Question 6.18: A Loan to Pay for a Small Wind Turbine. Suppose that a 900-W......
A Loan to Pay for a Small Wind Turbine. Suppose that a 900-W Whisper H900 wind turbine with 7-ft diameter (2.13 m) blade costs $1600. By the time the system is installed and operational, it costs a total of $2500, which is to be paid for with a 15-yr, 7 percent loan. Assuming O&M costs of $100/yr, estimate the cost per kWhr over the 15-year period if average windspeed at hub height is 15 mph (6.7 m/s).
The capital recovery factor for a 7%, 15-yr loan would be
CRF\left(0.07, \ 15 \ yr\right) \ = \ \frac{i\left(1 \ + \ i\right)^{n}}{\left(1 \ + \ i\right)^{n} \ – \ 1} \ = \ \frac{0.07\left(1 \ + \ 0.07\right)^{15}}{\left(1 \ + \ 0.07\right)^{15} \ – \ 1} \ = \ {0.1098}/{yr}which agrees with Table 5.5. So, the annual payments on the loan would be
A \ = \ P \ \times \ CRF\left(0.07, \ 15\right) \ = \ \$ 2500 \ \times \ {0.1098}/{yr} \ = \ {\$ 274.49}/{yr}The annual cost, including $100/yr of O&M, is therefore $274.49 + $100 = $374.49.
To estimate energy delivered by this machine in 6.7-m/s average wind, let us use the capacity factor approach (6.65):
CF \ = \ 0.087 \bar{V} \ – \ \frac{P_{R}}{D^{2}} \quad \left(\text{Rayleigh winds}\right) (6.65)
CF \ = \ 0.087 \bar{V} \ – \ \frac{P_{R}\left(kW\right)}{D^{2}\left(m^{2}\right)} \ = \ 0.087 \ \times \ 6.7 \ – \ \frac{0.90}{2.13^{2}} \ = \ 0.385The annual energy delivered (6.59)
\text{Annual energy} \ \left({kWh}/{yr}\right) \ = \ P_{R} \ \left(kW\right) \ \times \ 8760 \ \left({h}/{yr}\right) \ \times \ CF (6.59)
{kWh}/{yr} \ = \ 0.90 \ kW \ \times \ 8760 \ {h}/{yr} \ \times \ 0.385 \ = \ 3035 \ {kWh}/{yr}The average cost per kWh is therefore
\text{Average cost} \ = \ \frac{\text{Annual cost} \ \left({\$}/{yr}\right)}{\text{Annual energy} \ \left({kWh}/{yr}\right)} \ = \ \frac{{\$ 374.49}/{yr}}{3035 \ {kWh}/{yr}} \ = \ {\$ 0.123}/{kWh}That’s a pretty good price of electricity for a small system—cheaper than grid electricity in many areas and certainly cheaper than any other off-grid, home-size generating system.
| TABLE 5.5 Capital Recovery Factors as a Function of Interest Rate and Loan Term |
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| Years | 3% | 4% | 5% | 6% | 7% | 8% | 9% | 10% | 11% | 12% | 13% |
| 5 | 0.2184 | 0.2246 | 0.2310 | 0.2374 | 0.2439 | 0.2505 | 0.2571 | 0.2638 | 0.2706 | 0.2774 | 0.2843 |
| 10 | 0.1172 | 0.1233 | 0.1295 | 0.1359 | 0.1424 | 0.1490 | 0.1558 | 0.1627 | 0.1698 | 0.1770 | 0.1843 |
| 15 | 0.0838 | 0.0899 | 0.0963 | 0.1030 | 0.1098 | 0.1168 | 0.1241 | 0.1315 | 0.1391 | 0.1468 | 0.1547 |
| 20 | 0.0672 | 0.0736 | 0.0802 | 0.0872 | 0.0944 | 0.1019 | 0.1095 | 0.1175 | 0.1256 | 0.1339 | 0.1424 |
| 25 | 0.0574 | 0.0640 | 0.0710 | 0.0782 | 0.0858 | 0.0937 | 0.1018 | 0.1102 | 0.1187 | 0.1275 | 0.1364 |
| 30 | 0.0510 | 0.0578 | 0.0651 | 0.0726 | 0.0806 | 0.0888 | 0.0973 | 0.1061 | 0.1150 | 0.1241 | 0.1334 |