Renewable Energy xxx (2011) 1e10. Contents lists available at SciVerse ScienceDirect. Renewable Energy

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1 Renewle Energy xxx () e Contents lists ville t SciVerse ScienceDirect Renewle Energy journl homepge: Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior F. González-Longtt *, P. Wll, V. Terzij School of Electricl nd Electronic Engineering, The University of Mnchester, Mnchester M6 QD, United Kingdom rticle info strct Article history: Received My Accepted 9 August Aville online xxx Keywords: Wind frm Dynmic ehvior Stedy-stte ehvior Power system Stedy-stte wke effect The im of this pper is to evlute the impct of the wke effect on oth the stedy-stte opertion nd dynmic performnce of wind frm nd provide conclusions tht cn e used s thum rules in generic ssessments where the full detils of the wind frms re unknown. A simplified explicit model of the wke effect is presented, which includes: the cumultive impct of multiple shdowing, the effects of wind direction nd the wind speed time dely. The model is implemented in MATLAB Ò nd then integrted into power system simultion pckge to descrie the wke effect nd its impct on wind frm, prticulrly in terms of the wke coefficient nd overll ctive power losses. Results for two wind frm lyouts re presented to illustrte the importnce of wind turine spcing nd the directionlity of wind speeds when ssessing the wke effect during stedy-stte opertion nd dynmic ehvior. Ó Elsevier Ltd. All rights reserved.. Introduction Wind turines extrct energy from the wind to produce electricity; therefore, the wind leving the turine must hve lower energy content thn the wind upstrem of the turine []. As consequence, the wind downstrem of wind turine hs reduced speed nd is turulent; this downstrem wind is the wke of the turine. As the wind flow proceeds further downstrem this wke will egin to spred nd grdully return to free strem conditions. If wke intersects with the swept re of downwind turine the downwind turine is sid to e shdowed y the turine producing the wke. The two min effects of wke re: (i) reduction in the wind speed, which in turn reduces the energy production of the wind frm; (ii) n increse in the turulence of the wind, potentilly incresing the dynmic mechnicl loding on downwind turines. It is importnt to consider these wke effects in the design of wind frm in order to mximize the energy output nd lifetime of the mchines []. A lrge numer of numericl models, of vrying complexity, hve een developed to descrie wke. In generl, they cn e clssified s either explicit or implicit []. The explicit, or kinemtic, wke models re the erliest nd use self-similr velocity profiles determined semi-empiriclly [3e7]. The implicit wke models were developed s elorte lterntives to the explicit wke models. They re sed on pproximtions of either the Nviere- Stokes or vorticity trnsport equtions [8e]. The exct modeling of the wind speed distriution within wind prk is firly complicted tsk nd mny of the necessry prmeters re not routinely ville []. The choice of suitle model depends on three fctors: the desired computtionl time, the necessry ccurcy of prediction, nd the ville wind modeling prmeters. In this pper, simplified pproch for the simultion of the wke effect is presented nd used to investigte the impct of the wke effect upon the stedy-stte nd dynmic ehvior of wind frm. The model presented here only uses those prmeters nd dt tht re commonly ville for wind turine nd wind prk. Severl spects of the wke effect, such s: the cumultive impct of multiple shdowing, the effect of wind direction, nd the wind speed time dely re ll tken into considertion. The model hs een used to evlute the effect of turine wkes upon the energy production, clculted using severl indexes, of two different lyouts of wind frm. This pper is novelty ecuse it presents n exhustive nlysis of the effect of the wke effect on the stedystte nd dynmic ehvior. The conclusions presented here cn e used s thum rules in generic ssessments where the full detils of the wind frms re unknown.. Wke model * Corresponding uthor. Tel.: þ () E-mil ddresses: fglongtt@ieee.org, fglongtt@hotmil.com (F. González- Longtt). The development of models to descrie wind turine wkes egn in the 98s [] for the purpose of estimting the fll in 96-8/$ e see front mtter Ó Elsevier Ltd. All rights reserved. doi:.6/j.renene Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

