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ELEC0037 - Renewable and Sustainable Energy Systems
1. [16 marks] Solar parameters and Astronomy
a. Find a plot of AM0 and of AM1.5. What is the difference between the two graphs?
Show the region of light visible to our eyes on the two plots and indicate the IR and
UV regions. [3 marks]
Answer:
The difference is in meaning. The AM) is the spectrum above the atmosphere; AM1.5 is
within the atmosphere at the Earth’s surface.
b. On the summer solstice in London, is the Earth closest to the sun, farthest from the
sun, or neither? [2 marks]
c. What is a solstice and what is an equinox? [2 marks]
d. Through how many degrees does the Earth turn every hour? [2 marks]
e. What is the declination of the Earth?
i. Calculate the declination of the Earth from the ecliptic on the 100th day of the
year.
[3 marks]
f. What is the conventionally agreed upon solar radiant power intensity above the
atmosphere of the earth facing the sun?
g. Calculate the maximum solar insolation density (in kWh/m2-day), striking a flat
panel inclined at an angle of 25o in the hills surrounding Rio de Janeiro, Brazil on
June 15. [4 marks]
ELEC0037 - Renewable and Sustainable Energy Systems, UCL
Coursework #2 – Out of 100%
2/5
2. [7 marks] The Vestas V-164 wind turbine.
a. Calculate the power derived from an incoming wind speed of 10 m/s at the Betz limit
for the Vestas V-164 wind with a blade diameter of 164 meters. [4 marks]
b. Compare your figure with the nameplate capacity of 8 MW and suggest its
nameplate efficiency. [3 marks]
3. [12 marks] Two ships are in a race across the sea, a distance of 10,000 km.
The Magellan relies solely on its sails, averaging 8 kph on days when the wind is
blowing. The Wrangler uses a combination of sails and a solar powered engine. Due to
the extra weight, the Wrangler has an average speed of 6 kph on windy days (using
sails), and 5 kph on sunny days (using its engine). It cannot use both at the same time.
For both boats, the wind is blowing 75% of the time, while sunny days occur 70% of the
time, with no correlation to one another. Both ships are operational 24 hours a day.
a. Using a calculation or model you have designed, demonstrate who is expected
to win the race and by what margin. [4 marks]
b. Using a simple stochastic simulation, make a plot of the distance between the
boats over time, from start to finish. [4 marks]
c. If one were to run the simulation 100 times, roughly what percentage of the
time does Magellan win the race? [4 marks]
4. [18 marks] The tragedy of the commons
There is a sense that people tend not to care as fully as they should for environmental
assets that are shared equally among large numbers of people. Even when each individual
rationally maximizes his or her personal utility, u a situation develops creating a decrease in
overall utility for the total number of those using the resource. The images of trash-strewn
parks or over-used camp sites may spring to mind. The vast common of the earth’s
atmosphere may also be another important example.
We would like to understand the essence of this paradox using game theory.
Consider a highly simplified 2-player game, played over 2 periods – now (period 1), and in
the future (period 2). Each player is consuming a resource, say water from a well, of an
initial amount, y where there is no replenishing over time. If a situation arises where the
consumptions of players 1 (Jill) and 2 (Jack) in a given period is greater than the amount of
water in the well, they must simply split it. We can imagine an authority of some kind – the
water police – standing near the well taking initial bids for a certain number of buckets. If
the total number of buckets is less than the remaining volume, the police allows them to
extract the amount requested, if not – they must split what is left between them. Given this
ELEC0037 - Renewable and Sustainable Energy Systems, UCL
Coursework #2 – Out of 100%
3/5
simplified situation, we would like to understand the consumption strategy of each player
given they will maximise individual utility, u. We assume the consumption of players 1 and
2 in period 1 is given by c1 and c2.
a) Show that for the last period (the future), each player will get only
½[y- (c1+c2)]. [2 marks]
b) We will assume the utility function u has the form, ui = ln(ci). Does this function
exhibit a diminishing marginal utility as we expect it should?
[2 marks]
c) Now consider the 1st period. Player 1 must now choose the amount to consume in
period 1. She knows that the utility of her choice will depend on player 2’s choice,
since their collective consumption determines how much is left over at the start of
the final period. If she calls player 2’s chosen consumption c2*, show that her best
response to c2* is: c1(c2*)= (y -c2*)/2 [2 marks]
d) Assuming player 2’s response function is perfectly symmetric to player 1’s, find
c2(c1*). [2 marks]
e) With these response functions, a Nash equilibrium obtains when neither side can do
better in light of the other player’s best response. Find the Nash equilibrium
consumption levels in period 1 for both players. [2 marks]
f) What is the resultant utility for each player summed over both periods for this
consumption pattern? [2 marks]
g) Show that a socially optimal (maximum total utility over all players) consumption is
one that maximizes:
ln(c1) + ln(c2) + 2ln(½*[y- (c1+c2)]). [2 marks]
h) Find the optimal consumption pattern and discuss whether this is indeed a tragedy
of the commons. [2 marks]
i) Given the result above, what happens if player 1 decides she will be magnanimous,
and instead of choosing her u-optimising choice shown in iii), she instead chooses
the social welfare maximizing choice given in viii)?
[2 marks]
5. [12 marks] The polluter’s dilemma
a) The neighbouring villages of Quark and Meson lie deep within a valley, trapping all
the air pollution from either one, such that its damaging effects are shared equally.
