# State of the Art and Trends in Wind Resource Assessment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Anemometry and Remote Sensing

**Figure 1.**Photographs of (a) a tower-based measurement system with three redundant sets of cup anemometers at three heights, (b) a sodar unit on a trailer and equipped with a photovoltaic power system.

#### 2.1. Cup anemometry

_{d,v}and C

_{d,x}, respectively [5,8]:

_{0}is assumed, changing abruptly to U

_{1}=U

_{0}+ ΔU at a time t = 0, then the change in wind speed as a function of time is given by [5]

_{a}of the anemometer, which is only a property of the instrument itself and does not depend on the wind speed signal, as opposed to the time constant which is inversely proportional to the value of the final wind speed. In principle, the distance constant could be measured in a wind tunnel subjecting the anemometer to a step change, say from zero to a finite value of the wind speed. However, in practice, most of the increase in rotational frequency typically occurs during the first three full rotations, so the determination of L

_{a}from a fit of the experimental data to Equation (5) becomes very imprecise. Kristensen and Hansen [5] proposed an alternative to compare the field response of the cup anemometer under observation with an ultrasonic anemometer, the time response of which is significantly faster than the one of the cup anemometer. The cup anemometer transfer function, determined theoretically as

_{a}= 1.8 m was found [5]. A commercial anemometer may have larger values of up to about 3 m.

**Figure 2.**Illustration of the different response of a cup anemometer for positive and negative wind speed changes, respectively, for two values of the distant constant (1.8 m and 3 m).

#### 2.2. Ultrasonic sensors

_{12}and t

_{21}are the flight times from transceiver 1 to 2 and vice versa, and l is the distance between the transceivers; see Figure 3. Two or three orthogonal measurement paths can be combined in one instrument, in order to measure the full wind velocity vector. In principle, sonic anemometers have a series of advantages over cup anemometers, such as the absence of moving parts, allowing a faster response to fluctuations and avoiding overspeeding. Furthermore, their response is linear over a large range of frequencies, and the measurement is relatively independent of the flow properties, such as spatial and time variations, temperature, density etc. Finally, sonic anemometers are absolute instruments that do not require individual calibrations [7]. Some drawbacks do exist, such as the influence of the finite measurement path, path separation, and transducer shadows.

**Figure 3.**Geometry of ultrasonic transceiver arrangements (one measurement path). (a) Separate transmitters / receivers for each direction, (b) integrated transceiving units.

_{12}and t

_{21}can be calculated from [9,11]

_{p}(p,t) is the instantaneous wind speed at a position p between the path extremes p

_{1}and p

_{2}at a time t (see Figure 3) Both pulses are assumed to have been fired simultaneously. If the wind is constant in space and time (u

_{p}(p,t) = u), then equation (8) correctly predicts u

_{M}= u. In the case of a wind field varying in space and time, however, it can be expected that the fluctuations will contribute to the measured wind speed value. Using perturbation theory, Cuerva and Sanz-Andrés [11] derive an expression for the measured wind speed in the general case of a fluctuating wind field:

^{+}and F

^{–}account for the differences in traveling time:

_{0}

^{±}the travel times in either direction in the absence of the perturbation. As an example of the theory, Cuerva and Sanz-Andrés discuss the effect of an oscillating measurement mast in response to, e.g., vortex shedding. For an oscillating tower with an angular frequency ω the relative measurement error is found to be

_{0}is the time delay between the maximum of the oscillation and the pulse emission. Note that for typical values of the shedding frequency of a few Hz $\omega \text{}l/c1$, so that $\epsilon \text{}\omega \text{}l/c$ is a second order small quantity. Moreover, for a random phase distribution for the vortex shedding process, the average of the cosine function will be zero for typical data logger averaging intervals in the range of 1 to 10 minutes.

**k**with components k

_{1}, k

_{2}, and k

_{3}in the three orthogonal directions of the wind velocity coordinate system. Ψ

_{ij}(

**k**,t) are random functions which can be related to the spectral density tensor Φ

_{ij}(

**k**) by

**k**) is the three-dimensional spectrum which can be expressed as [12]

_{1}, can be obtained by integrating over the remaining coordinates:

_{ij}by setting

_{ij}

^{M}is the one-dimensional spectrum determined for the wind velocity as measured by the sonic anemometer. If the special case of an alignment of the measurement path with one of the components of the wind velocity, say x, is considered, then the corresponding transfer function R

_{11}can be calculated as follows [11]

_{∞}is the Mach number of the free-stream wind speed, l the length of the measurement path as before, and Z

_{B}is the non-dimensional delay time between pulse shots. For practical purposes, R

_{11}can be interpreted as a transfer function of the sonic anemometer [11,12], measuring the response of the instrument to a turbulent perturbation with wave number k

_{1}.