2 F. González-Longtt et l. / Renewle Energy xxx () e output power they cuse []. Some of these models include: Ainslie s model [], Frndsen s model [3], the Mosic Tile model [] nd Jensen s model [5]. In some studies it is necessry tht ny wke models used re: strightforwrd, dependent on reltively few wke mesurements nd economic in terms of the necessry computing power [6]. However, despite their reltive simplicity, these methods tend to give results tht re in resonle greement with the ville dt in the cse of single wke within smll wind frm, nd simple meteorologicl environment [,6e8]. In ddition, comprison of different wke models, presented in Ref. [7], does not suggest ny prticulr difference, in terms of ccurcy, etween the sophisticted nd simplified models. In this section the simple model for oth single wke nd multiple wkes is presented. The effects of the wind speed time dely re included in these models... Wke ehind n idelized wind turine Assuming n idelized turine, where the ir flow round nd ehind the turine is without rottion nd friction, it is possile to derive some importnt generl equtions descriing the wke wind speeds [5,9]. The simplified Bernoulli eqution stting tht the mechnicl energy, per unit mss, long stremline is conserved: rv þ p ¼ H () where r is the ir density, V is the wind speed, p is the pressure nd H is the totl energy (constnt long ny stremline). Applying the Bernoulli eqution for the wind just in front of nd just ehind the rotor llows the drop in pressure over the rotor plne to e clculted (see Fig. ): Dp ¼ r v v The xil thrust force (the force cting in the direction of the wind), denoted s T, is clculted using the pressure difference: T ¼ DpA (3) where Dp is the pressure difference nd A is the rotor re. Now defining n xil interference fctor,, which is used to define the rtio etween the reduction in speed t the plne of the lde disc (u) nd the undistured wind speed well upstrem of the turine (v ): u ¼ð Þv nd thus v ¼ð Þv () The sustitution of () for v in the thrust definition (3) yields: () Now defining thrust coefficient, C T ¼ ( ), the xil thrust force cn e expressed s: T ¼ ra C T v (6) Finlly, it is convenient to define the reltionship etween the downstrem wind velocity (v ) nd the free wind speed (v )in terms of the turine thrust coefficient (C T ): v p ¼ðÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi C v T The ove reltionship is sed on the ssumption of idelized downstrem wke expnsion; this is the most simple turine wke model nd is computtionlly very efficient... N.O. Jensen wke model The N.O. Jensen wke model is simple single wke model. The model is well documented in Ref. [6] nd it is sed on the ssumption of wke with linerly expnding dimeter [].... Single wke model If the ner field ehind wind turine is neglected the resulting wke ehind the wind genertor cn e treted s turulent wke. The spred of feture like this is such tht the liner dimension (rdius r) is proportionl to the down-wind distnce, x. A lnce of momentum gives (see Fig. ): pr u þ p r r (7) v ¼ pr v (8) Assuming liner expnsion of the wke, the pth tken y the wind tht hs pssed through the turine ldes is represented y cone. The rdius of this (shdow) cone r, cn e clculted using the following expression: r ¼ r þ x (9) The dimensionless sclr determines how quickly the wke expnds with distnce nd it is defined s: ¼ () z ln z A v v A( x) T ¼ rð Þv A (5) A u α v A A v u v r r = r +α x x p p p+δp p = p Fig.. Air flow ner n idelized turine: velocity nd pressure. This model llows some key wke equtions to e derived. Fig.. The N.O. Jensen wke model is simple single wke model tht ssumes liner expnsion of the wke cone. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

3 F. González-Longtt et l. / Renewle Energy xxx () e 3 where z is the hu height of the turine generting the wke nd z is constnt clled surfce roughness, which depends on the chrcteristics of the locl terrin. The sclr cn ssume mny vlues depending on the locl terrin nd/or wind climte conditions. For free wind, i.e. not ffected y ny upstrem turine ¼. is usully suitle, otherwise vlue of.8 could e used for [6]. The velocity in the wke t distnce x from the wind turine cn e otined y solving (8) in terms of v : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r v ¼ v þ v C T () r Wke strem ( ij) r x i θ A xi, A shdow, i r Rotor Swp Are z ij This eqution provides the wke speed t the down-wind loction s function of the incoming wind speed. L ij A... Multiple wke model The effect discussed in the previous section is single wke model. In wind frm every upstrem wind turine tht shdows studied downstrem turine will hve n effect on the studied turines performnce []. This multiple wke effect is shown in Fig. 