This problem set accompanies Chapter 3 (Wind Energy) of the textbook
Energy and the New Reality, Volume 2:
Carbon-Free Energy Supply
by
L. D. Danny Harvey
Department of Geography
University of Toronto
harvey@geog.utoronto.ca
Published by Earthscan (London, UK)
Homepage: www.earthscan.co.uk/?tabid=101807
Problem Set 2 – Wind Energy
Assigned:
Due:
Taken up:
Background – 1. How to Compute a Weighted Average
In this first part of this problem set, you will be computing a number of “mean” or average
quantities: the mean wind speed, given a probability distribution of wind speeds; the mean wind
power; the mean power output from a wind turbine; and the mean efficiency of a wind turbine. To
make sure that everybody is up to this task, we need to first review how to compute means or
averages.
If one has a series of numbers, computing the average is easy – one just adds up the numbers and
divides by the number of terms. Implicitly, what you are doing is giving each number a
“weighting” of 1.0, and dividing by the sum of all your weightings. More generally, however, we
might want to have certain numbers “count” for more when computing the average; that is, we
give some numbers a greater “weighting” than others. In this case, one multiplies each number by
its own weighting factor, sums up the products, and divides by the sum of the weighting factors.
Thus, if we have a series of numbers x1 to xn, and corresponding weighting factors p1 to pn, then
the weighted average or mean value of the xi, x , is given by

x

i n
p xi
i 1 i
i n
i 1
pi
Note that, in general, each pi could itself be the product of 2 or more factors.
When it comes to computing the mean wind speed, those wind speeds that occur more often
should be counted more strongly than those wind speeds that occur rarely. Of course, a precise
wind speed such as 2.0 m/s is really an infinitely thin point on the wind-speed axis, so the
probability of it occurring is infinitesimally small. Thus, the weighting factor should be the
probability of a wind speed occurring in some interval centered at the wind speed in question.
One interval should continue where the previous interval left off, so that you cover all the
possible wind speeds. That is, suppose we have wind speeds of 1, 2, 3, 4 m/s and so on. Then we
should use the probability of winds occurring between 0 and 1.5 m/s times a wind speed of 1 m/s,
the probability of occurring between 1.5 and 2.5 m/s times a wind speed of 2 m/s, and so on until
we have reached such a high wind speed that the probability of it being exceeded is negligible.
We add up the products of the probabilities times wind speeds, and divide by the sum of all the
probabilities, as in the above equation.
In this problem set, you will compute wind-speed probabilities from the Weibull distribution,
which directly gives probabilities for a wind-speed interval spanning a width of 1 m/s centered at
the wind speed in question. Thus, as long as you are computing wind speed probabilities at
increments of 1 m/s, you can just directly multiply each wind speed times the Weibull probability
when computing the mean wind speed.
The same approach would be used for computing the mean wind power and the mean power
output of a turbine – the wind power or turbine power at a series of wind speeds is weighted by
the probabilities of the corresponding wind-speed intervals. However, when computing the mean
turbine efficiency, weighting the efficiency at each wind speed by the probability of that wind
speed is not adequate. This is because, when we use efficiency in any calculations, we multiply it
by the wind power. Thus, the weighting factor for computing mean efficiency should be the
probability of the wind speed times the corresponding wind power – how much we should
“count” a given efficiency depends not only on how often the corresponding wind speed occurs,
but also by how much power is in that wind. This weighting factor is applied at each wind speed
used in computing the mean efficiency. As shown in Subsection 3.6.3 of the textbook,
multiplying the mean efficiency computed in this way times the mean wind power (weighted by
the wind speed probabilities) is mathematically equivalent to the mean turbine electricity output
(weighted by the wind speed probabilities).
Background – 2. How to Compute CO2 Emissions per kWh of Electricity Generated
For fossil fuel powerplants, the CO2 emission (expressed as a mass of C) per kWh of electricity
generated is equal to the fuel use per kWh generated times the emission factor, that is: GJ/kWh x
kgC/GJ
1 kWh of electricity is 1 kW x 1 hour = 1000 J/s x 3600 s = 3.6 MJ = 0.0036 GJ
(MJ = megajoule = 106 joules and GJ = gigajoule = 109 joules)
The amount of fuel required to generate 1 kWh of electricity is equal to 0.0036 divided by the
efficiency in generating electricity, because
Electrical energy produced = (fuel energy used) x (efficiency in generating electricity)
PART 1: Technical characteristics and performance of wind turbines
Question 1
In order to answer this question, students will need to meticulously read Sections 3.6 and 3.13 of
the textbook. The answers to all of the quantitative questions will go on your Excel template.
Please consolidate all of the written answers onto a single sheet.
(a) Using the formulae given in Box 3.1 of the textbook, compute the wind speed probability
distribution and the cumulative probability distribution in an Excel spreadsheet for the
Weibull scale factor c=8 m/s and the Weibull shape parameter k=1.6. Compute the
probabilities at wind speed intervals of 1 m/s, running from 0-25 m/s. Note: In Excel, you
have to type exp(-(u/c)^k) as exp(-((u/c)^k)) in order for Excel to give the right answer.
(b) Plot the wind speed distribution. For your distribution, what are the modal wind speed,
median wind speed, and mean wind speed? Plot the cumulative probability distribution on the
same graph, but using a secondary y axis. Explain in words what a cumulative probability
distribution is. How would you use the cumulative probability distribution to compute the
probability of a wind occurring within any given wind-speed interval?
(c) Assuming a temperature of 10C and a surface air pressure of 101.325 kPa (kilopascals),
compute the air density and from this compute the power density of the wind and the
intercepted wind power for each of the wind speeds used in (a). Show the air density that you
used to compute the wind power densities. Compute the mean wind power density.
(d) Compute the wind power density at the mean wind speed, and compare with the mean wind
power density computed in (c). Explain why these two are different.
(e) Below are power output data for an 1800-kW (1.8-MW) turbine, as provided by the
manufacturer. These data have already been entered into the Excel template that you will
receive, and correspond to an air density of 1.225 kg/m3. The turbine rotor swept area is
3845.5 m2. Using these data, (1) adjust the power output to correspond to the actual air
density, assuming that a change in air density does not change the efficiency of the turbine
and keeping in mind that the power output cannot exceed 1800 kW, irrespective of the air
density; (2) compute the efficiency of the turbine at each of the wind speeds used in (a)
(you’ll have to decide whether to use the published or adjusted power output); (3) compute
the mean efficiency of the turbine; and (4) compute the mean power output of the turbine, and
from this, the capacity factor for this turbine for the computed wind speed probability
distribution. Finally, plot the output data and efficiency on the same graph (as in Figure 3.10
of the textbook) (see the Excel Guidelines on how to get Excel to combine a bar chart and
line graph in the same figure).
Wind speed (m/s)
1
2
3
4
5
6
7
8
9
10
11
12
13
14-25
>25
Turbine Power Output
(kW)
0.0
0.0
15.3
51.8
116.8
208.8
338.7
510.1
725.6
991.0
1275.0
1549.0
1729.4
1800.0
0
(f) Confirm that your computed mean efficiency times the computed mean wind power density
times the rotor swept area is equal to the computed mean power production from the turbine.
PART 2: Economic of wind and hybrid wind-natural gas energy systems
Question 2
The cost recovery factor (CRF) that appears in the equations for the cost of electricity gives the
fraction of an initial investment (or borrowed amount of money) that needs to be paid back every
year so as to exactly pay back the initial investment plus interest over a period of N years. It is
given by
CRF 
i
1  (1  i )  N
where i is the annual interest rate expressed as a fraction (so 5%/yr is 0.05/yr). Rather than
accepting this on faith, I want you to see for yourself that this is true. Verify that this is indeed the
case for an interest rate of 5% and a 20-year equipment lifespan by keeping track of the amount
of money owed (the principle) at the start of each year, the interest for that year based on the
principle at the start of the year, and the principle at the end of year after adding in interest and
subtracting the fixed payment. To do this, use the part of the Excel worksheet set up for this
question – you need only type a few simple equations at the top of each column, then drag
everything down.
Question 3
Suppose that wind energy is going to be used with a natural gas powerplant as backup. Assume
that the average electricity demand is 65% of the peak demand. This means that the capacity
factor for the combined system (or with natural gas alone) is 0.65 (the capacity factor is the ratio
of average power output to peak output, and the output will vary so as to always equal the
demand). Assume the following for the wind and natural gas components of the hybrid system:
Wind turbine:
Total installed capital cost of $1500/kW, fixed annual operation and maintenance (O&M) equal
to 0.7% of the capital cost, variable O&M costs of 0.7 cents/kWh, a capacity factor of 0.3, an
availability of 95%, 3% miscellaneous losses, and 10% financing over a 20-year life.
