regi-03-04.pdf

Wind Energy -
Turbines, Speed and Energy
Calculations and Distributions
Lecture 3-4

• Wind turbine captures the wind’s kinetic energy in a rotor consisting of two or
more blades mechanically coupled to an electrical generator
• Turbine mounted on a tall tower to enhance the energy capture.
• Two distinctly different configurations for the turbine design,
• The horizontal axis configuration
• The vertical axis configuration
Wind Turbines

Vertical axis wind turbine showing major components
FIGURE Vertical-axis 33-m-diameter wind
turbine built and tested by DOE/Sandia
National Laboratory during 1994 in
Bushland, TX.

Vertical axis wind turbine showing major components

FIGURE Modern wind turbine for utility-scale power
generation.

History
• For thousands of years - Sailing boats
• For at least 3000 years - Windmills - for grinding grain or pumping water
• 13th century - horizontal-axis windmills- integral part of the rural economy
• Fell into disuse with cheap fossil-fuelled engines and then the spread of rural
electrification.
• Late 19th century - Use of windmills to generate electricity with - 12 kW DC
windmill generator
• For most 20th century - Little interest in using wind energy other than for battery
charging for remote dwellings. Quickly replaced once electricity grid came up.

Modern Power producing Wind Turbines
• 1931 - The 100 kW 30 m diameter Balaclava wind turbine (USSR)
• Early 1950s - The Andrea Enfield 100 kW 24 m diameter pneumatic design (UK).
• 1941 - Smith–Putnam wind turbine (1250 kW) (USA ,1941).
• 1956 -200 kW 24 m diameter Gedser machine (Denmark).
• 1950s and 1960s - Lightweight turbines (1950s/1960s. Prof. Hutter, Germany).
• 1963 - A 1.1 MW 35 m diameter turbine (Electricite´ de France)
• 1973- Interest in wind turbines rises with the price of oil rose dramatically.
• 1973 - Number of substantial Government-funded programmes of research,
development and demonstration.
• (USA, 1987)- WT prototype turbines- 38 m diameter 100 kW Mod-0 (USA,1975)
and 97.5 m diameter 2.5 MW Mod-5B

Research and Development
• R&D programmes also in the UK, Germany, Canada and Sweden.
• As the multi-megawatt prototypes were being constructed private companies, often with considerable
state support, were constructing much smaller, often simpler, turbines for commercial sale.
• (Mid 1980s) - Financial support mechanisms in California → small (100 kW) wind turbines
• ‘Danish’ wind turbine concept emerged of a three-bladed, stall-regulated rotor and a fixed-speed,
induction machine drive train (used in as large as 60 m in diameter and at ratings of 1.5 MW.
• As the sizes of commercially available turbines approached multi MW range, the concepts of variable-
speed operation, full-span control of the blades, and advanced materials are being used increasingly
by designers.
• The machines of Figures 1.1 and 1.2 are examples of this design.
• As the sizes of commercially available turbines approached multi MW range, the concepts of variable-speed
operation, full-span control of the blades, and advanced materials are being used increasingly by designers.

Figure 1
A 1.5 MW, 64 m
diameter Wind
Turbine
Variable-speed operation, full-span control of the blades, and
advanced materials used

Figure 2
A 750 kW, 48 m
dia. WT, Denmark.
Variable-speed operation, full-span control of
the blades, and advanced materials used –
Danish Design

Figure 3 Wind Farm of Variable-Speed Wind Turbines in Complex Terrain
A wind farm of direct-drive, variable speed wind turbines. Synchronous generator is coupled directly to
the aerodynamic rotor so eliminating the requirement for a gearbox.

Figure 4
1 MW Wind Turbine in
Northern Ireland
A more conventional, variable-speed
wind turbine that uses a gearbox

Figure 5 Wind Farm of Six
Pitch-regulated Wind
Turbines in Flat Terrain
A small wind farm of pitch-regulated
wind turbines, where full-span control
of the blades is used to regulate power.

Modern Wind Turbines
The power output, P, from a wind turbine is given by the well-known expression:
𝑃 =
1
2
𝜌𝐴𝑉3
𝐶𝑝 = 𝑃𝑤𝑖𝑛𝑑𝐶𝑝
𝜌 = density of air = 800 times less than that of water → large size of a WT.
A= swept area
𝑈= instantaneous wind velocity along direction of wind.
𝐶𝑝 = Power coefficient = fraction of the power in wind converted by WT into mechanical
power. (Theoretical max 0.593 Betz limit, lower practical values).
A 1.5 MW wind turbine - rotor > 60 m in dia.