_{11}has been plotted for different Mach numbers under the assumption of simultaneous emission of both measurement pulses, the influence of the Mach number on the instrument response is negligible for wind speeds in the interesting range, since even for a Mach number of 0.1 (equivalent to a wind speed of 34.5 m/s) the response function is essentially identical to the one for M

_{∞}= 0 (labeled as “Kaimal” in Figure 4). Not unexpectedly, for very large wind speed values the finite speed of sound becomes a limiting factor, reducing the spatial frequencies that can be resolved with the instrument. The fact that often the two opposite sound pulses are fired in a sequential manner, which gives rise to a finite delay time, does have an important impact on the instrument response. This can be seen in Figure 5, where the response function has been plotted for different values of the normalized delay time.

**Figure 4.**Transfer function of a sonic anemometer for different Mach numbers as a function of the dimensionless spatial frequency k

_{1}l. The non-dimensional delay time is Z

_{B}= 0 for all curves.

#### 2.3. Sonic detection and ranging (SODAR)

**so**nic

**d**etection

**a**nd

**r**anging) [13,14,15,16,17]. sodar was originally developed for atmospheric research [13,14,15] and air traffic safety but is increasingly deployed where a knowledge of the wind velocity profile up to greater heights than provided by anemometer towers (generally limited to 60 m or 80 m) is required, or the assessment of different locations within a vast wind project development is desirable. Commercially available systems tailored for wind resource assessment are capable of providing information on the atmospheric boundary layer up to heights of 200 m or 300 m, thereby covering the full range of heights swept by typical rotor blades. This type of information is particularly useful when dynamic structural analysis of the turbine rotor is performed, since both wind shear over the rotor diameter and the evolution of turbulence with height significantly impact on the prediction of rotor stress, and their precise knowledge avoids the use of sometimes oversimplifying assumptions.

**Figure 5.**Transfer function of a sonic anemometer as a function of the dimensionless spatial frequency k

_{1}l for different normalized delay times. M

_{∞}= 0.1 except for the Kaimal case (M

_{∞}= 0) shown for reference purposes.

_{0}the frequency at which the sonic beam is emitted and Δf the frequency shift registered upon receiving the backscattered signal, then the following relationships hold [16]:

_{x}and θ

_{y}are the tilt angles for the x- and y-beams respectively. The power levels of the backscattered signals are very low, as can be seen from the following equation, termed the sodar equation [18]:

_{R}and P

_{T}are the receive and transmit power, respectively, G is the antenna transmit efficiency, A

_{e}the effective receive area, σ

_{s}the cross section for turbulent scattering of sound in air per unit volume and unit solid angle, c the speed of sound, τ the pulse length, α the sound absorption coefficient and z the measurement height. The scattering cross section can be calculated from [19]

_{T}

^{2}is the structure function coefficient for temperature fluctuations defined as [20]

**r**and

**r**+

**r**

_{0}are the position vectors of two locations in the atmospheric boundary layer and the temperature structure function is assumed to depend only on the difference vector

**r**of the two positions.

**Figure 6.**Vertical wind speed and direction profiles as measured with a single frequency three-beam sodar device. Data courtesy of Atmospheric Research and Technology, Inc. [21].

**Figure 7.**Scatter plot showing the correlation of the sodar-measured (vector) wind speed at 50 m and the anemometer-measured (scalar) wind speed at 54 m above ground level. Courtesy of Atmospheric Research and Technology, Inc. Reproduced with permission [21].

#### 2.4. Light detection and ranging (lidar)

^{2}= 0.95 and higher), with better correlations for wind speed averages than for their standard deviations. Jaynes et al. [27] find the correlations are limited more by uncertainty sources at the reference tower than at the lidar, where most of the error sources could be traced back to interference with the tower structure. Due to encouraging initial experiences and despite its relatively high cost it can be expected that lidar units will play an increasingly important role for wind resource assessment and wind turbine performance certification.

