Volume 2 wind energy 2 04 – wind energy potential Volume 2 wind energy 2 04 – wind energy potential Volume 2 wind energy 2 04 – wind energy potential Volume 2 wind energy 2 04 – wind energy potential Volume 2 wind energy 2 04 – wind energy potential Volume 2 wind energy 2 04 – wind energy potential
2.04 Wind Energy Potential H Nfaoui, Mohammed V University, Rabat, Morocco © 2012 Elsevier Ltd All rights reserved 2.04.1 2.04.2 2.04.2.1 2.04.2.2 2.04.2.3 2.04.2.4 2.04.3 2.04.3.1 2.04.3.2 2.04.3.3 2.04.3.3.1 2.04.3.3.2 2.04.3.3.3 2.04.3.3.4 2.04.3.3.5 2.04.3.4 2.04.3.5 2.04.3.6 2.04.3.6.1 2.04.3.6.2 2.04.3.6.3 2.04.3.6.4 2.04.3.6.5 2.04.3.6.6 2.04.3.6.7 2.04.4 2.04.4.1 2.04.4.1.1 2.04.4.1.2 2.04.4.1.3 2.04.4.1.4 2.04.4.1.5 2.04.4.1.6 2.04.4.2 2.04.4.2.1 2.04.4.2.2 2.04.4.2.3 2.04.4.3 2.04.4.3.1 2.04.4.3.2 2.04.4.3.3 2.04.4.3.4 2.04.4.3.5 2.04.4.4 2.04.5 2.04.5.1 2.04.5.2 2.04.5.2.1 2.04.5.2.2 2.04.6 2.04.6.1 Introduction Wind Characteristics Origin of Wind Meteorology of Wind Wind Direction and Wind Velocity Fundamental Causes of Wind Wind Measurements Wind Speed Cup Anemometers Other Anemometers Pressure plate anemometer Pressure tube anemometer Hot-wire anemometer Laser Doppler anemometer Sonic anemometer Wind Direction Wind Rose Wind Speed Profiles Roughness classes and lengths Wind shear Turbulence Landscape without neutral stability Modified log law Nature of atmospheric winds Power spectrum of wind speed: Averaging periods Analysis of Wind Regimes Wind Speed Variation with Time Instantaneous phenomenon: Wind gust Diurnal breezes: Land–sea and valleys breezes Seasonal phenomenon Time distribution Calm wind spells Frequency distribution Mathematic Representation of Wind Speed Weibull distribution Weibull hybrid distribution Cumulative distribution Wind Resource Atlas Models Physical basis Stability model Roughness change model Shelter model Orographic model Wind Resource Atlas Dynamic Study of Wind Speed Stochastic Models for Simulating and Forecasting Wind Speed Time Series Bayesian Adaptive Combination of Wind Speed Forecasts from Neural Network Models Artificial neural network models Bayesian model averaging Wind Energy Wind Energy Production Comprehensive Renewable Energy, Volume doi:10.1016/B978-0-08-087872-0.00204-3 74 74 74 74 74 74 75 75 75 75 75 75 76 76 76 76 76 77 77 78 78 78 79 80 80 81 81 81 82 82 82 82 82 83 84 84 85 86 86 86 86 86 86 87 87 88 88 88 89 89 89 73 74 Wind Energy Potential 2.04.6.2 Wind Energy Potential 2.04.6.2.1 Power output from an ideal turbine: Betz limit 2.04.6.2.2 Axial momentum theory 2.04.7 Conclusions References Further Reading 90 90 90 91 91 92 2.04.1 Introduction The winds, in the macro-meteorological sense, are movements of air masses in the atmosphere These large scale-movements are generated primarily by temperature differences within the air layer due to differential solar heating Because the energy per unit surface area received from the sun depends on geographical latitude, temperature differences arise and hence pressure gradients which, together with the centripetal force and the Coriolis force associated with the rotation of the Earth, induce the movements of the air masses known as the gradient wind Solar radiation, evaporation of water, cloud cover, and surface roughness all play important roles in determining the conditions of the atmosphere The study of the interactions between these effects is a complex subject called meteorology, which is covered by many excellent books, for example, Météorologie Générale edited by Triplet and Roche [1] Therefore, only the part of meteorology concerning the flow of wind will be considered in this chapter 2.04.2 Wind Characteristics 2.04.2.1 Origin of Wind Wind is caused by differences in pressure When a difference in pressure exists, the air is accelerated from higher to lower pressure On a rotating planet the air will be deflected by the Coriolis effects, except exactly on the equator Globally, the two major driving factors of large-scale winds are the different heating between the equator and the poles and the effect of the Earth’s rotation Outside the tropics and upward from frictional effects of the surface, the large-scale winds tend to approach geostrophic balance Near the Earth’s surface, friction causes the wind to be slower than it would be otherwise Surface friction also causes winds to blow more inward into low pressure areas 2.04.2.2 