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Chapter 6

Q. 6.15

Q. 6.15

Annual Energy Delivered Using a Spreadsheet. Suppose that a NEG Micon 60-m diameter wind turbine having a rated power of 1000 kW is installed at a site having Rayleigh wind statistics with an average windspeed of 7 m/s at the hub height.
a. Find the annual energy generated.
b. From the result, find the overall average efficiency of this turbine in these winds.
c. Find the productivity in terms of kWh/yr delivered per m² of swept area.

TABLE 6.7 Examples of Wind Turbine Power Specifications
Manufacturer: NEG
Micon		 NEG
Micon		 NEG
Micon		 Vestas Whisper Wind
World		 Nordex Bonus
Rated Power (kW): 1000 1000 1500 600 0.9 250 1300 300
Diameter (m): 60 54 64 42 2.13 29.2 60 33.4
Avg. Windspeed
v (m/s) v(mph) kW kW kW kW kW kW kW kW
0 0 0 0 0 0 0.00 0 0 0
1  2.2 0 0 0 0 0.00 0 0 0
2  4.5 0 0 0 0 0.00 0 0 0
3  6.7 0 0 0 0 0.03 0 0 4
4  8.9 33 10 9 0 0.08 0 25 15
5 11.2 86 51 63 22 0.17 12 78 32
6 13.4 150 104 159 65 0.25 33 150 52
7 15.7 248 186 285 120 0.35 60 234 87
8 17.9 385 291 438 188 0.45 92 381 129
9 20.1 535 412 615 268 0.62 124 557 172
10 22.4 670 529 812 356 0.78 153 752 212
11 24.6 780 655 1012 440 0.90 180 926 251
12 26.8 864 794 1197 510 1.02 205 1050 281
13 29.1 924 911 1340 556 1.05 224 1159 297
14 31.3 964 986 1437 582 1.08 238 1249 305
15 33.6 989 1006 1490 594 1.04 247 1301 300
16 35.8 1000 998 1497 598 1.01 253 1306 281
17 38.0 998 984 1491 600 1.00 258 1292 271
18 40.3 987 971 1449 600 0.99 260 1283 259
19 42.5 968 960 1413 600 0.97 259 1282 255
20 44.7 944 962 1389 600 0.95 256 1288 253
21 47.0 917 967 1359 600 0.00 250 1292 254
22 49.2 889 974 1329 600 0.00 243 1300 255
23 51.5 863 980 1307 600 0.00 236 1313 256
24 53.7 840 985 1288 600 0.00 230 1328 257
25 55.9 822 991 1271 600 0.00 224 1344 258
26 58.2 0 0 0 0 0.00 0 0 0

Source: Mostly based on data in www.windpower.dk.

Step-by-Step

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a. To find the annual energy delivered, a spreadsheet solution is called for. Let’s do a sample calculation for a 6-m/s windspeed to see how it goes, and then present the spreadsheet results.
From Table 6.7, at 6 m/s the NEG Micon 1000/60 generates 150 kW.
From (6.45), the Rayleigh p.d.f. at 6 m/s in a regime with 7-m/s average windspeed is

f\left(v\right) \ = \ \frac{\pi \ v}{2 \bar{v}^{2}} \ \exp \ \left[-\frac{\pi }{4} \ \left(\frac{v}{\bar{v}}\right)^{2} \right] \ = \ \frac{\pi \ \cdot \ 6}{2 \ \cdot \ 7^{2}} \ \exp \ \left[- \frac{\pi}{4} \ \left(\frac{6}{7}\right)^{2}\right] \ = \ 0.10801

In a year with 8760 h, our estimate of the hours the wind blows at 6 m/s is

\text{Hours} \ @ 6 \ {m}/{s} \ = \ 8760 \ {h}/{yr} \ \times \ 0.10801 \ = \ 946 \ {h}/{yr}

So the energy delivered by 6-m/s winds is

\text{Energy} \ \left(@ 6 \ {m}/{s}\right) \ = \ 150 \ kW \ \times \ 946 \ {h}/{yr} \ = \ 141, \ 929 \ {kWh}/{yr}

The rest of the spreadsheet is given below. The total energy produced is 2.85 × 10^{6} kWh/yr.

b. The average efficiency is the fraction of the wind’s energy that is actually converted into electrical energy. Since Rayleigh statistics are assumed, we can use (6.48) to find average power in the wind for a 60-m rotor diameter (assuming the standard value of air density equal to 1.225 kg/m³):

\begin{matrix} \bar{P} \ = \ \frac{6}{\pi} \ \cdot \ \frac{1}{2} \rho A \bar{v}^{3} & = \ \frac{6}{\pi} \ \times \ 0.5 \ \times \ 1.225 \ \times \ \frac{\pi}{4} \left(60\right)^{2} \ \times \ \left(7\right)^{3} \\ & = \ 1.134 \ \times \ 10^{6} \ W \ = \ 1134 \ kW \quad \quad \quad \ \end{matrix}

In a year with 8760 h, the energy in the wind is

\text{Energy in wind} \ = \ 8760 \ {h}/{yr} \ \times \ 1134 \ kW \ = \ 9.938 \ \times \ 10^{6} \ kWh

So the average efficiency of this machine in these winds is

\text{Average efficiency} \ = \ \frac{2.85 \ \times \ 10^{6} \ {kWh}/{yr}}{9.938 \ \times \ 10^{6} \ {kWh}/{yr}} \ = \ 0.29 \ = \ 29 \%

c. The productivity (annual energy per swept area) of this machine is

\text{Productivity} \ = \ \frac{2.85 \ \times \ 10^{6} \ {kWh}/{yr}}{\left({\pi}/{4}\right) \ \cdot \ 60^{2} \ m^{2}} \ = \ 1008 \ {kWh}/{m^{2}} \ \cdot \ yr
Windspeed (m/s) Power (kW) Probability
f (v)		 Hrs/yr
at v		 Energy
(kWh/yr)
0 0 0.000 0 0
1 0 0.032 276 0
2 0 0.060 527 0
3 0 0.083 729 0
4 33 0.099 869 28,683
5 86 0.107 941 80,885
6 150 0.108 946 141,929
7 248 0.102 896 222,271
8 385 0.092 805 310,076
9 535 0.079 690 369,126
10 670 0.065 565 378,785
11 780 0.051 444 346,435
12 864 0.038 335 289,551
13 924 0.028 243 224,707
14 964 0.019 170 163,779
15 989 0.013 114 113,101
16 1000 0.008 74 74,218
17 998 0.005 46 46,371
18 987 0.003 28 27,709
19 968 0.002 16 15,853
20 944 0.001 9 8,709
21 917 0.001 5 4,604
22 889 0.000 3 2,347
23 863 0.000 1 1,158
24 840 0.000 1 554
25 822 0.000 0 257
26 0 0.000 0 0
Total: 2,851,109

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