#### 2.5. Tower shading effects

_{T}is the drag coefficient (dependent on both construction type and porosity of the towers), L is the horizontal tower face width, and R is the distance from the geometrical center of the tower. Expressions for C

_{T}can be obtained from national building codes, such as the Danish wind loading code [8], where the following expression has been proposed:

_{geo}is a drag factor taken as 4.4 for towers with square plane sections and sharp edges, 2.6 for square towers with round edges and 2.1 for triangular towers with round edges. Considering a conservative (high) value of C

_{T}= 0.6 and also a conservative value of L/R = 0.2 (i.e., corresponding to a boom length five times the face width of the tower) an upwind center-line velocity deficit of about 1.2% is obtained.

_{T}= 0.6 (calculated in this case directly from the Navier-Stokes equations) and a normalized inverse distance of d/R = 0.2 a velocity deficit of 1.6% is obtained.

**Figure 9.**Ratio of wind speed readings recorded by a pair of anemometers mounted on a lattice tower. The continuous line represents bin-averages for a bin width of 10°. Error bars correspond to ± one standard deviation.

## 3. Wind Shear

#### 3.1. General aspects

_{0}the local roughness length, κ the van Karman constant (usually taken as 0.4) and ψ

_{m}(z/L) the stability function accounting for different thermal conditions of the atmosphere known as stable, unstable and neutral. L is the so-called (Monin-) Obukhow length defined by [35]

_{v}is the virtual temperature defined as

_{v}is the mixing ratio and ε (=0.622) the ratio of the gas constants of air and water vapor and Q

_{v}the vertical heat flux. The stability function distinguishes between three stability regimes known as stable, unstable and neutral. Under neutral conditions the vertical heat flux vanishes and the stability function is zero (adiabatic case). In this case the vertical wind speed profile in the surface boundary layer becomes logarithmic. If a net upward heat flux occurs, then the Obukhov length is negative and the surface boundary layer is dominated by convection, giving rise to unstable conditions. Convection arises either as a consequence of the absorption of solar radiation by the ground or the advection of cold air over a warm surface; in both cases the result is an increased temperature lapse rate which drives convection. Stable conditions arise when the temperature decreases less with height than in the adiabatic case or even increases (in case of an inversion).

#### 3.2. Assessment of roughness length under neutral conditions

_{1}and v

_{2}are the average wind speed readings at the two heights z

_{1}and z

_{2}above ground level, and the wind speed ratio δ = v

_{2}/v

_{1}has been introduced. It is clear from a quick inspection of the formula that the result for ln(z

_{0}) will depend strongly on δ, given the fact that δ−1 is generally a small number. After some algebra it can be shown that the standard error for ln(z

_{0}) can be expressed as

_{12}is the theoretical wind speed ratio derived from the true roughness length z

_{0,true}

_{1}and r

_{2}the relative errors of the wind speed measurements at the two heights, and ρ

_{12}the correlation coefficient between the two sets of measurements. z

_{0,true}is of course not known a priori and is approximated by the value obtained from equation (37). For ideally correlated wind speed time series (ρ

_{12}= 1) at the two heights z

_{1}and z

_{2}and equal measurement errors (r

_{1}= r

_{2}) the square root in Equation (38) vanishes and therefore the standard error for ln(z

_{0}). In practice, due to the three-dimensional structure of the wind field and the effect of the tower and booms, the correlation is not perfect, however, ρ

_{12}values in the range of 0.990 to 0.995 are typical for differences in measurement height of 20 m. Anemometer uncertainties for individually calibrated instruments are of the order of 0.7%; uncertainties associated to tower wake effects are generally of the order of 0.1% and are therefore negligible compared to calibration uncertainties. From Figure 10 it can be seen how the relative error of ln(z

_{0}) varies as a function of ln(z

_{0}) for two different assumptions for the anemometer uncertainty (0.7% and 1.4%, respectively) and two values of the correlation coefficient ρ

_{12}(0.990 and 0.995, respectively). It is conspicuous that for a high degree of correlation (ρ

_{12}= 0.995) and an uncertainty representative of a calibrated anemometer (0.7%) the relative uncertainty of ln(z

_{0}) is of the order of 3%, except for rough terrain (z

_{0}~ 0.5 m) where σ

_{ln(z0)}/ln(z

_{0}) rises to about 5%. If uncalibrated anemometers are used and a lesser degree of correlation exists between the wind speed time series at the two heights used for extrapolation, then the relative error of ln(z

_{0}) can be of the order of 10%, especially for rough terrain. The uncertainty in the logarithmic roughness length translates into a corresponding uncertainty for the ratio γ

_{23}of the wind speeds at the hub height z

_{3}and z

_{2}:

**Figure 10.**Relative error of the natural logarithm of the roughness length as a function of the natural logarithm of the true roughness length for different anemometer uncertainties and correlation coefficients.