Meteorology of Wind The basic driving force of air movement is a difference in air pressure between two regions This air pressure is described by several physical laws One of these is the ideal gas law: pV ¼ nRT ½1 where p is pressure in Pascal (N m−2), V is volume (m3), n is the number of moles, R is the universal gas constant (8.3144 J K−1 mol−1), and T is absolute temperature (°K) 2.04.2.3 Wind Direction and Wind Velocity As the warm air expands and the cold air condenses, this process creates zones of relative high or low pressure Afterwards, the wind is fundamentally the movement of air created by these differences in pressure Theoretically, at the Earth’s surface, the wind blows from high-pressure areas toward low-pressure areas However, at the medium and higher latitudes, its direction is modified by the earth’s rotation The wind becomes parallel to isobaric lines instead of being perpendicular to them In the northern hemisphere, the wind rotates counter-clockwise round cyclonic areas and clockwise round anticyclonic areas In the southern hemisphere, these wind directions are reversed The wind direction is determined by the direction from which it blows For example, it is a westerly wind if the air blows from the West 2.04.2.4 Fundamental Causes of Wind There are two principal reasons for the movement and direction of motion of the Earth’s atmosphere – the unequal amounts of solar radiation received at different latitudes and the rotation of the earth Superimposed upon the general world wind circulation, due to these two factors, modifications arising from local disturbances, such as the tropical cyclone We have on a terrestrial scale regular pressure systems that produce important winds, called dominant winds or general circulation In practice, atmospheric circulation may be represented as it is shown in Figure and it is useful in identifying the most important global wind characteristics Wind Energy Potential North pole Polar circle 66° 75 18000 m Polar easterlies Storm Subtropical high pressure belt 30° Meridian cross section of the atmosphere showing the meridian circulation Westerlies N.E Equator (low pressure) Front Polar Trade 0° Doldrums Trade S.E Subtropical high pressure belt Winds 30° Polar circle Winds Westerlies Polar Front Storm 66° Polar easterlies 7500 m South pole Figure General world wind circulation [2] In each hemisphere, we can discern three more or less individualized cells: a tropical cell, a temperate cell, and a polar cell, which turn one against the other like cogs in a gear box The north and south tropical cells are separated from one another by the equatorial calm which is a low-pressure area and from the temperate cells by the subtropical high-pressure zones Actually, the sketch is not perfect The unequal heating of oceans and continents surface, relief, vegetation, and seasonal variations deform and modify the high- and low-pressure zones There are also atmospheric disturbances created by masses of cold air that move, from time to time, from the poles towards the equator Thus, the state of the atmosphere is continually evolving As a result, the most favorable area for wind energy production is situated on the continents near the seashores or in the sea 2.04.3 Wind Measurements 2.04.3.1 Wind Speed An anemometer is a device for measuring wind speed and is a common weather station instrument The term is derived from the Greek word ‘anemos’, meaning wind The oldest anemometer, invented in 1846, is the cup anemometer [3] 2.04.3.2 Cup Anemometers The cup anemometer consists of three or four hemispherical cups each mounted on one end of four horizontal arms, which in turn are mounted at equal angles to each other on a vertical shaft The airflow that pass the cups in any horizontal direction turns the cups in a manner that is proportional to the wind speed The three-cup anemometer was developed in 1926 It had a more constant torque and responded more quickly to gusts than the four-cup anemometer It was further modified in 1991 to measure both wind direction and wind speed Figure shows a cup-type anemometer used by most national weather service stations and airports [3, 5] 2.04.3.3 Other Anemometers Anemometer types include the propeller, cup, pressure plate, pressure tube, hot wire, Doppler acoustic radar, sonic, and so on The propeller and cup anemometers depend on rotation of a small turbine for their output, while the others basically have no moving parts 2.04.3.3.1 Pressure plate anemometer Another type of anemometer is known as a pressure plate or normal plate anemometer This is the oldest anemometer known having been developed in 1667 It uses the principle that the force of moving air on a plate being normal to wind is proportional to the area of the plate and to the square of wind speed The main application of this type of anemometer has been in gust studies because of its very short response time [3, 5] 2.04.3.3.2 Pressure tube anemometer Yet another type of anemometer is the pressure tube anemometer It is not used much in the field because of difficulties with ice, snow, rain, and the sealing of rotating joints However, it is often used as the standard in a wind tunnel when these difficulties are not present It has been known for almost two centuries that the wind blowing into the mouth of a tube causes an increase of pressure in the tube, and that an airflow across the mouth causes a suction [3, 5] 76 Wind Energy Potential Figure Cup anemometer [4] 2.04.3.3.3 Hot-wire anemometer The hot-wire anemometer uses a very fine wire electrically heated up to some temperature above the ambient Air flowing via the wire has a cooling effect on the wire As the electrical resistance of most metals is dependent upon the temperature of the metal, a relationship can be obtained between the resistance of the wire and the flow speed The hot-wire anemometer, while extremely delicate, has extremely high-frequency response and fine spatial resolution com pared to other measurement methods, and as such is almost universally employed for the detailed study of turbulent flows, or any flow in which rapid velocity fluctuations are of interest [3, 5] 2.04.3.3.4 Laser Doppler anemometer The laser Doppler anemometer uses a beam of light from a laser that is divided into two beams, with one propagated out of the anemometer Particulates flowing along with air molecules near where the beam exits reflect, or backscatter, the light back into a detector, where it is compared to the original laser beam When the particles are in great motion, they produce a Doppler shift for measuring wind speed in the laser light, which is used to calculate the speed of the particles, and therefore the air around the anemometer [3, 5] 2.04.3.3.5 Sonic anemometer The sonic anemometer was developed in the 1970s It uses ultrasonic sound waves to determine instantaneous wind speed by measuring how much sound waves traveling between a pair of transducers are sped up or slowed down by the effect of the wind Sonic anemometers can take measurements with very fine temporal resolution, 20 Hz or better, which makes them well suited for turbulence measurements The lack of moving parts makes it appropriate for long-term use in exposed automated weather stations and weather buoys where the accuracy and reliability of traditional cup-and-vane anemometers is adversely affected by salty air or large amounts of dust [3, 5] Two-dimensional (wind speed and wind direction) sonic anemometers are used in applications such as weather stations, ship navigation, wind turbines, aviation, and weather buoys 2.04.3.4 Wind Direction The wind vane, used for indicating wind direction, is one of the oldest meteorological instruments When mounted on an elevated shaft or spire, the vane rotates under the influence of the wind such that its center of pressure rotates to leeward and the vane points into the wind (Figure 3) Wind direction is measured in degrees from true north The word ‘vane’ comes from the Anglo-Saxon word ‘fane’ meaning ‘flag’ [3, 5] The wind direction is determined by the direction from which it blows It is a westerly wind if the air current blows from the West Modern aerovanes combine the directional vane with anemometer Colocating both instruments allows them to use the same axis and provides a coordinated read out 2.04.3.5 Wind Rose A wind rose is a graphic tool used by meteorologists to give a succinct view of how wind speed and direction are typically distributed at a particular location It summarizes the occurrence of winds at a location, showing their strength, direction, and frequency (Figure 4(a)) Wind Energy Potential 77 Figure Wind direction: vane and transmitter [5] N NORTH 20% 16% W 21.25% 12% 8% E 4% WEST EAST Wind speed (knots) S Percentage of calms within circle arcs