Modern wind turbine - trends
• Power coefficient = function of tip speed ratio (the ratio of rotor tip speed to free wind
speed) and is only a maximum for a unique tip speed ratio.
• Incremental improvement in 𝐶𝑝 and hence power - by detailed design changes of the
rotor and, by operating at variable speed over a range of wind speeds.
• Major increases in the output power - By increasing A and 𝑉
𝑤 by locating the
• WT on sites with higher average 𝑉
𝑤.
• Last 10 years -- continuous increase in A (proportional to 𝐷2
)
• Doubling of the rotor diameter → four-times increase in power output.
• Doubling of wind speed →eight-fold increase in power.
• Hence efforts to develop wind farms in high wind areas and optimal location.
• Use of very high towers - increase of wind speed with height.
• All modern electricity-generating WT use the lift force derived from the blades to drive
the rotor.

Modern wind turbine - trends
• High rotational speed of the rotor is desirable →reduced gearbox ratio.
• High rotational speed → leads to low solidity rotors.
• Solidity - the ratio of blade area/rotor swept area.
• Low solidity rotor acts as energy concentrator → reduced energy recovery period of a wind
turbine. (on a good site, is less than 1 year)
• energy recovery period - the period in which energy used to manufacture and install the wind
turbine is recovered (Musgrove in Freris, 1990).

AUTONOMOUS WIND ENERGY SYSTEMS.
- People in the world without access to mains electricity = 2 billion.
- WT with other generators, e.g., diesel engines, may in the future provide some of these people
with power.
- Autonomous power systems are extremely difficult to design and operate reliably, particularly in
remote areas of the world and with limited budgets.
A small autonomous AC power system
- has all the technical challenges of a large national electricity system
- due to the low inertia of the plant, requires a very fast, sophisticated control system to maintain
stable operation.
- Over the last 20 years there have been a number of attempts to operate autonomous wind–
diesel systems on islands throughout the world but with only limited success.
- This class of installation has its own particular problems.

OFFSHORE WIND TURBINES:
- Installation of offshore wind turbines is ongoing.
- The few offshore wind farms already installed are in rather shallow waters and
resemble land-based wind farms in many respects using medium sized wind
turbines.
- Very large wind farms with multi-megawatt turbines located in deeper water,
many kilometres offshore, are now being planned and these will be constructed
over the coming years.
- However, the technology of offshore wind-energy projects is still evolving

Speed and Power Relations
• The kinetic energy in air of mass “m” moving with speed V is given by the following in SI
units:
• Kinetic energy =
1
2
𝑚𝑉2
• P = mechanical power in the moving air
• ρ = air density, ൗ
𝑘𝑔
𝑚2
• A = area swept by the rotor blades, 𝑚2
• V = velocity of the air, m/s
• Power, P =
1
2
(𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑)𝑉2
• P =
1
2
(𝜌AV).𝑉2=
1
2
𝜌𝐴𝑉3 watts

Specific Power of site
• Two potential wind sites are compared in terms of the specific wind power
expressed in watts per square meter of area swept by the rotating blades.
• It is also referred to as the power density of the site, and is given by the following
expression.
• 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑡𝑒 =
1
2
𝜌𝑉3
𝑤𝑎𝑡𝑡𝑠 𝑝𝑒𝑟 𝑚2
𝑜𝑓 𝑡ℎ𝑒 𝑟𝑜𝑡𝑜𝑟 𝑠𝑤𝑒𝑝𝑡 𝑎𝑟𝑒𝑎
• It is the average power in the wind when average velocity is taken over a period
of time, say a year.

Power Extracted from the Wind
• The actual power extracted by the rotor blades is the difference between the
upstream and the downstream wind powers.
• 𝑃0 =
1
2
𝑚𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑠𝑒𝑐𝑜𝑛𝑑. {𝑉2
− 𝑉0
2
}
• 𝑃0= mechanical power extracted by the rotor, i.e., the turbine output power
• 𝑉 = upstream wind velocity at the entrance of the rotor blades
• 𝑉0
2
= downstream wind velocity at the exit of the rotor blades.
• 𝑀𝑎𝑠𝑠 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 = 𝜌. 𝐴.
𝑉+𝑉0
2

