (column centre locaton, z =1 m). Relaton ( 2 ) gve u te energy yeld of te wnd turbne at te centre of te roof: 1814 kw a year. Te mall peed on te roof n a cty ow tat t not neceary to ave a large out wnd peed at t locaton. Let u for ntance aume tat te rated wnd peed of 12 m/ equal

10 to te cut out wnd peed. A new calculaton wt relaton ( 2 ) gve u a new energy yeld of 1738 kw a year, only 4% le tan for te wnd turbne wt a cut out wnd peed of 2 m/. 2.4 EXAMPLE 2 Now uppoe tat we ave te ame buldng at te ame locaton but wtn a rougne of z =.3 m and we want to ntall te ame HAWT. We do not ave a tep n rougne n t cae o we put k ( x) = 6 m n relaton ( 2 ) and arrve at an undturbed wnd peed at roof egt of 5. m/. Toug a bt cloer to te eparaton bubble (ee fgure ( 3 )) we agan ue o a egt of 5 m above te roof but for z=.3 m te kew angleγ for ϕ = approxmately 2 o (ee ecton 1.2) and ould be taken nto account. Te nfluence on te HAWT power 3 approxmated wt te malgnment factor m = co γ were γ denote te kew angle (ee ecton 1.2). We wll ue te followng approxmaton for te malgnment of te flow on te HAWT. ϕ o γ o 3 m = co γ Table 3 Malgnment factor m ( = co 3 γ ) for kew angle γ for a HAWT at a roof of a buldng n urroundng rougne z =.3 m. Te ret of te malgnment factor found by ymmetry conderaton. Now, ummaton of te energy from te dfferent wnd drecton wt relaton ( 23 ) gve an energy yeld of 3757 [kw/year]. 2.5 EXAMPLE 3 Now uppoe tat we ave an H-Darreu ntead of te HAWT wt agan unform wnd from all drecton and a urroundng rougne lengt of z =.3 m. In tat cae Merten [ 7] owed an ncreae n power output. Let u approxmate t power ncreae a follow (ee relaton ( 7)). ϕ o γ o m v Table 4 Malgnment factor m v for kew angle γ for a VAWT at a roof of a buldng n urroundng rougne z =.3 m Te ret of te malgnment factor found by ymmetry conderaton. Te correpondng energy yeld of a VAWT wt te ame power curve (Fgure 6) a te HAWT now gve an energy yeld of 4361 kw/year. An ncreae of 16% compared to te HAWT wle te energy yeld at cty rougne te ame a te HAWT becaue te kew angle approxmately zero for tat cae. 3 CONCLUSIONS Te paper gve conderaton and decrbe ome tool tat can be ued to aure proper operaton of a roof wnd turbne. Furtermore t decrbe a frt ba for a calculaton procedure for predctng te energy yeld of a wnd turbne on a roof. Concernng te operaton, t ow tat te wnd condton at te majorty of locaton on te roof are very dfferent from te undturbed wnd condton. A a conequence te roof wnd turbne at toe locaton ould be too. At mot roof te, one can not mply ue a wnd turbne meant for rural area on te roof. Compared wt a HAWT, a VAWT can gve a larger energy yeld on buldng et n mall upwnd rougne. T caued by te ncreaed kewed-flow wnd peed acro te roof. Te wnd vector at te roof not orzontal but ntead kewed wt an angle to te

11 orzontal roof tat vare acro te roof. Te roof wnd turbne a to be utable for operaton n t flow. Becaue of te larger power output and energy yeld n kewed flow, te VAWT eem to be more utable on roof a compared to te HAWT. Te average wnd peed at te roof larger compared wt te mall undturbed wnd peed n te bult envronment. Nevertele, te wnd velocte above te roof are tll mall compared wt conventonal wnd turbne on tower n open urroundng. So, for roof turbne, g buldng are neceary to compenate te mall wnd peed n te bult envronment and produce an acceptable energy yeld. REFERENCES [ 1] Garratt, J.R., Te Internal Boundary layer a revew, Boundary Layer Meteorology, 5, 199, pp [ 2] Wood, D.H., Internal Boundary Layer growt followng a tep cange n rougne, Boundary Layer Meteorology, 22, 1982, pp [ 3] Smu, S., Scanlan, R.H., Wnd effect on tructure, Jon Wley & Son, New York, 1996 [ 4] Panofky, H.A., Dutton, J.A., Atmoperc turbulence, Model and Metod for Engneerng, Jon Wley & Son, 1984 [ 5] Wernga J., Rjkoort, P.J., Wndklmaat van Nederland, Staatutgeverj - Gravenage, 1983 [ 6] Merten, S., Performance of a g Tp Speed Rato H-Darreu n te kewed flow on a roof, ASME Wnd Energy Sympoum AIAA , 23, pp [ 7] Merten, S., Performance of an H-Darreu n te kewed flow on a roof, DOI: / , Journal of Solar Energy and Engneerng, 125, November 23, pp [ 8] Merten, S., Wnd Energy Converon n te bult envronment, keynote on te 1t SWH Internatonal Conference, Segova (Span), 23 [ 9] Burton, T., Sarpe, D., Jenkn, N., Boany, E., Wnd Energy Handbook, Jon Wley & Son, 21 [ 1]

Harmonic oscillator approximation

Harmonic oscillator approximation armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon