Another way of viewing the calculations of Example 12.7 is to obtain the pres­ent worth of each column and sum them to obtain the present worth of solar savings, using appropriate signs for each column. Therefore, the life cycle sav­ings (LCS) of a solar energy system over a conventional system is expressed as the difference between a reduction in the fuel costs and an increase in expenses incurred as a result of the additional investment required for the solar energy system, given by

Подпись:LCS = PjCF1FL – P2Cs


Pi = ratio of life cycle fuel cost savings to first-year fuel savings.

P2 = ratio of life cycle expenditure incurred from the additional investment to the initial investment.

The economic parameter P1 is given by

P1 = (1 – Cte) PWF(ne, iF, d) (12.32)


te = effective income tax rate.

Подпись: C = Подпись: 1 ^ commercial 0 ^ non-commercial Подпись: (12.33)

C = flag indicating whether the system is commercial or non-commercial,

For example, in the United States, the effective tax rate is given by Eq. (12.8). The economic parameter P2 includes seven terms:

1. Down payment, P21 = D

2. Life cycle cost of the mortgage principal and interest,

Подпись: P2,2 = (1 - D)PWF(«min,0, d)

PWF(nL,0, dm )

3. Income tax deductions of the interest,

Подпись: 1Подпись: PWF(Wi ,0, dm)Подпись: P2,3 = (1 - D)te jPWF(«mi„, dm, d) + PWF(«min,0, d) d

PWF(nL,0, dm) J

4. Maintenance, insurance and parasitic energy costs,

P2 4 = (1 – Cte)M1PWF(ne, i, d)

5. Net property tax costs,

P2,5 = tp (1 – te)V1PWF(ne, i, d)

6. Straight line depreciation tax deduction,


P2,6 = PWF(nm„,0, d)



image1693 image1694

Present worth of resale value,

Подпись: P2,1 + P2,2 Подпись: P2,3 + P2,4 + P2,5 image1697 image1698 Подпись: (12.34)

And P2 is given by


D = ratio of down payment to initial total investment.

M1 = ratio of first year miscellaneous costs (maintenance, insurance, and par­asitic energy costs) to the initial investment.

V1 = ratio of assessed value of the solar energy system in the first year to the

initial investment.

tp = property tax, based on assessed value.

ne = term of economic analysis.

n’mm = years over which depreciation deductions contribute to the analysis (usu­ally the minimum of ne and nd, the depreciation lifetime in years).

R = ratio of resale value at the end of its life to the initial investment.

It should be noted that, as before, not all these costs may be present, accord­ing to the country or region laws and regulations. Additionally, the contribu­tions of loan payments to the analysis depend on nL and ne. If nL < ne, all nL payments will contribute. If, on the other hand, nL > ne, only ne payments will be made during the period of analysis. Accounting for loan payments after ne depends on the reasoning for choosing the particular ne. If ne is the period over which the discounted cash flow is estimated without consideration for the costs occurring outside this period, nmin = ne. If ne is the expected operating life of the system and all payments are expected to be made as scheduled, nmin = nL. If ne is chosen as the time of sale of the system, the remaining loan principal at ne would be repaid at that time and the life cycle mortgage cost would consist of the present worth of ne load payments plus the principal balance at ne. The prin­cipal balance should then be deduced from the resale value.

Example 12.11

Repeat Example 12.7 using the P1, P2 method.


As noted in Example 12.7, the system is not income producing; therefore, C = 0. The ratio P1 is calculated with Eq. (12.32):

P = PWF(n, iF, d) = PWF(20, 0.09, 0.08) = 20.242

The various terms of parameter P2 are as follows:



P21 = D = 0.2



Подпись: +


PWF(20, 0, 0.08) PWF(20, 0, 0.07)

P2 4 = (1 – Cte)MjPWF(«e, i, d)

= (120/20,000)PWF(20, 0.05, 0.08) = 0.006 X 14.358 = 0.0861

P2,5 = tp (1 – te)VlPWF(ne, i, d)

= (300/20,000)(1 – 0.3) X 1 X PWF(20, 0.05, 0.08) = 0.015 X 0.7 X 14.358 = 0.151


P2,6 = -*■ PWF(nmin,0, d) = 0



From Eq. (12.31),

LCS = PxCfxFL – P2Cs

= 20.242 X 17.2 X 0.65 X 114.9 – 0.9911 X 20,000 = $6,180.50

This is effectively the same answer as the one obtained in Example 12.7.

As can be seen from this example, the Pb P2 method is quick and easy to carry out manually.

As also can be seen, Eqs. (12.32) and (12.34) include only PWF values and ratios of payments to initial investment of the system and do not include as inputs the collector area and solar fraction. Therefore, as P1 and P2 are indepen­dent of Ac and F, systems in which the primary design variable is the collector area, Ac, can be optimized using Eq. (12.31). This is analyzed in the following section.

Updated: August 29, 2015 — 5:35 pm