Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Survey of modelling methods for wind turbine wakes and wind farms

Survey of modelling methods for wind turbine wakes and wind farms   A. Crespo,* ETS de Ingenieros Industriales, UPM, Jose Gutierrez Abascal 2, E-28006 Madrid, Spain  J. Hernandez, ETS de Ingenieros Industriales, UNED, Ciudad Universitaria, E-28040 Madrid, Spain S. Frandsen, Risù National Laboratory, DK-4000 Roskilde, Denmark Key words: wind turbine wake; wind farm; dynamic loading; atmospheric turbulence This article provides an overview and analysis of different wake-modelling methods which may be used as prediction and design tools for both wind turbines and wind farms. We also survey the available data concerning the measurement of wind magnitudes in both single wakes and wind farms, and of loading effects on wind turbines under single- and multiple-wake conditions. The relative merits of existing wake and wind farm models and their ability to reproduce experimental results are discussed. Conclusions are provided concerning the usefulness of the different modelling approaches examined, and dif®cult issues which have not yet been satisfactorily treated and which require further research c are discussed. Copyright *1999 John Wiley & Sons, Ltd. Introduction Wind turbine wakes are an interesting topic of study, because the momentum de®cit and the increased level of turbulence created by turbines in a wind farm may cause a reduction in power output and unsteady loads on other machines. On the other hand, owing to the cost of land and civil works, wind turbines tend to be built as closely as possible to each other, and to this e€ect, Builtjes and Smit1 and Milborrow and Surman2 have provided guidelines for wind turbine spacing in a wind farm. The ®nal report of IEA, Annex IX on Wake E€ects,3 indicates that the experimental and analytical studies reported in the Annex point to signi®cant energy losses in arrays spaced at less than seven turbine diameters. Similarly, turbulence may increase in arrays, suciently to cause measurable damage due to fatigue and dynamic loads. Machines based on lift-like forces, associated with the generation of circulation, develop less intense wakes than drag-type units, which may be one reason, apart from other aerodynamic considerations, for preferring lift-type to drag-type machines.4 Nevertheless, although most modern machines are of the lift type, the wake e€ects that they produce are still important enough to be studied. In an early approach to the problem of modelling wind farms, it was assumed that when an area contained a large number of machines, the turbines acted as distributed roughness elements, and that they modi®ed the ambient atmospheric ¯ow (see reviews by Bossanyi et al.5 and Milborrow6). More recent work using this approach has been carried out by Frandsen7 and Emeis and Frandsen.8 This topic of turbines acting as distributed roughness elements will be treated in more detail in Section 2.   *Correspondence to: A. Crespo, ETS de Ingenieros Industriales, UPM, Jose Gutierrez Abascal 2, E-28006 Madrid, Spain. Contract/grant sponsor: European Union; Contract/grant number: JOU2-CT93-0350; JOU3-CT95-0089. CCC 1095±4244/99/010001±24 $17.50 Copyright # 1999 John Wiley & Sons, Ltd.  A. Crespo, J. Hernandez and S. Frandsen However, the most common approach to the problem, set in motion by the classical paper of Lissaman,9 considers each turbine wake of the farm individually and examines its interaction with and superposition on neighbouring ones. It thus calculates the detailed ¯ow ®eld and not the average distribution. Section 3 is dedicated to individual wake behaviour and Section 4 to wake superposition and the multiple-wake case that occurs in wind farms. Frandsen7 compared the results of both approaches in a particular example, and although further research along these lines was desirable, this, to the best of our knowledge, has not been carried out. Section 3 starts with a description of wake behaviour and continues with a discussion of the kinematic models, also known as explicit models, which have been used extensively because of their simplicity and low computational cost. They use self-similar velocity de®cit pro®les obtained from experimental and theoretical work on co-¯owing jets. The wake growth rate is calculated as being caused by the ambient turbulence, the turbulence created by the shear in the wake and that created by the turbine itself. The magnitude of the maximum velocity de®cit at each section is obtained from global momentum conservation, and the ground is taken into account by introducing an image machine. These methods provide acceptable results if the adjustable coecients are appropriate. Field models, also known as implicit models, calculate the ¯ow magnitudes at every point of the ¯ow ®eld. Field models require a substantially larger computer capacity than kinematic models, although their requirements are well within the capabilities of modern computers, not only in the case of single wakes but also for multiple wakes occurring in a wind farm, if appropriate simplifying assumptions are made (see below). The ®eld models give an acceptable representation of the ¯ow ®eld and a good insight into the processes governing wake development. Both kinematic and ®eld models use as starting or boundary conditions those at the end of the expansion region and the beginning of the near-wake region. If a uniform velocity de®cit is assumed at the initial cross-section, this de®cit can be estimated from the overall thrust on the machine; other possibilities are contemplated in Section 3. Kinematic and ®eld models do not directly take into account the bodily movement of the wake with the large atmospheric eddies, known as meandering. This is also examined in Section 3. An important issue in wind farm modelling is the interaction of several wakes and the way in which the velocity de®cits and turbulence created by each machine accumulate at locations where several wakes meet. Di€erent types of assumption are made regarding superposition rules, the most straightforward approach consisting of adding the velocity de®cits and turbulence kinetic energy. This and other alternative methods will be reviewed. The main problem is that any approach based on single-wake calculations will fail, because the ambient basic ¯ow in which the wake di€uses is to some extent also a€ected by the wakes of the upstream machines and will also be evolving. A more correct approach would be to solve the ¯ow equations for the whole wind park. At ®rst sight and from a practical point of view this would not seem feasible for wind farms with a large number of wind machines; however, if some simplifying assumptions are made, namely a parabolic approximation, the ®eld model for a single wake can be extended to the multiple-wake case and be practically solved with reasonable computer times, giving an acceptable degree of agreement with experimental measurements. This issue will be examined in more detail in Section 4. The problem of how to take into account terrain e€ects in wind farms placed in moderately irregular topography, not contemplated in Section 3 for single wakes, is also treated in Section 4. Although this article concentrates more on modelling aspects, the empirical or quasi-analytical expressions that may be used to estimate the downstream evolution of relevant wake parameters, such as velocity de®cit and turbulence intensity, are brie¯y reviewed in Section 5. Section 6 deals with the in¯uence of wake e€ects on wind turbine loading. We review available load measurements in wind turbines under di€erent types of wake conditions. Some existing models to estimate wake-loading e€ects are also described. In Section 7 we present what we think is the state of the art regarding wind turbine wakes and wind farm modelling, accompanied by the corresponding conclusions and recommendations. Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods 2. Turbines Acting as Distributed Roughness Elements The models of Templin,10 Newman,11 Crafoord12 and Moore,13 reviewed by Bossanyi et al.5 apply to in®nite clusters. They assume a logarithmic wind pro®le for the unperturbed wind, which includes ground roughness as a parameter. The presence of the turbines increases the value of the roughness, which is then calculated. From the modi®ed wind pro®le the wind velocity incident on each machine can be obtained and then the power produced can be calculated. Whereas the previous models assumed a single logarithmic pro®le, Emeis and Frandsen8 assumed that below hub height there is a logarithmic pro®le with the real ground roughness, and above hub height another pro®le with a roughness related to the drag of the machine; the pro®les match each other at hub height. Frandsen7 also applied a logarithmic pro®le above hub height and assumes the validity of a simpli®ed form of the geostrophic drag law, obtained from the Rossby number similarity theory. Bossanyi et al.5 explained how the previous models can be extended to the case of ®nite clusters. Schmid14 used results obtained by Taylor15 for a step change in roughness to calculate the friction velocity at each row of turbines. Crafoord,12 Moore13 and Musgrove16 considered a mixing layer of air above the ground and perform either a momentum or energy balance in this layer. It was assumed that sucient mixing occurs so that, by the time the next row of turbines is encountered, the velocity de®cit is averaged out across the whole mixing layer. The di€erence between the momentum (or energy) ¯uxes of two consecutive rows is due to the drag (or power extraction) of the turbine, the amount lost to the ground and the amount entrained from greater heights through mixing processes. The di€erence between the last two quantities was termed the replenishment rate by Bossanyi et al.,5 who discussed several hypotheses concerning the way to estimate the relevant parameters, in particular the mixing layer thickness and the replenishment rate, and compared the corresponding results. Although these models are not much used, they could be of interest to predict overall e€ects of large wind farms on wind characteristics. 3. Individual Wakes This section is dedicated to reviewing the models for individual wakes. It is of interest to present ®rst a general description, based on physical grounds, of the behaviour of the wake characteristics to be simulated with the models. In the second and third subsections the more simple kinematic models and the ®eld models respectively are described. The procedures and methods used by the di€erent models are presented and the results discussed. In the third subsection the results obtained with both kinematic and ®eld models are examined simultaneously and compared with experiments, in order to discuss the merits of both types of models. Finally, the meandering e€ect, which is not usually taken into account explicitly in the models, is examined in the last subsection. Description of the Wake Behaviour As the air approaches the wind turbine, its velocity decreases and the pressure increases. As it crosses the rotor, there is a sudden decrease in pressure. In the region immediately downstream of the rotor there are non-uniform de®cits of pressure and axial velocity, which are associated with the axial thrust, as well as an azimuthal component of velocity, which, in turn, is related to the torque on the machine. Vortex sheets, associated with the variation in circulation along the blades, are shed from their trailing edge and roll up in a short downstream distance, forming tip vortices that describe helical trajectories. When the inclination angle of the helix is small enough, the tip vortex can be interpreted as a cylindrical shear layer which separates the slow moving ¯uid in the wake from that on the outside. The velocity de®cit can be considered as induced by the vortices. The di€erence in pressure between the ¯uid behind the rotor and that on the outside is supported by the centrifugal force due to the curvature of the streamlines. As we Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen Figure 1. Schematic representation of a wind turbine wake move downstream, the cylindrical shear layer expands, the pressure increases and the velocity inside the wake decreases until ambient pressure is reached (Figure 1). According to the simple actuator disk theory, which assumes that ¯ow is ideal and that the shear layer is in®nitely thin, the velocity de®cit at the disk itself is half that in the expanded wake. Because of turbulent di€usion, the thickness of the shear layer increases with downstream distance, but if the length of this expansion region is suciently small, it may not be a bad approximation to consider that the thickness of the shear layer is small compared with its diameter. The length of this expansion region is about one turbine diameter. As we proceed further downstream, turbulent di€usion of momentum becomes the dominant mechanism. Turbulence production is more important in the shear layer, where the velocity gradients are larger. A well-de®ned ring-shaped domain where there is a high turbulence intensity is observed in this cylindrical shear layer, both experimentally17±23 and numerically.24±26 There are also signi®cant velocity gradients both inside the wake, since the velocity de®cits created by the turbine are not uniform, and in the atmospheric ¯ow, where the wind velocity changes with distance to the ground. Most of the turbulence that makes the wake di€use is, at this stage, probably created by the shear in the wake, mainly in the shear layer. However, the shear in the external atmospheric ¯ow also plays an important role, at least in the redistribution of the generated turbulence. As will be shown later, the turbulence of the ambient ¯ow is responsible for a non-uniform distribution of turbulence in the shear layer, where a maximum is observed in the upper part25,27 (Figure 2). Turbulent di€usion makes the shear layer thickness increase with downstream distance, and at a certain distance downstream (about two to ®ve diameters) the shear layer reaches the wake axis. This marks the end of the near-wake region. After the near-wake region there is a transition region leading to the far-wake region, where the wake is completely developed and, in the hypothetical absence of ambient shear ¯ow, it may be assumed that the perturbation pro®les of both velocity de®cit and turbulence intensity are axisymmetric and have selfsimilar distributions in the cross-sections of the wake. The only overall properties of the turbine that appear as parameters in these pro®les are the thrust on the turbine and the total turbulence kinetic energy produced by the rotor itself. This property of self-similarity of the velocity pro®les is the basis of the kinematic models describing wind turbine wakes. However, the presence of the ground and the shear of the ambient ¯ow invalidate the assumption of axial symmetry and, to some extent, the hypothesis of selfsimilarity. It has been observed both numerically and experimentally that the maximum turbulence intensity in the far wake is located above the turbine axis,21,28 and the point of maximum velocity de®cit is usually below the turbine axis.29±32 The maximum of turbulence intensity is about one turbine radius above the axis, and this is probably related to what happens in the near wake. Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods Figure 2. Vertical distribution of turbulence intensity normalized with its ambient value. Comparison of numerical 27 and experimental 21 results. 1D downstream, D ˆ 40 m, H ˆ 45 m. Notice peaks located at H ‡ D/2 and H À D/2 Kinematic Models for Single Wakes As already mentioned, kinematic models are based on self-similar velocity de®cit pro®les obtained from experimental and theoretical work on co-¯owing jets. The wake description does not consider the expansion region and gives di€erent types of pro®les for the near-wake, transition and far-wake regions. For the far wake these pro®les are self-similar and in the near wake there is usually a central core of constant velocity and diminishing radius; when this radius becomes zero, the near wake ends. Lissaman9 and Voutsinas et al.33 used the velocity pro®les proposed by Abramovich.34 Vermeulen35 used a Gaussian type of pro®le quite similar to that of Abramovich.34 Katic et al.36 simpli®ed the problem further and assumed a top-hat pro®le everywhere. More recently, Kiranoudis and Maroulis,37 based on a kinematic model similar to the previous ones, developed a `short-cut model of wind park eciency', giving simple analytical expressions of the eciency as functions of the farm and turbine characteristics. In these models the initial velocity de®cit is usually obtained from the thrust coecient of the machine. Voutsinas et al.38 related it to the power given by the machine, the advantage of which is that the power curve is usually more available than the thrust curve. In all the studies published to date, the reference value of the velocity de®cit at each section has been obtained from global momentum conservation, except in the case of Voutsinas et al.,38 who claimed that they obtained it from mass conservation, based on the fact that agreement with the results of Taylor39 was better. However, it is not clear what they mean by this and, in particular, how they take into account the mass entrainment through the lateral surface of their control volume. As a matter of fact, when applying the classical equation of momentum conservation (see e.g. Reference 35), it is implicitly assumed that mass is also conserved. Lissaman9 regarded the wake growth as being caused by the sum of the ambient turbulence and the turbulence created by the shear in the wake. Vermeulen35 added another term, the turbulence created by the turbine itself, although, in a later work based on the experimental results of Taylor,39 Voutsinas et al.33 Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen considered that this e€ect was negligible. Katic et al.36 assumed that the wake radius increases linearly with downstream distance; the proportionality constant must be adjusted by comparison with experiments. The ground e€ect is simulated by imaging techniques. Lissaman9 included a symmetrical turbine and added the velocity de®cits of both the real and image turbines, so that drag conservation is satis®ed. He pointed out that the ground surface can be treated exactly like the vertical-dividing plane between two adjacent identical rotors abreast. However, when there is ground, the total drag is not actually conserved because of friction; the three-dimensional models show that there is a slight decrease in total de®cit as the downstream distance increases. According to the image procedure, the velocity de®cit will be double that due to a single wake at the ground, whereas in reality any perturbation would be damped to zero. Crespo  et al.,29 Crespo and Hernandez40 and Kambezidis et al.41 use an antisymmetric wake so that velocity de®cits are subtracted and give zero perturbation at the ground; if it is considered that in reality the ambient velocity is not uniform and