represent 5% intervals 3–8 9–15 16–38 >39 Miles per hour Resultant vector 189° – 18% SOUTH > = 22 17–21 11–17 7–11 4–7 1–4 Calms: 2.59% Figure (a) Wind rose [6] (b) Wind rose [3] Historically, the wind rose was a predecessor of the compass rose, as there was no differentiation between a cardinal direction and the wind which blew from such a direction It was included on maps in order to let the reader know the characteristics of the eight major winds Using a polar coordinate system of gridding, the frequency of winds over a time period could be plotted by wind direction, with color bands showing wind ranges Presented in a circular format, the wind rose shows the frequency of winds blowing from particular directions The length of each ‘spoke’ around the circle is related to the frequency of time that the wind blows from a particular direction Each concentric circle represents a different frequency, emanating from zero at the center to increasing frequencies at the outer circles A wind rose plot may contain additional information, in that each spoke is broken down into color-coded bands that show wind speed ranges The wind rose typically uses 16 cardinal directions, such as North (N), NNE, NE, and so on There are a number of different formats that can be used to display wind roses A particular method for describing wind speed and direction is shown in Figure 4(b) 2.04.3.6 2.04.3.6.1 Wind Speed Profiles Roughness classes and lengths The shape of the terrain over which the wind is flowing will have a frictional effect upon the wind speed near the surface Both the height and the spacing of the roughness elements on the surface will influence the frictional effect on the wind A single parameter, the surface roughness length, z0, is used to express this effect Roughness classes (RCs) and roughness lengths are characteristics of the landscape used to evaluate wind conditions at a potential wind turbine site The roughness length is defined as the height above ground z0 in meters at which the wind speed is theoretically equal to zero [7] RCs are defined in terms of the roughness length in meters z0, according to 78 Wind Energy Potential V V10 α = 0,4 α V = V10 h 10 α = 0,1 10 20 30 40 50 60 Height (m) h 10 Figure Wind shear related to a reference height of 10 m, for various roughness heights, α [6] RC ẳ 1:699 823 015 ỵ RC ẳ 3:912 489 289 ỵ 2.04.3.6.2 ln z0 ị ln 150ị ln ðz0 Þ ln ð3:3333Þ for z0 ≤ 0:03 for z0 > 0:03 ½2 ½3 Wind shear The wind speed profile tends to a lower speed as it moves closer to the ground level This is designated as wind shear It is found that wind velocity increases with height above the ground according to formula [4] [2–8] The rate of change strongly depends upon the roughness of the terrain (Figure 5) V1 h1 ẳ ẵ4 h2 V2 where V1 is the wind velocity at some reference height h1 and V2 is the wind velocity at height h2 The constant α, Hellman exponent, depends on the nature of the surface and the stability of the air Thus, α may be defined as follows [8]: a ẳ 0:096ln z0 ịị ỵ 0:016ln z0 ịị2 ỵ 0:24 ẵ5 The variation of wind speed with height depends on the surface roughness and the atmospheric stability, and we could assume that wind speed grows logarithmically with height The wind speed at a certain height above ground can be also estimated as a function of height above ground z and the roughness length z0 in the current wind direction from the formula [7] z ln z0 V zị ẳ Vref ẵ6 zref ln z0 The reference speed Vref is a known wind speed at a reference height zref 2.04.3.6.3 Turbulence Wind flowing around buildings or over very rough surface locations exhibits rapid changes in speed and/or direction, called turbulence This turbulence decreases, for example, the power output of the windmill and can also lead to unwanted vibration of the machine (Figure 6) Hills can sometimes be used to give an enhancement of wind velocity over that on the surrounding plain The best hills are those having smooth sides and fairly steep conical shape (Figure 7) The effect of height above the top of the hill is complex and the previously quoted relationship cannot be used and it is necessary to measure wind speed on the considered site 2.04.3.6.4 Landscape without neutral stability The previous analysis assumed neutral atmospheric stability conditions meaning that a parcel of air is adiabatically balanced from a thermodynamic perspective Neutral