3, where the direction of free wind is shown for the undistured wind speed v. In this wind frm configurtion, the wind speed t wind turine j is ffected not only y the upstrem wind turine tht is directly in front of it, wind turine i, ut lso y other upstrem wind turines, such s, nd 3. In order to otin usle result for wind frms with mny turines, the effects of the multiple single wkes must e comined into single effect. A purely empiricl men is usully used to model the interction etween these multiple wkes. A detiled model of wke effects considers the shdowed res of the upstrem wind turines. This shdowing is mesure of the degree of overlp etween the re spnned y the wkes shdow cone (A shdow,i ) nd the re swept y the turine experiencing shdowing (A ); n exmple of which is depicted in Fig.. There re four distinct shdowing possiilities; nmely: complete shdowing, qusi-complete shdowing, prtil shdowing nd no shdowing. If the wind turines hve the sme dimeter (r ) then the re of the turine eing shdowed cn e clculted using the following sic trigonometric reltionships: A shdow;i ¼ r i xij cos " v j xij ¼ vi Xn p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi C T i ¼! L ij þ r r cos i xij r r i xij! Ashdow;i d! ij L ij d ij z ij r i xij A # () (3)..3. Wind speed dely etween two successive wind turines From the previous section, the reltionship etween the incoming wind speeds t the two units cn e otined sed on the wke flow. Fig. 5 gives digrm of the incoming wind speed for two successive units. The wke speed t the down-wind turine loction is function of the incoming wind speed t the upwind loction. A reltionship for the wind speed dely etween two successive wind turines (s ij ) tht re seprted y distnce x ij, cn e derived considering, v i, the incoming wind speed t turine i nd, v j, the incoming wind speed t wind turine j. Assuming the ir ccelertion etween the two successive wind turines, ij, is constnt then: d x dt v WT i Upwind turine v ¼ d v dt Wke Strem v i ( xij) v i WTj Downwind turine ¼ ij () The oundry conditions of the ove eqution () re: d ij Fig.. A detiled exmple of prtil shdowing tht cn e used to clculte the prt of turine s swept re tht is shdowed y nother turine s wke. The effect wke hs on turine cn e weighted using this vlue. α v x ij vi () t u () t vj () t uj () t i v i j WTi WTj 3 x ij Fig. 3. This exmple of multiple wkes shows why multiple wke models must comine the influence of every upstrem wke, which shdows turine, into single effect to otin usle result. Fig. 5. The cse of two successive turines is used to clculte the wind speed dely for the turines; tht is, the time dely etween wke eing creted t turine i nd the sme wke then reching the ldes of turine j. () wind turines; () wind turines. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

4 F. González-Longtt et l. / Renewle Energy xxx () e x t ¼ s ij ¼ xij vðt ¼ Þ ¼v i v t ¼ s ij ¼ vj (5) The incoming wind s flow time, or dely time, etween the two successive wind turines cn e clculted using the following eqution: s ij ¼ x ij v i þ v j (6) Finlly, the wind speed t the shdowed wind turine j is clculted in terms of the speed of the wind pproching the shdowing turine i. v j t þ sij ¼ v t þ sij þ ui ðtþv t þ sij ij (7) The coefficient ij is the shdowing fctor; defined s the swept re of turine j under the shdow of turine i, normlized to the swept re of turine j.... Wke comintion In order to otin usle result for wind frms with mny turines the effects of multiple single wkes must e comined into single effect. Wind turines in wind frm my experience vrying degrees of shdowing from upstrem turines. The result of eing exposed to mny single wkes, with different degrees of shdowing nd time dely, is non-uniform distriution of velocity tht my plce incresed mechnicl loding on the turine ldes. There re vriety of methods for comining these seprte wkes [,]: sum of squres of velocity deficits, energy lnce, geometric sum nd liner superposition. In this pper the lw of momentum conservtion is used to determine the resultnt wind speed t ech single wind turine loction. This pproch comines ech single wke into single equivlent wind speed, using the following reltionship [,6]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X n h i v j ðtþ ¼v j ðtþþ ij v j;k x kj ; t v j ðtþ u t k ¼ ksj (8) where v j (t) is the resultnt wind speed for n ritrry turine j, v j,k (x kj,t) is the wind speed pproching turine j (with turine k s the shdowing turine) nd v j is the incoming wind speed t j without shdowing. 