that it falls to zero at the ground, the perturbed velocity calculated will also be zero at the ground. However, although this alternative procedure eliminates the previously mentioned inconsistency that occurs near the ground, it is not clear that it will give a more valid result in the rest of the ¯ow ®eld, where the ground e€ect is not so dominant. Another procedure followed by Voutsinas et al.38 consists of superimposing the squares of velocity de®cits and taking into account the spatial variability of the incident velocity to estimate the location of the image turbine, although this method does not seem to be capable of overcoming the diculty. Indeed, the ground e€ect seems to be an intrinsic diculty of all the kinematic models that assume axial symmetry, and there is no satisfactory way in which they can deal with it. The ground e€ect can only be treated properly with 3D models. Recently, Larsen et al.42 proposed a simple analytical model, based on the classical wake theory as presented by Schlichting.43 The ¯ow is supposed to be axisymmetric, and a single self-similar velocity pro®le is assumed for the whole wake. The velocity de®cit decays with downstream distance as x À2/3, the turbulence intensity decays as x À1/3, as in References 25 and 27, and the wake width increases as x1/3. Compared with the previous kinematic models, this model only considers the far-wake region and the turbulence created by the shear. They compared the turbulence intensity and length scale calculated from their model with predictions based on empirical relations, and obtained relative di€erences lower than 5% in all cases where the downstream distance x is larger than two diameters. In spite of all the previous diculties, in many cases the kinematic models provide results that are in good agreement with the experimental measurements if appropriate values are chosen for the parameters appearing in them.17,44,45 Field Models for Single Wakes Sforza et al.46,47 described the wake using only the linearized momentum equation in the main ¯ow direction, with constant advective velocity, a constant eddy di€usivity and a parabolic approximation. For two-dimensional con®gurations they obtained analytical solutions with acceptable wake shapes. In the three-dimensional case they integrated the equation numerically using an alternating direction implicit (ADI) method. They made small-scale experiments and compared measured and calculated values for the velocity de®cit and wake growth as functions of downstream distance, obtaining agreement within 10% error, except for cases of high thrust loading, for which the error could reach 20%. This agreement was reasonably close considering the simplicity of the model. Besides, the tendencies were well predicted in all cases. A numerical model based on solving the ¯ow equations for wakes in neutrally strati®ed atmospheric boundary layers was proposed by Taylor,48 who considered an eddy viscosity gradient closure scheme. The wake e€ect was assumed to be small enough for the equations to be linearized around a basic ¯ow, and a boundary layer approximation was used. The model was two-dimensional and presented results integrated across turbine rows. Coriolis forces were retained and the pressure gradients were given by the geostrophic wind. However, this assumption cannot be justi®ed, because the length scale of the wake is not suciently large for the Coriolis forces to play a dominant role; indeed, they can be neglected and the Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods pressure ®eld will be that resulting from the momentum conservation in the wake. If a parabolic approximation is made, pressure variations across the wake can be neglected in the momentum equation for the main ¯ow direction, but not for the momentum components in the transverse direction, particularly when there is neither axial nor two-dimensional symmetry. Taylor48 compared his results with those of kinematic models and other models based on the assumption that the turbines act as distributed roughness, and with experimental results of Builtjes.49 Although there was reasonable agreement, Taylor48 admitted that the linear superposition of the e€ects of several rows of turbines may lead to low or even negative power outputs for the backmost rows. Liu et al.50 proposed another, three-dimensional, model which includes atmospheric stability e€ects. However, these authors neglected the di€usion due to turbulence originated in the wind turbine and the di€usion caused by the shear in the wake, and considered the turbulent viscosity and the di€usion coecients to be those of the unperturbed ¯ow. Along with Taylor,48 they retained Coriolis forces and assumed that the pressure gradients were given by the geostrophic wind. Ainslie51±53 developed a parabolic eddy viscosity model (EVMOD) which assumes axisymmetric wake ¯ow. Pressure variations are uncoupled in the analysis, and only the continuity and the axial momentum equations have to be solved. Consequently, the model is incapable of dealing with ground e€ects or with variations in ambient ¯ow conditions with height. The turbulent shear stresses are described using an eddy viscosity closure scheme in which the eddy viscosity is represented by a simple analytical form based on Prandtl's free shear layer model, but which also includes a contribution from ambient turbulence. This eddy viscosity is an average value over a cross-section, and variations in turbulent properties across the wake cannot be estimated from the model. At small downstream distances the eddy viscosity is modi®ed by an empirical ®lter function to account for the lack of equilibrium between the mean velocity ®eld and the developing turbulence ®eld. Several constants appear in the problem. They are adjusted by comparison with particular experiments, although their validity in more general situations is not clear. The model is fairly simple and gives reasonable results when compared with wind tunnel experiments. For large-scale experiments the results are corrected by taking into account meandering e€ects (which will be described at the end of this section). Albers et al.54 found that there was greater agreement between Ainslie's model and experimental results if the incident logarithmic pro®le was superimposed on the calculated axisymmetric wake. Luken and Vermeulen55 and Luken et al.32 used the experimental results from the TNO wind tunnel to validate Vermeulen's35 kinematic model (MILLY) and Ainslie's51 model (EVMOD) (see Figure 3). Although they found an acceptable degree of agreement, some aspects such as the downshift of the wake centreline, which appears in Figure 4, were not well predicted. To reproduce such e€ects, models which retain three-dimensional e€ects are needed. These will be described next. Crespo et al.29 developed the UPMWAKE model in which the wind turbine is supposed to be immersed in a non-uniform basic ¯ow corresponding to the surface layer of the atmospheric boundary layer; further  developments of the model are given by Crespo and Hernandez.56 The properties of the non-uniform incident ¯ow over the wind turbine are modelled by taking into account atmospheric stability, given by the Monin±Obukhov length, and the surface roughness. It is supposed that this basic ¯ow, described by analytical expressions obtained from theoretical considerations and experimental results given by Panofsky and Dutton,57 is perturbed by the wind turbine. The equations describing the ¯ow are the conservation equations of mass, momentum, energy, turbulence kinetic energy and dissipation rate of turbulence kinetic energy. The modelling of the turbulent transport terms is based on the k±e method for the closure of the turbulent ¯ow equations. This set of equations has been solved numerically using the SIMPLE algorithm proposed by Patankar and Spalding.58 Finite di€erence methods were used in the discretization of the equations. A parabolic approximation was made and the equations were solved numerically by using an ADI method. The developed wake model is three-dimensional, and pressure variations in the cross-section have to be retained in order to calculate transverse velocities. A simpli®ed version of UPMWAKE, which assumes that all the convection is due to the unperturbed ambient ¯ow, was presented by Crespo et al.29 This seemed a very attractive idea, because it was possible to retain the three-dimensional character of the problem and reduce the system of partial di€erential Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen Figure 3. Decay of maximum dimensionless velocity de®cit along wake, made dimensionless with wind velocity at hub height, as a function of downstream distance divided by turbine diameter. Comparison of wind tunnel measurements and results of di€erent wake models Figure 4. Vertical distribution of maximum dimensionless velocity de®cit along wake as a function of vertical distance divided by turbine diameter for several downstream sections. Comparison of wind tunnel measurements and results of wake models Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods equations from seven to three. However, this approximation can only be justi®ed very far downstream where the wake perturbation is small, and although in some cases the results obtained were in quite good agreement with the full model and with experiments, in others, particularly in the near wake, they were wrong. Most of the UPMWAKE calculations that have been published correspond to the seven-equation code.  Crespo and Hernandez40,56 and Crespo et al.28,30 compared UPMWAKE results with the results of wind tunnel experiments obtained by Luken et al.55 (see Figures 3 and 4) and those of ®eld experiments using full-scale machines.59 The wake model has also been validated numerically by using the general-purpose CFD PHOENICS code,24 and the corresponding results agree well with those of UPMWAKE. The code can predict e€ects such as the downward tilt of the wake centreline, as can be seen in Figure 4, the upward displacement of the point of maximum added turbulence kinetic energy, or the di€erent vertical and horizontal growths of the wake width, which have been con®rmed experimentally by other authors.3 Based on their experimental results, Helmis et al.60 attribute the downshift of the maximum velocity de®cit to the tower shadow rather than to the asymmetry due to the terrain. Some discrepancies between the UPMWAKE results and those of the experiments of Taylor et al.59 were found in the initial wake region, where the predicted velocity de®cits were smaller than the measured ones.  More recently, and based on the results of the code, Crespo and Hernandez25,27 have developed correlations to calculate the turbulence intensity in both the near and far wakes, and compared them with a great number of experiments (many of them compiled by Quarton61), both wind tunnel17,35,62±67 and ®eld20,68,69 experiments. The comparison was acceptable and demonstrated that UPMWAKE may be a useful tool for estimating turbulence characteristics. For the far wake, Frandsen et al.70 proposed correlations that give similar predictions for the decay in turbulence intensity with growing downstream  distance. Crespo and Hernandez25,27 also proposed a simple method for obtaining the turbulence spectra in the wake from the values of k and e obtained from UPMWAKE, and compared their results with the experiments of Hùjstrup,21 obtaining good agreement in some cases. The results of this procedure for calculating the spectra are compared with measurements made in Vindeby wind farm by Frandsen et al.70 and Crespo et al.;71 some of the results obtained for the turbulent length scale that are needed to estimate the spectrum seem to be smaller than those measured. Possible reasons for this discrepancy are, on the one hand, that UPMWAKE does not take into account the small-scale (large-frequency) turbulence originated by the boundary layers of the blades of the wind turbines, and, on the other hand, that the wind turbine is capable of responding to low-frequency ¯uctuations of wind speed and extracts energy from the wind in the low-frequency (large-scale) range,21 although this tendency may be reversed for wind speeds higher than that corresponding to the maximum power coecient, as measured by Papadopoulos et al.72 Larsen et al.42 proposed another procedure, in which the individual contributions of the di€erent scales to the spectrum are calculated and added; the details of the method are not included in the paper and will be published elsewhere. One aspect of modelling which has been insuciently treated is wake di€usion in non-neutral atmospheres. Crespo et al.29 showed that, as was to be expected, di€usion is inhibited in stable atmospheres and enhanced in unstable ones, although no experimental results were available for comparison. More recently, some interesting experimental results have been presented by Magnusson73 and Magnusson and Smedman,74 who found that for unstable strati®cation, with Richardson number (Ri) values smaller than À0.05, the velocity de®cit is independent of stability, while it increases linearly with Ri in the interval À0.05 5 Ri 5 0.05. They also found that, unlike in the experimental results of Luken et al.,32 there is an upshift of the point of maximum velocity de®cit. The calculations of Crespo et al.29 showed that in most cases there is a downshift, except for stable atmospheres, large surface roughness and a high level of turbulence kinetic energy created by the turbine. It would be of interest to compare the results of numerical models that can simulate atmospheric stability with these experiments. Smith and Taylor23 and, in more detail, Taylor26 presented a non-symmetric two-equation model that is in many ways similar to the three-equation model of Crespo et al.29. They neglected transverse velocities and just solve the momentum equation in the axial direction. To model the turbulent viscosity, they use a Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen k±L method where the turbulent length scale L is related to the width of the wake, obtained by ®tting a Gaussian pro®le to the calculated pro®le. The value of the dissipation rate of the turbulent kinetic energy, e, was obtained from an algebraic combination of k and L, and consequently a partial di€erential equation for e was not needed. The same type of problem previously mentioned for the three-equation version of UPMWAKE also appears in this case. The results obtained in a comparison with their windtunnel experimental results are very good, although a comparison with full-scale Nibe measurements showed that the model overestimates the values of the velocity de®cit. They attributed this discrepancy to meandering and obtained better agreement when they corrected for this e€ect using the method proposed by Ainslie,53 which will be discussed later. A starting Gaussian velocity de®cit pro®le is imposed, which would correspond to the end of the near wake (Figure 1). The results of the calculations show a clearly de®ned annular peak of turbulence intensity, as shown in Figure 2, although this peak should really be located in the near wake, upstream of the starting region (in general, as will be pointed out again later, application of the initial conditions of the wake is an uncertain aspect of all wake models). Although there is considerable scatter of the experimental results around the line of calculated values, the peak values of turbulence intensity are well de®ned and, in a cross-wind pro®le, are predicted within an error of 3%. In a vertical plane the turbulence intensity distribution is similar to that appearing in Figure 2, and the upper and lower peaks are predicted within errors lower than 0.1% and 5% respectively. Based on the model developed by Ainslie,53 the company Garrad and Hassan has developed the code EVFARM, described by Tindal et al.75 and Adams and Quarton.76 The code incorporates two alternative semiempirical models to calculate wake turbulence. One of them, described by Hassan,77 gives uniform turbulence in the wake, whereas the other, described by Luken et al.,32 takes into account radial variations in turbulence intensity. Adams and Quarton76 use both EVFARM and UPMWAKE codes in combination with machine load predictive tools to provide a method for fatigue load prediction. As part of this study, a comprehensive validation of both codes is made using the wind tunnel measurements of Hassan;77 good agreement is found for the velocity de®cit, which is underestimated by 2% and 3% by UPMWAKE and EVFARM respectively; the turbulence intensity is overestimated by 11% by EVFARM and underestimated by 17% by UPMWAKE. They also noted that there was better agreement with experiments if a downstream displacement of the origin is considered to account for development of the expansion region. The physical reason for this displacement is not clear, because even though the expansion region is located downstream of the rotor, the shear layer starts immediately behind the rotor. In the initial region of the wake some important discrepancies were also observed between the results of UPMWAKE and the Nibe measurements published by Taylor et al.59 By eliminating the boundary layer  approximation used in UPMWAKE, Crespo et al.28 and Crespo and Hernandez78 proposed an elliptic model to deal simultaneously with the axial pressure gradients and di€usion e€ects by retaining both the axial and transverse di€usion terms. Their model therefore describes both the evolution of the expansion region and the di€usion processes. No fundamental di€erences between the results of the elliptic and parabolic models were found, and displacement of the origin was apparently not necessary. Other elliptic models have also been proposed by Cleijne et al.79 and Ansorge et al.80 Any improvement in the agreement with experiments when comparing elliptic and parabolic codes is only slight and does not seem to justify the additional computational e€ort needed. Another reason for the discrepancies observed between models and experiments in the near wake may be the uncertainty involved in the initial velocity de®cit, assumed to be either uniform or of a prescribed shape (Gaussian in Reference 26) and obtained from the thrust coecient. More recently, Magnusson,81 using blade element theory and experimental results, investigated the in¯uence of the non-uniform incident wind and the yaw on the near-wake characteristics. Zervos et al.82 relate initial wake development to the aerodynamics of the rotor, using a vortex particle method governed by the vorticity transport equations and the Biot±Savart law. Although these authors do not need initial data to start calculating the wake, the validity of the solution is limited to the short initial expansion region where di€usion e€ects can be neglected. In general, non-uniform values of axial azimuthal velocity components at the end of the expansion region can be obtained using a classical blade element model, a strip model83 or even vortex Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods particle and lifting line methods, such as those proposed by Zervos et al.82 or Cleijne et al.79 