_{0}, covering a range of about 10

^{−4}to 2 × 10

^{−3}, depending on the specific assumptions. While σ

_{ln(z0)}/ln(z

_{0}) is approximately constant over many orders of magnitude, with an increase only for high values of the roughness length, the relative error of γ

_{23}increases monotonically with roughness. Even then, it remains at very small values for all parameter values studied. It should be mentioned, however, that while statistical errors have a relatively small impact on the wind speed prediction at hub height, systematic errors such as those caused by tower wake effects described above may lead to more severe errors in the prediction of wind speed at greater heights, so great care should be exercised when designing and installing sensors on a tower.

**Figure 11.**Relative error of the wind speed ratio γ

_{23}= v(z

_{3})/v(z

_{2}) for z

_{2}= 60 m and z

_{3}= 80 m as a function of the logarithmic roughness length.

_{0}.

_{0}, roughness height H, displacement height d and blending height z* for a series of homogeneous terrain types.

_{0}can therefore be determined from the observations of the turbulence intensity. It should be noticed, however, that often isolated surface obstacles such as trees, buildings and small hills that are large enough to create considerable wake turbulence, are present in the upwind fetch. If the distance between the obstacles is large enough (greater than about 15 times the obstacle height), the mean wind speed adapts quickly to the local terrain between the large roughness elements. However, the local value of z

_{0}obtained from these wind speed profiles should not be used to obtain the turbulence intensities because the large-scale wake turbulence created by these obstacles is quite persistent and takes a much longer distance to adapt to the local terrain [36]. The equivalent roughness length values z

_{0,equiv}to be used in this case for the determination of turbulence intensity are given in Table 2.

**Table 1.**Roughness parameters for homogeneous terrain classes [36].

Surface type | z_{0} (cm) | H (cm) | d (cm) | z* (cm) (20z_{0}) |
---|---|---|---|---|

Sea, loose sand and snow | 0.02 (U-dep.) | - | - | - |

Concrete, flat desert, tidal flat | 0.02-0.05 | - | - | 0.4-1.0 |

Flat snow field | 0.01-0.07 | - | - | 0.2-1.4 |

Rough ice field | 0.1-1.2 | - | - | 2.0-2.4 |

Fallow ground | 0.1-0.4 | - | - | 2.0-8.0 |

Short grass and moss | 0.8-3.0 | 2.5-5.0 | 1.3-3.5 | 16-60 |

Surface type | z_{0} (cm) | H (cm) | d (cm) | z* (cm) (1.5H) |

Long grass, and heather | 0.02-0.06 | 0.03-0.06 | 0.1-0.3 | 0.05-0.09 |

Low mature agricultural crops | 0.04-0.09 | 0.3-1.05 | 0.2-0.9 | 0.45-1.6 |

High mature agricultural crops (grain) | 0.12-0.18 | 1.0-2.6 | 0.6-1.5 | 1.5-3.9 |

Continuous bush land | 0.35-0.45 | 2.3-3.0 | 1.8-2.4 | 3.45-4.5 |

Mature pine forest | 0.8-1.6 | 10.0-27.0 | 10.0-17.0 | 15.0-40.5 |

Tropical forest | 1.7-2.3 | 20.0-35.0 | 27.0-31.0 | 30.0-52.5 |

Dense low buildings (suburb) | 0.4-0.7 | 5.0-8.0 | 3.5-5.6 | 7.5-12.0 |

Regular-built large town | 0.7-1.5 | 10.0-20.0 | 7.0-14.0 | 15.0-30.0 |

**Table 2.**Roughness parameters for heterogeneous terrain classes [36].

Surface type | z_{0,equ} (cm) | H (cm) | D (cm) | z* (cm) (1.5H) |
---|---|---|---|---|

Many fields ~200 m wide, separated by thin porous hedges | 0.09 | 4 | 0.5 | 6 |

Sparse bush area (~500 m radius) surrounded by fallow savanna | 0.17 | 2.3 | 0.9 | 3.5 |

