# Modeling Stochastic Wind Loads , on Vertical Axis Wind Turbines

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Modeling Stochastic Wind Loads , on Vertical Axis Wind Turbines

SANDIA REPORT Printed September SAND83–1909 ● Unlimited Release 1984 Modeling Stochastic Wind Loads , on Vertical Axis Wind Turbines Prepared by Sandia National Laboratories Albuquerque, New Mexico 87185 for the United States Department under Contract CIE-AC04-76DPO0789 and Livermore, of Energy California 94550 ● UC–60 Issued by .%rdia NationalLaboratories, operatedforthe United States DepartmentofEnergyby SandiaCorporation. NOTICE Thisreportwas preparedasan accountofwork sponsoredby an agencyoftheUnitedStatesC,overnment. 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Printed intheUnitedStatesofAmerica Available from NationalTechnicalInformation Service U.S.DepartmentofCommerce 5285PortRoyalRoad Springfield, VA 22161 NTIS pricecedes Printedcopy A19 Microfiche copy AO1 MODELING STOCHASTIC WIND LOADS ON VERTICAL AXIS WIND TURBINES Paul S. Veers Applied Mechanics Division 1524 Sandia National Laboratories Albuquerque, New Mexico Abstract The Vertical AXIS Wind Turbine (VAWT) is a machine which extracts energy Since random turbulence is always present, the effect of from the wind. This this turbulence on the wind turbine fatigue life must be evaluated. problem 1s approached by numerically simulating the turbulence and calcu– latlng, In the time domain, the aerodynamic loads on the turbine blades. These loads are reduced to the form of power and cross spectral densities The relawhich can be used In standard linear structural analysls codes. tive Importance of the turbulence on blade loads 1s determined. The most common des]gn for Vertical Axis Wind Turb]nes (VAWT’S) was first patented in 1931 by Th]s “egg beater” shaped Darrleus, a Frenchman. machine consists of one or more blades with air– foil cross sections attached to the top and bottom of a central shaft, or tower. To m]nlmize bending stresses In the blades while the turbine rotates, the blades usually have a characteristic tropos– Torque is klen, or “splnnlng rope,” shape. produced when the turb]ne rotor turns In the wind. Fig. 1 shows the 17 meter research VAWT at Sandla National Laboratories, Albuquerque, NM This turb]ne has the most common configuration of VAWT’S currently being built, two blades w]th a central tower and guy cables supporting the top of the rotor. The aerodynamic analysls of the VAWT 1s compli– Since the VAWT blade cated by several factors rotates through 360 degrees relatlve to the lncl– dent wind during each rotation, the rate of change much greater than of angle of attack 1s often But experienced ]n other airfoil applications. because the speed of the blade 1s normally much greater than the w]nd speed, the angle of attack varies between posltlve and negative values around zero degrees. The higher the w]nd speed, the greater the angle of attack excursions. When the winds become h]gh enough, the angles of attack become large enough that the alrfoll begins to stall dynamically. Research In VAWT aerodynamics 1s currently lnvestlgatlng both dynamic stall and Aerodynrunlc analys]s using p]tch rate effects. the streamtube momentum balance approach was the earnest and slm lest approach used to est]mate Y VAWT performance The vortex llftlng llne ap– preach has produced some better results, in an average sense, over a wider range of wlnd2conexpensive d]tlons, but 1s computatlonally Both methods suffer from the lack of a valldated method of predicting dynemlc stall. Except for emplrlcal llft and drag curves from wind tunnel testing, accurate pltchlng airfoil, dynamic stall models do not exist. Most VAWT’S experience stall over a s]gnlflcant portion of the normal operat]ng The loss of llft due to stall in high range winds 1s considered benef]clal because It llmlts the maximum power that the drive train must trans– mlt, thereby holding down the turbine cost. The structural analysls of the VAWT must deal with Corlolls the fact that the structure is rotating. and centrifugal effects make the modes of vlbra– tlon complex, but the system remains l]near. Free v]bratlon analysls and aero lastlc analysis 5 are well developed and valldated The forced v]bratlon analysis of the VAWT rotor ]s developed, but agreement w]th field data from operating turbines 1s sometimes poor, especlall~ In the high w]nd speed, significant stall, regime It is not clear whether the source of the error 1s In yet the structural analysls or ]n the calculation of The relatively good agree– the aerodynem]c loads ment of the structural analysis to the data gathered at low wind speeds would seem to indicate that the problem may l]e in the aerodynamics. 