This problem asks you to calculate the NPV (net present value) of a PV installation as a function of the simple payback timeTand of the system lifetimet. Note thatTand NPV depend on the solar energy, and on the cost of grid power, in a given location.

For simplicity we will assume that: - The role of inflation has been factored out, so thatthe annual income is constant. -We pay back the capital cost without using a loan. (You could include a loan in the calculation, but then the result would depend on the assumed loan rate and period.) -The income from the system is simply (the energy generated) x (the cost of grid power). (In a full calculation, we would need to subtract out fixed costs, maintenance costs, taxes.)

For a fixed annual incomeA[$/yr] and a given discount rater[fraction/yr], the NPV is#$%

= &!"##$&

However, we need to pay back the capital costC[$] which is incurred att= 0, and therefore is not discounted. Assume that during the firstTyears of operation, whereTis the simple payback time,T=C/A, all profits are used internally to repayC. (As a simplification we are not discounting the internal payback.) Then

#$%

=+&!"##$'

a.Carry out the integral and show the formula.

b. AssumeT=5years,t=25years,andr=0.06/yr.WhatistheNPVasafractionofC?

c. The plot from the McKinsey Quarterly is copied below. We will only make use of the solar energy yield (X-axis, annual kWh / kW peak) and the average cost of grid electricity (Y-axis, cents per kWh). Assume that we install a system rated at 10 kW peak and that the cost of installed PV is $2.00 / W.Calculate the annual income A for the United Kingdom, Brazil and India (base).

d. Note that the United Kingdom and Brazil have a similar cost of electricity but a big difference in solar energy; and that Brazil and India (base) have similar solar energy but a big difference in the cost of electricity.For each location, calculate the NPV as a fraction of C. User= 0.06 / yr.