The first example is about the fuel cost of a non-solar or conventional energy system. It examines the time value of an inflating fuel cost.
Calculate the cost of fuel of a conventional (non-solar) energy system for 15 years if the total annual load is 114.9 GJ and the fuel rate is $17.2/GJ, the market discount rate is 7%, and the fuel inflation rate is 4% per year.
The first-year fuel cost is obtained from Eq. (12.4) as
CL = CFL JLdt = 17.2 X 114.9 = $1,976.30
(because the total annual load is given, the integral is equal to 114.9 GJ).
The fuel costs in various years are shown in Table 12.4. Each year’s cost is estimated with Eq. (12.13) or from the previous year’s cost multiplied by
Table 12.4 Fuel Costs in Various Years for Example 12.5
(1 + i). Each value for the present worth is estimated by the corresponding value of the fuel cost using Eq. (12.11).
An alternative method is to estimate PWF(n, i, d) from Eq. (12.18) or Appendix 8 and multiply the value with the first year’s fuel cost as follows. From Eq. (12.18),
Present worth of fuel cost = 1,976.3 X 11.5752 = $22,876
As can be seen, this is a much quicker method, especially if the calculations are done manually, and the same result is obtained but the intermediate values cannot be seen.
Although in the previous example, a fixed fuel inflation rate is used for all years, this may vary with time. In the case of a spreadsheet calculation, this can be easily accommodated by having a separate column representing the fuel inflation rate for each year and using this rate in each annual estimation accordingly. So, in this case, either the same value for all years or different values for each year can be used without difficulty. These estimations can also be performed with the help of the PWF, as shown in Example 12.5, but as the number of different rates considered increases, the complexity of the estimation increases, because the PWF needs to be calculated for every time period the rate changes, as shown in the following example.
Calculate the present worth of a fuel cost over 10 years if the first year’s fuel cost is €1,400 and inflates at 8% for 4 years and 6% for the rest of the years. The market discount rate is 7% per year.
The problem can be solved by considering two sets of payments at the two inflation rates. The first set of five payments has a first payment that is €1,400 and inflates at 8%. Therefore, from Eq. (12.18),
Thus, the present worth of the first set is 1,400 X 4.7611 = €6,665.54.
The second set starts at the beginning of the sixth year and, for this period, i = 6% per year. To find the initial payment for this set, €1,400 is inflated four times by 8% and one time by 6%. Therefore,
Initial payment for the second set is 1,400(1.08)4(1.06) = €2,018.97.
As before, for the second series of payments,
The second set of payments needs to be discounted to the present worth by
So the answer is the sum of the present worth of the two sets of payments: 6,665.54 + 6,602.11 = €13,267.65.