The blade element and strip methods include the e€ect of drag on each blade section, and this can be used to estimate the dissipated power and the turbulence kinetic energy produced, whereas vortex particle methods, which do not include aerodynamic losses, do not have this possibility. Cleijne et al.79 attempted to combine ®eld and vortex particle models in such a way that boundary conditions for the ®eld model are obtained from the vortex particle method; although there is better agreement with the experimental results in some cases, the complications and additional computing costs involved do not appear to be justi®ed. An alternative approach that requires less computing capacity is the multiparametric wake model of Voutsinas et al.,84,85 which was further developed by Cleijne et al.79 and Voutsinas et al.86 This model divides the wake into the rotor region, the near-wake region and the far-wake region, and applies a vortex particle method in the rotor region, a ®eld model in the near-wake region and, in the far-wake region, explicit self-similar expressions similar to those used in the kinematic models. Di€erent assumptions are made to match the di€erent regions. The method was partially successful in simulating the experimental results of the Nibe turbines given in Reference 39. Later, Magnusson et al.87,88 applied the model to reproduce the experimental results of Alsvik wind farm. The agreement between experimental and model results for the velocity de®cit is qualitatively reasonable, with relative errors of less than 25% and a scatter of experimental results of similar order. However, the comparison of turbulence characteristics (turbulence kinetic energy and Reynolds stress) is poorer, with relative errors that can even reach 200%, although tendencies are well predicted. All the previous models solve the Reynolds-averaged turbulence ¯ow equations and use a closure scheme, based on zero-, one- or two-equation models, to calculate the turbulence transport terms. An eddy viscosity is used in all cases which implicitly assumes an isotropic turbulence ®eld. Transport equations for the Reynolds stresses have only been used occasionally to calculate this type of wake. Ansorge et al.80 used a Reynolds stress turbulence model based on the commercial code PHOENICS and obtained reasonable results, although the computational e€ort may still be considered too great from an engineering point of view. Neither is it clear whether there was improved agreement with experiments. In general, the ®eld models give an acceptable representation of the ¯ow ®eld and a better insight into the processes governing wake development than the kinematic models. Meandering of the Wake In general, ®eld models show better agreement with wind tunnel experiments than with ®eld experiments, one reason being the meandering of the wake. When there is meandering, turbines can be signi®cantly misaligned most of the time. Whale et al.89 found signi®cant di€erences between the experimental results obtained in a wind tunnel and those of large-scale tests, although they point to other factors being partially responsible besides wake meandering, such as the scale e€ect and the in¯uence of terrain. The individual wakes calculated by both kinematic and ®eld models do not directly take into account eddies that are large in comparison with the size of the wake and which can move it bodily, a phenomenon known in studies of atmospheric dispersion as meandering. Nor is this e€ect usually included in wind tunnel tests. The maximum velocity de®cit will be smaller than that predicted by the theoretical models or wind tunnel tests, and, in addition, velocity ¯uctuations may appear that can be interpreted as an additional contribution to the turbulence kinetic energy. Baker and Walker,68 Ainslie52 and Taylor26 took meandering into account by assuming that the large eddies increase in size linearly with downstream distance x and in proportion to the standard deviation of the wind direction, sy . However, Hogstrom È È et al.20 argue that this is not the correct approach, because sy is caused by eddies of all sizes, including those that are smaller than the wake diameter, and take for their analysis a value of 0.053x for the size of large eddies, based on the results of some oil-fog experiments. Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen 4. Wind Farm Models This section is dedicated to study wake e€ects in wind farms, where there are usually many machines located in irregular terrain. The ®rst subsection is dedicated to study the interaction of several wakes, and the next one to the topographic e€ects. In this last subsection, methods to take into account simultaneously wake and topographic e€ects are also reviewed. A brief review is also made of o€shore wind farms, which are nowadays of great interest, although this subject is so broad that it will have to be treated in a separate review. Interaction of Several Wakes A wind farm consists of many wind turbines whose wakes can interact and whose turbines may be a€ected by the wakes of several machines located upstream. Wind farm codes usually rely on the results of singlewake calculations and make superposition assumptions to take into account the combined e€ect of di€erent wakes. The linear superposition of the perturbations created by wakes of di€erent machines in a wind farm model was ®rst used by Lissaman9 in a classical paper, although this assumption fails for large perturbations as it overestimates velocity de®cits and could lead to the absurd result of negative velocities when many wakes superimpose. Instead, Katic et al.36 assumed linear superposition of the squares of the velocity de®cits. In this case the cumulative e€ect, when there are many wakes, will be smaller than that calculated for linear superposition, and, in general, this assumption provides better agreement with experimental results than the linear superposition. The corresponding code, named PARK, was applied by Beyer et al.90 for the optimization of wind farm con®gurations using genetic algorithms. Voutsinas et al.38,86 formulated an explicit energy equation, also used by Kiranoudis and Maroulis,37 by assuming the total energy loss at each point of the ¯ow ®eld due to the presence of di€erent machines to be equal to the sum of the individual energy losses due to each machine. In this way they obtained the velocity ®eld and then calculated the incident velocity on each machine by taking the average over the turbine disk. To evaluate the individual energy losses of each wake, they considered the di€erence between the wake velocity and the in¯ow velocity on the machine that creates the wake, whereas Katic et al.36 considered the di€erence between ambient and wake velocity. For small velocity de®cits both methods should give similar results. Based on the idea that wake turbulence should increase di€usion when the wakes superpose, Beyer et al.91 proposed a modi®cation of the parameters describing single-wake development. However, although they obtained acceptable agreement with experimental results in the Hamswehrum wind farm, no systematic procedure is provided of how to apply the method to other con®gurations. Smith and Taylor23 found, for a particular experimental con®guration of two machines in a row, that the wake velocity of the downstream machine recovers more rapidly than the one upstream, so that, at the same relative position, the velocity de®cit is smaller in the downstream machine wake. This result contradicts the qualitative behaviour predicted by the two previous superposition assumptions and may be explained by the turbulence levels and shear stress pro®les generated by the upstream machine, which enhance momentum di€usion, leading to a faster recovery in the downstream machine. Stefanatos et al.92 also showed experimentally that the linear superposition of wakes provides a poor approximation of the ¯ow. By making a number of crude assumptions concerning the momentum transfer within the downstream wake that is imbedded in the upstream wake, Smith and Taylor23 were able to formulate a semiempirical superposition law that works quite well. However, it is cumbersome and can only be applied for the interaction of the wakes of two turbines in a row. For small velocity de®cits the method reduces to the linear superposition assumption, but it is not clear what the limit is for the quadratic superposition assumption to be recovered. Voutsinas et al.38,86 claimed that their explicit energy equation gives similar results to this method but do not give any physical explanation. When there are many turbines in a line, it has been observed experimentally93 that while the ®rst turbine produces full power, there is a signi®cant decrease in power in the second turbine, with practically no further loss in successive machines. Based on these observations and on the results of the calculations of Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods Figure 5. Velocity distribution in wake region of seven turbines forming a row. Comparison of measurements and results of several wind farm codes. Measurements and con®guration correspond to the Zeebrugge wind farm.93,94 Uref is the unperturbed upstream velocity at hub height. The UPMPARK calculations have been performed for two stability situations: neutral atmosphere (Monin±Obukhov length, L ˆ I ) and stable atmosphere (L ˆ 100 m, hub height 31 m) Crespo et al.,28 van Leuven93 assumed in his farm model (WINDPARK) that a given turbine is only a€ected by the wake of the closest upstream turbine, obtaining good agreement in comparison with measurements made at the Zeebrugge wind farm93,94 (see Figure 5). In this ®gure are also shown the results of calculations with UPMPARK,95 PARK36 and FARMS, based on the kinematic model MILLY.96,97 Regarding the increase in turbulence intensity which occurs when there are many turbines in a line, Builtjes and Vermeulen98 carried out an experimental investigation in a wind tunnel with wind turbine simulators and found that turbulence intensity reached an equilibrium value after three to four rows of turbines. They also observed that turbulence intensity had a maximum in the second row of turbines, where it was higher than the equilibrium value. Luken99 proposed a simple correlation to calculate the equilibrium value of turbulence intensity as a function of turbine spacing. Crespo et al.28 applied their elliptic model for studying the interaction of the wakes from two turbines in two con®gurations: abreast and in a line. There was good agreement with experimental results, and when other superposition assumptions were compared, it was found that the linear superposition worked well for the two machines abreast, in which velocity de®cits in the interference region are small. However, for the two turbines placed in a row the linear superposition overestimated the velocity de®cit, as was to be expected. The previously mentioned model of van Leuven93 which considers that only the wake of the closest turbine upstream acts on a given turbine also agrees well with the elliptic model. When the results of the elliptic model of Crespo et al.28 are considered, it can be observed that the truly elliptic e€ects, such as axial pressure variations, only occur very close to the turbine, so that the parabolic approximation may be a suitable approach for studying wake interactions over most of the region where this interaction occurs. Moreover, to extend the fully elliptic code to a wind farm consisting of many machines, besides consuming a lot of calculation time, would require very powerful computers and would therefore be of little practical interest for modelling wind farms. Because of this, Crespo et al.95,100 have developed a code, UPMPARK, extending the parabolic UPMWAKE code for a single wake to the case of Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen a park with many machines. No assumptions are required regarding the type of superposition or the type of wake to be used, as all the wakes and their interactions are e€ectively calculated by the code. A brief description of UPMPARK follows. The conservation equations solved are the same as those for the single-wake code UPMWAKE, as speci®ed in Reference 56, and turbulence is closed using a k±e model. The wakes of the machines di€use in an ambient ¯ow that represents the surface layer of the atmospheric boundary layer, in which instability e€ects are retained by means of the Monin±Obukhov length. For uniform terrain this ambient basic ¯ow is the same over the whole wind farm, although the code could also handle moderate terrain irregularities, using a superposition assumption for the e€ects of terrain and wakes101 that will also be reviewed below. At in®nity, in regions not perturbed by the wind turbines, and in the upstream section, boundary conditions are imposed that correspond to an unperturbed ambient ¯ow. As we progress downwind in the numerical marching procedure associated with the parabolic model, each turbine found at any crosssection of the farm acts as a source (or sink) of the three velocity components, k and e. The number of grid points should be large enough to contain the whole cross-section of the park and to consider that the lateral boundaries are at in®nity. As the code is parabolic, there is no limit to the downstream distance, except for the fact that if the wakes di€use very much, the number of grid points may not be large enough to apply the boundary conditions at in®nity. The case of wind turbines in a row is particularly suited for this code, which has been validated by comparison with measurements made on wind farms in Zeebrugge, Sexbierum and Vindeby and on the Nibe wind farm. Adams and Quarton76 extended the UPMWAKE model to wind farms using a procedure quite similar to that of UPMPARK, and compared the results obtained with this extended version of UPMWAKE and EVFARM (also extended to wind farms) with the wind tunnel experiments of Hassan77 for double-wake cases, and found a degree of agreement smaller than for the single-wake case. EVFARM predicted the velocity de®cit within 1% error for the near wake and within 10% for the far wake, whereas UPMPARK underpredicts the velocity de®cit by 33% for the near wake and by 5% for the far wake. Topographic Effects Usually, wind farm models make the assumption that the terrain is ¯at and that the unperturbed wind velocity is uniform, an assumption which is not reasonable in many cases of interest, since, as is well known, terrain irregularities can be used to enhance or concentrate wind power. Studies on the in¯uence of ambient ¯ow on the wake development are scarce. John and Schobeiri102 made an experimental study of the in¯uence of streamline curvature and longitudinal positive pressure gradient on the development of the wake of a cylinder in a two-dimensional curved channel, observing that the wake decay is slower with the positive pressure gradient than with a zero pressure gradient. The e€ect of curvature on mean velocity de®cit distribution is small, whereas it strongly a€ects the Reynolds stress distribution, particularly in the inner half of the channel. For terrains that are moderately complex, the simple procedure of adding the velocity perturbations of  the wake and terrain should give an approximate ¯ow ®eld; this procedure was applied to the Ampurdan wind farm.31,40 Measured and calculated values of power output of the wind farm as a function of wind direction were compared. A scatter of measurements of the order of 20% was found and their average was predicted with an error of less than 10%. A similar procedure was used by Adams and Quarton76 and by van Leuven93 to take into account the interaction of an obstacle and turbine wakes in the Zeebrugge wind farm. However, in all the above cases there were simultaneous interactions of terrain and several wakes, which raises some uncertainty about the validity of the results, since, as is well known, the linear superposition of several wind turbine wakes overestimates the velocity de®cit, as indicated in the previous section. Crespo et al.101 studied the Monteahumada wind farm, in which the velocity irregularities of the terrain and the velocity de®cit created by a single wake interact and are both of a similar order of magnitude; this con®guration is thus appropriate to examine the validity of the assumption of linear superposition of wake and terrain e€ects. Although the data were few and not easy to interpret, the study Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods shows that for a moderately irregular terrain the linear superposition of wake and terrain e€ects gives good results (with relative errors of the order of 10% and less than 20%), whereas for the interaction of two wakes with perturbations of a similar order of magnitude this assumption is less valid. Voutsinas et al.38 described a procedure to take into account non-uniformity in wind velocity and the curvature of the streamlines in wind farms with small terrain irregularities, which is similar to the linear superposition; some sample calculations were made, but no experiments were presented to validate the method. Stefanatos et al.103 developed another model which assumed that the vorticity fed by the wind turbine into its wake follows the streamlines of the unperturbed wind ¯ow ®eld, and compared the results with the experimental ones, obtaining a good degree of agreement in the upper part of the wake (within 5%). However, in the lower part of the wake the calculated velocity de®cits exhibit a downward displacement of one-tenth to one turbine diameter. Taylor and Smith104 made measurements in a wind tunnel which showed that the changes in the wake characteristics due to topography may be important. Hemon et al.105 made a theoretical study of how terrain may modify turbine aerodynamics and near-wake characteristics. Second-order corrections to the linear superposition of terrain and wake e€ects were made by van Oort et al.106 using PHOENICS. They found that terrain irregularity creates additional turbulent di€usion near the ground, which diminishes the wake e€ect; on the other hand, above the apex of a hill the streamlines concentrate, thereby increasing wake e€ects. Crespo et al.,100 Gunther et al.107 and Ansorge È et al.80 also used the commercial code PHOENICS to model the interaction of wakes with obstacles and terrain irregularities. Stefanatos et al.103,108 and Helmis et al.60 give some guidelines, obtained from their experimental results in both wind tunnel and large-scale tests, to study the interaction between wake and terrain. A related problem arises in o€shore wind farms where, when the wind blows from land to sea, there is an internal boundary layer, whose development is superposed on that of the wakes, as mentioned by  Crespo and Gomez-Elvira.109 As the surface roughness of the sea is usually much smaller than the corresponding roughness on land (see e.g. Reference 110), it is to be expected that wind velocity will be greater and turbulence intensities lower than for equivalent inland stations. Consequently, turbulent di€usion of the wake will also be lower and wake e€ects will probably be more persistent