stability is a reasonable assumption in high wind when shearing forces rather than buoyancy Wind Energy Potential 79 Wind ≈2 H H ≈ 20 H 2H Figure Zone of turbulence over a small building [8] High wind at low altitude High wind Inverse flow High wind Inverse flow perturbations Figure Acceleration of wind over hills [6] forces are dominant However, the atmosphere is rarely neutral and the buoyancy forces usually predominate over the shear forces as noticed from the scattering of the measurements at neighboring data collection points 2.04.3.6.5 Modified log law The logarithmic wind profile equation in the lowest 100 m may still be used under nonequilibrium conditions with some appropriate modifications [7]: 80 Wind Energy Potential Vzị ẳ Vfr ln k ! z z −Ψ L z0 ½7 where Vfr is friction velocity, k is von Karman constant (k = 0.4), L is Monin–Obukhov length, Ψ is correction term, and z0 is roughness length The Monin–Obukhov length L is a scaling parameter that depends upon the heat flux at the ground surface q0, and is given by [7] Lẳ T0 cp Vfr3 kgq0 ẵ8 where T0 is ground surface absolute temperature (°K), q0 is ground surface heat flux, cp is heat capacity of the air at constant pressure, and g is gravity’s acceleration 2.04.3.6.6 Nature of atmospheric winds To be able to understand and predict the performance, for example, of wind turbines, it is essential that the designer has knowledge of the behavior and structure of the wind itself that will vary from time to time and from site to site dependent on the climate and topography of the region, the surface condition of the terrain around the site, and various other factors Since the early 1970s, significant progress has been made in our understanding of the structure of the wind, in our ability to predict the conditions likely to be experienced at a site and to assess the suitability of a site for generating power from wind Figure shows typical anemograph records of wind speed monitored at three heights on a tall mast during strong wind conditions These records demonstrate the main characteristics of the flow in the region near the ground The turbulent fluctuations are random in character and they are not always governed by deterministic equations, but we will need to use statistical techniques This short-spanned change in wind speed is primarily due to the local geographic and weather effects 2.04.3.6.7 Power spectrum of wind speed: Averaging periods In order to separate the short-period fluctuations of wind speed due to mixing from the long-term changes associated with � Thus it is defined by [7]: macroscale meteorological phenomena, we introduce the time average mean wind speed, V to ỵ T ẳ Vtịdt V T T to ẵ9 where V(t) is the instantaneous wind speed component along the average wind direction at time, t, and we must define the time interval, T, over which the average is taken Figure shows the frequency variation of kinetic energy of the horizontal wind speed near the ground as measured at Brookhaven, New York [9] This spectrum shows the long-term distribution of energy An important feature of all long-term spectra is the gap between periods of about 10 and h where the spectrum contains very little energy The significance of the spectral gap is that, if the averaging period for the mean wind speed is chosen to lie within this range, the synoptic variations can be separated from those due to turbulence It has been suggested that a good averaging period for defining mean wind speeds lies between 20 and h Wind speed (m/s) 20 150 m 15 50 m 10 10 m Figure Simultaneous recordings of wind speed at three heights [9] Time (min) Wind Energy Potential 81 Power spectral density n.S (n) (m/s)2 10–3 1000 10–2 100 day 10–1 0.2 10 0.5 1 0.5 0.2 Semi-diurnal 10 20 0.1 Spectral gap Macrometeorological range 50 100 0.02 0.005 1000 cycles/h 0.001 h 5s Micrometeorological range Figure Spectrum of horizontal wind speed at Brookhaven National Laboratory [9] 2.04.4 Analysis of Wind Regimes Here we will discuss the wind pattern as such and its characterization by numbers and graphs We assume that a set of short-time measurements from a meteorological station is available The reliability of the data is not questioned here 2.04.4.1 2.04.4.1.1 Wind Speed Variation with Time Instantaneous phenomenon: Wind gust The lower region of the atmosphere is known as the planetary boundary layer and the movement of the air is