2 Fig. 1 Darrleus Vertical AXIS Wind Turb]ne at Sand]a National Laboratories, Albuquerque, NM (VAWT) Another possible source of error In previous load calculations 1s the assumption of a steady w]nd. Until now, all the aerodynamic loads have been The calculated based on a constant emblent wind. loads which result from this calculation are shown In Fig. 2. Tangential forces are def]ned as the component In the dlrect]on tangent to the path of the blade as It rotates and normal forces are normal to this path. The tangential forces pro– duce a net torque about the central tower while When a the normal forces produce no net energy. constant wind lS assumed, the forces repeat exThe frequency actly for each rotor revolution. content of these forces 1s therefore limlted to Integer mult;ples of the turbine rotational speed (abbreviated as “per rev” frequencies). This approach will produce zero excltat]on at any frequency other than the per rev frequencies. This lS not totally unreasonable because the per rev frequency content of the loads 1s produced by the mean wind and the rotation of the rotor, and ]s therefore large compared to the turbulence However, data collectinduced, stochastic loads ed from VAWT’S operat]ng in high winds have shown slgnlf]cant response at the structural resonant frequencies as well as at the per rev frequencies. The spectral content of the blade stress response 3 could never be predicted with shown in Flg loads calculated assuming e steady wind. By calculating the aerodynamic loads due to a turbulent wind, the loads will contain all fre– These quencles, not only the per rev frequencies. stochastic loads are described by their power spectral densities (psd’s) and cross spectral Because the relationship densltles (csd’s). between incident wind speed and blade forces is nonlinear, even before the onset of stall, the blade loads are calculated in a time marching manner But since the structural response to the stochastic loads is linear, the psd’s and csd’s will be sufficient input for structural analysis. J’urbulent Wind Model -1.64 00 \ I 30 20 10 TURBINE 49 The first step in a time domain calculation of the the stochastic blade loads IS to produce a numeriThe cal model of the atmospheric turbulence. turbulence model must posses frequencies of interest for structural analysls (O – 10 per rev). In a NASA summary of a mospheric environments by k Frost, Long and Turner , the followlng fOrm for the psd of atmospheric turbulence is suggested; REVOLUTIONS clfi[ln(]O/zo+ l)ln(h/zo+ 1)]–1 (1) S(Q) = 1 + c2[hti ln(lO/zo+ 1)/~ ln(h/zo+ 1)] 5/3 where u = frequency h = height ~ = mean wind z Lo 30 20 TURBINE Fig. 2 BC032 1 roughness coefficient 40 Clx REVOLUTIONS Dlmenslonless Blade Loads calculated assuming a steady wind, (a) normal to the blade path, (b) tangential to the blade path. 12(D 1- = surface speed at h = 10m and c are constants which differ for each c1 These constants spatial ~omponent of turbulence. are given as; .oll~ 0.0 0 above ground CHANNEL I I 13 I =i Cly = 12.3, C2X = 192. = Clz = 4.0, c 2y = 0“5’ C2Z = 70. 8.0 for the longitudinal direction, for the lateral direction and for the vert)cal direction. An example turbulence psd at a reference height of 30ft (lOm) with a surface roughness coeff]clent of .1 1s shown :n Fig. 4. For the wind velocity a lo’s so a w ld : 0. Io”z 4{D 10-‘: n -i)mo 1.6 3.2 4.8 6.4 8.0 f 1 10-’ 10° FREQUENCY (HZ) FREQUENCY Fig. 3 Power spectral density (psd) of the stress in a VAWT blade measured during operation In high winds. The turbine rotational frequency 1s 0.8 Hz, Fig. 4 [ Id Psd of turbulence using Eq. (l); h=lom, Zo=0.1,V=15m/s. 3 Geussian time ser)es described b~ this psd can be ct.talned by the followlng method” Represent the and power In a narrow band, &@, of the psd by s,n~ cosine components, with random phase, .gt the rcntral frequency and sum these Inputs over al 1 the frequencies of Interest. Such a veloclty time hlstor’y 1s g]ven by, where a = decay coeff]c]ent ~ = frequvncv l,r = distance ~ between po]nts 1 and j = mean wind speed [1 x V(l)=v, (AJ S]Ilti,t+ B COSU (2) ~ t) 1 j=l Ili ‘-”—~ 1 Wht’re r, s —— = magnitude of the psd at frequency J $, = a uniformly distributed random on the Interval zero to 2n, L! J n o ;— ----—r—00 100 variable 300 400 60.0 TIME(sEcoNtIs) (a) The central llmlt. theorem guarantees convergence Of ~(t) to a Gaussian form as the number of slnllsoidal corrponents at different frequencies with random phase becomes large, This ]s equlv– aler,t to band l]m] ted, f]ltered wh]te no]se If the’ frequency spacing is regular and beg]ns at zero. the compuiatlon of the time series can be tif<omp] l~l)ed w th 8 d]screte Inverse Fourier -1 trensform .