Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: THEend8_
Design & Computing CENG10014
UOB Confidential
Design and Computing-Wind Power Calculations and Costings-updated Feb 2020
Overview
This supplement to the Nordic Wind farm project provides equations and calculations to work out
theoretical power output of a turbine as well as wind gradients. There is also more detailed cost
information.
Calculating Maximum Theoretical Power Output of a Wind Turbine
The maximum theoretical power output of a wind machine is 16/27 times the rate at which kinetic
energy of the air arrives at the effective disk area of the machine (see Betz's law [1]). Assuming no
efficiency losses, the maximum theoretical power output P (in Watts) is:
Where:
ρ is the air density(kg/m3)
v is wind velocity (m/s)
A is effective area of the rotors/disk (m2).
Wind Gradient
Wind turbine operation is affected by wind gradient. Vertical wind-speed profiles result in different
wind speeds at the blades nearest to the ground level compared to those at the top of blade travel
which results in asymmetric load. Therefore, the wind gradient can create a large bending moment
in the shaft of a two bladed turbine when the blades are vertical. The polynomial variation in wind
speed with height can be defined (relative to wind measured at a reference height of 10 meters) as:
where:
vw(h)= velocity of the wind [m/s], at height, h [m]
v 10= velocity of the wind [m/s], at height, h10 = 10m
α= Hellmann exponent
The Hellmann exponent depends upon turbine location and the shape of the terrain on the ground,
as well as the stability of the air. Examples of values of the Hellmann exponent are given in the table
below:
Table 1. Approximate Hellman exponent for different terrain types
location α
Unstable air over rocky/mountain terrain 0.06
Unstable air over open plains and hilly areas 0.11
Unstable air over plains and inhabited areas 0.27
Stable air over open plains, coast, wetland 0.40
Design & Computing CENG10014
Limitations and losses
Considerations need to be made about efficiency as a theoretical power output is not achievable in
real world conditions. Betz’s law takes into account that wind passes through the turbine (not all
wind is captured) and if a wind turbine was 100% efficient-there would be no wind behind the
turbine. For most wind turbines rotor efficiency starts to drop after 15m/s and the max power
generated saturates-the cut off wind speed is 25m/s where no power is generated. See fig 1.
Fig 1. Power output vs wind speed
It can be assumed that the bearings and transmission of a wind turbine are around 95% efficient, the
generator is 70% efficient and the rotors are 40% efficient. Overall turbine efficiency is 30-40%
You can find the approx. value of total efficiency of a wind turbine as follows:
μ = (1 - kₘ) * (1 - kₑ) * (1 - ke,t) *(1 - kt) * (1 - kw) * Cₚ
where:
Cₚ is the turbine efficiency. It must be lower than the Betz limit (59.3%), and is typically between 30-
40%
kw are the wake losses due to neighbouring turbines and the terrain topography, typically 3-10%
kₘ are the mechanical losses of the blades and gearbox, typically 0-0.3%
kₑ are the electrical losses of the turbine, typically 1-1.5%
ke,t are the electrical losses of transmission to grid, typically 3-10%
kt is the percentage of time out of order due to failure or maintenance, typically 2-3%
μ is the efficiency
To find the actual wind turbine power, multiply the efficiency by the theoretical wind power
available:
Poutput = μ * Pwind
Design & Computing CENG10014
Cost Information
The manufacturing and construction of a wind turbine costs around £750,000 per MW and it can be
assumed that the turbine costs are around 70% of the total project and infrastructure costs. This
value can be scaled up for multiple turbines. The operation and maintenance (O&M) costs of a single
wind turbine can be approximated from the table below:
Table 2. Approximate O&M costs for different wind turbine power outputs
Maximum Power Output of Turbine (MW) Annual O&M cost (£)
0.5 20k
4.5 200k
8 400k
Again, these values can be scaled up for multiple turbines. It can be assumed that a long term
warranty and maintenance package can last between 5 to 15 years (depending on turbine size).
These O&M cost values also cover business rates and insurance.
The Return on Investment of a windpower scheme depends on the net income received and the
capital costs of the project. Determining the income is complicated, it depends on:
-the proportion of electricity exported to the grid vs electricity consumed on site
-the current and future cost of using electricity imported from the national grid on site
-as well as the current and future price paid for electricity sold to the grid from the wind system.
These figures are complicated because of the uncertainty in the rising cost of electricity, in particular
non-domestic electricity that has risen by 9% per year between 2004 and 2019.
Future changes to electricity prices may not follow historical price rises and can be hard to predict
because they are heavily influenced by fossil-fuel prices and government policy changes in response
to market conditions and carbon reduction commitments.
It is approximated that annual electricity price rises for domestic use is 3% per year. Since 2017, the
UK government has since increased its commitment to carbon reduction which would mean that
investment in low carbon energy will inevitably have to increase beyond that-and it is possible that
electricity prices may rise or fall in future. It can be assumed that electricity price changes up to 2030
are consistent.
To calculate the revenue you would expect from a wind turbine or wind turbine development. It can
vary based on the electricity tariff – and thus the earnings per one kWh generated by the turbine or
turbine development. Assuming a UK tariff of 14p/kWh, the following calculation can be used:
Revenue = tariff * Poutput
Design & Computing CENG10014
Table 3. shows what the Internal Rate of Return (IRR) could be for a range of wind turbines with
annual mean wind speeds ranging from 5 to 8.5 m/s, where 100% of electricity is consumed on site
and there is an assumed annual price rise of the cost of electricity of 3%.
Table 3. IRR 100% on-site consumption, annual electricity price rise 3%
Maximum Power Output per turbine IRR’s for sites with for the following annual average wind speeds [m/s]:
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5
100 kW 4% 7% 9% 11% 13% 14% 15% 16%
800 kW 13% 17% 20% 24% 27% 30% 34% 36%
1 MW 14% 17% 21% 24% 27% 30% 34% 37%
4.5 MW 15% 19% 24% 28% 33% 37% 42% 46%
8 MW 23% 28% 33% 38% 43% 47% 51% 55%
Another scenario is that 50% of electricity is consumed on-site (to meet local energy needs around
the site) and 50% is exported to the national grid. It can be assumed that:
-the majority of electricity demand is from households
-that there are 11,500 homes in Orkland
-a typical UK home uses around 3700kWh per year.
This IRR is shown in table 4.
Design & Computing CENG10014
Table 4. IRR 50% on-site consumption, annual electricity price rise 3%
Maximum Power Output IRR’s for sites with for the following annual average wind speeds:
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5
100 kW 1% 4% 6% 7% 9% 10% 11% 12%
800 kW 9% 12% 15% 18% 20% 22% 24% 26%
1 MW 10% 13% 15% 18% 21% 23% 25% 27%
4.5 MW 11% 14% 18% 21% 25% 28% 31% 35%
8 MW 17% 21% 25% 28% 32% 35% 38% 41%
Finally, if it is not possible to consume a significant proportion of electricity on-site, then 90-100% of
the electricity must be exported to the grid. A typical initial export price of 6.5 p/kWh, which
increases annually by an inflation rate of 2%, is used for this Rate of Return illustration in table 5.:
Table 5. IRR 100% exported to the grid, annual export price rise of 2%
Maximum Power Output IRR’s for sites with for the following annual average wind speeds: