Site Assessment and Installation - Small Wind Turbines: Analysis, Design, and Application - page 258

function out = opt_ht(m,U10, a, b, U0)
out = fzero(@(h) m*a/b/h+ m-1 ? U0/U10*(10/h)^m,30);
end

where the value ''30'' in the fzero reference is the initial value passed to the root
finding algorithm. The results are shown in Fig. 12.4 plotted against m. Although
the results are specific to one turbine and its towers, similar calculations for other
turbines yield similar results. From Table 1.3 , the lowest value of m in Fig. 12.4 is
for a very smooth surface, probably not attainable on land, and the largest value
corresponds to city centres. For small m the optimum heights are obviously not
physical and those for large m and low U 10 are also questionable because they
represent extrapolations of the data in Table 12.2 . Nevertheless, it is clear that for
rougher sites with low 10 m wind speeds, the optimum tower heights can be very
large. It is noted that some manufacturers provide only one tower height of around
10 m.
Typical distributions of Eq. 12.2 with height are plotted in Fig. 12.5 for
U 10 = 4 and 5 m/s. In some cases, such as for m = 0.15 and U 10 = 5 m/s, there is
little change with h, indicating that only a small penalty would be paid for using
non-optimum heights. In other cases, such as m = 0.3 for the same U 10 , h [ 20 m
is required for the same conclusion to apply.
So far, the discussion of siting has concentrated on optimising output per unit
cost by finding the windiest location and height. This parameter, however, is not
necessarily the only one to be optimised and there may be constraints such as
visual impact and noise as mentioned above. Windfarm layout is often treated as a
multi-dimensional optimisation problem in reducing noise, maximising power
production, minimising installation cost etc., e.g. Kusiak and Song [ 1 ]. Such
studies are not common for small turbine installations, but Professor Ferrer-Marti

Fig. 12.5 Variation of tower
height for Skystream 2.4 kW
turbine for varying 10 m
wind speeds and power law
exponent. The origin for the
vertical co-ordinate is zero

m = 0 .3

m = 0.25
m = 0.2

m = 0.15

U 10 = 4.0 m/s

m = 0.1

m = 0. 3

m = 0.25

m = 0.2

m = 0.15

U 10 = 5.0 m/s

m = 0.1

10 15 20 25 30 35

Tower height (m)