Natural gas powerplant:
Assume a capital cost of $750/kW, 15% backup, fixed annual O&M equal to 2.5% of the capital
cost, variable O&M costs of 0.7 cents/kWh, and 10% financing over a 40-year lifespan (with the
15% backup, we can assume the availability to be 100%). As well, assume an efficiency of 55%
and a fuel cost of $6/GJ. Ignore variable O&M.
Do the following:
(a) Compute the direct cost of electricity provided by the wind turbine alone, based on the wind
turbine capacity factor given above.
(b) Compute the cost (from Eq. 3.17) and CO2 emission (kg C/kWh) for electricity from natural
gas alone (the C emission factor for natural gas is given in the Excel spreadsheet). Since the
natural gas power plant output can go up and down as needed to meet fluctuating demand, the
capacity factor in this case is assumed to be equal to the average electricity demand as a
fraction of the peak demand (in reality, there would be some down-time for maintenance or
repairs). The Excel template directs you to compute the individual components (capital cost,
O & M, fuel) as separate steps and then to add them up, rather than doing it all in one step, so
that the marker can see where you went wrong (if you go wrong) and so that you can see the
relative importance of the different terms.
(c) Consider a hybrid system, in which 1 kW of wind capacity in combined with 1 kW of natural
gas capacity with the normal backup (assumed to be 15%). Compute the fraction of electricity
supplied by wind and the fraction supplied by natural gas, assuming (1) that any wind
electricity generated can be used, (2) that the average demand is 0.65 times the total demand
(i.e., 0.65 kW for every kW of peak demand), and (3) the difference between the average
(d)
(e)
(f)
(g)
(h)
(i)
demand and the average amount of power supplied by wind is the average output of the
natural gas part of the hybrid system.
Compute the extra cost of electricity supplied by the hybrid system compared to natural gas
power alone. Do this in two ways: (1) by taking the average of the wind and natural gas costs
computed in (a) and (b), weighted by their fractional contributions to the total electricity
supply (as computed in (c)); and (2) based on the total revenue required per year to pay back
the initial total capital cost (wind + NG), the total fixed O&M costs, total amount of
electricity supplied per year, the fraction of the electricity supplied by the natural gas
component, the variable O&M costs, and the fuel cost, using Eq.(3.16) of the textbook (but
ignoring any wind capacity credit for now).
The two methods for computing the cost give different answers (even if you have made no
mistakes). This is because there is a conceptual flaw in using approach (1). What is the flaw?
Compute the average CO2 emission (kg C/kWh) for the hybrid system.
Compute the cost of avoided CO2 emission ($/tonne C) in the hybrid system, based on the
difference in the cost of electricity per kWh in the two systems and the difference in CO2
emission between the two systems (the CO2 emission for the turbine can assumed here to be
zero, although some people might include emissions entailed during the construction of the
turbine and averaged over all of the electricity generated during the lifetime of the turbine,
but then energy used in constructing the natural gas powerplant and in finding, extracting and
transmitting natural gas should be included too).
What is the cost of avoided CO2 emission if the price of natural gas is $12/GJ instead of
$6/GJ?
Compute the cost of electricity in the hybrid system if the wind turbine is given a capacity
credit equal to its capacity factor (as directed to in the Excel spreadsheet, show the separate
capital cost, O&M and fuel cost contributions to the total cost of electricity). By roughly what
fraction has the total cost of electricity been reduced?
In computing the cost of the hybrid system, we assume that the efficiency for generating
electricity from natural gas to be the same as in the pure natural gas system. Based on a careful
reading of the draft textbook, under what conditions is this a reasonable assumption?
Question 4
Consider a site where the average power density of the wind at the turbine hub height is 500 W
m-2. Assume a rotor diameter of 90 m, an average efficiency of 25%, and that turbines are laid out
in a wind farm on a rectangular grid with a spacing equal to 7 times the rotor diameter in both
directions. What is the average wind farm electricity production per square kilometer of land
area? (First, try to think of a simple way to figure out what the land area is per turbine.) What
would be the dimensions of a square area of land needed to produce the average power of a
typical large nuclear or fossil power plant, namely, 3 GW? If electricity sells for 8 cents/kWh and
the landowners are paid 2% of this, what is the annual revenue that the landowners would earn
per hectare of land (1 ha=100 m by 100 m)? By comparison, typical net income for farmers in
many countries is $100-200/ha per year, but of course highly variable and sometimes negative!