retarded by frictional forces and large obstructions on the surface of the Earth, as well as by the Reynolds stresses produced by the vertical exchange of momentum due to turbulence The turbulence, which may be mechanical and/or thermal in origin, also causes rapid fluctuations in the wind velocity over a wide range of frequencies and amplitudes, commonly known as gusts (Figure 10) It is often useful to know the maximum wind gust that can be expected to occur in any given time interval This is usually represented by a gust factor G, which is the ratio of a wind gust to the hourly mean wind speed G is obviously a function of the turbulence intensity, and it also clearly depends on the duration of the gust Thus, the gust factor for a s gust will be larger than for a s gust, since every s gust has within it a higher s gust [10] Time-varying wind speeds and, more specifically, the uncertainty over what wind speed will be during the next hour or by hour for the next day is a challenge for wind turbine designers For example, a wind gust that rapidly changes the wind turbine’s output necessitates a control system that adjusts the rotor speed and thus optimizes the turbine’s power output for slow wind speed variation and attenuates high-frequency wind gust effects to reduce the resulting fatigue V in m/s 40 30 20 10 Figure 10 Wind gusts from Orkney (UK) [2] Time in sec 82 Wind Energy Potential (a) day (b) night Figure 11 Diurnal wind 2.04.4.1.2 Diurnal breezes: Land–sea and valleys breezes At the sea shore the General Circulation Patterns have superimposed on them an airflow from the sea to the land at night, and in the reverse direction during the day (Figures 11(a) and 11(b)) These sea/land breezes can be quite strong and can dominate the wind pattern A rather similar situation to the sea breeze can arise between valleys and mountains 2.04.4.1.3 Seasonal phenomenon The monthly variation of wind speed depends essentially on geographical location and only meteorological measurements can give information about these variations But, generally, the yearly variations are repetitive with a good precision 2.04.4.1.4 Time distribution Plotting the monthly average of each hour of the day shows the diurnal fluctuations of the wind speed in that particular month (Figure 12) The major reason for the velocity variation here is the difference in temperature between the sea and land surface It should be noted that in this specific case, the diurnal variation can be advantageous for wind energy generation as we may need more power during the daytime hours than at night In a similar manner, the monthly average can be plotted to show the monthly fluctuations of the wind speed, compared with the annual average wind speed (Figure 13) Knowledge of this variation of velocity at a potential wind site is essential to ensure that the availability of power matches the demand 2.04.4.1.5 Calm wind spells According to the Beaufort wind scale at a standard altitude of 10 m above an open, even surface, the action of wind is calm when the wind speed is less than 0.3 m s−1 and smoke rises vertically [1, 4] 2.04.4.1.6 Frequency distribution Another type of information that can be extracted from the time distribution of the data is the distribution of periods with low wind speed In other words, how often did it happen that the wind speed was lower than, for example, m s−1 during 12 h or during days This type of information is valuable for the calculation of the size of storage devices Apart from the distribution of the wind speeds over a day or a year, it is important to know the number of hours per month or per year during which the given wind speeds occurred, that is, the frequency distribution of the wind speed (Figure 14) The maximal value of this histogram corresponds to the most frequent wind speed It is often important to know the number of hours that a windmill, for example, will not run or the time fraction that a windmill produces more than a given power In this case, it is necessary to add the number of hours in all intervals above the given wind Wind Energy Potential 83 10 V (m/s) V = 7.4 m/s 0 10 12 14 16 18 20 22 24 Hours Figure 12 Diurnal pattern of the wind speed at Praia airport for a selected month [8] 10 V (m/s) V = 6.6 m/s J F M A M J J A S O N D Month Figure 13 Monthly average wind speeds at Praia airport for a selected month [8] speed The result is