Pasture and some narrow woods | 0.25 | 20 | 7 | 30 |

Some narrow woods in open fields | 0.3–0.4 | 10 | 5 | 15 |

Scrawny 1.5 m wide trees on savanna (~20 m spacing) | 0.4 | 8 | 5 | 12 |

Tiger bushes ~20 m wide with ~50 m bare interspaces | 0.5 | 3 | 2 | 4.5 |

Regular bush land | 0.43 | 2.3 | 1.8 | 3.5 |

Woods and some pasture fields | 0.85 | 20 | 12 | 30 |

Scrawny 2 m wide trees on savanna (~10 m spacing) | 0.9 | 9.5 | 7 | 15 |

Woods with some large clearings | 1.3 | 10 | 5 | 15 |

Broken forest | 1.2 | 20 | 6 | 30 |

#### 3.3. The effects of atmospheric stability

_{m}(z/L) = 0. It should be noted, that the departure from the logarithmic profile is generally small and a fit of equation (42) to the measured wind speed data is unlikely to provide the correct value of the Obukhov length, unless a range of vertical wind speed values, such as from SODAR measurements, is available. In the common wind resource assessment practice, however, measurements are performed at a few discrete heights (typically 40 m, 50 m, and 60 m) and the measured wind speed values generally adjust well to a logarithmic profile of the form

_{0,eff}is an effective roughness length, incorporating the effects of both surface roughness and atmospheric stability. As shown in Figure 12 where the values of the apparent roughness or equivalently, the exponents α of a power-law fit of the form v(z) = Cz

^{α}with a constant C have been plotted for a sample case, z

_{0,eff}generally shows a pronounced daily profile, corresponding to the variation from stable stratification during nighttime hours (corresponding to high effective roughness lengths) and very low values during daytime hours when convection and the corresponding vertical momentum transfer drastically reduces the wind shear at typical measurement heights.

**Figure 12.**Daily variation of apparent surface roughness for a six-month measurement period compared to the corresponding variation in relative turbulence intensity and fluctuations in wind direction.

_{0,eff}from measurements of the wind speed at typical monitoring heights of 40 m, 50 m, and 60 m for a range of values of the true roughness length z

_{0}. The results have been plotted in Figure 13.

_{0,eff}equals 0.1m during evening and nighttime hours and drops to about 10

^{−4}m during the daytime hours, with the transition occurring in about two hours during the morning and some three hours in the evening. As evidenced by the graphs for turbulence intensity TI ($={\sigma}_{v}/\overline{v}$) and a corresponding measure for the variation of the wind direction ($={\sigma}_{\theta}/\overline{\theta}$), this reduction in apparent roughness is a direct consequence of the onset of convection during the morning hours, triggered by the absorption of sunlight on the ground. Clearly, the use of an average roughness length to extrapolate the wind speed readings to hub height would under-predict the hub height wind speed during nighttime hours and over-predict it during the daytime. If the wind speed measurements are conducted at 40 m, 50 m and 60 m and the turbine hub height is 80 m, then sufficient accuracy may be obtained if extrapolation is performed for each time step t

_{i}using the effective roughness length z

_{0,eff}(t

_{i}). If the gap to be bridged by the extrapolation is higher, it may be advisable to determine the Monin-Obukhov length L at every time step for a more accurate prediction of hub height wind speed. The vertical heat flux required for the determination of L can be calculated from

_{p}the specific heat of air. The measurement of Q

_{v}requires an anemometer with a short length constant and a data logger capable of calculating the instantaneous product w’T’ before averaging.

**Figure 13.**Effective or apparent roughness length as a function of the stability parameter 10 m/L for different values of the true roughness length, both under stable and unstable conditions. Inset: Magnified portion of the negative range of 10 m/L (unstable stratification).

_{v}from

_{0}= 100W/m

^{2}. Evidently, Equation (48) represents a gross oversimplification of the problem, so care has to be taken before using this approach for vertical extrapolation.

## 4. Long-Term Assessment and MCP (Measure-Correlate-Predict) Methods

#### 4.1. General approach

_{site}and v

_{met}of the site under development and a suitable reference station, respectively [42,43,44,45]:

_{site,long}(v

_{site}) from the knowledge of the corresponding distribution at the reference site. Due to the conservation of probability we have

#### 4.2. Linear methods

^{2}‑value and standard errors for each of the fit parameters and the predicted wind speed. While the standard statistical tests for significance and the calculated standard error provide a useful first estimate, a low standard error is no guarantee for an accurate prediction of the long term wind resource (as characterized by mean wind speed, mean wind power density, wind speed distribution and related parameters), since the fit parameters in Equation (53) may change over time, responding to intra- and inter-annual variations of the regional wind flow pattern.