downstream. Wake e€ects in o€shore wind farms obtained from both experiments and numerical models are reported by Frandsen et al.70 and Crespo et al.,71 and their e€ect on fatigue loading by Frandsen.111 5. Quasi-analytical and Semiempirical Expressions to Describe Wake Evolution In many cases it is of interest for the designer to have, as an alternative to numerical models, analytical expressions which can estimate the order of magnitude and the tendencies of the most important parameters characterizing wake evolution. However, this issue will not be examined in detail in this article, which is more concerned with aspects of modelling. Regressions or correlations of this type were obtained by di€erent authors to describe single-wake behaviour: see References 20, 32, 40 and 112±114 for the velocity de®cit and the width of the wake, and References 20, 25, 27, 42, 61, 113 and 115 for turbulence intensity. Taylor26 performed a parametrization of the calculated wake magnitudes as functions of several dimensionless input parameters; however, the results were represented in graphic form and no regressions were made. The case of wind clusters is covered in a review by Luken,99 who proposed a correlation for the equilibrium value of the turbulence intensity reached in a row of turbines, using the experimental results of Builtjes and Vermeulen.98 This point is discussed in more detail by Frandsen et al.70 who presented correlations giving values of the average velocity, turbulence intensity, turbulence scale and width of the wake at di€erent positions of each machine in a row as functions of their operating characteristics. These correlations are obtained by making the best ®t with numerical results from UPMPARK, and are validated by comparison with measurements made in Vindeby wind farm. Most of the above studies express wake di€usion as a function of downstream distance made dimensionless with turbine diameter, x/D, dimensionless turbine height H/D, thrust coecient CT and ambient Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen turbulence intensity. Instead of using x/D, Magnusson112 and Magnusson and Smedman113,114 expressed wake di€usion as a function of the transport time t ˆ x/u (where u is the local incident velocity), made dimensionless with t0 , the transport time where the near wake ends. In turn, t0 is made dimensionless with the rotational frequency f. Implicitly, then, the authors were considering another parameter, the tip speed ratio pfD/u. The e€ect of atmospheric stability (expressed in terms of the Richardson number) on velocity de®cit decay was taken into account in the correlation given by Magnusson and Smedman.74 All these correlations have been compared with some experimental results and show at least an acceptable degree of agreement, although more work is needed to carry out a full comparison both among these correlations and with experimental results. 6. Loading under Wake/Wind Farm Conditions As already mentioned, there is large body of data available in the literature which has been obtained from measurements made of the ¯ow characteristics of wind turbine wakes, and considerable e€orts have been made to develop numerical models, whose results are in many cases in good agreement with the experimental results. However, although such wake measurements and modelling have focused on energy production and loads, relatively few direct measurements of structural loads under wake conditions have been made, despite the shortcomings of the international standard on loads and safety116 and the lack of guidelines on how to take loads into account. Dynamic and Fatigue Loading The most serious expected structural e€ect on a wind turbine which is in the wake of a neighbouring machine is fatigue. The combined e€ect of increased turbulence, wind speed de®cit and changes in turbulence structure causes dynamic loading which may excite the wind turbine structure. The apparent e€ect on, for example, ¯apwise bending moments depends primarily on the distance to the neighbouring machine. As mentioned in Section 3, two wake regions can be considered: the near wake, which extends over three rotor diameters, and the far wake. The three-diameter length is chosen as the distance below which the e€ect on a machine which is partly in the wake of another (half-wake load case) is clearly visible on, for example, blade loads. Near Wake Some of the ®rst measurements of fatigue loading in wakes were made in the early 1990s by Vùlund117 and Stiesdal.118 The measurements made by Vùlund117 on a 250 kW machine, which was in the wake of a turbine of similar size placed two rotor diameters upstream, showed an increase in the standard deviation of ¯apwise bending moment of approximately 100% relative to the unobstructed case. Measurements also showed that loads were largest when the machine was exposed to half-wake conditions. The concept of equivalent load, which is the amplitude of a sinusoidal load with a ®xed frequency that would generate the same fatigue damage as the actual (random) load, was introduced by Stiesdal118 to provide a more precise fatigue measurement than the standard deviation. Here the separation between the 450 kW machine on which measurements were made and the wake-generating machine was 2.5D. The equivalent ¯apwise load increase was about 100% at 12 m s À1 and lower for high wind speeds. When loading under wake conditions was weighted with non-wake operation, the average e€ect was much smaller (10%±20% in terms of equivalent load). Far Wake Frandsen and Christensen119 made measurements in the large wind turbine array Nùrrekaer Enge II in Denmark, consisting of 42 machines of 300 kW, separated by 6D±8D. Two turbines in opposite corners Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods Figure 6. Schematic sketch of Vindeby wind farm were instrumented to measure stresses in towers and blades. To the north of the wind farm there is open water and the surrounding terrain is farmland. For northerly winds one of the instrumented machines is exposed to the low turbulence of the sea and the other one to the turbulence environment of the interior of the wind farm. For southerly winds one instrumented machine is exposed to the wind farm turbulence while the other is open to the free (ambient) ¯ow o€ the farmland. In both cases there was a clear increase in the equivalent load when the instrumented machine was in the direct wake of a neighbouring machine, but the integrated e€ect of wake and non-wake operation was signi®cantly smaller. It was found that the integrated e€ect in terms of fatigue loading for the higher ambient turbulence (southerly wind) was insigni®cant. For low ambient turbulence (northerly wind) the added loading in terms of equivalent load due to wake e€ects of the wind farm was approximately 10%. Frandsen and Thomsen120 carried out similar measurements in the Danish o€shore Vindeby wind farm. The wind farm, which is represented schematically in Figure 6, consists of 11 machines of 450 kW which are arranged in two rows, each machine and each row separated by approximately 8D. Two instrumented masts provided measurements of ambient and wake ¯ow parameters. The geometry of the wind farm and the instrumentation allowed for both single- and multiple-wake measurements. Data, including equivalent loads, were recorded for 2 years. An example of the measured data is shown in Figure 7, where equivalent widths of the ¯apwise bending moment of a blade are plotted against wind direction. Each data point represents 0.5 h of operation. For a wind direction of around 2558 the machine rows are aligned with the wind. As can be seen, despite the fairly large spacing between wind turbines, the fatigue load increase in the wake is signi®cant, about 80%. The data shown for other wind directions demonstrate that there is no appreciable di€erence between single- and multiple-wake loads. In Figure 7. Equivalent load widths of ¯apwise bending moment of instrumented units 4 W and 5E in Vineby wind farm Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen Reference 120 a model for adding up the wake e€ects was devised to reproduce the integrated e€ect of the individual wakes. As in the case of Nùrrekaer Enge II, the integrated e€ect was much lower. It was also estimated that the integrated fatigue loading would be approximately 20% lower in o€shore conditions than in onshore sites. The Sexbierum wind farm in the Netherlands consists of 18 units of 300 kW, placed in three rows, each with six machines.121 The machines are spaced 5D or 10D apart and the rows are separated by 8D. Six meteorological towers provided information on ¯ow characteristics. As in the Vindeby wind farm, it was found that equivalent loads under single- and multiple-wake conditions were very similar. Measurements have been made on one wind turbine in the experimental Alsvik wind farm in Sweden,122 which has four machines sited so that the instrumented unit is exposed to 5D, 7D, or 9.5D single-wake loads, depending on wind direction. The terrain is smooth and the ambient turbulence low. The wind farm layout is unique in o€ering many experimental possibilities. It was found that when the instrumented machine is within 7D to 9.5D downstream of another machine, there are no signi®cant half-wake loads. However, the e€ect on a machine which is only partly in the wake of another one is di€erent in the 5D situation. It was also found that under full-wake conditions the equivalent load is increased by 10% at 9.5D and up to 45% at 5D. The Kappel wind farm in Denmark consists of 24 units of 400 kW, sited in a row on a westerly shoreline, with 3.7D separations. Thomsen et al.123 discussed whether fatigue load increased (in the integrated sense) under wake conditions, arguing that the load increase would be smaller for high-wind cases, as wake e€ects are lower in such conditions. Nevertheless, the experimental evidence pointed to a signi®cant increase in the equivalent load, which represents a fatigue loading. Modelling As mentioned above, there have only been limited e€orts made towards directly estimating wake loading. In References 75 and 120 the task was solved by altering the ( free ¯ow) design turbulence. Thus the change in design turbulence accounted for intermittent wake conditions with high turbulence, smaller turbulence scale and half-wake cases. Tindal et al.75 presented an elaborate scheme of how to summarize the numerous load cases. The reader is referred to the report for more details. In Reference 120 a simpli®ed model of taking the increase in fatigue loading into account was discussed and, to some extent, veri®ed by the data presented. Similar models have previously been applied124 and di€erent expressions have been proposed for the `virtual' or `ecient' turbulence intensity that will result in the same fatigue damage occurring as in the ¯ow-wise complicated environment of a wind farm array. This turbulence intensity depends on the geometry of the array, the thrust coecient of the wind turbines, the separation of wind turbines and the ambient turbulence. Despite the mentioned complexity of the ¯ow in wakes/wind farms, the model proposed by Frandsen and Thomsen120 has been demonstrated to work for turbine separations larger than 3D±4D and, with some modi®cations, also for smaller separations. Extreme Loading Extreme loads are to be expected occasionally during the lifetime of a wind turbine. Extreme loading may occur either under common wind conditions with the wind turbine in operation or under extreme wind conditions (e.g. 50 years return-period wind gust). Loads on an operating machine increase with increasing wind speed up to a certain velocity, above which they may still continue to increase, depending on the turbine design.120 At wind speeds higher than about 15 m s À1, wake e€ects rapidly diminish, since the thrust coecient decreases, making the wind farm more `transparent'. Under extreme wind conditions the wind turbines will be parked, with little or no wake e€ects from neighbouring machines. However, extreme loading may also occur during non-extreme wind conditions, since (1) the typical blade tip speed of large machines is about 60 m s À1 and the loads may be correspondingly high, (2) large yaw errors generate large dynamic response which may also cause large extremes, and (3) transient loads caused, for Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods example, by emergency stops may signify the largest loads for one or more machine components in the machine's lifetime. This case has not been satisfactorily dealt with in the literature. 7. State of the Art and Conclusions There are many di€erent models which simulate the behaviour of wind turbine wakes and wind farms. Most of them are based on a deterministic simulation which takes into account each individual wind turbine. Models considering the turbines as distributed roughness elements are not much used now, although they may be of interest in the future to predict overall changes in wind characteristics originated by the large wind farms which are likely to be established. Many of the models proposed show an acceptable degree of agreement with the experiments with which they are compared. However, the assumptions and coecients that are chosen are such that the agreement with some particular experiments may be good, although the overall validity has not been checked in more general situations. The models which depend on the least simplifying assumptions are better suited in dealing with di€erent con®gurations and in reproducing wake development in more detail. For example, an axisymmetric model will never be able to reproduce the peak in turbulence intensity in the upper part of the shear layer in the near wake. In general, the more complicated models are more likely to accurately reproduce ¯ow ®eld characteristics, although in many cases the physical reasons for the hypotheses used, particularly in those aspects related to turbulence modelling, are not always clear. The classical wind farm model relies on an individual wake model, usually a kinematic model, and some sort of superposition assumption. In general, the superposition assumptions cannot be justi®ed from a physical point of view and may even lead to absurd or contradictory results; the corrections and alternatives necessary to handle such physically unrealistic situations are either unjusti®able or dicult to implement. Although it is a generally held opinion that ®eld models are too complicated and that they are impossible to extend to a wind farm consisting of many turbines, we do not think that this is still necessarily the case. For example, the UPMPARK model, which retains all the characteristics of one of the most complete non-symmetric k±e wake models, UPMWAKE, can be successfully run on a workstation in reasonably short times for large wind farms such as Sexbierum or Taendpibe, and even on a PC for wind farms with a smaller number of machines. The most important simplifying assumption used by UPMPARK is the parabolization of the mathematical problem in the main wind direction. While greater emphasis used to be directed to calculations of velocity de®cits and park eciency in terms of energy production, calculations are nowadays more orientated to other issues, such as estimating magnitudes related to the structural and fatigue behaviour or ¯uctuations in the electrical energy produced by machines a€ected by upstream wakes. To estimate these magnitudes, it is necessary to know the turbulence characteristics of the ¯ow (turbulence intensity, correlations and spectrum) and wind shear data, which clearly cannot be provided by simple kinematic models. There is experimental evidence that wake e€ects may signi®cantly increase the equivalent load experienced by a turbine under wake conditions, although further research is still necessary for a more precise estimation of their in¯uence in di€erent operational conditions. However, measurements also show that the integrated loading e€ect of wake and non-wake operation may be, in some circumstances, signi®cantly smaller than that measured under continuous wake conditions. It has also been found that under multiple-wake conditions the equivalent loads due to di€erent wakes are not additive. Only limited e€orts have been made towards developing predictive models to estimate the loading increase due to wake e€ects. An issue of some importance is the non-isotropic nature of the turbulence of ambient atmospheric ¯ow, in contrast with the more isotropic turbulence in the wakes. This problem cannot be dealt with by the k±e or eddy viscosity wake models. The use of Reynolds stress models involves greatly increased mathematical Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen diculties and we do not think it can be treated by the engineering codes in general use. Alternative strategies should be explored. One of the most important diculties that has not been treated satisfactorily is the choice of appropriate input parameters to de®ne ambient unperturbed ¯ow, particularly in complicated terrains. Usually, a comparison with wind tunnel experiments is reasonably straightforward, but when ®eld experiments are compared, there are many diculties, and e€ects such as meandering have never been satisfactorily modelled. The results obtained from experimental and modelling studies for terrains of varying roughness and the appearance of internal boundary layers, such as those observed in wind farms located near the coast to o€shore, should be incorporated into the description of ambient ¯ow. For a terrain that is moderately irregular, UPMPARK assumes a superposition of the perturbations due to the wakes and those of the terrain, which are estimated either from measurements or from codes such as WASP. However, for complicated topography this approach may not work. A code which simultaneously takes into account terrain and wind turbine wakes would be too dicult to apply and still contain many uncertainties regarding the appropriate boundary conditions. Some work has been done, but more is needed, towards estimating the local e€ects of interference of single wakes and terrain irregularities. The problem is that it is dicult to envisage solutions and we will always be solving particular problems that, at most, could only point to general tendencies. A possible alternative for cases in which turbine spacing is small compared with the characteristic length of variation in terrain irregularities could be to treat the problem as one of ¯ow over an irregular terrain of changing roughness, as indicated in Section 2. Although considerable progress has been made since 1992, some of the conclusions and recommendations presented here are of a similar nature to those formulated in the ®nal report of IEA, Annex IX on Wake E€ects.3,125 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Wind Energy Wiley

Survey of modelling methods for wind turbine wakes and wind farms

Wind Energy , Volume 2 (1) – Jan 1, 1999

Loading next page...
 
/lp/wiley/survey-of-modelling-methods-for-wind-turbine-wakes-and-wind-farms-vpF2y7lusq

References (44)

Publisher
Wiley
Copyright
Copyright © 1999 John Wiley & Sons, Ltd.