the duration distribution (Figure 15(a)) The flatter the duration curve, that is, the longer one specific wind speed persists, the more constant the wind regime is The steeper the duration curve, the more irregular the wind regime is In some cases it is preferred to plot the time during which the wind speed was smaller than a given wind speed, and when this is plotted versus the wind speed, a cumulative distribution results (Figure 15(b)) 2.04.4.2 Mathematic Representation of Wind Speed In order to predict the output of a wind generator system by using simulation methods, it is necessary to have a large series of measurements for the considered site The real data are variable To be usable, this enormous volume of data must be reduced without losing any information One way to achieve this is the statistical treatment of data For instance, one may use mathematical functions that approach the velocity frequency data in a histogram as closely as possible In this respect, much attention has been given to the Weibull function, since it gives a good match with the experimental data But, in some cases the Rayleigh distribution, a special case of the Weibull distribution, assuming k = 2, is preferred 84 Wind Energy Potential 140 120 Hours per month 100 80 60 40 20 10 12 14 V (m/s) Figure 14 Velocity frequency data for Praia airport for a selected year [8] (a) (b) 14 700 12 600 Hours per month V (m/s) 10 400 300 200 100 500 0 100 200 300 400 500 600 700 Hours per month 10 12 14 16 V (m/s) Figure 15 Histogram of the duration distribution (a) and the cumulative distribution (b) for Praia (during a selected month) [8] 2.04.4.2.1 Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution It is named after Waloddi Weibull who described it in detail in 1951 [11] It is mostly used to describe wind speed distributions, and it is important in weather forecasting due to its flexibility The presentation of wind data makes use of the Weibull distribution as a tool to present the experimental frequency distribution of wind speed in a compact form The two-parameter Weibull distribution is expressed mathematically as [8, 12] " # k V k k V pVị ẳ exp ½10 c c c where p is the frequency of occurrence of wind speed V The two Weibull parameters thus defined are usually referred to as the scale parameter c (m s−1) and the shape parameter k (dimensionless) (Figure 16) 2.04.4.2.2 Weibull hybrid distribution This method is used also, if we neglect the frequent calms It is a little complex to use, but the results are often comparable to those obtained by Weibull distribution and sometimes better It must be solved by an iterating method It is given by the following eqn [13]: Wind Energy Potential 85 0.11 0.10 Actual data 0.09 0.08 0.07 0.06 Weibull 0.05 0.04 0.03 0.02 0.01 10 12 14 16 18 20 22 24 Figure 16 Actual wind data and weighted Weibull density function [5] 20 6B 6B B k¼6 6B 4@ N X ni Vik ln ðVi Þ C B C B C B C−B N X A @ k ni Vi N X 13 − ni Vi C7 C7 C7 N C A5 ẵ11 and 6 Cẳ6 N X ni Vik 7 7 N K ½12 where N is the total number of observations of no zero wind speed 2.04.4.2.3 Cumulative distribution The cumulative distribution function F(V) indicates the time fraction or probability that the wind speed V is smaller than or equal to a given wind speed V: FVị ẳ PV Vị ẵ13 The probability density function, represented in our case by the velocity frequency curve: f ðV Þ ẳ dF Vị dV ẵ14 or v FVị ẳ f VịdV ẵ15 and the cumulative distribution function (on the basis of Weibull distribution) is given by " # V k FV ị ẳ exp c ½16 The velocity duration function S(V), defined as the time fraction or probability that the wind speed V is larger than a given wind speed V′, can be written as SVị ẳ 1FVị ẳ PV > Vị ẵ17 86 Wind Energy Potential 2.04.4.3 Wind Resource Atlas Models The topography of the site can greatly affect the wind at a specific location The wind speed will tend to accelerate up hill and decelerate down hill If the slopes are too steep, however, the wind may ‘separate’ from the terrain and produce some damaging excess turbulence and lower mean wind speed Careful siting is therefore essential Some major progress has been made in computation of the local topographical effects by the Danish National Laboratory, RISO, who have published a very ‘user-friendly’ topographical wind flow model ‘WASP’ [13], which allows relatively inexpert users access to a very powerful tool of