• The mechanical power extracted by the rotor, which is driving the electrical
generator, is therefore:
• The above expression can be algebraically rearranged:
• 𝑃0 =
1
2
[𝜌. 𝐴.
(𝑉+𝑉0)
2
].(𝑉2 − 𝑉0
2
)
• 𝑃0 =
1
2
𝜌. 𝐴. 𝑉3
1+
𝑉0
𝑉
(1−
𝑉0
𝑉
2
)
2
• The power extracted by the blades is customarily expressed as a fraction of the
upstream wind power as follows:
• 𝑃0 =
1
2
𝜌𝐴𝑉3
𝐶𝑝
• 𝑤ℎ𝑒𝑟𝑒 𝐶𝑝 =
1+
𝑉0
𝑉
(1−
𝑉0
𝑉
2
)
2
Wind turbine power calculation

• 𝐶𝑝 is the fraction of the upstream wind power, which is captured by the rotor
blades.
• 𝐶𝑝 is called the power coefficient of the rotor or the rotor efficiency.
• For a given upstream wind speed, the value of 𝐶𝑝 depends on the ratio of the
down stream to the upstream wind speeds, that is ( ൗ
𝑉0
𝑉)
• It has the maximum value of 0.59 when the ( ൗ
𝑉0
𝑉) is one-third which is also called
the Getz limit.
• The maximum power is extracted from the wind at that speed ratio, when the
downstream wind speed equals one-third of the upstream speed.

Rotor efficiency versus tip speed ratio
FIGURE Rotor efﬁciency vs. Vo/V ratio has a single maximum.

Rotor efficiency versus tip speed ratio for rotors with different numbers of
blades

Why need to control turbine speed?
• Why control speed of turbine rotor?
• Fixed Speed Wind Turbine (FSWT)
• Variable Speed Wind Turbine (VSWT)
• Gen-1 FSWT with SCIG (Geared, two speed). Problem with reactive support.
• Gen-2 VSWT with WRIG and Rotor Resistance Control and DFIG. Losses in
former and advantages of DFIG for pf control. Limited speed range.
• Gen-3 VSWT with Electrically excited SG and Full-rated PE for wider speed
range.
• Gen-4 VSWT with PMSG and Direct drive (gearless design) for slower speed
alternators and thus elimination of gear box.

• 𝐶𝑝 has the maximum value of 0.59 when
𝑉0
𝑉
ratio is one third.
• Under such condition,
• 𝑃𝑚𝑎𝑥 =
1
2
𝜌𝐴𝑉3
𝐶𝑝
• There is a relation between rotor tip speed and wind speed which achieves
maximum value of Cp when pitch angle is kept constant.
• Since 𝐶𝑝 ranges from 0.4 – 0.5 for modern high speed two-speed turbines and
between 0.2 and 0.4 for slow speed turbines with more blades.
• Taking and average value of 0.5, the approximate maximum power output
becomes
• 𝑃𝑚𝑎𝑥 =
1
4
𝜌𝑉3

Rotor Swept Area
• As seen in the power equation, the output power of the wind turbine varies
linearly with the rotor swept area
• For a horizontal wind turbine, rotor swept area is given as,
𝐴 =
𝜋
4
𝐷2
where D is the rotor diameter.
For Darrieus vertical axis wind turbines, the rotor swept area is calculated by the
formula given below.
𝐴 =
2
3
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑟𝑜𝑡𝑜𝑟 𝑤𝑖𝑑𝑡ℎ 𝑎𝑡 𝑐𝑒𝑛𝑡𝑒𝑟 . (𝐻𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑜𝑡𝑜𝑟)

Air density variation with pressure and temperature
• Air Density
• The wind power varies linearly with the air density sweeping the blades. The
air density ρ varies with pressure and temperature in accordance with the gas
law:
• 𝜌 =
𝑝
𝑅.𝑇
• where p = air pressure
• T = temperature on the absolute scale
• R = gas constant.
•The air density at sea level at 1 atm (14.7 psi) and 60 ⁰F is 1.225 ൗ
𝑘𝑔
𝑚3.
•Using this as reference air density is corrected for site specific temperature and
pressure.

• The temperature and the pressure both in turn vary with the altitude.
• Their combined effect on the air density is given by the following equation, which
is valid up to 6,000 meters (20,000 feet) of site elevation above the sea level:
• where 𝐻𝑚 is the site elevation in meters.
• 𝜌 = 𝜌0 − 1.194 × 10−4 𝐻𝑚
The temperature varies with the elevation
▪ 𝑇 = 15.5 −
19.83 𝐻𝑚
3048
⁰C
Air density variation with height

Global Wind Patterns
• The global wind patterns are created by uneven heating and the spinning of the
earth.
• The warm air rises near the equator, and the surface air moves in to replace the
rising air. As a result, two major belts of the global wind patterns are created.
• The wind between the equator and about 30° north and south latitudes move east to
west. These are called the trade winds because of their use in sailing ships for trades
• The prevailing winds move from west to east in two belts between latitudes 30° and
60°north and south of the equator. This motion is caused by circulation of the trade
winds in a closed loop.
• The speed is measured with an instrument called anemometer, which comes in
several types
• The wind direction is measured with an instrument called the weather vane.