# of the complex series, —-+ 200 40(17 - ——-—--”~ I I .. v(t) = v +.x -1 (Aj + }Bj) (3) x j=l If the three directional components of the turbu– lence are assumed to be uncorrelated, each com– ponent can be Independently generated using the above method The usual approach to transforming th]s temporal representation Into a spatial var]a– tlon IS to use Taylor’s frozen turbulence hypo– thesis Using this approach, It 1s assumed that all turbulence IS propagated in the longltudlnaJ dlrectlon at the mean wind speed F]g 5 shows a single point wind simulation time series where the horizontal longltudlnal, and lateral veloclty components have been resolved Into magnitude and dlrect]on. A s]mple approach to simulating a full three dlmens]onal field of w]nd ]s to assume that the w)nd speed and dlrectlon In a plane normal to the mean wind dlrectlon are constant. This IS obvl– OUSIV quite crude, but can be adequate If the region of Interest In this plane 1s relatively small and the w nd IS h)ghly correlated over that i region Frost has an est]mate of spat]al coher– ence of the form, 7:J = exp(–a u Ar/~) (4) 4 -400 --–— -—T--— 00 100 (b) F]g. 5 ~o~o,o TIME(SECONDS) Example simulated, single point, wind veloc]ty time series, (a) w]nd speed, wind dlrectlon (b) The suggested value for the decay coefficient, “a, “ 1s 7.5 for both lateral and vertical sepa– rations. This expression lndlcates that the size of the reg]on of slgnlflcant correlation 1s a For function of frequency and mean wind speed. the 17m diameter VAW’i’s, the most common built to date , the coherence over the range of most ener– get]c frequencies IS greater than .5. This Implles that the correlation over most of the swept area of the turbine rotor IS qu]te high. For the larger rotors of the future, or the cur– rent large d]ameter propeller type turbines, the one d)mens]onal wind varlat]on may not be ade– quate A fully can be three–dimensional spatially simulated by producing time varying wind series of the wind speed at several points which lie in a plane perpendicular to the mean wind direction, as shown schematically in Fig. 6. The longitudinal varia– tions are, as before, produced by Taylor’s frozen turbulence hypotheses, The velocity time history at a particular point 1s reproduced at a down– stream point at a later time. The time shift is equal to the distance between the points divided by the mean wind speed. The wind time series at each point )s a random process which is correlated to wind at every other point. AS shown by Eq. 4, the level of correlation depends on the distance between the po:nts, the mean wind speed and its frequency. z L Y x The csd’s are, in general, complex fmctlons of frequency. Eq. 5 relates only the magnitude of the csd’s to the coherence and the psd’s and is therefore insufficient to completely define the spectral matrix. Analytical expressions for the csd’s have been suggested by Frost, but they are very complicated functions Involving modified Bessel functions of fractional order, If the csd’s are approximated as real functions, Eqs. 1, 4 and 5 can be used to define the spectral matrix. Since there is usually no relat]ve phase between the winds at two points that are at the same height above the ground, the csd between these points is real. For points with some vertical separation, there 1s a tendency for the winds at the higher elevation to lead the lower elevation winds , wh]ch results in a complex csd. However, this tendency is not strong and the magnitude of the lmaglnary part is usually much smaller than the real part. The csd’s are therefore asstuned 0 be entirely real and approximated by Eq, 5. v~(t) A The method of converting the spectral matrix into a vector of tlm realizations of the random Define a lower triangular PrOcess fo~lows~ser’es matrix of transfer functions, [H], such that [H]*[H]T = [s] Fig, 6 Schematic of a wind simulation with varla– Correlated tions in three dimensions. single point time series are generated at several Input points. Physically, [H] 1s a transfer matrix from uncorrelated white noise to the correlated spectra of the wind at the Input po)nts. In general, both [s] and [H] are complex, but for the degenerate case of a real [S], [H] is real also. The general case of a complex [s] is derived in Ref. 7, Because [H] IS lower triangular, lt 1s completely defined by Eq. 6. The elements of [H] can be determined recursively. H 1/2 11 = ’11 H ~. = H Since the wind speed at each point can be repre– sented by Gaussian random processes, the total input wind field can be represented as a random process vector, {V]. The second order statistics of {V} can be defined by the spectral matrix [s]. The element In the i–th row and j-th column of [s] is the cross spectral density (csd) between the The diagonal wind speeds at input points i and j The terms are the psd’s of the Input wind speeds. csd’s are related to the coherence function and the psd”s by the expression, s.. l] 2 = 7:, Sli s.. 22 32 S 31 /H1l = (S32-H31 . Sii = psd at point i. H21)/H22 H 11 = (s,, - H,12)1’2 x j =1 j (5) –1 = (Slj - H,k HJk)/Hj J z k=l function points (7) i-1 11 Sij = csd between 1/2 . . H 7;, = coherence = (S22 - H212) ’31 = H S21/H11 JJ where (6) i and j Define [X] as a diagonal matrix of complex frequency domain representations of Independent white The noise Gaussian signals with unit variance. 