3. Implementtion of the wke model The im of this pper is to evlute the effects which wkes hve upon the stedy-stte nd dynmic ehvior of wind frms. For this purpose the wke model riefly outlined ove hs een developed, firstly, s Mtl Ò [3] progrm. This progrm ws used to directly evlute the effects in stedy-stte nd lter improvement in the progrm llowed the inclusion of the time dely effects of wkes. The fetures developed in Mtl Ò were lso integrted into the DIgSILENTÔ [] simultion pckge in order to perform nlysis of the dynmic ehvior of wind frm. This section outlines the clcultion method nd the quntittive indictor used to evlute the overll impct of the wke effect on the output power of the wind frm. 3.. Outline of the clcultion method The wke model used here requires site pln of the wind frm with the coordintes of ech wind turine loction within the frm. Wke Coefficient (p.u.) Wke Coefficient (p.u.).9.8 D 6D 8D D Angle (deg).7.6 D 6D.5 8D D Angle (deg) Fig. 6. The effect of tower spcing upon the wke coefficient cn e seen. Tower spcing vlues re defined in terms of the turine rotor dimeter, D. Two rry lyouts re considered; regulr grids of nd turines. () Tower spcing: D. () Tower spcing: D. The following wind turine detils re lso necessry: geometricl chrcteristics (rdius nd hu height), the power coefficient curve C p (v w ) nd the thrust coefficient curve C T (v w ). The clcultion procedure is s follows: Wke Coefficient (p.u.) Wke Coefficient (p.u.) x x 6x6 8x8 x Angle (deg) x x 6x6 8x8 x Angle (deg) Fig. 7. The effect of the numer of wind turines upon the wke coefficient cn e seen. Regulr rrys with etween nd wind turines re used. The tower spcing vlues re defined in terms of the rotor dimeter, D. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

5 F. González-Longtt et l. / Renewle Energy xxx () e 5 Wke Coefficient (p.u.) The wind speed clcultion is initited with the turine positioned t the most upstrem position.. Clculte the wind speeds downstrem of this turine for ll downwind turine positions. 3. Clculte the wind speed for ll downstrem turine positions reltive to the free wind speed v (t).. If the downstrem turine is in prtil wke reduce the velocity ccording to the relevnt shdowing fctor ( ij ). 5. Clculte the resultnt wind speed in the downstrem turine, comining ll single wkes into uniform wind speed ccording to the lw of momentum conservtion. 6. Continue with the next turine (using step ), y summing the lw of momentum conservtion. 3.. Quntifying the wke effect The wke coefficient (c wke ) ws used to evlute the overll impct of the wke effect on the output power of the wind frm. It hs een defined s follows: c wke ¼ x x.6 6x6.5 8x8 x Tower spcing (rotor dimeters) Fig. 8. The decrese in the wke coefficient s tower spcing flls, due to incresed wke interction, cn e seen. Regulr rrys with,, 6 6, 8 8 nd turines re used. totl output with wke effect totl output power neglecting wke effect (9) This coefficient comines the locl effect of the interctions etween every individul wke t ech individul turine into single mesure of the wkes effect on the wind frm output.. Influence of wind direction nd wind frm lyout In order to evlute the effect of the wkes upon the stedystte output power of wind frm some tests hve een performed. The tests descried in this section del with the effect the wind direction nd wind frm lyout, specificlly the numer of turines nd their spcing, hve upon the wke coefficient. The wind direction nd wind frm lyout hve n importnt effect on wind frm performnce. This is ecuse they modify the orienttion nd loction of the wke cones; therefore, chnges in wind direction or wind frm lyout produce chnges in the wkes interction (overlpping re) nd consequently their effects on the power output of ech individul turine. In order to evlute the effect of wind direction upon wind frm performnce the wke coefficient hs een plotted in terms of the wind direction etween nd 9 for squre rry with uniform spcing. Fig. 6 shows how chnges in the wind direction modify the wke coefficient for severl different vlues of tower spcing (in terms of the rotor dimeter, D) nd for two wind frm rrys; one rry hs sixteen turines in grid whilst the other hs one hundred turines in grid. In oth rrys the turines re regulrly spced so the spcing etween ech turine in the sme row nd ech row of turines is the sme. The lowest vlues of wke coefficient occur for the smllest ngles ( e9 ) nd the lrgest ngles (85 e9 ); under these conditions the wkes of wind turines in the sme row, or column respectively, will produce either complete or qusi-complete shdowing. A similr condition is found gin for wind directions