ISSN
1095-4244
eISSN
1099-1824
DOI
10.1002/(SICI)1099-1824(199901/03)2:1<1::AID-WE16>3.0.CO;2-7
Publisher site
See Article on Publisher Site

Abstract

  A. Crespo,* ETS de Ingenieros Industriales, UPM, Jose Gutierrez Abascal 2, E-28006 Madrid, Spain  J. Hernandez, ETS de Ingenieros Industriales, UNED, Ciudad Universitaria, E-28040 Madrid, Spain S. Frandsen, Risù National Laboratory, DK-4000 Roskilde, Denmark Key words: wind turbine wake; wind farm; dynamic loading; atmospheric turbulence This article provides an overview and analysis of different wake-modelling methods which may be used as prediction and design tools for both wind turbines and wind farms. We also survey the available data concerning the measurement of wind magnitudes in both single wakes and wind farms, and of loading effects on wind turbines under single- and multiple-wake conditions. The relative merits of existing wake and wind farm models and their ability to reproduce experimental results are discussed. Conclusions are provided concerning the usefulness of the different modelling approaches examined, and dif®cult issues which have not yet been satisfactorily treated and which require further research c are discussed. Copyright *1999 John Wiley & Sons, Ltd. Introduction Wind turbine wakes are an interesting topic of study, because the momentum de®cit and the increased level of turbulence created by turbines in a wind farm may cause a reduction in power output and unsteady loads on other machines. On the other hand, owing to the cost of land and civil works, wind turbines tend to be built as closely as possible to each other, and to this e€ect, Builtjes and Smit1 and Milborrow and Surman2 have provided guidelines for wind turbine spacing in a wind farm. The ®nal report of IEA, Annex IX on Wake E€ects,3 indicates that the experimental and analytical studies reported in the Annex point to signi®cant energy losses in arrays spaced at less than seven turbine diameters. Similarly, turbulence may increase in arrays, suciently to cause measurable damage due to fatigue and dynamic loads. Machines based on lift-like forces, associated with the generation of circulation, develop less intense wakes than drag-type units, which may be one reason, apart from other aerodynamic considerations, for preferring lift-type to drag-type machines.4 Nevertheless, although most modern machines are of the lift type, the wake e€ects that they produce are still important enough to be studied. In an early approach to the problem of modelling wind farms, it was assumed that when an area contained a large number of machines, the turbines acted as distributed roughness elements, and that they modi®ed the ambient atmospheric ¯ow (see reviews by Bossanyi et al.5 and Milborrow6). More recent work using this approach has been carried out by Frandsen7 and Emeis and Frandsen.8 This topic of turbines acting as distributed roughness elements will be treated in more detail in Section 2.   *Correspondence to: A. Crespo, ETS de Ingenieros Industriales, UPM, Jose Gutierrez Abascal 2, E-28006 Madrid, Spain. Contract/grant sponsor: European Union; Contract/grant number: JOU2-CT93-0350; JOU3-CT95-0089. CCC 1095±4244/99/010001±24 $17.50 Copyright # 1999 John Wiley & Sons, Ltd.  A. Crespo, J. Hernandez and S. Frandsen However, the most common approach to the problem, set in motion by the classical paper of Lissaman,9 considers each turbine wake of the farm individually and examines its interaction with and superposition on neighbouring ones. It thus calculates the detailed ¯ow ®eld and not the average distribution. Section 3 is dedicated to individual wake behaviour and Section 4 to wake superposition and the multiple-wake case that occurs in wind farms. Frandsen7 compared the results of both approaches in a particular example, and although further research along these lines was desirable, this, to the best of our knowledge, has not been carried out. Section 3 starts with a description of wake behaviour and continues with a discussion of the kinematic models, also known as explicit models, which have been used extensively because of their simplicity and low computational cost. They use self-similar velocity de®cit pro®les obtained from experimental and theoretical work on co-¯owing jets. The wake growth rate is calculated as being caused by the ambient turbulence, the turbulence created by the shear in the wake and that created by the turbine itself. The magnitude of the maximum velocity de®cit at each section is obtained from global momentum conservation, and the ground is taken into account by introducing an image machine. These methods provide acceptable results if the adjustable coecients are appropriate. Field models, also known as implicit models, calculate the ¯ow magnitudes at every point of the ¯ow ®eld. Field models require a substantially larger computer capacity than kinematic models, although their requirements are well within the capabilities of modern computers, not only in the case of single wakes but also for multiple wakes occurring in a wind farm, if appropriate simplifying assumptions are made (see below). The ®eld models give an acceptable representation of the ¯ow ®eld and a good insight into the processes governing wake development. Both kinematic and ®eld models use as starting or boundary conditions those at the end of the expansion region and the beginning of the near-wake region. If a uniform velocity de®cit is assumed at the initial cross-section, this de®cit can be estimated from the overall thrust on the machine; other possibilities are contemplated in Section 3. Kinematic and ®eld models do not directly take into account the bodily movement of the wake with the large atmospheric eddies, known as meandering. This is also examined in Section 3. An important issue in wind farm modelling is the interaction of several wakes and the way in which the velocity de®cits and turbulence created by each machine accumulate at locations where several wakes meet. Di€erent types of assumption are made regarding superposition rules, the most straightforward approach consisting of adding the velocity de®cits and turbulence kinetic energy. This and other alternative methods will be reviewed. The main problem is that any approach based on single-wake calculations will fail, because the ambient basic ¯ow in which the wake di€uses is to some extent also a€ected by the wakes of the upstream machines and will also be evolving. A more correct approach would be to solve the ¯ow equations for the whole wind park. At ®rst sight and from a practical point of view this would not seem feasible for wind farms with a large number of wind machines; however, if some simplifying assumptions are made, namely a parabolic approximation, the ®eld model for a single wake can be extended to the multiple-wake case and be practically solved with reasonable computer times, giving an acceptable degree of agreement with experimental measurements. This issue will be examined in more detail in Section 4. The problem of how to take into account terrain e€ects in wind farms placed in moderately irregular topography, not contemplated in Section 3 for single wakes, is also treated in Section 4. Although this article concentrates more on modelling aspects, the empirical or quasi-analytical expressions that may be used to estimate the downstream evolution of relevant wake parameters, such as velocity de®cit and turbulence intensity, are brie¯y reviewed in Section 5. Section 6 deals with the in¯uence of wake e€ects on wind turbine loading. We review available load measurements in wind turbines under di€erent types of wake conditions. Some existing models to estimate wake-loading e€ects are also described. In Section 7 we present what we think is the state of the art regarding wind turbine wakes and wind farm modelling, accompanied by the corresponding conclusions and recommendations. Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods 2. Turbines Acting as Distributed Roughness Elements The models of Templin,10 Newman,11 Crafoord12 and Moore,13 reviewed by Bossanyi et al.5 apply to in®nite clusters. They assume a logarithmic wind pro®le for the unperturbed wind, which includes ground roughness as a parameter. The presence of the turbines increases the value of the roughness, which is then calculated. From the modi®ed wind pro®le the wind velocity incident on each machine can be obtained and then the power produced can be calculated. Whereas the previous models assumed a single logarithmic pro®le, Emeis and Frandsen8 assumed that below hub height there is a logarithmic pro®le with the real ground roughness, and above hub height another pro®le with a roughness related to the drag of the machine; the pro®les match each other at hub height. Frandsen7 also applied a logarithmic pro®le above hub height and assumes the validity of a simpli®ed form of the geostrophic drag law, obtained from the Rossby number similarity theory. Bossanyi et al.5 explained how the previous models can be extended to the case of ®nite clusters. Schmid14 used results obtained by Taylor15 for a step change in roughness to calculate the friction velocity at each row of turbines. Crafoord,12 Moore13 and Musgrove16 considered a mixing layer of air above the ground and perform either a momentum or energy balance in this layer. It was assumed that sucient mixing occurs so that, by the time the next row of turbines is encountered, the velocity de®cit is averaged out across the whole mixing layer. The di€erence between the momentum (or energy) ¯uxes of two consecutive rows is due to the drag (or power extraction) of the turbine, the amount lost to the ground and the amount entrained from greater heights through mixing processes. The di€erence between the last two quantities was termed the replenishment rate by Bossanyi et al.,5 who discussed several hypotheses concerning the way to estimate the relevant parameters, in particular the mixing layer thickness and the replenishment rate, and compared the corresponding results. Although these models are not much used, they could be of interest to predict overall e€ects of large wind farms on wind characteristics. 3. Individual Wakes This section is dedicated to reviewing the models for individual wakes. It is of interest to present ®rst a general description, based on physical grounds, of the behaviour of the wake characteristics to be simulated with the models. In the second and third subsections the more simple kinematic models and the ®eld models respectively are described. The procedures and methods used by the di€erent models are presented and the results discussed. In the third subsection the results obtained with both kinematic and ®eld models are examined simultaneously and compared with experiments, in order to discuss the merits of both types of models. Finally, the meandering e€ect, which is not usually taken into account explicitly in the models, is examined in the last subsection. Description of the Wake Behaviour As the air approaches the wind turbine, its velocity decreases and the pressure increases. As it crosses the rotor, there is a sudden decrease in pressure. In the region immediately downstream of the rotor there are non-uniform de®cits of pressure and axial velocity, which are associated with the axial thrust, as well as an azimuthal component of velocity, which, in turn, is related to the torque on the machine. Vortex sheets, associated with the variation in circulation along the blades, are shed from their trailing edge and roll up in a short downstream distance, forming tip vortices that describe helical trajectories. When the inclination angle of the helix is small enough, the tip vortex can be interpreted as a cylindrical shear layer which separates the slow moving ¯uid in the wake from that on the outside. The velocity de®cit can be considered as induced by the vortices. The di€erence in pressure between the ¯uid behind the rotor and that on the outside is supported by the centrifugal force due to the curvature of the streamlines. As we Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen Figure 1. Schematic representation of a wind turbine wake move downstream, the cylindrical shear layer expands, the pressure increases and the velocity inside the wake decreases until ambient pressure is reached (Figure 1). According to the simple actuator disk theory, which assumes that ¯ow is ideal and that the shear layer is in®nitely thin, the velocity de®cit at the disk itself is half that in the expanded wake. Because of turbulent di€usion, the thickness of the shear layer increases with downstream distance, but if the length of this expansion region is suciently small, it may not be a bad approximation to consider that the thickness of the shear layer is small compared with its diameter. The length of this expansion region is about one turbine diameter. As we proceed further downstream, turbulent di€usion of momentum becomes the dominant mechanism. Turbulence production is more important in the shear layer, where the velocity gradients are larger. A well-de®ned ring-shaped domain where there is a high turbulence intensity is observed in this cylindrical shear layer, both experimentally17±23 and numerically.24±26 There are also signi®cant velocity gradients both inside the wake, since the velocity de®cits created by the turbine are not uniform, and in the atmospheric ¯ow, where the wind velocity changes with distance to the ground. Most of the turbulence that makes the wake di€use is, at this stage, probably created by the shear in the wake, mainly in the shear layer. However, the shear in the external atmospheric ¯ow also plays an important role, at least in the redistribution of the generated turbulence. As will be shown later, the turbulence of the ambient ¯ow is responsible for a non-uniform distribution of turbulence in the shear layer, where a maximum is observed in the upper part25,27 (Figure 2). Turbulent di€usion makes the shear layer thickness increase with downstream distance, and at a certain distance downstream (about two to ®ve diameters) the shear layer reaches the wake axis. This marks the end of the near-wake region. After the near-wake region there is a transition region leading to the far-wake region, where the wake is completely developed and, in the hypothetical absence of ambient shear ¯ow, it may be assumed that the perturbation pro®les of both velocity de®cit and turbulence intensity are axisymmetric and have selfsimilar distributions in the cross-sections of the wake. The only overall properties of the turbine that appear as parameters in these pro®les are the thrust on the turbine and the total turbulence kinetic energy produced by the rotor itself. This property of self-similarity of the velocity pro®les is the basis of the kinematic models describing wind turbine wakes. However, the presence of the ground and the shear of the ambient ¯ow invalidate the assumption of axial symmetry and, to some extent, the hypothesis of selfsimilarity. It has been observed both numerically and experimentally that the maximum turbulence intensity in the far wake is located above the turbine axis,21,28 and the point of maximum velocity de®cit is usually below the turbine axis.29±32 The maximum of turbulence intensity is about one turbine radius above the axis, and this is probably related to what happens in the near wake. Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods Figure 2. Vertical distribution of turbulence intensity normalized with its ambient value. Comparison of numerical 27 and experimental 21 results. 1D downstream, D ˆ 40 m, H ˆ 45 m. Notice peaks located at H ‡ D/2 and H À D/2 Kinematic Models for Single Wakes As already mentioned, kinematic models are based on self-similar velocity de®cit pro®les obtained from experimental and theoretical work on co-¯owing jets. The wake description does not consider the expansion region and gives di€erent types of pro®les for the near-wake, transition and far-wake regions. For the far wake these pro®les are self-similar and in the near wake there is usually a central core of constant velocity and diminishing radius; when this radius becomes zero, the near wake ends. Lissaman9 and Voutsinas et al.33 used the velocity pro®les proposed by Abramovich.34 Vermeulen35 used a Gaussian type of pro®le quite similar to that of Abramovich.34 Katic et al.36 simpli®ed the problem further and assumed a top-hat pro®le everywhere. More recently, Kiranoudis and Maroulis,37 based on a kinematic model similar to the previous ones, developed a `short-cut model of wind park eciency', giving simple analytical expressions of the eciency as functions of the farm and turbine characteristics. In these models the initial velocity de®cit is usually obtained from the thrust coecient of the machine. Voutsinas et al.38 related it to the power given by the machine, the advantage of which is that the power curve is usually more available than the thrust curve. In all the studies published to date, the reference value of the velocity de®cit at each section has been obtained from global momentum conservation, except in the case of Voutsinas et al.,38 who claimed that they obtained it from mass conservation, based on the fact that agreement with the results of Taylor39 was better. However, it is not clear what they mean by this and, in particular, how they take into account the mass entrainment through the lateral surface of their control volume. As a matter of fact, when applying the classical equation of momentum conservation (see e.g. Reference 35), it is implicitly assumed that mass is also conserved. Lissaman9 regarded the wake growth as being caused by the sum of the ambient turbulence and the turbulence created by the shear in the wake. Vermeulen35 added another term, the turbulence created by the turbine itself, although, in a later work based on the experimental results of Taylor,39 Voutsinas et al.33 Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen considered that this e€ect was negligible. Katic et al.36 assumed that the wake radius increases linearly with downstream distance; the proportionality constant must be adjusted by comparison with experiments. The ground e€ect is simulated by imaging techniques. Lissaman9 included a symmetrical turbine and added the velocity de®cits of both the real and image turbines, so that drag conservation is satis®ed. He pointed out that the ground surface can be treated exactly like the vertical-dividing plane between two adjacent identical rotors abreast. However, when there is ground, the total drag is not actually conserved because of friction; the three-dimensional models show that there is a slight decrease in total de®cit as the downstream distance increases. According to the image procedure, the velocity de®cit will be double that due to a single wake at the ground, whereas in reality any perturbation would be damped to zero. Crespo  et al.,29 Crespo and Hernandez40 and Kambezidis et al.41 use an antisymmetric wake so that velocity de®cits are subtracted and give zero perturbation at the ground; if it is considered that in reality the ambient