PART 3:
Question 5
Oversizing wind farms to provide more reliable electricity supply
For this question, we are interested in the economics of a wind farm that might consist of dozens
or hundreds of wind turbines, but we will do all our calculations in relation to one single turbine,
the 1.8-MW turbine considered in Q1. The output data for this turbine are found in the worksheet
for Q1 (this time we’ll ignore any differences in air density).
Compute the wind speed probabilities and the mean wind speed where indicated. In columns G to
I, enter the electricity outputs times the indicated scaling factor, but don’t let the output with
scaling factor > 1 to exceed 1.8 MW, which is the assumed transmission line capacity per turbine
when there is no oversizing (and which is held fixed).
Next, compute the mean constrained output with the different scaling factors. Then compute two
capacity factors for each scaling factor: one based on the constrained output divided by the scaled
wind turbine capacity (for example, when the scaling factor is 2, the scaled wind turbine capacity
if 3.6 MW), and the other based on the constrained output divided by the transmission line
capacity (which is always 1.8 MW).
Next, compute the fraction of potential electricity generation that is wasted due to limited
transmission capacity.
Next, compute the separate wind farm and transmission line contributions to the cost of
electricity, and add the two. Then, compute the average transmission loss, the total cost of
electricity taking into account the transmission loss, and the contribution of transmission loss
alone to the cost of electricity. Comment on what you find.
Finally, copy the completed worksheet and change the Weibull scale factor c from 9.0 m/s to 5.0
m/s. This gives a mean wind speed of 4.4 m/s, and so might correspond to the typical lousy wind
resource near major demand centres. One would certainly not build long transmission lines to
transmit electricity from such a poor resource to the demand centres, but one might make use of
the local wind resource. The wind turbine costs (including transport and installation) near a major
demand centre might also be less costly than wind turbines in a distance location. Thus, the
interesting comparison is between costs with c=5 m/s, capital cost of $1200/kW rather than
$1500/kW, no oversizing, and no transmission cost or loss (representing the local wind farm) and
the various oversized cases with c=9 m/s. Compare the cost of electricity and the capacity factor
for the local wind farm and for the oversized, distant wind farms.
Question 6
In question 5 we explored the effect of oversizing of the wind farm relative to the transmission
capacity on the fraction of wind energy potential from the wind farm that is wasted (due to
potential output occasionally exceeding the transmission-line capacity).
Even if the wind farm is not oversized relative to the transmission link, if the wind farm size were
increased such it could provide an annual number of kWhs equal to the total annual demand of
the region that it serves, there would be times when the wind output exceeded the demand and
would therefore be wasted (see Fig 3.40 for the wasted wind fraction as a function of output as a
fraction of annual demand for the specific wind and demand variation found in Denmark). The
wastage of wind in effect reduces the capacity factor of the wind farm, and thereby increases the
cost per kWh. However, if a use could be found for the excess wind and it were sold, the
economic penalty could be reduced (if the electricity has to be sold at a discount) or eliminated (if
no discount is necessary). Another option would be to store the wind energy in various ways, as
discussed in the text, but there would be an economic cost to the storage medium and due to
losses incurred while charging and discharging the storage medium.
In this question, you are going to explore the economics of using deeply discounted but
intermittent wind to desalinate water by reverse osmosis. Background information is found in
Subsection 3.14.3 of the textbook. You will also explore the economics of wind farms dedicated
to reverse osmosis desalination but oversized relative to the peak desalination power requirement.
Because winds tend not to be great in many of those parts of the world where there are water
shortages (such as the regions around 30°N and 30°S, although Somalia is an exception, as shown
in the supplemental figures in the powerpoint file that accompanies Chapter 3), we will not
assume very large wind speeds in this case.
For both cases, assume that the capital cost of desalination equipment is $1100 per m3/day of
desalination capacity, that the electricity requirement is 3 kWh/m3, that the plant is financed at
3% real interest rate over 20 years, and that there is a fixed annual O&M cost equal to 2% of the
initial capital cost. Ignore chemicals, variable operation and maintenance, and labour costs, as
these are all relatively small compared to the capital and energy costs.