etween nd 8 nd the wke coefficient vlues re the second lowest vlues. This is ecuse whilst the downwind turines will still e under complete or qusi-complete shdowing, s in the cse of ( e9 ;85 e9 ), the pprent distnces etween turine nd those turines shdowing it re lrger. The vrition in the wind prk performnce for wind directions etween these zones of high shdowing is etter. The wke coefficient hs ripple; the mplitude of which is dependent on tower spcing whilst the frequency is determined y the numer of turines. An increse in the numer of wind turines in the wind prk decreses the wke coefficient when the spcing is mintined. Given fixed numer of wind turines inside the prk smller spcing etween the turines increses the shdow interference compromising the performnce of the wind frm. Fig. 7 plots the vrition of the wke coefficient with wind direction for wind frms with different numers of turines nd two vlues of turine spcing (D nd D). From this the dependence of the wke coefficient upon wind direction is cler. In ddition, the chnges in the wke coefficient tend to e lrger for smller vlues of tower spcing. Chnges in the wind direction hve more significnt effect on the wke coefficient in tightly pcked wind frms thn in wind frms with incresed turine spcing. If the numer of turines mx % clm 5 Vlue (m/s) 5 J F M A M J J A S O N D A men min % 56% 8% Fig. 9. The monthly sttistics for the wind speed nd wind speed frequency rose of the yer long wind regime used in the long term simultions. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

6 6 F. González-Longtt et l. / Renewle Energy xxx () e 6 6 Output Power(p.u).8.6. v φ ΔyΔ Δy Time (dys) 3 Δx Arry I: Regulr 3 Δx Arry II: Irregulr Fig.. The output power for n idel wind frm lyout, which experiences no wke effects, exposed to the wind regime descried in Appendix. Fig.. The two non-idel wind frm lyouts compred in these simultions. () Fixed direction t. () Considering chnges in the direction. increses then the men vlue of the wke coefficient flls; s does the mgnitude of the chnges in the wke coefficient s the ngle vries. The effect of the numer of wind turines, nd the spcing of those turines, upon the wke coefficient is relevnt when ttempting to select n pproprite lyout for proposed wind frm. Fig. 8 illustrtes the degrdtion of performnce when wind turines re too close together. This degrdtion is more significnt for n rry with more turines ecuse their will e more wke cones shdowing ech turine. 5. Stedy-stte (long term effects) The long term effects of wkes in wind frm re evluted for one wind frm with 6 wind turines ( MW turines, see detils in Appendix) using 365-dy time series of wind speed nd wind direction. The men wind speed is m/s nd the previling wind direction is from, some sttisticl detils out these time series re shown in Fig. 9 (more detils in Appendix). Initilly, ll wind turines re plced in stright line tht is perpendiculr to the previling wind direction. This wind frm lyout effectively elimintes the wke effect for the wind regime considered. Fig. shows the output power in per unit sed on the nominl power of the wind frm; during the time period of the evlution the wind frm would hve produced rted power for pproximtely dys/yer. The wke coefficient for this wind frm geometry is the mximum possile (C wke ¼. p.u.) ecuse no wke interctions occurred. The proility distriution function (PDF) is plotted in Fig. ; the est-fit Weiull distriution shows tht the output power flls within the rnge of.975e. p.u. for.97% of the time. This design provides high proility (93.%) of generting Proility Density (%) 8 6 Actul Dt Best-fit Weiull Distriution k =.966 p.u c = Output Power (p.u) Fig.. The proility density function of the output power considering n idel wind frm lyout tht experiences no wke effects. more thn 75% of rted power during yer nd genertion ove this level occurs for n verge of 6.7 h/dy. In order to evlute the effect of the wind direction nd lyout of wind frm upon overll performnce two different configurtions were used (Fig. ): () Arry I: regulr distriution with equllyspced wind turines every -rotor-dimeters (D D), nd () Arry II: the wind frm hs tower spcing of -rotor-dimeters long its rows nd 7-dimeter spcing etween rows (D 7D) in configurtion of stggered towers s shown in Fig. The wke coefficient nd proility distriution functions for Arry I re shown in Figs. 3 nd respectively for two wind regimes; the first regime hs fixed direction of whilst the second hs vrile direction. The wind turine lyout of Arry I provides the mximum shdowing possile when the wind