Solar thermal collectors and applications

Time value of money

It must be noted that a sum of money at hand today worth less than the same sum in the future, because the money at hand can be invested at some compounding interest to generate a bigger sum in the future. Therefore, a sum of money or cash flow in the future must be discounted and worth less at present-day value. A cash flow F occurring N years from now can be reduced to its present value P by:

F

(1 + d)N where d is the market discount rate (%).

Similarly, the amount of money needed to purchase an item is increasing because the value of money is decreasing. With an annual inflation rate i, a purchase cost C at the end of year N will become a future cost F according to:

F = C(1 + if2 (94)

4.7.1. Method description

In general, the PW (or discounted cost) of an investment or cost C at the end of year N, at a discount rate of d and interest rate of i is obtained by combining Eqs. (93) and (94) as:

C(1 + if-1 (1 + d)N

Eq. (95) can easily be incorporated into a spreadsheet with the parameters d and i entered into separate cells and referencing them in the formulae as absolute cells. In this way a change in either d or i will cause automatic recalculation of the spreadsheet.

The fuel savings are obtained by subtracting the annual cost of the conventional fuel used for the auxiliary energy from the fuel needs of a fuel only system. The integrated cost of the auxiliary energy use for the first year, i. e. solar back up, is given by the formula:

Caux = CFA Qaux dt (96)

0

The integrated cost of the total load for the first year, i. e. cost of conventional fuel without solar, is:

Cload = CFLQload dt (97)

0

where CFA and CFL are the cost rates for auxiliary energy and conventional fuel, respectively.

Such analysis is performed annually and the following are evaluated:

• Fuel savings;

• Extra mortgage payment;

• Extra maintenance cost;

• Extra parasitic cost;

• Extra tax savings;

• Solar savings.

In some countries some other costs may be present, i. e. extra property tax to cover the new system. In this case these costs should be considered as well. The word ‘extra’ appearing in some of the above items assumes that the associated cost is also present for a fuel-only system and therefore only the extra part of the cost incurred by the installation of the solar system should be included. The inflation, over the period of economic analysis, of the fuel savings is estimated by using Eq. (94) with i equal to the fuel inflation rate. The parasitic cost is the energy required to power auxiliary items like
the pump, fan and controllers. This cost is also increased at an inflational rate over the period of economic analysis by using Eq. (94) with i equal to the annual increase of electricity price.

The mortgage payment is the annual value of money required to cover the funds borrowed at the beginning to install the system. This includes interest and principal payment. The estimation of the annual mortgage payment can be found by dividing the amount borrowed by the present worth factor (PWF). The PWF is estimated by using the inflation rate equal to zero (equal payments) and with the market discount rate equal to the mortgage interest rate. The PWF can be obtained from tables or calculated by the following equation [97]:

PWF = M1 -(rf (98)

where d is the interest rate, and N is the number of years (equal instalments).

Solar savings for each year are the sums of the items as:

Solar savings = Extra mortgage payment + Extra maintenance cost + Extra parasitic cost + Fuel savings + Extra tax savings (99)

Actually, the savings are positive and the costs are negative. Finally, the PW of each year’s solar savings is determined by using Eq. (95). The results are estimated for each year. These annual values are then added up to obtain the LCS according to the equation:

Solar savings
(1 + d)N