Case 1: Deeply discount the cost of wind electricity at times of excess wind
Compute the cost of desalinated water ($/m3) for the following three cases:
1. Electricity cost = 8 cents/kWh and desalination capacity factor (CFd) = 0.95 (there is
always some down time) (this could correspond to purchase from the regular grid supply)
2. Electricity cost = 8 cents/kWh and CFd = 0.25 (we assume that the peak output from the
wind farm matches the electrical input capacity of the desalination equipment, and that on
average 25% of the wind output is excess but is sold at a non-discounted price to the
desalination operation, so that CFd = the electricity fraction that would otherwise be
wasted)
3. Electricity cost = 1 cent/kWh and CFd = 0.25 (all of the excess output from the wind farm
is used by the desalination operation, so that none is wasted, but the electricity has to be
sold at a deep discount).
For case (3), compute the impact on the cost of electricity to the other customers of selling
electricity to the desalination operation at only 1 cent/kWh (assuming that the cost to other
customers is increased so as to compensate for the discount offered for desalination).
Write out the formulae that you used in your calculations. In light of your answers, comment on
the economics of using desalination as a flexible load to absorb excess wind electricity. Provide a
brief explanation for your results. What would be the likely effect of intermittent operation on the
maintenance costs and longevity of the desalination equipment?
Case 2: Run the desalination equipment with an oversized wind farm
Instead of considering a distant wind farm that is oversized relative to the transmission link to
some urban demand centre (as in question 5), we can consider a wind farm that is adjacent to a
large desalination operation in a coastal desert region but that is oversized relative to the
desalination capacity. With oversizing of a distant wind farm, oversizing increases the wind farm
capital-cost component of the total cost of electricity but reduces the transmission capital-cost
component. Here, oversizing will increase the wind farm contribution to the cost of desalination,
but will reduce the contribution of the capital cost of the desalination equipment to the final cost
of water because the equipment will be operated with a larger capacity factor. You are to compute
the cost of water using electricity from wind farms using two different methods.
Method One: Use the cost of electricity and the wind farm capacity factors as computed in
Question 5 with c=5 m/s for the different wind farm scaling factors as input to the cost
calculation, along with the capital cost of the desalination equipment, electricity requirement,
CRF and O&M factors given above. Lay out your calculations as indicated in the Excel template
that is sent with this file.
Method Two: The cost of electricity used in Method One is just the cost required to pay back the
investment in the wind farm (with interest, and including O&M), but if the windfarm is built
solely to supply the desalination equipment, we do not need to explicitly consider the cost of
electricity. Rather, we can just consider the total annual revenue requirements and the total water
production for the different wind farm scaling factors.
For method two, consider a desalination plant with a capacity of 1000 m3/day. If the desalination
plant ran at full capacity and the wind turbine that powers the desalination plant ran at full
capacity, what would be the required wind turbine capacity (kW)?
Having worked out this wind turbine capacity, enter the total required wind turbine investment
cost in the Excel template where indicated for scaling factors ranging from 1.0 to 4.0, assuming a
fixed turbine unit cost of $1200/kW (as in the previous calculations). Also enter the desalination
plant investment cost (which is the same for all wind farm scaling factors). From these, compute
the annual revenue requirements. Using the appropriate capacity factors, compute the amount of
water generated per year, and from that, the average cost of water. Your costs should be exactly
the same as those computed by method one.
Note: this is obviously an idealized and simplified calculation, but it’s a good way to get started
thinking about this problem, and the results do provide some insights. In the real world,
calculations like this would serve to indicate whether or not it is worth the much greater effort
that would be required to do a more realistic set of calculations. For starters, if there is a strong
seasonal variation in the average winds, one might need to build facilities to store excess water
produced during the windy season for use during the less windy season.
Appendix: How to combine a bar chart and line graph in the same figure in Excel
Highlight the column with the 25 power outputs (corresponding to wind speeds from 0-25 m/s),
the do Insert – Chart, and select the vertical-bar chart as the chart type. Then go to “Chart” and
then “Add data”, and select the column with the efficiency values. As soon as you have added
that data, it does not appear. Go to View – Toolbars – Chart, and select series 2, then go to
Format – Axis, and select secondary axis. Now the efficiency data will appear as vertical bars but
on the right hand axis. Highlight that data series, then go to Chart – Type, and change the chart
type to a smooth line graph – the chart type will be changed only for that data series. Then adjust
the upper limit of the left and right vertical axis if necessary, and the spacing between the tick
points, so that the ticks on both axis will line up. Finally, allow for horizontal lines between the
ticks, as in Figure 3.10 (using the Grid line option).