Global
Wind
Patterns
Source: https://en.wikipedia.org/wiki/Prevailing_winds

Measurement of wind speed & direction
(Cup anemometer and Weather vane)
https://www.metoffice.gov.uk/weather/guides/observations/how-we-measure-
wind#:~:text=The%20instruments%20used%20to%20measure,%2D1%20%3D%201.15%20mph).

Measurement of wind speed
(Optical sensor)
FIGURE Optical wind speed sensor construction. (From Georgia Institute of Technology, Atlanta, GA

Measurement of wind speed (Ultrasonic sensor)
https://www.biral.com/product/ultrasonic-anemometer-3d-4-3830-20-340/#product-overview

Measurement of wind direction (Weather Vane)
https://www.kintech-engineering.com/catalogue/wind-vanes/vector-w200p/

Wind Speed Distribution and assessment of Power potential of a site
• Power has a cubic relationship with wind speed.
• Hence the wind speed is the most critical data needed to appraise the power
potential of a candidate site
• The wind is never steady at any site. It is influenced by the weather system, the
local land terrain, and the height above the ground surface.
• The annual mean speed needs to be averaged over 10 or more years. Such a long
term average raises the confidence in assessing the energy-capture potential of a
site
• The short term, say one year, data is compared with a nearby site having a long
term data to predict the long term annual wind speed at the site under
consideration.
• This is known as the “measure, correlate and predict (MCP)” technique.

Wind Speed Distribution and
Assessment of Energy /Power potential of a site
• Because wind is driven by the sun and the seasons, the wind pattern generally
repeats over a period of 1 yr.
• The wind site is usually described by the speed data averaged over calendar
months.
• Sometimes, the monthly data is aggregated over the year for brevity in reporting
the overall “windiness” of various sites.
• Wind speed variations over the period can be described by a probability
distribution function.

Annual and Seasonal Variations
• Year-to-year variation in annual mean wind speeds is hard to predict
• But wind speed variations during the year can be well characterized in terms of a probability distribution.
• The variation in wind speed is best described by the Weibull probability distribution function h with two
parameters, the shape parameter k, and the scale parameter c.
• The probability of wind speed being v during any time interval is given by the following:
• ℎ 𝑣 =
𝑘
𝑐
𝑣
𝑐
𝑘−1
𝑒−(
𝑣
𝑐
)𝑘
for 0 < v < ∞
• In the probability distribution chart, h over a chosen time period is plotted against v, where h is defined as
follows:
• ℎ =
𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑤𝑖𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 𝑖𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑣 𝑎𝑛𝑑 (𝑣+∇𝑣)
∇𝑣
• By definition of the PDF, the probability that the wind speed will be between zero and infinity during the
entire chosen time period is unity, i.e.:
• ‫׬‬
0
∞
ℎ∆𝑣 = 1

FIGURE 7 Weibull probability distribution function with scale parameter c = 10 and shape parameters k =
1, 2, and 3.

Weibull distribution
• Figure 7 is the plot of h vs. v for three different values of k in Equation.
• The curve with k = 1 has a heavy bias to the left, where most days are windless (v = 0).
• The curve with k = 3 looks more like a normal bell shaped distribution, where some days have
high wind and an equal number of days have low wind.
• The curve with k = 2 is a typical wind speed distribution found at most sites.
• In this distribution, more days have speeds lower than the mean speed, whereas a few days have
high wind.
• The value of k determines the shape of the curve and hence is called the shape parameter.
• The Weibull distribution with k = 1 is called the exponential distribution, which is generally used in
reliability studies.
• For k > 3, it approaches the normal distribution, often called the Gaussian or the bell-shaped
distribution.

FIGURE 8 Weibull probability distribution with shape parameter k = 2 and scale parameter c ranging from 8
to 16 mph. For higher c curves shift to the right to higher wind speeds. Higher the c, greater the number of days that
have high wind speeds. Hence c called scale parameter.