_{ij}running over site direction indices i and met direction indices j and subjected to the condition ${\sum}_{j}{Z}_{ij}=1$. The sectorized site wind speed data are then correlated with the met data by either

_{i}, c

_{i}) are the linear fit parameters obtained for concurrent wind speed data in the angular sector i. In the first approach, the sectorized regression relationships contributing to a given site wind direction sector j are weight-averaged, while in the second one, the met wind speeds themselves are averaged before applying the sector regression relation for the site wind direction sensor j. The authors state that the second approach yields better back-prediction results for their cases. They then go on to demonstrate their method for one site pair with 100 observation days and a pronounced “channeling” effect, i.e., data showing a systematic veer between the bearings of met and site towers. While the overall predicted wind speed was found to be almost identical for a standard linear regression, a veer-correcting MCP technique and the matrix-averaging method proposed by the authors, the angular average wind speeds were found to be better predicted by the new method. No details on the intervals used for obtaining the sectorized linear fit parameters and the back-prediction period were given however, and the authors recognize that the concurrency period for the one period studied was too short to make statements about the performance of the method under intra- or inter-annual fluctuations.

**Figure 14.**Daily profiles for a prospective at 60 m height above ground and downscaled to 10 m using hourly wind shear coefficients, as well as the 10 m met station profile.

_{0}= b and a

_{1}= m we have

**Figure 15.**Regression plots for hourly wind speed values. (a) Correlation of 60 m site data with 10 m met data. (b) Correlation between the downscaled (10 m) site data and the 10 m met data.

^{2}-value for the original two data sets was 0.63, after downscaling the R

^{2}-value increased to 0.70. Moreover, the vertical axis intercept decreased significantly from 3.57 to 1.79, resulting in a greatly reduced wind speed distribution narrowing effect.

#### 4.2. Non-linear approaches

**Figure 16.**Mean wind speed and wind power density metrics (=predicted/true wind speed and wind power density, respectively) for the 10 pairs of sites [46].

#### 4.3. The choice of reference data

**Figure 17.**Evolution of the standard error of an MCP-derived wind speed prediction as a function of the averaging interval compared to the standard error obtained by adding normally distributed noise.

## 5. Extreme Winds

#### 5.1. Types of extreme wind

#### 5.2. Extreme wind statistics in well-behaved climates

_{E}(v) as the probability that the wind speed v will be exceeded during a one-year period. Closely related with P

_{E}is the return period T

_{R}defined by [54,55,56]

_{R}can be interpreted as the average number of years that have to pass before such an event arises. Note that the definition above applies in principle to any suitable average value of the wind speed, but 10‑minute averages v

_{10 min}(sometimes referred to as sustained wind speeds) and 3-second averages v

_{3s}(generally termed gusts) are frequent choices. Electronic data loggers generally record both the 10-minute average and the highest 3s-gust that occurred during the 10-minute averaging interval. Clearly, it is very important to state whether a given return period refers to a sustained wind speed value or a gust, since the gust by definition is higher than the average. Another common choice for a representative averaging interval is the concept of the annual fastest-mile wind speed used in the US. The fastest-mile wind is the peak wind speed averaged over a distance of one mile passing through the anemometer.

_{E}is then calculated from [56] (see also [54] for an earlier approach):

_{R}one can now plot the inverse of the cumulative distribution function, the so-called percent point function, vs. the measured extreme values of the wind speed and obtain the best linear fit to the data. From equation (62) we have

_{R}, i.e., ln(–ln(1–1/T

_{R})) should be a linear function of the extreme wind speed V. Often, the highest yearly wind speed V

_{year}is used as the independent variable; T

_{R}is then specified in years. At wind farm development sites, on the other hand, often only short-term records are available. Then, using monthly maximal sustained wind speeds can be used and the return period T

_{R}has to be specified in months.

_{R})) = –lnT

_{R}we can calculate the extreme wind speed corresponding to a return period T

_{R}

_{R}is again in years. Similarly, an expression for the standard deviation of the largest yearly wind speeds within the return period can be calculated [33]

_{R}has to be replaced by 12 T

_{R}in both formulas. In the former discussion we have implicitly assumed that the extreme wind speed observations are independent, which is a good approximation if annual or monthly values are considered.

## 6. Modeling of wind flow

#### 6.1. General considerations

_{0}, y

_{0}). Given that model, in principle one point measurement is enough to generate a full map f(x, y). In practice, however, at a greater distance from the reference location the model may yield results with little accuracy, so additional measurements at locations (x

_{i}, y

_{i}) may be required. Clearly, if measurements at a great number of locations are available, then no mathematical model may be required and an interpolation between the measurement locations may be sufficient. This situation is typical of geophysical mapping studies, where a series of spatially distributed point measurements are combined into a map by using weighted interpolation techniques like universal Kriging [59].