velocity is not uniform and that it falls to zero at the ground, the perturbed velocity calculated will also be zero at the ground. However, although this alternative procedure eliminates the previously mentioned inconsistency that occurs near the ground, it is not clear that it will give a more valid result in the rest of the ¯ow ®eld, where the ground e€ect is not so dominant. Another procedure followed by Voutsinas et al.38 consists of superimposing the squares of velocity de®cits and taking into account the spatial variability of the incident velocity to estimate the location of the image turbine, although this method does not seem to be capable of overcoming the diculty. Indeed, the ground e€ect seems to be an intrinsic diculty of all the kinematic models that assume axial symmetry, and there is no satisfactory way in which they can deal with it. The ground e€ect can only be treated properly with 3D models. Recently, Larsen et al.42 proposed a simple analytical model, based on the classical wake theory as presented by Schlichting.43 The ¯ow is supposed to be axisymmetric, and a single self-similar velocity pro®le is assumed for the whole wake. The velocity de®cit decays with downstream distance as x À2/3, the turbulence intensity decays as x À1/3, as in References 25 and 27, and the wake width increases as x1/3. Compared with the previous kinematic models, this model only considers the far-wake region and the turbulence created by the shear. They compared the turbulence intensity and length scale calculated from their model with predictions based on empirical relations, and obtained relative di€erences lower than 5% in all cases where the downstream distance x is larger than two diameters. In spite of all the previous diculties, in many cases the kinematic models provide results that are in good agreement with the experimental measurements if appropriate values are chosen for the parameters appearing in them.17,44,45 Field Models for Single Wakes Sforza et al.46,47 described the wake using only the linearized momentum equation in the main ¯ow direction, with constant advective velocity, a constant eddy di€usivity and a parabolic approximation. For two-dimensional con®gurations they obtained analytical solutions with acceptable wake shapes. In the three-dimensional case they integrated the equation numerically using an alternating direction implicit (ADI) method. They made small-scale experiments and compared measured and calculated values for the velocity de®cit and wake growth as functions of downstream distance, obtaining agreement within 10% error, except for cases of high thrust loading, for which the error could reach 20%. This agreement was reasonably close considering the simplicity of the model. Besides, the tendencies were well predicted in all cases. A numerical model based on solving the ¯ow equations for wakes in neutrally strati®ed atmospheric boundary layers was proposed by Taylor,48 who considered an eddy viscosity gradient closure scheme. The wake e€ect was assumed to be small enough for the equations to be linearized around a basic ¯ow, and a boundary layer approximation was used. The model was two-dimensional and presented results integrated across turbine rows. Coriolis forces were retained and the pressure gradients were given by the geostrophic wind. However, this assumption cannot be justi®ed, because the length scale of the wake is not suciently large for the Coriolis forces to play a dominant role; indeed, they can be neglected and the Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods pressure ®eld will be that resulting from the momentum conservation in the wake. If a parabolic approximation is made, pressure variations across the wake can be neglected in the momentum equation for the main ¯ow direction, but not for the momentum components in the transverse direction, particularly when there is neither axial nor two-dimensional symmetry. Taylor48 compared his results with those of kinematic models and other models based on the assumption that the turbines act as distributed roughness, and with experimental results of Builtjes.49 Although there was reasonable agreement, Taylor48 admitted that the linear superposition of the e€ects of several rows of turbines may lead to low or even negative power outputs for the backmost rows. Liu et al.50 proposed another, three-dimensional, model which includes atmospheric stability e€ects. However, these authors neglected the di€usion due to turbulence originated in the wind turbine and the di€usion caused by the shear in the wake, and considered the turbulent viscosity and the di€usion coecients to be those of the unperturbed ¯ow. Along with Taylor,48 they retained Coriolis forces and assumed that the pressure gradients were given by the geostrophic wind. Ainslie51±53 developed a parabolic eddy viscosity model (EVMOD) which assumes axisymmetric wake ¯ow. Pressure variations are uncoupled in the analysis, and only the continuity and the axial momentum equations have to be solved. Consequently, the model is incapable of dealing with ground e€ects or with variations in ambient ¯ow conditions with height. The turbulent shear stresses are described using an eddy viscosity closure scheme in which the eddy viscosity is represented by a simple analytical form based on Prandtl's free shear layer model, but which also includes a contribution from ambient turbulence. This eddy viscosity is an average value over a cross-section, and variations in turbulent properties across the wake cannot be estimated from the model. At small downstream distances the eddy viscosity is modi®ed by an empirical ®lter function to account for the lack of equilibrium between the mean velocity ®eld and the developing turbulence ®eld. Several constants appear in the problem. They are adjusted by comparison with particular experiments, although their validity in more general situations is not clear. The model is fairly simple and gives reasonable results when compared with wind tunnel experiments. For large-scale experiments the results are corrected by taking into account meandering e€ects (which will be described at the end of this section). Albers et al.54 found that there was greater agreement between Ainslie's model and experimental results if the incident logarithmic pro®le was superimposed on the calculated axisymmetric wake. Luken and Vermeulen55 and Luken et al.32 used the experimental results from the TNO wind tunnel to validate Vermeulen's35 kinematic model (MILLY) and Ainslie's51 model (EVMOD) (see Figure 3). Although they found an acceptable degree of agreement, some aspects such as the downshift of the wake centreline, which appears in Figure 4, were not well predicted. To reproduce such e€ects, models which retain three-dimensional e€ects are needed. These will be described next. Crespo et al.29 developed the UPMWAKE model in which the wind turbine is supposed to be immersed in a non-uniform basic ¯ow corresponding to the surface layer of the atmospheric boundary layer; further  developments of the model are given by Crespo and Hernandez.56 The properties of the non-uniform incident ¯ow over the wind turbine are modelled by taking into account atmospheric stability, given by the Monin±Obukhov length, and the surface roughness. It is supposed that this basic ¯ow, described by analytical expressions obtained from theoretical considerations and experimental results given by Panofsky and Dutton,57 is perturbed by the wind turbine. The equations describing the ¯ow are the conservation equations of mass, momentum, energy, turbulence kinetic energy and dissipation rate of turbulence kinetic energy. The modelling of the turbulent transport terms is based on the k±e method for the closure of the turbulent ¯ow equations. This set of equations has been solved numerically using the SIMPLE algorithm proposed by Patankar and Spalding.58 Finite di€erence methods were used in the discretization of the equations. A parabolic approximation was made and the equations were solved numerically by using an ADI method. The developed wake model is three-dimensional, and pressure variations in the cross-section have to be retained in order to calculate transverse velocities. A simpli®ed version of UPMWAKE, which assumes that all the convection is due to the unperturbed ambient ¯ow, was presented by Crespo et al.29 This seemed a very attractive idea, because it was possible to retain the three-dimensional character of the problem and reduce the system of partial di€erential Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen Figure 3. Decay of maximum dimensionless velocity de®cit along wake, made dimensionless with wind velocity at hub height, as a function of downstream distance divided by turbine diameter. Comparison of wind tunnel measurements and results of di€erent wake models Figure 4. Vertical distribution of maximum dimensionless velocity de®cit along wake as a function of vertical distance divided by turbine diameter for several downstream sections. Comparison of wind tunnel measurements and results of wake models Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods equations from seven to three. However, this approximation can only be justi®ed very far downstream where the wake perturbation is small, and although in some cases the results obtained were in quite good agreement with the full model and with experiments, in others, particularly in the near wake, they were wrong. Most of the UPMWAKE calculations that have been published correspond to the seven-equation code.  Crespo and Hernandez40,56 and Crespo et al.28,30 compared UPMWAKE results with the results of wind tunnel experiments obtained by Luken et al.55 (see Figures 3 and 4) and those of ®eld experiments using full-scale machines.59 The wake model has also been validated numerically by using the general-purpose CFD PHOENICS code,24 and the corresponding results agree well with those of UPMWAKE. The code can predict e€ects such as the downward tilt of the wake centreline, as can be seen in Figure 4, the upward displacement of the point of maximum added turbulence kinetic energy, or the di€erent vertical and horizontal growths of the wake width, which have been con®rmed experimentally by other authors.3 Based on their experimental results, Helmis et al.60 attribute the downshift of the maximum velocity de®cit to the tower shadow rather than to the asymmetry due to the terrain. Some discrepancies between the UPMWAKE results and those of the experiments of Taylor et al.59 were found in the initial wake region, where the predicted velocity de®cits were smaller than the measured ones.  More recently, and based on the results of the code, Crespo and Hernandez25,27 have developed correlations to calculate the turbulence intensity in both the near and far wakes, and compared them with a great number of experiments (many of them compiled by Quarton61), both wind tunnel17,35,62±67 and ®eld20,68,69 experiments. The comparison was acceptable and demonstrated that UPMWAKE may be a useful tool for estimating turbulence characteristics. For the far wake, Frandsen et al.70 proposed correlations that give similar predictions for the decay in turbulence intensity with growing downstream  distance. Crespo and Hernandez25,27 also proposed a simple method for obtaining the turbulence spectra in the wake from the values of k and e obtained from UPMWAKE, and compared their results with the experiments of Hùjstrup,21 obtaining good agreement in some cases. The results of this procedure for calculating the spectra are compared with measurements made in Vindeby wind farm by Frandsen et al.70 and Crespo et al.;71 some of the results obtained for the turbulent length scale that are needed to estimate the spectrum seem to be smaller than those measured. Possible reasons for this discrepancy are, on the one hand, that UPMWAKE does not take into account the small-scale (large-frequency) turbulence originated by the boundary layers of the blades of the wind turbines, and, on the other hand, that the wind turbine is capable of responding to low-frequency ¯uctuations of wind speed and extracts energy from the wind in the low-frequency (large-scale) range,21 although this tendency may be reversed for wind speeds higher than that corresponding to the maximum power coecient, as measured by Papadopoulos et al.72 Larsen et al.42 proposed another procedure, in which the individual contributions of the di€erent scales to the spectrum are calculated and added; the details of the method are not included in the paper and will be published elsewhere. One aspect of modelling which has been insuciently treated is wake di€usion in non-neutral atmospheres. Crespo et al.29 showed that, as was to be expected, di€usion is inhibited in stable atmospheres and enhanced in unstable ones, although no experimental results were available for comparison. More recently, some interesting experimental results have been presented by Magnusson73 and Magnusson and Smedman,74 who found that for unstable strati®cation, with Richardson number (Ri) values smaller than À0.05, the velocity de®cit is independent of stability, while it increases linearly with Ri in the interval À0.05 5 Ri 5 0.05. They also found that, unlike in the experimental results of Luken et al.,32 there is an upshift of the point of maximum velocity de®cit. The calculations of Crespo et al.29 showed that in most cases there is a downshift, except for stable atmospheres, large surface roughness and a high level of turbulence kinetic energy created by the turbine. It would be of interest to compare the results of numerical models that can simulate atmospheric stability with these experiments. Smith and Taylor23 and, in more detail, Taylor26 presented a non-symmetric two-equation model that is in many ways similar to the three-equation model of Crespo et al.29. They neglected transverse velocities and just solve the momentum equation in the axial direction. To model the turbulent viscosity, they use a Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen k±L method where the turbulent length scale L is related to the width of the wake, obtained by ®tting a Gaussian pro®le to the calculated pro®le. The value of the dissipation rate of the turbulent kinetic energy, e, was obtained from an algebraic combination of k and L, and consequently a partial di€erential equation for e was not needed. The same type of problem previously mentioned for the three-equation version of UPMWAKE also appears in this case. The results obtained in a comparison with their windtunnel experimental results are very good, although a comparison with full-scale Nibe measurements showed that the model overestimates the values of the velocity de®cit. They attributed this discrepancy to meandering and obtained better agreement when they corrected for this e€ect using the method proposed by Ainslie,53 which will be discussed later. A starting Gaussian velocity de®cit pro®le is imposed, which would correspond to the end of the near wake (Figure 1). The results of the calculations show a clearly de®ned annular peak of turbulence intensity, as shown in Figure 2, although this peak should really be located in the near wake, upstream of the starting region (in general, as will be pointed out again later, application of the initial conditions of the wake is an uncertain aspect of all wake models). Although there is considerable scatter of the experimental results around the line of calculated values, the peak values of turbulence intensity are well de®ned and, in a cross-wind pro®le, are predicted within an error of 3%. In a vertical plane the turbulence intensity distribution is similar to that appearing in Figure 2, and the upper and lower peaks are predicted within errors lower than 0.1% and 5% respectively. Based on the model developed by Ainslie,53 the company Garrad and Hassan has developed the code EVFARM, described by Tindal et al.75 and Adams and Quarton.76 The code incorporates two alternative semiempirical models to calculate wake turbulence. One of them, described by Hassan,77 gives uniform turbulence in the wake, whereas the other, described by Luken et al.,32 takes into account radial variations in turbulence intensity. Adams and Quarton76 use both EVFARM and UPMWAKE codes in combination with machine load predictive tools to provide a method for fatigue load prediction. As part of this study, a comprehensive validation of both codes is made using the wind tunnel measurements of Hassan;77 good agreement is found for the velocity de®cit, which is underestimated by 2% and 3% by UPMWAKE and EVFARM respectively; the turbulence intensity is overestimated by 11% by EVFARM and underestimated by 17% by UPMWAKE. They also noted that there was better agreement with experiments if a downstream displacement of the origin is considered to account for development of the expansion region. The physical reason for this displacement is not clear, because even though the expansion region is located downstream of the rotor, the shear layer starts immediately behind the rotor. In the initial region of the wake some important discrepancies were also observed between the results of UPMWAKE and the Nibe measurements published by Taylor et al.59 By eliminating the boundary layer  approximation used in UPMWAKE, Crespo et al.28 and Crespo and Hernandez78 proposed an elliptic model to deal simultaneously with the axial pressure gradients and di€usion e€ects by retaining both the axial and transverse di€usion terms. Their model therefore describes both the evolution of the expansion region and the di€usion processes. No fundamental di€erences between the results of the elliptic and parabolic models were found, and displacement of the origin was apparently not necessary. Other elliptic models have also been proposed by Cleijne et al.79 and Ansorge et al.80 Any improvement in the agreement with experiments when comparing elliptic and parabolic codes is only slight and does not seem to justify the additional computational e€ort needed. Another reason for the discrepancies observed between models and experiments in the near wake may be the uncertainty involved in the initial velocity de®cit, assumed to be either uniform or of a prescribed shape (Gaussian in Reference 26) and obtained from the thrust coecient. More recently, Magnusson,81 using blade element theory and experimental results, investigated the in¯uence of the non-uniform incident wind and the yaw on the near-wake characteristics. Zervos et al.82 relate initial wake development to the aerodynamics of the rotor, using a vortex particle method governed by the vorticity transport equations and the Biot±Savart law. Although these authors do not need initial data to start calculating the wake, the validity of the solution is limited to the short initial expansion region where di€usion e€ects can be neglected. In general, non-uniform values of axial azimuthal velocity components at the end of the expansion region can be obtained using a classical blade element model, a strip model83 or even vortex Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods particle and