computational fluid dynamics Used under the correct conditions this tool can produce reliable results for local flows and has been used extensively by wind energy community 2.04.4.3.1 Physical basis The layer closest to the ground is called the atmospheric boundary layer It extends up to about 100 m on clear nights with low wind speeds and up to more than km on a fine summer day The lowest part of this layer is called the surface layer, which is sometimes defined as a fixed fraction, say 10% of the boundary layer depth For the purpose of climatology relevant to wind power utilization, we can neglect the lowest wind speeds, so only situations where the atmospheric boundary layer extends to approximately km and surface-layer physics apply in the lowest 100 m of the layer are of concern [13, 14] The wind in the atmospheric boundary layer can be considered to arise from pressure differences caused mainly by the passing of high- and low-pressure systems As the boundary layer structure has a rather rapid response to changes in pressure forcing, an approximate balance is found between the pressure gradient force and the frictional force at the surface of the earth This balance can be theoretically derived under idealized conditions of stationarity, homogeneity, and barotropy [13, 15] 2.04.4.3.2 Stability model The stability modifications of the logarithmic wind profile are often neglected in connection with wind energy, the justification being the relative unimportance of the low wind speed range The model can treat stability modifications as small perturbations to a basic neutral state [13, 15] 2.04.4.3.3 Roughness change model If the driving force of the wind or the geostrophic wind is the same over the area under consideration, it is possible to still use a modified logarithmic law to describe the profile The effect of this is to alter the stability of the wind profile and the vertical movement of the air becomes more important Note that the logarithmic wind profile applies only if the upwind terrain is reasonably homogeneous If this is not the case, deviation will be observed and it is not possible to assign a unique roughness length to the terrain Even though ‘effective’ roughness lengths can be assigned by different methods, these will depend on the height observation An exception to this is the effective roughness length implicitly defined by the geostrophic drag law [13] The relations describing the vertical wind profile in neutral conditions within the boundary layer are [13, 16] the logarithmic wind profile law z VÃ V ẳ ln ẵ18 k z0 the geostrophic drag law, G(m s−1): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 VÃ Gẳ ln A ỵ B jf jz0 ẵ19 where V* is the friction velocity (m s−1), z0 is surface roughness (m), f is Coliolis’ parameter (1 s−1), k is von Karman’s constant, A is the empirical constant, A = 1.8, and B is the empirical constant, B = 4.5 2.04.4.3.4 Shelter model The frictional effect of a land surface is caused by drag on surface – mounted obstacles ranging from individual sand grains, grass, leaves, and so on, to large trees and buildings Their collective effect is modeled through the surface roughness length Close to an individual obstacle, at distances comparable to the height of the obstacle and at heights likewise comparable to the height of the obstacle, the wind profile is perturbed, particularly in the downstream wake, and the object must be treated separately In the wake immediately behind a blunt object, such as a row of trees or a house (less than five object heights downstream and at heights less then twice the height of the object), the details of the object exert a critical influence on the effects The wake behind a building depends, for example, on the detailed geometry of the roof and the incidence angle of the wind, to mention two parameters Besides, wakes from other nearby objects may interfere, causing the problem to become very complicated [13] 2.04.4.3.5 Orographic model Like the change-of-roughness and shelter models, the orographic model is used to correct measured wind data for the effect of local terrain inhomogeneities; in the present case, this means differences in terrain height around the meteorological stations [16] Wind Energy Potential 87 m/s >6.0 5.0–6.0 4.5–5.0 3.5–4.5 6.0 4.5–6.0 39 Miles per hour Resultant vector 189° – 18% SOUTH > = 22 17 21 1 1–1 7 7–1 1 4–7 1–4 Calms: 2. 59%