#### 6.2. Linearized models

_{1}(x, y, 0) is the outer region pressure at Ξ = z/L =0 and the coordinates X and Y have been normalized to an appropriate horizontal length scale L, i.e., X = x/L, Y = y/L. Z = (z–z

_{s})/l is the normalized vertical distance with respect to the local terrain height defined by

_{0}is the roughness length of the terrain, assumed constant in the original model [62], although subsequent modifications [63] allow for horizontal variations of surface roughness. As stated earlier, the flow in the outer region is modeled as unviscous potential flow, leading to the following equation for the pressure

_{0}is the modified Bessel function and $\tilde{f}$ is the Fourier transform of the normalized terrain topography f(X, Y).

#### 6.3. Roughness modeling in linearized models

_{01}to z

_{02}the IBL height h can be defined at a distance x downstream from the change as:

_{0}= max(z

_{01}, z

_{02}). The change in surface friction velocity between the two regions is then related to the internal boundary layer height by:

_{1}(z) is the upstream boundary layer profile, and u

_{1}(z) is the profile at a distance x downstream of the change, we have [64]

_{2}(z)/u

_{1}(z) is assumed. In practice there will be multiple changes in roughness upstream and these can be simply accumulated as:

_{2}(z)/u

_{1}(z) evolves as a function from the distance of a roughness discontinuity, where the upstream roughness length was z

_{01}= 0.01 m and the downstream value z

_{02}= 0.1 m. The horizontal distance increment was 500 m, and adjacent curves have been offset horizontally by 0.1 for clarity. For the purposes of comparison, the function 0.3 h(x) (where h is the internal boundary layer height) has been overlaid after appropriate scaling. It can be seen from the figure how the slope of the vertical wind speed curve continuously evolves as one moves away from the roughness change in the downstream direction.

**Figure 19.**Evolution of the ratio of the vertical wind speed profiles before and after a change in roughness length as a function of the distance from the discontinuity. The wind profiles have been offset horizontally for clarity. The evolution of the internal boundary layer has been shown for comparison.

#### 6.4. Full CFD models and turbulence modeling

_{T}the turbulent viscosity. Often, the k-ε model is used to provide an expression for the turbulent viscosity in terms of k and its dissipation rate ε, as well as transport equations for k and ε themselves [67,69]:

c_{μ} | σ_{k} | σ_{ε} | c_{ε1} | c_{ε2} | |
---|---|---|---|---|---|

Standard k-ε model | 0.09 | 1.0 | 1.3 | 1.44 | 1.92 |

Modified for ABL modeling | 0.0324 | 1.0 | 1.85 | 1.44 | 1.92 |

**Figure 21.**Normalized wind speed over the Askervein Hill for section A (a) and AA (b). Reproduced with permission [68].

#### 6.5. Roughness modeling in CFD

_{s}and the friction velocity v

_{τ}:

^{+}and y

^{+}:

_{s}

^{+}in the following way:

_{S}is often taken as 0.5 if k-ε turbulence models are used. Commercial CFD codes like Fluent first compute the friction velocity v

_{τ}which is used to evaluate the dimensionless roughness height that is used to calculate ΔB and finally the velocity at the wall adjacent cell by means of equation (87). In order to apply the wall roughness model to wind flow modeling Crasto in his Ph.D. thesis [69] proposes an approximate expression relating the roughness length z

_{0}with the parameters of the wall roughness model described above:

_{0}using the value for K

_{S}suggested by equation (89). He then proceeds to obtain refined values for the effective roughness constant Ks, maintaining a value of Cs = 1 in all cases, in order to reproduce the desired value of the roughness length z

_{0}. In his simulations, the desired logarithmic velocity profile is specified at the inlet of the simulation domain, together with analytical expressions for the turbulent kinetic energy and its dissipation rate. The roughness length of the fully developed atmospheric boundary layer is then evaluated at the outlet of the simulation domain.