lifting line methods, such as those proposed by Zervos et al.82 or Cleijne et al.79 The blade element and strip methods include the e€ect of drag on each blade section, and this can be used to estimate the dissipated power and the turbulence kinetic energy produced, whereas vortex particle methods, which do not include aerodynamic losses, do not have this possibility. Cleijne et al.79 attempted to combine ®eld and vortex particle models in such a way that boundary conditions for the ®eld model are obtained from the vortex particle method; although there is better agreement with the experimental results in some cases, the complications and additional computing costs involved do not appear to be justi®ed. An alternative approach that requires less computing capacity is the multiparametric wake model of Voutsinas et al.,84,85 which was further developed by Cleijne et al.79 and Voutsinas et al.86 This model divides the wake into the rotor region, the near-wake region and the far-wake region, and applies a vortex particle method in the rotor region, a ®eld model in the near-wake region and, in the far-wake region, explicit self-similar expressions similar to those used in the kinematic models. Di€erent assumptions are made to match the di€erent regions. The method was partially successful in simulating the experimental results of the Nibe turbines given in Reference 39. Later, Magnusson et al.87,88 applied the model to reproduce the experimental results of Alsvik wind farm. The agreement between experimental and model results for the velocity de®cit is qualitatively reasonable, with relative errors of less than 25% and a scatter of experimental results of similar order. However, the comparison of turbulence characteristics (turbulence kinetic energy and Reynolds stress) is poorer, with relative errors that can even reach 200%, although tendencies are well predicted. All the previous models solve the Reynolds-averaged turbulence ¯ow equations and use a closure scheme, based on zero-, one- or two-equation models, to calculate the turbulence transport terms. An eddy viscosity is used in all cases which implicitly assumes an isotropic turbulence ®eld. Transport equations for the Reynolds stresses have only been used occasionally to calculate this type of wake. Ansorge et al.80 used a Reynolds stress turbulence model based on the commercial code PHOENICS and obtained reasonable results, although the computational e€ort may still be considered too great from an engineering point of view. Neither is it clear whether there was improved agreement with experiments. In general, the ®eld models give an acceptable representation of the ¯ow ®eld and a better insight into the processes governing wake development than the kinematic models. Meandering of the Wake In general, ®eld models show better agreement with wind tunnel experiments than with ®eld experiments, one reason being the meandering of the wake. When there is meandering, turbines can be signi®cantly misaligned most of the time. Whale et al.89 found signi®cant di€erences between the experimental results obtained in a wind tunnel and those of large-scale tests, although they point to other factors being partially responsible besides wake meandering, such as the scale e€ect and the in¯uence of terrain. The individual wakes calculated by both kinematic and ®eld models do not directly take into account eddies that are large in comparison with the size of the wake and which can move it bodily, a phenomenon known in studies of atmospheric dispersion as meandering. Nor is this e€ect usually included in wind tunnel tests. The maximum velocity de®cit will be smaller than that predicted by the theoretical models or wind tunnel tests, and, in addition, velocity ¯uctuations may appear that can be interpreted as an additional contribution to the turbulence kinetic energy. Baker and Walker,68 Ainslie52 and Taylor26 took meandering into account by assuming that the large eddies increase in size linearly with downstream distance x and in proportion to the standard deviation of the wind direction, sy . However, Hogstrom È È et al.20 argue that this is not the correct approach, because sy is caused by eddies of all sizes, including those that are smaller than the wake diameter, and take for their analysis a value of 0.053x for the size of large eddies, based on the results of some oil-fog experiments. Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen 4. Wind Farm Models This section is dedicated to study wake e€ects in wind farms, where there are usually many machines located in irregular terrain. The ®rst subsection is dedicated to study the interaction of several wakes, and the next one to the topographic e€ects. In this last subsection, methods to take into account simultaneously wake and topographic e€ects are also reviewed. A brief review is also made of o€shore wind farms, which are nowadays of great interest, although this subject is so broad that it will have to be treated in a separate review. Interaction of Several Wakes A wind farm consists of many wind turbines whose wakes can interact and whose turbines may be a€ected by the wakes of several machines located upstream. Wind farm codes usually rely on the results of singlewake calculations and make superposition assumptions to take into account the combined e€ect of di€erent wakes. The linear superposition of the perturbations created by wakes of di€erent machines in a wind farm model was ®rst used by Lissaman9 in a classical paper, although this assumption fails for large perturbations as it overestimates velocity de®cits and could lead to the absurd result of negative velocities when many wakes superimpose. Instead, Katic et al.36 assumed linear superposition of the squares of the velocity de®cits. In this case the cumulative e€ect, when there are many wakes, will be smaller than that calculated for linear superposition, and, in general, this assumption provides better agreement with experimental results than the linear superposition. The corresponding code, named PARK, was applied by Beyer et al.90 for the optimization of wind farm con®gurations using genetic algorithms. Voutsinas et al.38,86 formulated an explicit energy equation, also used by Kiranoudis and Maroulis,37 by assuming the total energy loss at each point of the ¯ow ®eld due to the presence of di€erent machines to be equal to the sum of the individual energy losses due to each machine. In this way they obtained the velocity ®eld and then calculated the incident velocity on each machine by taking the average over the turbine disk. To evaluate the individual energy losses of each wake, they considered the di€erence between the wake velocity and the in¯ow velocity on the machine that creates the wake, whereas Katic et al.36 considered the di€erence between ambient and wake velocity. For small velocity de®cits both methods should give similar results. Based on the idea that wake turbulence should increase di€usion when the wakes superpose, Beyer et al.91 proposed a modi®cation of the parameters describing single-wake development. However, although they obtained acceptable agreement with experimental results in the Hamswehrum wind farm, no systematic procedure is provided of how to apply the method to other con®gurations. Smith and Taylor23 found, for a particular experimental con®guration of two machines in a row, that the wake velocity of the downstream machine recovers more rapidly than the one upstream, so that, at the same relative position, the velocity de®cit is smaller in the downstream machine wake. This result contradicts the qualitative behaviour predicted by the two previous superposition assumptions and may be explained by the turbulence levels and shear stress pro®les generated by the upstream machine, which enhance momentum di€usion, leading to a faster recovery in the downstream machine. Stefanatos et al.92 also showed experimentally that the linear superposition of wakes provides a poor approximation of the ¯ow. By making a number of crude assumptions concerning the momentum transfer within the downstream wake that is imbedded in the upstream wake, Smith and Taylor23 were able to formulate a semiempirical superposition law that works quite well. However, it is cumbersome and can only be applied for the interaction of the wakes of two turbines in a row. For small velocity de®cits the method reduces to the linear superposition assumption, but it is not clear what the limit is for the quadratic superposition assumption to be recovered. Voutsinas et al.38,86 claimed that their explicit energy equation gives similar results to this method but do not give any physical explanation. When there are many turbines in a line, it has been observed experimentally93 that while the ®rst turbine produces full power, there is a signi®cant decrease in power in the second turbine, with practically no further loss in successive machines. Based on these observations and on the results of the calculations of Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods Figure 5. Velocity distribution in wake region of seven turbines forming a row. Comparison of measurements and results of several wind farm codes. Measurements and con®guration correspond to the Zeebrugge wind farm.93,94 Uref is the unperturbed upstream velocity at hub height. The UPMPARK calculations have been performed for two stability situations: neutral atmosphere (Monin±Obukhov length, L ˆ I ) and stable atmosphere (L ˆ 100 m, hub height 31 m) Crespo et al.,28 van Leuven93 assumed in his farm model (WINDPARK) that a given turbine is only a€ected by the wake of the closest upstream turbine, obtaining good agreement in comparison with measurements made at the Zeebrugge wind farm93,94 (see Figure 5). In this ®gure are also shown the results of calculations with UPMPARK,95 PARK36 and FARMS, based on the kinematic model MILLY.96,97 Regarding the increase in turbulence intensity which occurs when there are many turbines in a line, Builtjes and Vermeulen98 carried out an experimental investigation in a wind tunnel with wind turbine simulators and found that turbulence intensity reached an equilibrium value after three to four rows of turbines. They also observed that turbulence intensity had a maximum in the second row of turbines, where it was higher than the equilibrium value. Luken99 proposed a simple correlation to calculate the equilibrium value of turbulence intensity as a function of turbine spacing. Crespo et al.28 applied their elliptic model for studying the interaction of the wakes from two turbines in two con®gurations: abreast and in a line. There was good agreement with experimental results, and when other superposition assumptions were compared, it was found that the linear superposition worked well for the two machines abreast, in which velocity de®cits in the interference region are small. However, for the two turbines placed in a row the linear superposition overestimated the velocity de®cit, as was to be expected. The previously mentioned model of van Leuven93 which considers that only the wake of the closest turbine upstream acts on a given turbine also agrees well with the elliptic model. When the results of the elliptic model of Crespo et al.28 are considered, it can be observed that the truly elliptic e€ects, such as axial pressure variations, only occur very close to the turbine, so that the parabolic approximation may be a suitable approach for studying wake interactions over most of the region where this interaction occurs. Moreover, to extend the fully elliptic code to a wind farm consisting of many machines, besides consuming a lot of calculation time, would require very powerful computers and would therefore be of little practical interest for modelling wind farms. Because of this, Crespo et al.95,100 have developed a code, UPMPARK, extending the parabolic UPMWAKE code for a single wake to the case of Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen a park with many machines. No assumptions are required regarding the type of superposition or the type of wake to be used, as all the wakes and their interactions are e€ectively calculated by the code. A brief description of UPMPARK follows. The conservation equations solved are the same as those for the single-wake code UPMWAKE, as speci®ed in Reference 56, and turbulence is closed using a k±e model. The wakes of the machines di€use in an ambient ¯ow that represents the surface layer of the atmospheric boundary layer, in which instability e€ects are retained by means of the Monin±Obukhov length. For uniform terrain this ambient basic ¯ow is the same over the whole wind farm, although the code could also handle moderate terrain irregularities, using a superposition assumption for the e€ects of terrain and wakes101 that will also be reviewed below. At in®nity, in regions not perturbed by the wind turbines, and in the upstream section, boundary conditions are imposed that correspond to an unperturbed ambient ¯ow. As we progress downwind in the numerical marching procedure associated with the parabolic model, each turbine found at any crosssection of the farm acts as a source (or sink) of the three velocity components, k and e. The number of grid points should be large enough to contain the whole cross-section of the park and to consider that the lateral boundaries are at in®nity. As the code is parabolic, there is no limit to the downstream distance, except for the fact that if the wakes di€use very much, the number of grid points may not be large enough to apply the boundary conditions at in®nity. The case of wind turbines in a row is particularly suited for this code, which has been validated by comparison with measurements made on wind farms in Zeebrugge, Sexbierum and Vindeby and on the Nibe wind farm. Adams and Quarton76 extended the UPMWAKE model to wind farms using a procedure quite similar to that of UPMPARK, and compared the results obtained with this extended version of UPMWAKE and EVFARM (also extended to wind farms) with the wind tunnel experiments of Hassan77 for double-wake cases, and found a degree of agreement smaller than for the single-wake case. EVFARM predicted the velocity de®cit within 1% error for the near wake and within 10% for the far wake, whereas UPMPARK underpredicts the velocity de®cit by 33% for the near wake and by 5% for the far wake. Topographic Effects Usually, wind farm models make the assumption that the terrain is ¯at and that the unperturbed wind velocity is uniform, an assumption which is not reasonable in many cases of interest, since, as is well known, terrain irregularities can be used to enhance or concentrate wind power. Studies on the in¯uence of ambient ¯ow on the wake development are scarce. John and Schobeiri102 made an experimental study of the in¯uence of streamline curvature and longitudinal positive pressure gradient on the development of the wake of a cylinder in a two-dimensional curved channel, observing that the wake decay is slower with the positive pressure gradient than with a zero pressure gradient. The e€ect of curvature on mean velocity de®cit distribution is small, whereas it strongly a€ects the Reynolds stress distribution, particularly in the inner half of the channel. For terrains that are moderately complex, the simple procedure of adding the velocity perturbations of  the wake and terrain should give an approximate ¯ow ®eld; this procedure was applied to the Ampurdan wind farm.31,40 Measured and calculated values of power output of the wind farm as a function of wind direction were compared. A scatter of measurements of the order of 20% was found and their average was predicted with an error of less than 10%. A similar procedure was used by Adams and Quarton76 and by van Leuven93 to take into account the interaction of an obstacle and turbine wakes in the Zeebrugge wind farm. However, in all the above cases there were simultaneous interactions of terrain and several wakes, which raises some uncertainty about the validity of the results, since, as is well known, the linear superposition of several wind turbine wakes overestimates the velocity de®cit, as indicated in the previous section. Crespo et al.101 studied the Monteahumada wind farm, in which the velocity irregularities of the terrain and the velocity de®cit created by a single wake interact and are both of a similar order of magnitude; this con®guration is thus appropriate to examine the validity of the assumption of linear superposition of wake and terrain e€ects. Although the data were few and not easy to interpret, the study Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods shows that for a moderately irregular terrain the linear superposition of wake and terrain e€ects gives good results (with relative errors of the order of 10% and less than 20%), whereas for the interaction of two wakes with perturbations of a similar order of magnitude this assumption is less valid. Voutsinas et al.38 described a procedure to take into account non-uniformity in wind velocity and the curvature of the streamlines in wind farms with small terrain irregularities, which is similar to the linear superposition; some sample calculations were made, but no experiments were presented to validate the method. Stefanatos et al.103 developed another model which assumed that the vorticity fed by the wind turbine into its wake follows the streamlines of the unperturbed wind ¯ow ®eld, and compared the results with the experimental ones, obtaining a good degree of agreement in the upper part of the wake (within 5%). However, in the lower part of the wake the calculated velocity de®cits exhibit a downward displacement of one-tenth to one turbine diameter. Taylor and Smith104 made measurements in a wind tunnel which showed that the changes in the wake characteristics due to topography may be important. Hemon et al.105 made a theoretical study of how terrain may modify turbine aerodynamics and near-wake characteristics. Second-order corrections to the linear superposition of terrain and wake e€ects were made by van Oort et al.106 using PHOENICS. They found that terrain irregularity creates additional turbulent di€usion near the ground, which diminishes the wake e€ect; on the other hand, above the apex of a hill the streamlines concentrate, thereby increasing wake e€ects. Crespo et al.,100 Gunther et al.107 and Ansorge È et al.80 also used the commercial code PHOENICS to model the interaction of wakes with obstacles and terrain irregularities. Stefanatos et al.103,108 and Helmis et al.60 give some guidelines, obtained from their experimental results in both wind tunnel and large-scale tests, to study the interaction between wake and terrain. A related problem arises in o€shore wind farms where, when the wind blows from land to sea, there is an internal boundary layer, whose development is superposed on that of the wakes, as mentioned by  Crespo and Gomez-Elvira.109 As the surface roughness of the sea is usually much smaller than the corresponding roughness on land (see e.g. Reference 110), it is to be expected that wind velocity will be greater and turbulence intensities lower than for