5 correlated vector of wind then be calculated from {v] random processes =L%-l([H][X]){lI can misrepresenting physical anced by the statistical of this approach. (8) &dvnemic where {II IS a column vector of ModeL ones Multiplying [H] by [X] gives a random phase at each frequency to each column of [H]. The multi– pllcatlon in (8) by the vector of ones has the effect of summing the rows of the Inverse Fourier transform of [H][X]. Once [H] has been calcu– lated, new unique realizations of the process can be formed simply by introducing a new [X] Into Eq. 8. The vector of wind turbulence components, {V~, has zero mean and describes only the turbulent part of the wind. The mean wind, wh]ch may vary with height to Include wind shear effects, 1s added to the turbulent part. The main dlff]culty in Implementing this procedure ls the size of the matrices Involved. For N Input po]nts, the matrices [s], [H] and [X] are NxN. Each element in the matrices 1s a function of frequency wh]ch must be defined at each discrete Input frequency. If the number of Input points and discrete frequencies 1s not kept to a mlnlmum, the size of the problem can easily exceed the core storage space on a main frame computer. In spite of the size of the problem, the use of the fast Fourier transform makes the computation relatively fast. Example problems which would exceed the core storage on a Cray were run on a VAX 11/780 in a few minutes. In summary, a full field of turbulent wind can be simulated with varlat]ons In one to three spatial The one dimensional variations are directions. produced by creating a wind time ser)es at a point and assuming that the w:nd speed 1s constant In a plane perpendicular to the mean wind dlrec– tlon The longitudinal varlatlon 1s achieved by aPPIYlng Taylor’s frozen turbulence hypothesis. A three dimensional wind field can be simulated by producing an array of correlated single point time The need for the three dimensional repre– series. sentatlon depends on the size of the wind field of Larger fields have lower spatial cor– Interest turbu– relatlon and require a multldlmenslonal If the different components of the lence model. turbulence are uncorrelated, they can each be created Independently by repeating the procedures outl]ned above. The main difference between this turbulence slmu– latlon and other methods currently being developed (Ref. 8, 9, 10) 1s that in this simulation the turbulence IS based solely on Its first and second order statistical moments w)thout any reference to There 1s no the fluld dynamics of the wind. guarantee that this simulated flow field will be continuous , compatible or physically realizable. However, by matching Its f]rst two stat]stlcal moments , the turbulence properties which have the most s:gnlflcant Impact on the blade loads and therefore the structural response of the wind The r]sk of turb]ne are accurately represented. 6 flow properties 1s bal– accuracy and slmpllclty Various approaches to calculating the aerodynamic forces on a VAWT blade have been developed in the past The earnest, and simplest, ]nvolves Since then, streamtube momentum balance methods vortex llftlng llne codes have been written which are more ac urate over a wider range of operating 5 wind speeds The cost of this generality IS a tremendous Increase In computational time needed to calculate the blade loads. A random blade load s]mulatlon necessitates a computatlonally fast method of converting instantaneous winds into To reduce the statistical error, the blade loads. loads must be calculated repeatedly and the psd and csd estimates averaged. The streamtube momen– tum balance method 1s by far the fastest Subdlvldlng the swept area of the available. turbine rotor Into a bundle of streamtubes creates a ready made computational grid. Each streamtube can be used as a locatlon to Input correlated The wind single point wind speed time series. speed at each point In the streamtube can then be tracked as lt passes through the rotor The basic aerodynamic model used for this analysis was developed by Strickland. The method, which is comprehensively described in Ref. 11, WI1l be outllned briefly here. The Idea 1s to subdivide the