4. Net present value of PV vs. location: This problem asks you to calculate the NPV (net present value) of a PV installation as a function of the simple payback time T and of the system lifetime 7. Note that T and 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 that the 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 income A [$/yr] and a given discount rater [fraction/yr], the NPV of income is NPV = ct=T Ae-rt dt t=0 However, we need to pay back the capital cost C [$] which is incurred at t = 0, and therefore is not discounted. Assume that during the first T years of operation, where T is the simple payback time, T = C/A, all profits are used internally to repay C. (As a simplification we are not discounting the internal payback.) Then t=T NPV = Ae-rt dt t=T a. Carry out the integral and show the formula. b. Assume T = 5 years, t = 25 years, and r = 0.06/yr. What is the NPV as a fraction of C? 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. Thus, the economics of solar depend on the product of insolation and grid power cost! 40 Denmark 35 Germany California (Tier 4) Italy 30 Spain $ Japan 25 United Kingdom Australia Brazil France New York 20 India (peak rate) 15 New Jersey Pennsylvania Georgia Florida California (base) USA Ohio Texas 10 China Canada South Africa South Korea $ 5 India (base) 0 700 800 900 1,000 1,100 1,200 1,300 1,400 1,500 1,600 1,700 1,800 1,900 Solar-energy yield, specific annual yield in kilowatt hour/kilowatt peak (kWh/kWp)