equivalent inland stations. Consequently, turbulent di€usion of the wake will also be lower and wake e€ects will probably be more persistent downstream. Wake e€ects in o€shore wind farms obtained from both experiments and numerical models are reported by Frandsen et al.70 and Crespo et al.,71 and their e€ect on fatigue loading by Frandsen.111 5. Quasi-analytical and Semiempirical Expressions to Describe Wake Evolution In many cases it is of interest for the designer to have, as an alternative to numerical models, analytical expressions which can estimate the order of magnitude and the tendencies of the most important parameters characterizing wake evolution. However, this issue will not be examined in detail in this article, which is more concerned with aspects of modelling. Regressions or correlations of this type were obtained by di€erent authors to describe single-wake behaviour: see References 20, 32, 40 and 112±114 for the velocity de®cit and the width of the wake, and References 20, 25, 27, 42, 61, 113 and 115 for turbulence intensity. Taylor26 performed a parametrization of the calculated wake magnitudes as functions of several dimensionless input parameters; however, the results were represented in graphic form and no regressions were made. The case of wind clusters is covered in a review by Luken,99 who proposed a correlation for the equilibrium value of the turbulence intensity reached in a row of turbines, using the experimental results of Builtjes and Vermeulen.98 This point is discussed in more detail by Frandsen et al.70 who presented correlations giving values of the average velocity, turbulence intensity, turbulence scale and width of the wake at di€erent positions of each machine in a row as functions of their operating characteristics. These correlations are obtained by making the best ®t with numerical results from UPMPARK, and are validated by comparison with measurements made in Vindeby wind farm. Most of the above studies express wake di€usion as a function of downstream distance made dimensionless with turbine diameter, x/D, dimensionless turbine height H/D, thrust coecient CT and ambient Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen turbulence intensity. Instead of using x/D, Magnusson112 and Magnusson and Smedman113,114 expressed wake di€usion as a function of the transport time t ˆ x/u (where u is the local incident velocity), made dimensionless with t0 , the transport time where the near wake ends. In turn, t0 is made dimensionless with the rotational frequency f. Implicitly, then, the authors were considering another parameter, the tip speed ratio pfD/u. The e€ect of atmospheric stability (expressed in terms of the Richardson number) on velocity de®cit decay was taken into account in the correlation given by Magnusson and Smedman.74 All these correlations have been compared with some experimental results and show at least an acceptable degree of agreement, although more work is needed to carry out a full comparison both among these correlations and with experimental results. 6. Loading under Wake/Wind Farm Conditions As already mentioned, there is large body of data available in the literature which has been obtained from measurements made of the ¯ow characteristics of wind turbine wakes, and considerable e€orts have been made to develop numerical models, whose results are in many cases in good agreement with the experimental results. However, although such wake measurements and modelling have focused on energy production and loads, relatively few direct measurements of structural loads under wake conditions have been made, despite the shortcomings of the international standard on loads and safety116 and the lack of guidelines on how to take loads into account. Dynamic and Fatigue Loading The most serious expected structural e€ect on a wind turbine which is in the wake of a neighbouring machine is fatigue. The combined e€ect of increased turbulence, wind speed de®cit and changes in turbulence structure causes dynamic loading which may excite the wind turbine structure. The apparent e€ect on, for example, ¯apwise bending moments depends primarily on the distance to the neighbouring machine. As mentioned in Section 3, two wake regions can be considered: the near wake, which extends over three rotor diameters, and the far wake. The three-diameter length is chosen as the distance below which the e€ect on a machine which is partly in the wake of another (half-wake load case) is clearly visible on, for example, blade loads. Near Wake Some of the ®rst measurements of fatigue loading in wakes were made in the early 1990s by Vùlund117 and Stiesdal.118 The measurements made by Vùlund117 on a 250 kW machine, which was in the wake of a turbine of similar size placed two rotor diameters upstream, showed an increase in the standard deviation of ¯apwise bending moment of approximately 100% relative to the unobstructed case. Measurements also showed that loads were largest when the machine was exposed to half-wake conditions. The concept of equivalent load, which is the amplitude of a sinusoidal load with a ®xed frequency that would generate the same fatigue damage as the actual (random) load, was introduced by Stiesdal118 to provide a more precise fatigue measurement than the standard deviation. Here the separation between the 450 kW machine on which measurements were made and the wake-generating machine was 2.5D. The equivalent ¯apwise load increase was about 100% at 12 m s À1 and lower for high wind speeds. When loading under wake conditions was weighted with non-wake operation, the average e€ect was much smaller (10%±20% in terms of equivalent load). Far Wake Frandsen and Christensen119 made measurements in the large wind turbine array Nùrrekaer Enge II in Denmark, consisting of 42 machines of 300 kW, separated by 6D±8D. Two turbines in opposite corners Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods Figure 6. Schematic sketch of Vindeby wind farm were instrumented to measure stresses in towers and blades. To the north of the wind farm there is open water and the surrounding terrain is farmland. For northerly winds one of the instrumented machines is exposed to the low turbulence of the sea and the other one to the turbulence environment of the interior of the wind farm. For southerly winds one instrumented machine is exposed to the wind farm turbulence while the other is open to the free (ambient) ¯ow o€ the farmland. In both cases there was a clear increase in the equivalent load when the instrumented machine was in the direct wake of a neighbouring machine, but the integrated e€ect of wake and non-wake operation was signi®cantly smaller. It was found that the integrated e€ect in terms of fatigue loading for the higher ambient turbulence (southerly wind) was insigni®cant. For low ambient turbulence (northerly wind) the added loading in terms of equivalent load due to wake e€ects of the wind farm was approximately 10%. Frandsen and Thomsen120 carried out similar measurements in the Danish o€shore Vindeby wind farm. The wind farm, which is represented schematically in Figure 6, consists of 11 machines of 450 kW which are arranged in two rows, each machine and each row separated by approximately 8D. Two instrumented masts provided measurements of ambient and wake ¯ow parameters. The geometry of the wind farm and the instrumentation allowed for both single- and multiple-wake measurements. Data, including equivalent loads, were recorded for 2 years. An example of the measured data is shown in Figure 7, where equivalent widths of the ¯apwise bending moment of a blade are plotted against wind direction. Each data point represents 0.5 h of operation. For a wind direction of around 2558 the machine rows are aligned with the wind. As can be seen, despite the fairly large spacing between wind turbines, the fatigue load increase in the wake is signi®cant, about 80%. The data shown for other wind directions demonstrate that there is no appreciable di€erence between single- and multiple-wake loads. In Figure 7. Equivalent load widths of ¯apwise bending moment of instrumented units 4 W and 5E in Vineby wind farm Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen Reference 120 a model for adding up the wake e€ects was devised to reproduce the integrated e€ect of the individual wakes. As in the case of Nùrrekaer Enge II, the integrated e€ect was much lower. It was also estimated that the integrated fatigue loading would be approximately 20% lower in o€shore conditions than in onshore sites. The Sexbierum wind farm in the Netherlands consists of 18 units of 300 kW, placed in three rows, each with six machines.121 The machines are spaced 5D or 10D apart and the rows are separated by 8D. Six meteorological towers provided information on ¯ow characteristics. As in the Vindeby wind farm, it was found that equivalent loads under single- and multiple-wake conditions were very similar. Measurements have been made on one wind turbine in the experimental Alsvik wind farm in Sweden,122 which has four machines sited so that the instrumented unit is exposed to 5D, 7D, or 9.5D single-wake loads, depending on wind direction. The terrain is smooth and the ambient turbulence low. The wind farm layout is unique in o€ering many experimental possibilities. It was found that when the instrumented machine is within 7D to 9.5D downstream of another machine, there are no signi®cant half-wake loads. However, the e€ect on a machine which is only partly in the wake of another one is di€erent in the 5D situation. It was also found that under full-wake conditions the equivalent load is increased by 10% at 9.5D and up to 45% at 5D. The Kappel wind farm in Denmark consists of 24 units of 400 kW, sited in a row on a westerly shoreline, with 3.7D separations. Thomsen et al.123 discussed whether fatigue load increased (in the integrated sense) under wake conditions, arguing that the load increase would be smaller for high-wind cases, as wake e€ects are lower in such conditions. Nevertheless, the experimental evidence pointed to a signi®cant increase in the equivalent load, which represents a fatigue loading. Modelling As mentioned above, there have only been limited e€orts made towards directly estimating wake loading. In References 75 and 120 the task was solved by altering the ( free ¯ow) design turbulence. Thus the change in design turbulence accounted for intermittent wake conditions with high turbulence, smaller turbulence scale and half-wake cases. Tindal et al.75 presented an elaborate scheme of how to summarize the numerous load cases. The reader is referred to the report for more details. In Reference 120 a simpli®ed model of taking the increase in fatigue loading into account was discussed and, to some extent, veri®ed by the data presented. Similar models have previously been applied124 and di€erent expressions have been proposed for the `virtual' or `ecient' turbulence intensity that will result in the same fatigue damage occurring as in the ¯ow-wise complicated environment of a wind farm array. This turbulence intensity depends on the geometry of the array, the thrust coecient of the wind turbines, the separation of wind turbines and the ambient turbulence. Despite the mentioned complexity of the ¯ow in wakes/wind farms, the model proposed by Frandsen and Thomsen120 has been demonstrated to work for turbine separations larger than 3D±4D and, with some modi®cations, also for smaller separations. Extreme Loading Extreme loads are to be expected occasionally during the lifetime of a wind turbine. Extreme loading may occur either under common wind conditions with the wind turbine in operation or under extreme wind conditions (e.g. 50 years return-period wind gust). Loads on an operating machine increase with increasing wind speed up to a certain velocity, above which they may still continue to increase, depending on the turbine design.120 At wind speeds higher than about 15 m s À1, wake e€ects rapidly diminish, since the thrust coecient decreases, making the wind farm more `transparent'. Under extreme wind conditions the wind turbines will be parked, with little or no wake e€ects from neighbouring machines. However, extreme loading may also occur during non-extreme wind conditions, since (1) the typical blade tip speed of large machines is about 60 m s À1 and the loads may be correspondingly high, (2) large yaw errors generate large dynamic response which may also cause large extremes, and (3) transient loads caused, for Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999) Survey of Wake-modelling Methods example, by emergency stops may signify the largest loads for one or more machine components in the machine's lifetime. This case has not been satisfactorily dealt with in the literature. 7. State of the Art and Conclusions There are many di€erent models which simulate the behaviour of wind turbine wakes and wind farms. Most of them are based on a deterministic simulation which takes into account each individual wind turbine. Models considering the turbines as distributed roughness elements are not much used now, although they may be of interest in the future to predict overall changes in wind characteristics originated by the large wind farms which are likely to be established. Many of the models proposed show an acceptable degree of agreement with the experiments with which they are compared. However, the assumptions and coecients that are chosen are such that the agreement with some particular experiments may be good, although the overall validity has not been checked in more general situations. The models which depend on the least simplifying assumptions are better suited in dealing with di€erent con®gurations and in reproducing wake development in more detail. For example, an axisymmetric model will never be able to reproduce the peak in turbulence intensity in the upper part of the shear layer in the near wake. In general, the more complicated models are more likely to accurately reproduce ¯ow ®eld characteristics, although in many cases the physical reasons for the hypotheses used, particularly in those aspects related to turbulence modelling, are not always clear. The classical wind farm model relies on an individual wake model, usually a kinematic model, and some sort of superposition assumption. In general, the superposition assumptions cannot be justi®ed from a physical point of view and may even lead to absurd or contradictory results; the corrections and alternatives necessary to handle such physically unrealistic situations are either unjusti®able or dicult to implement. Although it is a generally held opinion that ®eld models are too complicated and that they are impossible to extend to a wind farm consisting of many turbines, we do not think that this is still necessarily the case. For example, the UPMPARK model, which retains all the characteristics of one of the most complete non-symmetric k±e wake models, UPMWAKE, can be successfully run on a workstation in reasonably short times for large wind farms such as Sexbierum or Taendpibe, and even on a PC for wind farms with a smaller number of machines. The most important simplifying assumption used by UPMPARK is the parabolization of the mathematical problem in the main wind direction. While greater emphasis used to be directed to calculations of velocity de®cits and park eciency in terms of energy production, calculations are nowadays more orientated to other issues, such as estimating magnitudes related to the structural and fatigue behaviour or ¯uctuations in the electrical energy produced by machines a€ected by upstream wakes. To estimate these magnitudes, it is necessary to know the turbulence characteristics of the ¯ow (turbulence intensity, correlations and spectrum) and wind shear data, which clearly cannot be provided by simple kinematic models. There is experimental evidence that wake e€ects may signi®cantly increase the equivalent load experienced by a turbine under wake conditions, although further research is still necessary for a more precise estimation of their in¯uence in di€erent operational conditions. However, measurements also show that the integrated loading e€ect of wake and non-wake operation may be, in some circumstances, signi®cantly smaller than that measured under continuous wake conditions. It has also been found that under multiple-wake conditions the equivalent loads due to di€erent wakes are not additive. Only limited e€orts have been made towards developing predictive models to estimate the loading increase due to wake e€ects. An issue of some importance is the non-isotropic nature of the turbulence of ambient atmospheric ¯ow, in contrast with the more isotropic turbulence in the wakes. This problem cannot be dealt with by the k±e or eddy viscosity wake models. The use of Reynolds stress models involves greatly increased mathematical Copyright # 1999 John Wiley & Sons, Ltd. Wind Energ., 2, 1±24 (1999)  A. Crespo, J. Hernandez and S. Frandsen diculties and we do not think it can be treated by the engineering codes in general use. Alternative strategies should be explored. One of the most important diculties that has not been treated satisfactorily is the choice of appropriate input parameters to de®ne ambient unperturbed ¯ow, particularly in complicated terrains. Usually, a comparison with wind tunnel experiments is reasonably straightforward, but when ®eld experiments are compared, there are many diculties, and e€ects such as meandering have never been satisfactorily modelled. The results obtained from experimental and modelling studies for terrains of varying roughness and the appearance of internal boundary layers, such as those observed in wind farms located near the coast to o€shore, should be incorporated into the description of ambient ¯ow. For a terrain that is moderately irregular, UPMPARK assumes a superposition of the perturbations due to the wakes and those of the terrain, which are estimated either from measurements or from codes such as WASP. However, for complicated topography this approach may not work. A code which simultaneously takes into account terrain and wind turbine wakes would be too dicult to apply and still contain many uncertainties regarding the appropriate boundary conditions. Some work has been done, but more is needed, towards estimating the local e€ects of interference of single wakes and terrain irregularities. The problem is that it is dicult to envisage solutions and we will always be solving particular problems that, at most, could only point to general tendencies. A possible alternative for cases in which turbine spacing is small compared with the characteristic length of variation in terrain irregularities could be to treat the problem as one of ¯ow over an irregular terrain of changing roughness, as indicated in Section 2. Although considerable progress has been made since 1992, some of the conclusions and recommendations presented here are of a similar nature to those formulated in the ®nal report of IEA, Annex IX on Wake E€ects.3,125

Journal

Wind EnergyWiley

Published: Jan 1, 1999

There are no references for this article.