rotor Into streamtubes along the mean wind direction. As a blade passes through a stream– tube , the flow IS retarded. The amount of retardation ]s related to an “Interference factor.” The retardation 1s related to the net stresmwlse force by conservation of momentum wlthln the streemtube Because of the Interdependence of streamwlse force and angle of attack, the force Once the iteration calculation 1s lteratlve converges , the angle of attack lS found and the coefficients of normal and tangential force are Interpolated from a table of values. The code used In this study neglects both Reynolds number and dynamic stall effects. The results reported here are for low wind speeds where these effects are relatively small. Fig. 7 shows a typ]cal streamtube as it passes through a VAWT rotor. As shown In the plan view, each streamtube is divided Into three regions by upstream, Intersections with the blade path. The wind speed lnslde the rotor and downstream. 1s reduced In each reg]on by successive blade Each streamtube has three mean wind passages The speeds which correspond to the three regions. mean wind speed Inside the rotor, ~ , 1s reduced from the mean free stream veloclty,r~m, by the Interference factor which is calculated by as– sumlng a steady wind. Although they propagate at the reduced mean wind speed, Vr, the turbulence components transverse to the mean wind dlrectlon are assumed undiminished as they pass through the rotor The Instantaneous wind speed at a given blade element 1s determined in the following manner. ,– ––L .-L L. .._ ---.2 -. For each Streamtube , tne wlnas wnlcn nave IJM>SCU through the plane just upwind of the rotor (see The length Fig. i’) in the recent past are stored. of time, At, requ]red to traverse the distance from the upwind plane to the current posltlon of the blade element ]s calculated by, ...I. --W,, c, c a= 0< R–rslnO b=O O<e<n II 2r sln .5 (9) VuPWIND PLANE t a v e < Zn T< 8 < 2n The wind speed that passed through the upwind plane at a time, At, earner ]s then determined For the downstream passage of the blade through the streamtube, the upwind wind speed IS reduced by twice the mean upwind streamtube interference factor An Instantaneous Interference factor and angle of attack are calculated and the Instan– taneous forces on the blade are determined. I R i b BLADE ELEMENT FLIGHT PATH Ik STREAMTUSE PLAN VIEW Results and Dlscusslon If a steady wind is assumed, the calculated forces on the blade are Identical for each rotation. Plots of the normal and tangential dimensionless forces near the turbine equator In a steady wind are shown in Flg 2. The force components are defined ]n a reference frame fixed to the VAWT blade. The normal force changes direction as the blade passes through upwind and downwind orlen– tatlons. The tangential force 1s almost always posltlve, only going to zero, or sll,ghtly nega– tlve, when the blade chord 1s al)gned with the wind The steady wind loads cons]st entirely of frequencies which are Integer multlples of the turbine rotating frequency (per rev frequencies). Table 1 contains the sine and cosine coefficients of a Fourier series representation of these steady wind loads for a 50 ft (15m) dlmneter turbine in a .?1mph wind. VERTICAL AXIS TABLE / 1 A NORMAL STREAMTUBE TANGENTIAL PER REV FREQ Cos SIN 1 2 3 4 5 –2.21 1.37 216 –.036 – 073 18.8 2.15 1.32 .302 148 / Clr R (h v DOWNSTREAM Cos SIN –.001 – .926 –1 .58 –.043 –.192 –.008 – .342 .261 – .063 .014 ‘RDTOR EOUATOR Table VIEW Fig. ‘7 The path of a typical streemtube with The mean respect to the VAWT rotor. freestream wind speed, ~m, 1S reduced tO ~r by the first blade passage and reduced again by the second blade passage. 1 Cosine and sine coefficients of a Fourier series representation of the VAWT loads calculated assuming a steady wind. (~= 21 mph). Fig. 6 shows plots of the normal and tangential components of the blade forces. These are the force components at the turbine equator calculated using a one dimensional turbulent wind simulation. The basic form of the steady loads 1s retained, but a stochastic component caused by the turbu– Ience has been added to the underlying deter– However, due to the nonlinear ministlc part. relationship between the incident wind and blade loads, this 1s not simply a superposition of the loads due to the steady wind and the loads due to the turbulent component of the wind. 7 Standard random data analysls can not be applled to this data until the determ]nlstlc part has been removed S]nce the determ]nlstlc part can be represented by a sum of frequency components at the per rev frequencies, It can be removed by The Buys-Bal ~~t fllterlng those frequencies filter 1s perfectly su]ted to this purpose Fig. 9 contains psd’s of the normal