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 . 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[$/yr] and a given discount rate r [fraction/yr], the NPV of income is NPV=t=0t=Aertdt However, we need to pay back the capital cost C[$] 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 NPV=t=Tt=Aertdt a. Carry out the integral and show the formula. b. Assume T=5 years, =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 10kW peak and that the cost of installed PV is $2.00/W. Calculate the annual income A for the United Kingdom, Brazil and India (base). d. Note that the United Kingdom and Brazil have a similar cost of electricity but a big difference in solar energy; and that Brazil and India (base) have similar solar energy but a big difference in the cost of electricity. For each location, calculate the NPV as a fraction of C. Use r=0.06/yr. Thus, the economics of solar depend on the product of insolation and grid power cost! This problem asks you to calculate the NPV (net present value) of a PV installation as a function of the simple payback time T and of the system lifetime . Note that T and NPV depend on the solar energy, and on the cost of grid power, in a given location. For simplicity we will assume that: - The role of inflation has been factored out, so that the annual income is constant. - We pay back the capital cost without using a loan. (You could include a loan in the calculation, but then the result would depend on the assumed loan rate and period.) - The income from the system is simply (the energy generated) x (the cost of grid power). (In a full calculation, we would need to subtract out fixed costs, maintenance costs, taxes.) For a fixed annual income A[$/yr] and a given discount rate r [fraction/yr], the NPV of income is NPV=t=0t=Aertdt 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 NPV=t=Tt=Aertdt a. Carry out the integral and show the formula. b. Assume T=5 years, =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 10kW 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. Use r=0.06/yr. Thus, the economics of solar depend on the product of insolation and grid power cost