direction is fixed equl to. As consequence of this the wind frm performnce is lower in this scenrio (Fig. 3). The verge output power is.85% less thn the no-wke wind frm cse; with minimum wke coefficient of.697 p.u. The rted power is only expected during 8 dys/yer with dily verge of 8.7 h/dy. When the effect of wind direction is considered in the wke model the verge power output of Arry I increses y 8.5% nd the time t rted output increses to 68 dys/yer; the minimum wke coefficient is.736 p.u. This improvement is cused y the reduction of the shdowing fctor s Wke Coefficient (p.u) Wke Coefficient (p.u) Time (dy) Time (dy) Fig. 3. The wke coefficient for Arry I when exposed to the wind speed of the yer long wind regime with oth fixed,, () nd ctul () wind direction. () Fixed wind direction t. () Considering wind direction. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

7 F. González-Longtt et l. / Renewle Energy xxx () e 7 Proility Density (%) 8 6 Actul Dt Best-fit Weiull k =.88 p.u c = Output Power (p.u) Proility Density (%) 8 6 Actul Dt Best-fit Weiull k =.93 p.u c = Output Power (p.u) Proility Density (%) 8 6 Actul Dt Best-fit Weiull k =.9 p.u c = Output Power (p.u) Proility Density (%) 8 6 Actul Dt Best-fit Weiull k =.96 p.u c = Output Power (p.u) Fig.. The proility density function of the output power of Arry I when exposed to the wind speed of the yer long wind regime with oth fixed,, () nd ctul () wind direction. () Fixed direction t. () Considering chnges in the direction. Fig. 6. The proility density function of the output power of Arry II when exposed to the wind speed of the yer long wind regime with oth fixed,, () nd ctul () wind direction. consequence of etter interction etween the wkes for different wind directions. In this prticulr cse the verge wind direction is 7.36 with rther lrge stndrd devition of 58.69, this is more fvorle thn the scenrio with fixed direction of for the geometric configurtion of Arry I; s is shown in Figs. 3 nd. The wke coefficient for Arry II is shown in Fig. 5. It is evident tht the est wke coefficient ehvior occurs when the wind direction is considered nd, despite the minimum wke coefficient (.698 p.u) eing otined in this cse, the verge vlue is.998 p.u. If wind direction is not included the verge output is 3.77% less thn when direction is considered. The men wke coefficient is only.957 p.u. nd the minimum vlue is.866 p.u.. The proility density function of the output power is shown in Fig. 6. Arry II exhiits superior ehvior when the chnges in wind direction re included with rted power output for dys/ yer t n verge of. h/dy. If fixed direction wind is considered the numer of dys/yer t rted power is only 7. Finlly, n overll picture of the power nd energy production for ech cse is shown in Tle. An rry without wke effects is fvorle condition in terms of the totl power production, nd mechnicl stress on individul wind turines. In this cse the verge output power otined ws highest when wke effects did not occur; however, this design is neither techniclly nor economiclly fesile. When ccounting for wke effects the output Wke Coefficient (p.u) Wke Coefficient (p.u) Time (dy) Fig. 5. The wke coefficient for Arry II when exposed to the wind speed of the yer long wind. () Fixed wind direction t. () Considering wind direction. of wind frm sed on Arry II is the highest; in terms of oth men nd minimum power output, nd nnul energy output. A etter view of the rted power nd energy delivered for the different cses is evluted considering the cpcity fctor (C F ); nd from the results presented, the superior performnce of Arry II (.5) is evident. The offsetting, or stggering, of one row of towers ehind nother, provides etter wke coefficient ecuse it reduces the mount of wke interction within the rry. This mens tht significnt increse in the nnul energy output of the sme set of turines, exposed to the sme wind regime, hs een chieved simply y djusting the rry lyout in n ttempt to reduce the wke effects occurring within the rry. 6. Dynmic (short-term effect) The wke effect is fctor during every time-scle of power system ehvior. An ssessment of the impct of wkes upon the dynmic ehvior of single wind frm is performed through time domin simultions using the wke model presented in this pper; with the inclusion of the wind time dely etween wind turines inside the wind frm. The vriility of wind speed hs een considered y using 6 s time series of wind dt, with n verge vlue of.678 m/s nd stndrd devition of.688 (more detils in Appendix), for the simultions. Wind direction is considered fixed