FIGURE 9 Rayleigh distribution of hours per year compared with measured wind speed distribution at St. Annes
Head, Pembrokeshire, U.K.

Raleigh distribution
• At most sites, wind speed has the Weibull distribution with k = 2, known as the Rayleigh
distribution.
• The actual measurement data taken at most sites compare well with the Rayleigh distribution, as
seen in Figure 9.
• The Rayleigh distribution is then a simple and accurate enough representation of the wind speed
with just one parameter, the scale parameter c.
k = 1 makes it the exponential distribution,
ℎ = 𝜆𝑒−𝜆𝜈where 𝜆 =
1
𝑐
k = 2 makes it the Rayleigh distribution
ℎ = 2𝜆2𝑣𝑒−(𝜆𝑣)2
k > 3 makes it approach a normal bell-shaped distribution.
Most wind sites have a scale parameter ranging from 10-20 mph (5-10 m/sec) and the shape
parameter from 1.5-2.5 (rarely 3)

FIGURE 10 Weibull distributions of hours per year with three different shape parameters k = 1.5, 2, and 3, each
with scale parameters c = 10, 15, and 20 mph. Figure shows no. of hours on y-axis and wind speed on x-axis.
Three graphs with different scale parameters c=10,15 and 20 mph and three shape parameter k ranging from
1.5-2.5.

FIGURE 11 3-d Weibull distribution plots of
hours per year with three different shape
parameters k = 1.5, 2, and 3, each with scale
parameters c = 10, 15, and 20 mph.
• Figure depicts the same plots in three-
dimensional h–v–k space.
• Shows the effect of k in shifting the shape
from the bell shape in the front right-hand side
(k = 3) to the Rayleigh and to flatten shapes as
the value of k decreases from 3.0 to 1.5.
• Also observe that as c increases, the
distribution shifts to higher speed values.

Mean and Mode wind speed
Terms associated with wind speed.
• Mode speed is defined as the speed corresponding to the hump in the distribution
function. This is the speed of the wind most of the time.
• Mean speed over the period is defined as the total area under the h–v curve integrated
from v = 0 to ∝ and divided by the total number of hours in the period (8760 if the period
is 1 year).
• The annual mean speed is the weighted average speed given by:
• 𝑉
𝑚𝑒𝑎𝑛 =
1
8760
‫׬‬0
∞
ℎ𝑣d𝑣
• For c and k values in the range found at most sites, the integral expression can be
approximated to the Gamma function:
• 𝑉
𝑚𝑒𝑎𝑛 = 𝑐 (1 +
1
𝑘
)
• For the Rayleigh distribution with k = 2, the Gamma function can be further
approximated to the following:
• 𝑉
𝑚𝑒𝑎𝑛 = 0.90𝑐

Root mean cube speed
• Wind power is proportional to the cube of the speed, and the energy collected over the year is
the integral of ℎ𝑣3
𝑑𝑣
• We, therefore, define the root mean cube or the rmc speed in a manner similar to the root mean
square (rms) value in alternating current (AC) electrical circuits:
• 𝑉
𝑟𝑚𝑐 =
3 1
8760
‫׬‬
0
∞
ℎ𝑣3
d𝑣
• The rmc speed is useful in quickly estimating the annual energy potential of the site. Using 𝑉
𝑟𝑚𝑐 in
the equation gives the annual average power generation in W/m2:
• 𝑃𝑟𝑚𝑐 =
1
4
𝜌𝑉
𝑟𝑚𝑐
3
• Annual energy production potential = 8760 𝑃𝑟𝑚𝑐
• Importance of rmc speed can be seen from Table 1, where wind power density at three sites with
same average wind speed of 6.3 m/sec are compared. San Gorgonio site has 66 % higher power
density than Culebra site because of different shape factors k and hence different rmc speeds
though having the same average wind speed.

Why take rmc speed?
The important difference between the mode, the mean, and the rmc speeds is illustrated in Table 3.2.
The values of the three speeds are compiled for four shape parameters (k = l.5, 2.0, 2.5, and 3.0) and three scale
parameters (c = 10, 15, and 20 mph).
The upstream wind power densities are calculated using the respective speeds in the wind power equation 𝑃 =
Τ
1
2 𝜌𝐴𝑉3
W/m2 using the air mass density of 1.225 kg/m3.