Each village can choose to consume at a high or low rate. The net utility to each
ELEC0037 - Renewable and Sustainable Energy Systems, UCL
Coursework #2 – Out of 100%
4/5
village has two terms: one due to the immediate benefit of the consumption, and
another due to the damage caused by the pollution. The utility benefit owing to
consumption at high and low levels is 6 and 3 respectively. The waste damage to the
utility created by each village at these levels is -2 and -1 respectively. The damage is
however aggregated throughout the valley, so for example if Quark chooses high and
Meson chooses low, the utility damage is -3 for both. The net utility positions for
Quark and Meson in this case are 3 and 0. Create a normal form depiction of this
game and determine if there are any Nash equilibria.
[4 marks]
b) As time goes on the pollution problem becomes more acute for Quark and Meson.
Now a high consumption choice results in a waste damage of -3. Everything thing
else is as before. Create a new normal form depiction for this game and discuss any
expected changes to the equilibrium choices of the two villages.
[4 marks]
c) Discuss these games in terms of payoff and risk dominance and explore potential
implications for similar real-world systems. Do you expect the nature of the
equilibria to persist regardless of the pollution levels associated with the high
consumption rate? [4 marks]
6. [ 11 marks] A home in San Diego, CA has the electrical load curve shown in figure 2.1.
They want to install solar panels on the roof.
Figure 2.1: The Electrical Load Curve
(a) What is an electrical load curve; that is, describe the meaning behind Figure 1. [3
marks]
(b) Calculate the daily energy consumption and the average power consumption by the
people who live here. [3 marks]
ELEC0037 - Renewable and Sustainable Energy Systems, UCL
Coursework #2 – Out of 100%
5/5
(c) Suppose these homeowners wanted to remain on the electrical grid and use solar
panels without battery storage.
• Over which hours of the day can this home in San Diego acquire energy from the
sun (assuming cloudless conditions on the Equinox).
• How much energy do the panels need to supply over that period? [3 marks]
(d) The solar panels will be placed in a fixed position using the roof. Assuming
cloudless conditions on the Equinox, suggest a tilt angle and azimuth for the solar
panels and give a physical or mathematical reason for your choice. [2 marks]
7. [12 marks] Beating Betz’ Limit?
A scientist claims to have cracked Betz limit by using a series of perfectly positioned Betz
extractors. It is a purely theoretical idea. His idea is to place n perfect Betz extractors in
series, each one ideally placed downstream behind the other.
a) Calculate the new overall extraction efficiency for the n Betz extractors in series. [2
marks]
b) What is the final wind velocity downstream of the nth extractor? [2 marks]
c) What is the overall extraction efficiency as n approaches infinity? [2 marks]
d) What is the final velocity of the wind in this limit? [2 marks]
e) What are the flaws in this approach in practice? [2 marks]
f) Describe why such an approach is uneconomical even for n=2? [2 marks]
8) [9 marks] Altus optimus
A new wind farm manufacturer is undertaking to optimise the size of its wind turbine
generator (WTG). At a blade diameter of 131m and a hub height of 84m, the Altus1 WTG is
so far their prize model. Like all Altus WTG, it delivers a Betz limit-like power curve over all
hub height wind speeds it intercepts. When one writes the power output as a function of
wind speed, one finds, for all of their turbines, that P = af(v), where f(v) is a function of v
only. The constant, a is the same for all their WTG. The designers want to reconfigure the
Altus to deliver maximal payback economics. The 131m model costs 3m EUR. Another
version, the 100m version (with hub height = 60m) costs 1.948m EUR. The Altus 1 delivers
3MW of power when its hub height wind speed reaches 7 m/s.
The velocity profile, v(z) fits the functional description given in your lectures:
ELEC0037 - Renewable and Sustainable Energy Systems, UCL
Coursework #2 – Out of 100%
6/5
® vz =V ln
z - d
z0
æ
èç
ö
ø÷
for z > d and z-d > z0
Where V is a reference speed, z0 is a characteristic roughness length, and d is an offset.
Here they are given by 3 m/s, 11m, 10m respectively. We may also assume the long-term
average speed at z is given by vz.
We assume the WTG runs non-stop, round the clock at the output level, P(v) for the average
speed at the hub height in question. The price of electricity ($/MWh) is also assumed to be
constant. The engineers at Altus have agreed that to simplify their design, they can set all
hub heights to half the rotor diameter plus some headroom, which they are holding
constant at 10m. They also know that the total cost of a WTG is proportional to the hub
height (when rotor diameter is constant), while it is proportional to the square of the rotor
diameter (when hub height held constant). Their goal is to find the hub height and rotor
diameter that minimises the payback period.
a) Why is the constant, a the same for all their WTG? [1 marks]
b) What is the value of a? [1 marks]
c) Derive an expression, with all relevant constants, for the total cost, TC of the Altus
WTG’s as a function of hub height, h. [2 marks]
d) Derive an expression for the function you are minimising in its simplest form.
[2 marks]
e) Find the hub height and diameter that minimizes the payback time for the Altus.
[3 marks]
----------------- L7 students only beyond this point ---------------------------------------------------------
9) [8 marks] L7 MEng ONLY! Utility re-imagined
It is said that people will live more happily if they enjoy the pursuit in itself, as opposed to
the payoffs associated with the achievement of the end goal. The main reason for this is
that we cannot control final outcomes, although we can control what we decide to pursue
at any given point in time.
Laertes has the ambition to make it to the NBA. He has one try-out in 1,000 days that will
determine his fate. On any given day the probability of failure is 0.001, at which point he
ELEC0037 - Renewable and Sustainable Energy Systems, UCL
Coursework #2 – Out of 100%
7/5
will decide NOT to try out (eg, due to injury, major defeat, etc). At the end of 1000 days, if
he has not encountered a failure event, he has succeeded!
a) Determine the aggregate probability of success after t days
[2 marks]