during the 6 s simultion ut multiple simultions re performed to llow severl different wind directions to e considered. The wind frm consists of 6 constnt-speed wind turines ( MW ech, detils in Appendix) connected through mediumvoltge underground distriution system with four feeders. Two step up trnsformers in prllel feed the output power of the wind frm into n externl grid. The network shown in Fig. 7 hs een used s the test network. The wind frm lyouts previously defined s Arry I nd II, in Section 5, hve een used for comprehensive evlution of the Tle Power nd energy output for the rrys considered. Vrile No wke Arry I Arry II Wind direction Wind direction Fixed Vrile Fixed Vrile Men output power (MW) Minimum power (MW) Energy output (GW h/yer) C F Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

8 8 F. González-Longtt et l. / Renewle Energy xxx () e Externl grid v θ 66 kv kv kv Fig. 7. Schemtic lyout of the distriution system supporting the wind frm; the collecting point is susttion. The specific network for ech rry lyout is designed using the ctul distnces etween the turines in tht lyout. impct of the wke effect upon the wke coefficient nd ctive power losses in the wind frm. The internl network hs een designed considering the ctul distnces etween the wind turines in the ech of the rry lyouts (Fig. ). Figs. 8 nd 9 depict the vrition of the wke coefficient over the 6 s simultion period for five seprte wind directions for oth Arry I nd Arry II. These figures show tht the wind direction hs significnt influence upon the impct of the wke effect on the output power nd ctive power losses of the wind frm. Arry I exhiits worse ehvior, in terms of wke coefficient, t thn Arry II ecuse the lyout of Arry II mximizes the numer of wind turines with undistured wind for this wind direction. Whilst the sme design feture ensures tht the opposite is true for wind direction of. Wke Coefficient (p.u) Losses (p.u) x Fig. 9. The influence of wind direction on the vrition in the wke coefficient nd power losses of Arry II is cler. The wind direction is fixed for ech simultion nd rnge of vlues from to re used. 3 3 Wke Coefficient (p.u) Losses (p.u) x Fig. 8. The influence of wind direction on the vrition in the wke coefficient nd power losses of Arry I is cler. The wind direction is fixed for ech simultion nd rnge of vlues from to re used. 3 3 Wke Coefficient (p.u) Losses (p.u) x Arry I Arry II No wke Arry I Arry II Fig.. A comprison of the wke coefficient nd power losses for no wke rry; Arry I nd Arry II show the improved performnce s the wke effects re reduced. A fixed wind direction of is used for the entire simultion nd therefore the wke coefficient of the idel rry is lwys. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

9 F. González-Longtt et l. / Renewle Energy xxx () e 9 Delys in Arry I increse s the wind direction chnges etween nd nd the mximum delys re reched t 5. This is ecuse the incresed pprent distnce etween the wind turines increses the time delys etween ech turine; which in turn reduces the interctions etween the wkes. The lyout of Arry II provides the sme tower spcing long its rows s Arry I, however the spcing etween ech row is lrger nd therefore for smll chnges in the wind direction the ehvior of ech rry is similr. The comprison of the short-term ehvior of the wke coefficient for Arry I nd Arry II, depicted in Fig. for fixed wind direction of, shows tht Arry II offers superior performnce. Active power losses inside Arry II re higher thn Arry I, due to the incresed spcing etween rows. However, the mximum difference is less thn.% of the rted power output. Power (kw) Wind speed (m/s) Fig. A.. Power curve (P), power coefficient (C p ) nd thrust coefficient (C T )ofmw turine. P Ct Cp.5 Cp nd Ct 7. Conclusion The primry impct of the wke effect is to distur the flow of the wind inside wind frm. As consequence of single wke or interction etween multiple seprte wkes the wind downstrem from wind turine hs reduced speed nd ecomes highly turulent. This reduces the energy production of the wind frm when compred to tht expected if free wind existed throughout the frm. In this pper simple explicit model for wkes ws presented, nd used to evlute the impct of the wke effect on oth the stedy-stte opertion nd dynmic performnce of wind frm. Simultions of the effect of wkes upon wind frm opertion indicte tht the rry efficiency depends upon the spcing etween turines nd the nture of the wind regime they re exposed to, in terms of the forcing conditions nd distriution of wind directionlity nd speeds. Wind conditions depend