Influence of k and c on the Mode, Mean, and RMC
Speeds and the Energy Density
Observations from the c = 15 rows of Table 2: Assumption rmc gives most accurate results.
• For k = 1.5, the power density using the mode speed is 230 as against the correct value of 4134
W/m2 using the rmc speed. The ratio of incorrect /correct value of the power density is 18 (huge
error when using mode in place rmc).
• For k = 2, the power densities using the mode and rmc speeds are 731 and 2748 W/m2,
respectively, in the ratio of 1 to 3.76. The corresponding power densities with the mean and the
rmc speeds are 1439 and 2748 W/m2 in the ratio of 1 to 1.91. (use of mean instead mode gives
lesser error)
• For k = 3, the power densities using the mode and rmc speeds are 1377 and 2067 W/m2
respectively, in the ratio of 1 to 1.50. The corresponding power densities with the mean and the
rmc speeds are 1472 and 2067 W/m2, in the ratio of 1 to 1.40. (error in using mean or mode
instead of rmc reduces further as k increases)

Influence of k and c on the Mode, Mean, and RMC
Speeds and the Energy Density
Observations from the last column in Table 2
• Last column gives yearly energy potentials of the corresponding sites in khr per year per square
meter of the blade area for the given k and c values calculated for Cp of 50%, which is the
practical maximum achievable.
• Regardless of the shape and the scale parameters, use of the mode or the mean speed in the
power density equation introduces a significant error in the annual energy estimate, sometimes
off by several folds, making the estimates completely useless.
• Only the rmc speed in the power equation always gives the correct average power over a
period.

Illustration of error in energy calculation when mean speed in
place of rmc speed
Average wind speed = (0+10+20+30)/4=15 mph
Annual wind energy using average speed = 153
=3375 units
Root mean cube wind speed =
3
(03 + 103 + 203 + 303=
3
9000=20.8009
Annual wind energy using rmc speed = 20.80093
= 9000 units
Ratio of Energy potential estimated correctly using rmc speed to that calculated using average
speed=
9000
3375
= 2.67. Error =(9000-3375)/9000=62.5 %

FIGURE 12 Annual frequency distributions of hours vs. wind speed and energy density per year with c = 10
and k = 2 (Rayleigh distribution).

ENERGY DISTRIBUTION
• If we define the energy distribution function:
• 𝑒 =
kWh contribution in the year by the wind between 𝑣 and (𝑣+∆𝑣)
∆𝑣
• then, for the Rayleigh speed distribution (k = 2), the energy distribution would look like the
shaded curve in Figure 12.
• For wind speed curve, mode = 5.5 m/sec and mean = 6.35 m/sec.
• Because of the cubic relation with speed, the maximum energy contribution comes from the wind
speed at 9.45 m/sec. Mode of the energy contribution curve = 9.45 m/sec.
• Above this speed, although 𝑉3
continues to increase in a cubic manner, the number of hours at
those speeds decreases faster than 𝑉3
. The result is an overall decrease in the yearly energy
contribution.
• Hence it is advantageous to design the wind power system to operate at variable speeds in order
to capture the maximum energy available during high-wind periods rather than designing it for
the mean wind, mode or rmc wind speed to use the maximum energy available during the high
wind periods.

Figure 13 - Rayleigh distributions of hours vs. wind speed and energy per year with c = 15
and k = 1. 5.

FIGURE 14 Rayleigh distributions of hours vs. wind speed and energy per year with k = 2 and c = 10,
15, and 20 mph.

ENERGY DISTRIBUTION
• Figure 3.13 is a similar chart showing the speed and energy distribution functions for a shape
parameter of 1.5 and a scale parameter of 15 mph.
• The mode speed is 10.6 mph, the mean speed is 13.3 mph, and the rmc speed is 16.5 mph. The
energy distribution function has the mode at 28.5 mph. That is, the most energy is captured at
28.5-mph wind speed, although the probability of wind blowing at that speed is low.
• Comparing Figure 3.12 and Figure 3.13, we see that as the shape parameter value decreases
from 2.0 to 1.5, the speed and the energy modes move farther apart.
• On the other hand, as the speed distribution approaches the bell shape for k > 3, the speed and
the energy modes get closer to each other.
• Figure 14 compares the speed and the energy distributions with k = 2 (Rayleigh) and c = 10, 15,
and 20 mph.
• As seen here, the relative spread between the speed mode and the energy mode remains about
the same, although both shift to the right as c increases.

