## 813 Equivalent Cash Flows and Levelizing

It is convenient to express a series of payments that are irregular or variable as equivalent equal payments in regular intervals; in other words, one replaces nonuniform series by equivalent uniform or level series. This technique is referred to as levelizing. It is useful because regularity facilitates understanding and planning. To develop the formulas, one must calculate the present value P of a series of N equal annual payments A. If the first payment occurs at the end of the first year, its present value is A/(1 + rd). For the second year it is A/(1 + rd)2, etc. Adding all the present values from year 1 to N gives the total present value

This is a simple geometric series, and the result is readily summed to

For zero discount rate this equation is indeterminate, but its limit rd ^ 0 is A/N, reflecting the fact that the N present values all become equal to A in that case. Analogous to the notation for the present worth factor, the ratio of A and P is designated by

This is called the capital recovery factor and is plotted in Figure 8.3. For the limit of long life, it is worth noting that (A/P,rd,N) ^ rd if rd > 0. The inverse is known as the series present worth factor since P is the present value of a series of equal payments A.

With the help of the present worth factor and capital recovery factor, any single expense Cn that occurs in year n, for instance, a major repair, can be expressed as an equivalent annual expense A that is constant during each of the N years of the life of the system. The present value of Cn is P = (P/ F,rd,n) Cn and the corresponding annual cost is

A very important application of the capital recovery factor is the calculation of loan payments. In principle, a loan can be repaid according to any arbitrary schedule, but in practice, the most common arrangement is based on constant payments in regular intervals. The portion of A due to interest varies, but to find the relationship between A and the loan amount L, one need not worry about that. First consider a loan of amount Ln that is to be repaid with a single payment Fn at the end of n years. With n years of interest at loan interest rate rl, the payment must be

 ✓ ✓ 5 / / / ✓ / 1 V 10 0 0 0 1 0 1 The capital recovery factor (A/P,r,N) as function of rate r and number of years N. FIGURE 8.3 The capital recovery factor (A/P,r,N) as function of rate r and number of years N. Comparison with the present worth factor shows that the loan amount is the present value of the future payment Fn, discounted at the loan interest rate. A loan that is to be repaid in N equal installments can be considered as the sum of N loans, the nth loan to be repaid in a single installment A at the end of the nth year. Discounting each of these payments at the loan interest rate and adding them gives the total present value, which is equal to the total loan amount This is just the series of the capital recovery factor. Hence, the relationship between annual loan payment A and loan amount L is Now the reason for the name capital recovery factor becomes clear: it is the rate at which a bank recovers its investment in a loan. Some payments increase or decrease at a constant annual rate. It is convenient to replace a growing or diminishing cost with an equivalent constant or levelized cost. Suppose the price of energy is p,, at the start of the first year, escalating at an annual rate re while the discount rate is rd. If the annual energy consumption Q is constant, then the present value of all the energy bills during the N years of system life is assuming the end-of-year convention described above. As in Eq. 8.3, a new variable rde is introduced and defined by which allows writing Pe as Pe = (P/Ar^N) Q Pe Since (A/P,rd,N) is the inverse of (A/P,rd,N), this can be written as If the quantity in brackets were the price, this would be just the formula without escalation. Let us call this quantity the levelized energy price pe: This allows calculation of the costs as if there were no escalation. Levelized quantities can fill a gap in our intuition which is ill prepared to gauge the effects of exponential growth over an extended period. The levelizing factor levelizing factor=

## Solar Stirling Engine Basics Explained

The solar Stirling engine is progressively becoming a viable alternative to solar panels for its higher efficiency. Stirling engines might be the best way to harvest the power provided by the sun. This is an easy-to-understand explanation of how Stirling engines work, the different types, and why they are more efficient than steam engines.

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