on nture nd cn therefore not e modified; however, the wind frm lyout cn e modified to reduce the influence of the wke effect nd therey optimize wind frm performnce. When ttempting this modifiction the geometric distriution of wind turines inside the wind frm is the most sensile design prmeter to djust. Both the stedy-stte nd dynmic simultions performed here hve demonstrted tht considering the wke effect when developing turine lyout cn offer considerle improvements in the power output of wind frm. Furthermore the dynmic simultions performed hve demonstrted tht even if the wke influenced wind frm lyout increses the distnce etween the turines the resulting increse in electricl losses will e miniml compred to the increse in the power drwn from the wind. Appendix Tle A. Chrcteristics of MW wind turine. Description Prmeter Vlue Hu height h 78 Rotor dimeter r 8 Are swept A wt 536 m Cut-in speed V in m/s Cut-out speed V out 5 m/s Stnding trust coefficient C t,s.58 Fig. A.. Wind speed nd direction time series for long term evlution (365 dys). Wind Speed (m/s) References Fig. A.3. Wind speed time series for short-term evlution (6. s). [] Koch F, Gresch M, Shewreg F, Erlich I, Bchmnn U. Considertion of wind frm wke effect in power system dynmic simultion. In: Power tech, 5 IEEE Russi; 5. p. e7. [] Kirnoudis CT, Mroulis ZB. Effective short-cut modelling of wind prk efficiency. Renewle Energy 997;:39e57. [3] Lissmn PBS. Energy efficiencies of ritrry of wind turines. Journl of Energy 979;3. [] Vermenulen PEJ. An experimentl investigtion of wind-turines wkes. In: 3rd Interntionl symposium on wind energy systems, Copenhgen, Denmrk; 98. p. e3. [5] Milorrow DJ. The performnce of rrys of wind turines. Journl of Wind Engineering nd Industril Aerodynmics 98;5. [6] Ktic I, Højstrup J, Jensen NO. A simple model for cluster efficiency. In: Europen wind energy conference nd exhiition, Rome; 986. p. 7e. [7] Voutsins S, Rdos K, Zervos A. On the nlysis of wke effects in wind prks. Wind Engineering 99;. [8] Smith D, Tylor GJ. Further nlysis of turine wke development nd interction dt. In: 3th BWEA conference, Swnse, Wles; 99. p. 8e9. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53

10 F. González-Longtt et l. / Renewle Energy xxx () e [9] Zervos A, Huerson S, Hemon A. Three-dimensionl free-wke clcultion of wind turine wkes. In: Interntionl conference on wind frms, Leeuwrden, Netherlnds; 987. p. 5e9. [] Zervos A, Huerson S, Hermom A. Three-dimensionl free-wke clcultion of wind turine wke. Journl of Wind Engineering nd Industril Aerodynmics 988;7:65. [] Ainslie JF. Clculting the flow field in the wke of wind turines. Journl of Wind Engineering nd Industril Aerodynmics 988;7:3e. [] Mortensen NG, Herthfield H, Lndergg H. Getting strted with WASP 7.. In: Risø-I-53(EN);. [3] Frndsen S, Brthelmie R, Pryor S, Rthmnn O, Lrsen S, Højstrup J, et l. Anlyticl modeling of wind speed deficit in lrge offshore wind frms. Wind Energy 6;9:39e53. [] Rthmnn O., Frndsen S.T., Brthelmie R.J. Wke modeling for intermedite nd lrge wind frms. Europen Wind Energy Conference nd Exhiition, Miln, My 7, p 8. [5] Jensen NO. A note on wind genertor interction. Risø Ntionl Lortory; 983. [6] Brthelmie RJ, Hnsen K, Frndsen ST, Rthmnn O, Schepers JG, Schlez W, et l. Modeling nd mesuring flow nd wind turine wkes in lrge wind frms offshore. Wind Energy 9;:3e. [7] Brthelmie RJ, Folkerts L, Lrsen GC, Rdos K, Pryor SC, Frndsen ST, et l. Comprison of wke model simultions with offshore wind turine wke profiles mesured y sodr. Journl of Atmospheric nd Ocenic Technology 5;3. [8] Mgnussin M, Smedsm AS. Air flow ehind wind turines. Journl of Wind Engineering nd Industril Aerodynmics 999;8:69e89. [9] Andersen PS, Kre U, Lundsger P, Petersen H. Bsis mteril for wind turine design. Risø Ntionl Lortory; 98. [] Sørensen T, Thøgersen ML, Nielsen P. Adpting nd clirtion of existing wke models to meet the conditions inside offshore wind frms. Alorg: EMD Interntionl A/S; 8. p. 53. [] Zhng XY, Wng WQ. Wind frm nd wke effect modeling for simultion of studied power system. In: 9 IEEE/PES power systems conference nd exposition, vols. e3; 9. p. 3e8. [] Lnge B, Wldl HP, Brthelmie R, Gil A, Heinemnn D. Modelling of offshore wind turine wkes with the wind frm progrm FLP. Wind Energy 3;6: 87e. [3] MATLAB, (R9) ed. Ntick, Msschusetts: The MthWork, Inc.; 9. [] DigSILENT PowerFctory,..59. ed. Gomringen, Germny: DIgSILENT GmH;. Plese cite this rticle in press s: González-Longtt F, et l., Wke effect in wind frm performnce: Stedy-stte nd dynmic ehvior, Renewle Energy (), doi:.6/j.renene..8.53