EFFECT OF HUB HEIGHT
• The wind shear at a ground-level surface causes the wind speed to increase with height in
accordance with the following expression:
• 𝑉2 = 𝑉1(
ℎ2
ℎ1
)∝
• Where 𝑉1 = wind speed measured at reference height ℎ1
• 𝑉2= wind speed estimated at height h2
• ∝= ground surface friction coefficient.
• The friction coefficient ∝ is low for smooth terrain and high for rough ones.
• The values of ∝ for typical terrain classes are given in Table 3.3, and their effects on the wind
speed at various heights are plotted in Figure 3.15.
• It is noted that the offshore wind tower, being in low-α terrain, always sees a higher wind speed
at a given height and is less sensitive to tower height.

Figure 15: Wind speed variations with height over different terrain. Smooth, low-friction terrain with low α develops a
thinner layer of slow wind near the surface and high wind at heights.

Figure 16: Wind speed variations with height measured at Merida airport, Yucatan, in Mexico. (From Schwartz, M.N.
and Elliott, D.L., “Mexico Wind Resource Assessment Project,” DOE/NREL Report No. DE95009202, National
Renewable Energy Laboratory, Golden, Colorado, March 1995.)

• Wind speed does not increase with height indefinitely, even at a slower rate.
• The data collected at Merida airport in Mexico show that typically wind speed increases with
height up to about 450 m and then decreases (Figure 3.16)
• The wind speed at 450-m height can be 4 to 5 times greater than that near the ground surface.
• Modern wind turbines operate on increasingly taller towers to take advantage of the increased
wind speeds at higher altitudes. Disadvantage: Very little is known about the turbulent wind
patterns at these heights, which can damage the rotor.
• Of particular interest on the Great Plains, where many wind farms will be located, are the high
level wind flows called nocturnal jets that dip close to the ground at night, creating violent
turbulence.
• Engineers at the National Wind Technology Center (NWTC) have been measuring higher-altitude
wind patterns and developing simulation models of the turbine’s interaction with turbulent wind
patterns to develop designs that can prevent potential damage to the rotor.

FIGURE 3.17 Annual average wind power density in W/m2 in the U.S. at 50-m tower height. (From DOE/NREL.)

FIGURE 3.27 Wind-monitoring sites in India.
(From Tata Energy Research Institute, New
Delhi, India.)

FIGURE 3.28 Lamda wind farm in Gujarat, India. (From Vestas Wind Systems, Denmark. With permission.)

WIND RESOURCE:
- The power and energy available in the wind varies as the cube of the wind speed, so
understanding of the characteristics of the wind resource is critical to all aspects of wind energy
exploitation.
o Identification of suitable sites
o Predictions of the economic viability of wind farm projects
o Design of wind turbines themselves.
o Understanding their effect on electricity distribution networks and consumers.
- Most striking characteristic of the wind resource - its variability.
- The wind is highly variable, both geographically and temporally.
- Variability persists over a very wide range of scales (in space and time).
- Importance of variability - amplified by the cubic relationship to available energy.
- On a large scale, spatial variability - the fact that there are many different climatic regions in the
world, some much windier than others.
- These regions are largely dictated by the latitude, which affects the amount of insolation.

Wind resource
- Within any one climatic region, there is a great deal of variation on a smaller scale, largely dictated
by physical geography – the proportion of land and sea, the size of land masses, and the presence of
mountains or plains for example.
- The type of vegetation may also have a significant influence through its effects on the absorption or
reflection of solar radiation, affecting surface temperatures, and on humidity.
- More locally, the topography has a major effect on the wind climate. More wind is experienced on
the tops of hills and mountains than in the lee of high ground or in sheltered valleys, for instance.
- More locally still, wind velocities are significantly reduced by obstacles such as trees or buildings.
- At a given location, temporal variability on a large scale means that the amount of wind may vary
from one year to the next, with even larger scale variations over periods of decades or more.
- These long-term variations are not well understood, and may make it difficult to make accurate
predictions of the economic viability of particular wind-farm projects, for instance.
- On time-scales shorter than a year, seasonal variations are much more predictable, although there
are large variations on shorter time-scales still, which although reasonably well understood, are
often not very predictable more than a few days.