and tangential forces near the equator of a VAWT blade after the per rev frequency content has been extracted with The sharp downward spikes the Buys–Ballot filter. In the spectra of Fig. 9 indicate where the per rev components have been removed. -1.6I 0.0 1 10 20 30 As may be seen by comparing Fig. 9 with Fig. 4, the low frequency content of the tangential forces 1s a qua.s–static reflection of the energy present The blade load psd’s have much In the turbulence. more high frequency content than would be expected There are two by examlnlng the turbulence psd. F~rst, the motion of the blade reasons for this through the alr causes the frequency content of the wind that the blade sees to shift to higher This IS much llke the situation frequencies. where the frequency content of the loads on a car travellng over a rough road depend on the speed of The more energetic low frequencies of the car the wind are transformed Into higher frequency loads Second, relatively small changes in the Incident wind at the blade can cause large changes in the Induced angle of attack, and therefore large changes In the load. This magnifies the low energy, h]gh frequency part of the turbulence spectrum 40 (a) TURBINE REVOLUTIONS 04, I I’A -OIL—-------00 20 10 30 Table 2 shows the dimensionless forces due to a turbulent wind which has been separated Into determlnlstlc and random parts by the Buys–Ballot The deterrnln]stlc part 1s written in filter The and cosine coefficients. terms of sine “ZRANDOM” IS the percentage of the total variance of the blade load In a one per rev band around each per rev frequency due to the random part. At low frequency, the random part 1s comparable to the determlnlstlc per rev’s, but lt dominates the loads at the higher frequencies. 40 (b) TURBINE Fig. 8 REVOLUTIONS Dlmenslonless Blade Loads calculated using a turbulent w,nd, (a) normal to the blade path, (b,) tangent. to the blade path TABLE 2 TANGENTIAL NORMAL _——— PER REV FREQ Cos –2.24 1 24 260 1 2 3 4 004 – 084 5 Table SIN ZWNDOM Cos 2 18.7 2 20 131 .225 .129 %RLNDOM SIN ————— ——— — 17 67 ?-i 98. 99. 029 –1 52 –.101 – 211 – 019 – 891 –.387 ,212 – 004 031 69 45 89. 90 99 The frequency content of the blade loads, calculated using a tur– bulent wind (~ = 21 mph), 1s dlv]ded Into one per rev wide bands around each per rev frequency. The determlnlstlc per rev fre– quencles are written In terms of cosine and sine coefficients. The random part 1s written as a percentage of the total variance in each frequency band. 10-’ TABLE 3 — — NORMAL PER REV FREQ 10-: Cos 1 SIN ————— –3.62 –1 90 153 1 69 237 2 3 4 5 10-2 TANGENTIAL 27.6 8.82 3 56 –1 24 .508 Cos SIN 1.32 –1 .60 –2 17 – 931 –.569 –.508 .816 –1 .25 –.391 1.03 10-; j 10-2 ( 10-’ I 1 10-’ FREQUENCY Fig. 9 Table 3 I Id (CYCLES PER REVOLUTION) mEQuENcy 10-’ I lcf 1 1 I& Id I (HCLES PER REVOLUTION Cosine and sine coefficients of a Fourier series representation of the VAWT loads calculated assuming a steady wind. (~= 34mph) the same mean wind speed as above 1s shown In Table 4. The magnitudes of the components are aPPrOxlmate Since the aerodynamic stall model IS not exact. However, the trend ]n the data ]s apparent Although the random part of the loads 1s not much different than It was for the case of low w]nds without stall (higher at some per revs and lower at others), the per rev components are noticeably lower for the turbulent wind case, For this mean wind speed, Table 5 shows the percentage difference In the rms determlnlst]c loads for the two models For high w]nds, the loads due to the steady mean wind are not the same as the mean loads due to a turbulent wind. Structural response calculations using the steady wind load estimates have had a tendency to match field data quite well at low wind speeds and deviate at F]g higher wind speeds. 10 1s a comparison of measured VAWT blade stress and estimated stress Power spectral dens it]es (psd’s) of the blade loads calculated using a turbulent wind The per rev frequency content has been removed so only the random part remains, (a) normal force psd, (b) tangential force psd. ● FFIVD Predictions + Blade 1 Oata x Blade $ J ● 2 Oata + $~ L* An Important difference between the steady and the i,urbulent wind loads 1s the change in the deter– As indi– minlstlc sine and cosine coefficients. cated by Tables I and 2, this difference is greatest for the tangential forces and at higher If the relat)onshlp between wind and frequencies blade loads were llnear, the coefficients would be The fact that they are ldentlcal In both cases. very close lnd]cates that, for this example, where stall is not occurring (~ = 21 mph), the non– IInearltles are relatively small. If a higher wind case 1s examined (~ = 34 mph), the change In the per rev components 1s more The sine and cosine coefficients for dramatic. loads calculated assuming a steady wind which causes a slgnlflcant amount of stall are shown in l’able 3. The results from a turbulent wind with I 10 20 Iifndspeed Fig. 10 30 40 (mph) Comparison of measured VAWT stress response and stresses calculated using The analysls code is steady wind loads. “FFEVI).” Data were collected from both blades of the turbine. 