_{0}in CFD simulations, no information about the development of boundary layer as the fluid moves along the simulation domain is obtained from his results. Moreover, cell height has only been considered as far as the selection of cell size depending on the desired roughness class is concerned. In order to further investigate these issues the authors conducted a series of numerical 2D simulations for a flat surface with constant wall roughness parameters using the commercial CFD solver Fluent. As opposed to Crasto, they did not specify the desired vertical wind speed profile at the domain inlet, but rather specified a constant inlet velocity and allowed the vertical profile to converge. Cell heights were taken to be 1, 2.5, 5, 10, and 15, respectively, where K

_{S}was varied in a geometric progression between 0.00625 and 25.6. The maximum value of K

_{S}in each of the simulations was limited by the height of the cell adjacent to the surface.

_{S}an approximately linear relationship between z

_{0}and K

_{S}exists, while for large values of K

_{S}, z

_{0}converges to an asymptotic value given by the height of the cell adjacent to the surface. Interestingly, z

_{0}was found to scale with the cell height as illustrated by the inset of where z

_{0}/h has been plotted against K

_{S}, showing that for small values of K

_{S}all scaled curves coincide, before they level off at higher K

_{S}values.

**Figure 22.**Relation between roughness length z

_{0}and the dimensionless roughness height K

_{s}for different heights of the grid cell adjacent to the surface. Inset: Roughness length normalized by the cell height.

_{S}the scaled roughness length z

_{0}/h is a function of K

_{S}alone. It is also evident, however, that the wall roughness model is only appropriate as long as the cell height is large compared to the roughness length, imposing a certain trade-off between the modeling of small topographic features and surface roughness.

#### 6.6. The effect of topography on wind shear

_{m}= 0.09, C

_{1-e}= 1.444, C

_{2e}= 1.92, Energy Prandtl Number = 0.85, TDR Prandtl Number = 1.3, TKE Prandtl Number = 1, Wall Prandtl Number = 0.85. The total domain length was 15 km, and the length of the mesa was fixed at 500 m.

_{S}has been plotted for a location close to the leading edge of the mesa, the wind speed increases rapidly with height, reaching a maximum at a height of approximately 150 m for most values of the roughness parameter and then decreases to converge against the geostrophic value of the wind speed.

_{S}. These apparent values of the roughness length have been plotted in Figure 23(b) for different locations on the mesa as a function of the slope parameter S and a fixed value of the roughness parameter K

_{S}. The free-stream roughness length was z

_{0}= 0.8 m in all cases. As shown by Figure 23(b) the apparent roughness length is lower at the leading edge than at the corresponding symmetric positions near the trailing edge. Moreover, a significant reduction of apparent roughness with increasing values of the slope parameter is evident from the figure. Also, while at small values of S all curves tend to converge, at high values a more substantial differentiation between the leading and trailing edge situations occurs. The general decrease in wind shear for wind flow over the mesa can be understood in terms of a compactation of the flow lines as the air is forced to flow uphill. Note that this effect is less pronounced on the trailing edge, where the flow lines tend to detach from the surface, particularly at higher values of the slope parameter.

**Figure 23.**Vertical wind shear profiles for wind flow over symmetric mesa structures as determined with a commercial CFD tool (Fluent). (a) Vertical profiles at the leading edge of the mesa for different values of the wall roughness parameter H (=K

_{S}). (b) Effective roughness length at different positions on the mesa as a function of the average edge slope S. The free-stream roughness length was fixed at z

_{0}= 0.8 m.

**Figure 24.**Surface wind speed map (10 m) for Southern Texas and Northeastern Mexico created from the North American Mesoscale (NAM) Model output for one year.

#### 6.7. Mesoscale modeling

_{air-sea}, where sea breeze conditions were considered to occur if ΔT

_{air-sea}> x °C. As shown in Figure 27, the average wind speed difference for the period studied (May 1st through July 31st, 2008) was close to zero in both cases, with no difference occurring between sea breeze and non‑sea breeze conditions, showing that the numerical weather models adequately predicts both situations.

**Figure 25.**Boxplot showing the differences between measured and predicted wind speed values for the TCOON site “Bob Hall Pier” for different wind speed bin intervals. The measured (TCOON) wind speed histogram is shown for comparison.

**Figure 26.**Boxplot showing the differences between measured and predicted temperature values for the TCOON site “Bob Hall Pier” for different air temperature bin intervals. The measured (TCOON) temperature histogram is shown for comparison.

**Figure 27.**Boxplots for overall wind speed differences of the measured and the predicted values for both sea breeze conditions (left box of each graph) and non-sea breeze conditions.

## 7. Summary and Conclusions

## Acknowledgements

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Probst, O.; Cárdenas, D.
State of the Art and Trends in Wind Resource Assessment. *Energies* **2010**, *3*, 1087-1141.
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