Fig 6 - Wind Spectrum Farm Brookhaven Based on Work by van der Hoven (1957)

Van der Haven (1957)
- Van der Hoven (1957) constructed a wind-speed spectrum from long- and short-term records at
Brookhaven, New York, showing clear peaks corresponding to the synoptic, diurnal and turbulent effects
referred to above.
- Of particular interest is the so-called ‘spectral gap’ occurring between the diurnal and turbulent peaks,
showing that the synoptic (large scale) and diurnal (daily) variations can be treated as quite distinct from
the higher-frequency fluctuations of turbulence. There is very little energy in the spectrum in the region
between 2 h and 10 min.
- These ‘synoptic’ variations are associated with the passage of weather systems.
- Depending on location, there may also be considerable variations with the time of day (diurnal variations)
which again are usually fairly predictable.
- On these time-scales, the predictability of the wind is important for integrating large amounts of wind
power into the electricity network, to allow the other generating plant supplying the network to be
organized appropriately (operation, demand generation matching, gen planning).
- On still shorter time-scales of minutes down to seconds or less, wind-speed variations known as
turbulence can have a very significant effect on the design and performance of the individual wind
turbines, as well as on the quality of power delivered to the network and its effect on consumers (power
quality).

Geographical Variation in the Wind Resource
- The winds are driven almost entirely by the sun’s energy, causing differential surface heating.
- The heating is most intense on land masses closer to the equator, and obviously the greatest
heating occurs in the daytime, which means that the region of greatest heating moves around
the earth’s surface as it spins on its axis.
- Warm air rises and circulates in the atmosphere to sink back to the surface in cooler areas.
- The resulting large-scale motion of the air is strongly influenced by coriolis forces due to the
earth’s rotation.
- The result is a large-scale global circulation pattern.
- Certain identifiable features of this such as the trade winds and the ‘roaring forties’ are well
known.
- The non-uniformity of the earth’s surface, with its pattern of land masses and oceans, ensures
that this global circulation pattern is disturbed by smaller-scale variations on continental
scales.
- These variations interact in a highly complex and nonlinear fashion to produce a somewhat
chaotic result, which is at the root of the day-to-day unpredictability of the weather in particular
locations.

Other effects
- Clearly though, underlying tendencies remain which lead to clear climatic differences
between regions.
- These differences are tempered by more local topographical and thermal effects.
- Hills and mountains result in local regions of increased wind speed. This is partly a result
of altitude – the earth’s boundary layer means that wind speed generally increases with
height above ground, and hill tops and mountain peaks may ‘project’ into the higher
wind-speed layers.
- It is also partly a result of the acceleration of the wind flow over and around hills and
mountains, and funnelling through passes or along valleys aligned with the flow.
- Equally, topography may produce areas of reduced wind speed, such as sheltered
valleys, areas in the lee of a mountain ridge or where the flow patterns result in
stagnation points.
- Thermal effects may also result in considerable local variations.
- Coastal regions are often windy because of differential heating between land and sea.

Other effects
- While the sea is warmer than the land, a local circulation develops in which surface air flows
from the land to the sea, with warm air rising over the sea and cool air sinking over the land.
- When the land is warmer the pattern reverses. The land will heat up and cool down more rapidly
than the sea surface, and so this pattern of land and sea breezes tends to reverse over a 24 h
cycle.
- These effects were important in the early development of wind power in California, where an
ocean current brings cold water to the coast, not far from desert areas which heat up strongly by
day.
- An intervening mountain range funnels the resulting air flow through its passes, generating
locally very strong and reliable winds (which are well correlated with peaks in the local electricity
demand caused by air-conditioning loads).
- Thermal effects may also be caused by differences in altitude. Thus cold air from high mountains
can sink down to the plains below, causing quite strong and highly stratified ‘downslope’ winds.
- The high-frequency wind fluctuations known as turbulence, which are crucial to the design and
operation of wind turbines and have a major influence on wind turbine loads.
- Extreme winds are also important for the survival of wind turbines.

Long-term Wind speed Variations
- There is evidence that the wind speed at any particular location may be subject to very
slow long-term variations. Although the availability of accurate historical records is a
limitation, careful analysis by, for example, Palutikoff, Guo and Halliday (1991) has
demonstrated clear trends. Clearly these may be linked to longterm temperature
variations for which there is ample historical evidence.
- There is also much debate at present about the likely effects of global warming, caused
by human activity, on climate, and this will undoubtedly affect wind climates in the
coming decades.
- Apart from these long-term trends there may be considerable changes in windiness at a
given location from one year to the next. These changes have many causes.
- They may be coupled to global climate phenomema such as El Nino, changes in
atmospheric particulates resulting from volcanic eruptions, and sunspot activity, to
name a few.
- These changes add significantly to the uncertainty in predicting the energy output of a
wind farm at a particular location during its projected lifetime.

Reference
• Chapter 3 Wind and Solar Power systems- Patel, Taylor and Francis
• Chapter 3 Wind Energy Handbook – Tony Burton et al. Wiley

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