9 response using stea~y wind Input as reported by The overpredictlon of the Lobltz and Sullivan stress response in high winds IS consistent with the overpredictlon of blade forces which results from assuming a steady wind. The coherence is generally low at high frequency A high coherence and during changes in the phase is generaly found near per rev frequencies However, high coherence tend to repeat at fre– quency Intervals somewhat wider than one per rev. The cause of this effect IS not clear, e \\hough 1 t has been found Independently by Anderson , using a frequency domain approach. The correlations and phase relationships between the loads at different points on the rotor are The complex csd can be contained In the csd’s. expressed in terms of its magnitude and phase. The level of correlation ]s estimated using Eq. 5 by calculating the coherence function from the csd The phase and coherence between ]den– magnitude. tlcal points on opposite blades of a two bladed rotor are shown in Fig. 11. Fig. 12 shows the phase and coherence between loads on the same blade but separated by a distance of one fifth of the rotor height. Both Figs. 11 and 12 are for a 50ft (15m) dlemeter turbine using a one dlmen– s]onal wind simulation at low wind speeds where there is no stall. The coherence funct]on ls a measure of the correlation of the calculated The coherence 1s also useful as an estl– loads mate of the statistical significance of the calcu– A coherence near one Indicates a lated phase. consistent phase relat]onshlp between the loads at that frequency while a coherence near zero indl– cates that the calculated phase has no meaning. Larger diameter turbines were also examined with The only dlffer– the one d]menslonal wind model. ences were an overall reduction in the coherence and a sl]ght increase in the percent random con– The psd’ s and trlbut]on for the larger turbines. phase relationships remained essentially the same. Another factor which Increases the random contri– butlon IS the amount of atmospheric turbulence. This 1s controlled In the model by ad]ustlng the surface roughness coeff]clent (zo). Summarv and Conclusions A method for simulating a single po]nt w]nd time ser]es ]s outl]ned. This method produces a Gauss Ian random process with the specified tur– One dimensional bulence power spectral dens]ty. var]at]ons are obtained by using Taylor’s frozen turbulence hypothesis and a single point wind time TABLE 4 NORMAL Cos SIN — –3 40 –1 43 –.069 762 1 2 3 4 5 Table TANGENTIAL — PER REV FREQ 248 4 %RANOOM Cos SIN 7ZRANDOM —— 1.32 –1.36 –1.57 –.936 055 .202 – 607 – 776 – :348 .246 76 ‘?3 85 81 97 ——— 1 10 0.35 3 33 – 111 .153 50 ‘iZ 95 99. z? The frequency content of the blade loads, calculated using a tur– bulent wind (~ = 34 mph), is divided Into one per rev wide bands around each per rev frequency. The determlnlstlc per rev fre– quencies are written In terms of cosine and sine coefficients. The random part is written as a percentage of the total variance In each frequency band. TABLE 5 NORMAL PER REV FREQ Steady 1 19 7 6 38 2.52 1.48 .396 2 3 4 5 Table 10 5 Turbulent 19 3 5 99 2.36 .545 .206 TANGENTIAL Z dlff –2.0 –6.1 –6.5 –63.2 –46 .0 Steady 1.00 1.27 1.77 .714 .632 Turbulent — .994 1.05 1.24 .706 .178 Z dlff — –5.6 –17.3 –30.0 –1 1 –78 .6 The deterministic rms variation at each per rev frequency is shown for loads calculated in a 34 mph mean wind speed assuming both The percentage difference between them steady and turbulent winds. is also shown. 180.0 q 90.0 E 0.0 4 1 10-’ (a) 1 10° 1 I Id (a) mmumcy FRi3quENcy(CYCLESPER REV) (cycLEsPER REV) 10 [ 1.0 A A I 1 08 0.8 W u z w $ z 0 u 0.6 0,4 0.6 04 0.2 ~ 0.2 p, :11 :’,L FREQUENCY Id 10° 10-’ (b) (b) (cYcIXs PER REV) rREfauENcY (CYCLES pER REV) 1800 1800 90,0 w w 90.0 s n. 0.0 0,0 1 10-’ (c) 1 10° 10-’ 1 10’ (c) 10’ 10° FREQUENCY (cycLEspER REV) rREqumfcY (CYCLES PFR REV) l.o 0,80.6 0.60.4 0.4- 0.2 0.2- ~ ,,,,,,,M 10-’ ,!, L 10° ld FREQUENCY (CYCLES PER REV) iwmumcy (CYCLES pER REV) Fig. Fig. 11 Estimates of phase and coherence between the loads at the equator of the two blades of a VAWT rotor; (a) and (b) are for the normal forces, (c) and (d) are for the tangential forces. 12 Estimates of phase and coherence between the loads at two different points, separated by a distance of one fifth of the rotor height, on the same blade; (a) and (b) are for the normal forces, (c) and (d) are for the tangential forces. 