Geoscience Reference
In-Depth Information
designed to produce power because the power gener-
ated would be low and uneconomical. Above the cut-in
wind speed, the instantaneous power output increases
roughly proportionally to the cube of the wind speed.
At the
rated wind speed
,the power output reaches the
rated power (Figure 13.14a). The power output stays at
the rated power for all higher wind speeds until the
cut-
out wind speed
is reached, at which point the power
output drops to zero to prevent damage to the turbine. A
turbine can survive wind speeds up to the
destruction
wind speed
,which is often 50 to 60 m s
−
1
.Forcompar-
ison, wind speeds in a Category 4 hurricane are 58.3 to
69 m s
−
1
.
If a wind turbine ran for a full year at its rated power,
its energy output would be
P
r
H
.However, because wind
speeds are generally lower than the rated wind speed,
wind turbines realize only a fraction of their maximum
energy output during the year. The fraction of the maxi-
mum energy that a wind turbine produces during a year
is the
capacity factor
of the turbine. A simple, yet accu-
rate (within 1-3 percent for most and
Example 13.1
Calculate the range of capacity factors of a 5-MW
turbine with a 126-m rotor diameter operating in
7to8.5ms
−1
annually averaged wind speeds.
Solution
Substituting
P
r
=
126 m, and
V
m
=
7-8.5 m s
−1
into Equation 13.2 gives capacity fac-
tors ranging from 0.294 to 0.425. Thus, during
the year, this turbine produces 29.4 to 42.5 per-
cent of its maximum possible energy output.
5,000 kW,
D
=
The
system efficiency
of a wind turbine is the ratio
of energy delivered to raw energy produced by the tur-
bine; it accounts for array, transmission, and distribution
losses.
Array losses
are energy losses resulting from
decreases in wind speed that occur at a large wind farm
when upstream turbines extract energy from the wind,
reducing the wind speed slightly for downstream tur-
bines. Upstream turbines also create vortices or ripples
(wakes) that can interfere with downwind turbines. The
greater the spacing between wind turbines, the lower the
array losses due to loss of energy in the wind, vortices,
and wakes. In a wind farm with an array of turbines, the
spacing area
(m
2
) occupied by one turbine is typically
10 percent for
all turbines tested) equation for the capacity factor of a
wind turbine is
<
D
2
CF
=
0
.
087
×
V
m
−
P
r
/
(13.2)
where
V
m
is the mean annual wind speed in units of
ms
−
1
,
P
r
is the rated power in kW, and
D
is the tur-
bine rotor diameter in m (Masters, 2004; Jacobson and
Masters, 2001). The units used must be those speci-
fied here. Units do not equate because the equation is
empirical. The mean annual wind speed in Equation
13.2 differs from the instantaneous wind speed used in
Figure 13.14a. The mean wind speed used in Equation
13.2 is the mean of a
Rayleigh probability distribu-
tion
of wind speeds, which is a probability distribution
of wind speeds that looks similar to a bell curve but
skewed toward higher wind speeds (Figure 13.14b).
The Rayleigh probability distribution is a specialized
case of the
Weibull probability distribution
of wind
speeds,
A
t
=
4
D
×
7
D
or
A
t
=
3
D
×
10
D
(13.4)
where
D
is the turbine rotor diameter (m). The first
dimension is the distance between turbines in a row, and
the second dimension is the distance between rows in
the predominant direction of the wind. Spacing between
wind turbines is not wasted space. It is often used for
agriculture, rangeland, or open space. Over the ocean,
it is open water.
With the spacing configurations in Equation 13.4,
array losses are generally 5 to 20 percent (Masters,
2004). Frandsen (2007), for example, examined wind
speed data from the Norrekaer Enge II wind farm in
Denmark, which consists of seven rows of six 300-KW
turbines. Spacing within rows was 7
D
to 8
D
and between
rows was 6
D
.Wind speeds averaged over each successive
row were derived from power output. The data indicated
that, despite wind speed reductions past turbine rows 1
and 2, wind speeds in subsequent turbine rows stayed
constant or increased compared with row 3. The overall
reduction in wind speed between rows 1 and 7 was only
7 percent, representing an array efficiency of 93 percent.
Immediately downwind of an individual turbine (in
the
turbine's wake
), wind speeds first decrease and then
increase, eventually converging into the background
exp
k
c
k
−
1
c
k
c
v
=
−
f
(
)
(13.3)
where
f
(
)isthe fractional occurrence of instantaneous
wind speed
v
(m s
−
1
)a
m
ong all wind speeds,
k
is an
v
2
V
m
/
√
integer, and
c
=
.ForaRayleigh distribution,
k
2. Winds are generally Rayleigh in nature, so mea-
surements of a wind speed frequency distribution over
a year often result in a Rayleigh distribution (Archer
and Jacobson, 2005).
=
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