1.Suppose the probability density function (PDF) for wind speed is given as
a) Find the value of for this to be a legitimate PDF. β¨ b) Find the cumulative distribution function (CDF) for the wind speed.
2.A homeowner considers purchasing a rooftop PV system that costs $11,000. Assume the only costs for those PVs are the annual loan payments on a $11,000, rate 4.5%, 10-year loan.
a)Find the levelized cost of electricity (LCOE, $/kWh) of this system. β¨ b)Suppose the PV system generates constant power and delivers 6,000 kWh of electricity per year. It works 6 hours a day and 200 days a year. The PV system is used to power a β¨ 50-gallon electric water heater. If the efficiency of the tank is 90% (i.e., 10% of heat loss), how long would the PV system take to heat the full tank of water from 10βπΆ to 50βπΆ ? (Recall that 3412 Btu = 1kWh and 1 gallon of water weighs 8.35 lbs) β¨ c)By using the LCOE, find the cost of a 12-gallon, 110βπΉ shower if the cold-water supply temperature is 55βπΉ.
3. Consider a 0.015 solar cell with the equivalent circuit shown below, where the parallel resistance of π π = 5 πΊ. The reverse saturation current is
and at an insolation of 1-sun the short-circuit is At 20βπΆ, with an output voltage of 0.5V, find the following:
a)The load current and the power consumed by the load. β¨ b)The efficiency of the solar cell.
4.Consider a wind turbine with a cut-in wind speed of 4 m/s, a rated wind speed of 14 m/s, and a cut-out wind speed of 26 m/s. If the wind speed satisfies a Weibull distribution with the shape parameter k=2, and an average speed of 10 m/s.
a)For how many hours per year will the turbine be shut down because of excessively high- speed winds? β¨ b) For how many hours per year the turbine has no power output because the wind speed is lower than the cut-in speed? β¨ c)If this is a 1.5-MW turbine, how much energy (kWh/yr) would be produced for winds blowing above the rated wind speed? β¨ d)This wind turbine is used for a pumped storage system; i.e., pumping water from a lake to a pond at an elevation of 200m above. Assume the pump has 75% efficiency and the pond is 5000 big. Find the change in height of the water level in the pond if the turbine keeps working at its rate power for one day.
5. A small-scale wind turbine with the rotor diameter 1.5m uses a 500-Watt DC motor as a generator.
a) Find the capacity factor for the machine if it delivers 100 kWh in a 30- day month. β¨ b) Under the standard condition of 15βπΆ temperature and 1 atm pressure, how fast β¨ would the wind have to blow for the turbine to generate half of its rate power if the β¨ machine is 30% efficient at that point? β¨ c) If the tip-speed-ratio (TSR) is 6, find the gear ratio if the generator needs to turn at β¨ 1110 rpm to deliver 0.25kW at the wind speed obtained in question (b). β¨ d) Suppose the turbine is deployed in a new environment of 70βπΉ and 1.5 atm. Repeat β¨ (b) and (c).