11 series A three dimensional wind simulation 1s produced by creating single point s]mulatlons at a number of Input points. These t]me series are random and partially correlated In a way that matches the emplrlcal coherence function suggested by Frost 4 Lobltz. D. W., and Sulllvan, W N., “A Comparison of Flnlte Element Predictions and Experimental Data for the Forced Response of the DOE 100kW Vertical AXIS W]nd Turbine, ” SAND82-2584 (Albuquerque, NM. Sandla National Laboratories, February, 1984). The simulated wind turbulence ]s tracked as lt passes through the rotor by using a multlple streamtube representation of the flow. B1 &de loads are calculated using a momentum balance Before reducing apprOach developed by Strickland. the blade load time series to the form of psd’s and csd’s, the random and determlnlst]c parts are separated using the Buys–Ballot filter, 5 Frost , Walter, Long, D. H. , and Turner, R.E> “Englneer]ng Handbook on the Atmospheric Environment Guldellne for Use Wind Turbine Generator Development, ” NASA Technical Paper 1359, December, 1979 The load psd’s Ind]cate that the turbulent energy In the wind ]s shifted to higher frequencies In the blade loads This ]s due both to the motion of the blade through the turbulence and the magnl– fylng effect of small variations In angle of The percentage of the total variance of attack the load that 1s random lS comparable to the per rev contr)butlon at lower frequency, but lt doml– nates the loads at h]gher frequency. This random percentage Increases with turbine SIZ? and amount of turbulence The most Important result is that when steady winds are assumed, the high wind per rev loads are overpredlcied. This IS consistent with the di ference between actual measured stresses and stresses calculated using the steady w]nd mode The difference between the determlnlstlc per rev components )n steady and turbulent winds IS due to the nonl)nearlty In the relationship between angle of attack and blade load, especially in h]gh winds where stall in occurring. The actual magn]tude of the difference has not been determined since the aerodynamic stall model 1s approximate. More accurate estimates of the stochastic loads In high winds require a valldated dynemlc stall model References 12 1. Templln, R. J , “Aerodynamic Performance Theory for the NRC Vertical–Axis Wind Turbine, ” National Research Council of Canada Report LTR–LA–160, June, 1974. 2 Strickland, J H , Smith, T , and Sun, K Vortex Model of the Darrleus Turbine. An Analytical and Experimental Study,” Sandla SAND81-7017 (Albuquerque, NM National Laboratories, June, 1981). 3. Carrie, T. G., Lobltz. D W., Nerd, A. R., Watson, R. A., “F~nite Element Analysls and Modal Testing of a Rotating Wind Turbine, ” Sandla SAND82-0345 (Albuquerque, Nh. National Laboratories, October, 1982). “A In 6 Shlnozuka, M., “Simulation of Multlvarlate and Multldlmenslonal Random Processes,r’ Journal of the Acoustical Society of America, Vol. 49, No. 1 (part 2), p. 357, 1971. 7 Smallwood, D. O., “Random Vibration Testing of a Single Test Item with a Multlple Input Control System,” Proceedings of the Institute of Environmental Sc]ences, April, 1982. 8 Thresher, R. W., Honey, W E., Smith, C. E., Jafarey, N., Lin, S.–R., “Modellng the Reponse of Wind Turbines to Atmospheric Turbulence, ” U. S. Department of Energy Report DOE/ET/23144-81/Z, August, 1961. 9 W E,, Hershberg, Thresher, R. W, , Honey, E. L , Lln, S.–R., “Response of the MOD–OA Wind Turbine Rotor to Turbulent Atmospheric Winds,’< U. S. Department of Energy Report DOE/RL/10378–82/1, October, 1983. 10 Flchtl> G. H., “COvariance Statistics of Turbulence Veloclty Components for Wind Energy Conversion System Design – PNL–3499 Homogeneous , Isotropic Case,” (Paclflc Northwest Laboratories, Rlchland, WA) , September, 1983. 11 A Strickland, J. H , “The Darr eus Turbine. Performance Predlctlon Model Using Multlple Streamtubes,” SAND75-0431 (A buquerque, NM. October, 1975). Sandla National Laboratories 12 Koopmans, L. H. , The %ectral .knalVSIS of Time Series, Academic press, New York, 1974. 13 Anderson, M. B., Slr Robert McAlplne & Sons, Ltd. , London, UK, private correspondence, May, 1984. DISTRIBUTION: Canadian Standards 178 Rexdale Blvd. Rexdale, Ontario CANADA M9W 1R3 Attn: Tom Watson Aerolite, Inc. S50 Russells Mills South Attn: Road 02748 Dartmouth, MA R. K. St. Aubin Association Professor V. A. L. Chasteau School of Engineering University of Auckland Private Bag Auckland, NEW ZEALAND Alcoa Technical Center (4) Aluminum Company of America 15069 AICOa Centert PA Attn : D. K. Ai J. T. Huang J. 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