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| Fig. 1: Global Wind Energy Installed Capacity. [1,3] (Image Source: A. Sheoran) |
In 2023, the installed capacity of wind power increased by a record 116 GW (figure 1). With over 1000 GW of installed capacity generating over 2300 terawatt-hours (8.3 × 1018 Joules), wind power was the largest source of variable renewable electricity in 2023. [1] Further, continued additions to this installed capacity will be the key to achieving the target of tripling renewable energy capacity by 2030 which countries agreed to during COP28. [2,3]
The sun is the ultimate source of energy driving atmospheric circulation. Solar flux at the earth's distance from the sun is around φ0 = 1370 W m-2. [4] Taking earth's radius to be R = 6378 km, we find that the average flux deceived by the earth is
| Φ | = | πR2 4πR2 |
× Φ0 | = | 1370 W m-2 4 |
= | 350 W m-2 |
and the total power power received by the earth is
| πR2Φ0 | = | π × (6.378 × 106 m)2 × 1370 W m-2 | = | 1.75 × 1017 Watts |
This reflects a limit on solar energy-based processes driving atmospheric circulations. To keep the atmosphere in motion, a part of the above energy goes into nullifying the dissipative effects of turbulence and friction in the atmosphere and at the earths surface. [4] Studies show that around 2% of the total solar flux is used to maintain the ongoing earth's wind systems. [4] This amounts to around 7 watts per square meter. For the whole earth it is around 0.02 × 1.75 × 1017 W = 3.5 × 1015 W.
Now, windmills can be installed within the boundary layer at the earth's surface. Studies reveal that around 35 percent (near earth factor) of the total dissipation occurs within 1 km of earth's surface. [4] This amounts to around 2.5 W m-2. For Earth as a whole, this amounts to 1.23 × 1015 watts. For the whole year, the solar energy deposition is
| 1.75 × 1017 W × 3600 s h-1 × 24 h d-1 × 365 d | = | 5.52 × 1024 J |
The boundary layer value is 0.02 × 0.35 = 0.007 of this number, or 3.86 × 1022 J.
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| Table 1: NREL classification of wind power. [5] |
Wind energy generation should not lead to a significant change in air circulation. While there are lots of estimates, it can be assumed that at a maximum of 10 percent of the near-surface dissipation can be extracted safely. [4] This amounts to around .25 watts per meter square. The total available amount for the earth as a whole will thus be 1.3 × 1014 watts. The available yearly energy amounts to 4 × 1021 joules. This should be compared with the world's total primary energy consumption which was around 6.0 × 1020 joules in 2021. [3] Further, the total electricity generated stood at around 28466.3 Terawatt-hours (1.02 × 1020 joules) in 2021. [3]
Let ρ represents the density of the air in kg m-3, A the area swept out by the blades in m2, and v the wind velocity in m/sec. The total power contained in the wind resource (PW) depends on these factors.
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(1) |
Power Coefficient (CP) represents the ratio of power extracted by the turbine to the total contained in the wind resource (PT/PW). Thus
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(2) |
Betz's law places a physical limit on the value of Cp. The maximum possible CP is 16/27. The reason why higher, for example, 100%, efficiency is not possible is due to the fluid mechanical nature of wind. If 100% of the kinetic energy was extracted, then the flow of air would stop. The maximum extraction efficiency is achieved at the optimum balance of the largest wind slowdown that still maintains sufficiently fast flow past the turbine. [5] Table 1 shows wind power classes measured at 50 m above ground according to NREL. Another important variable is the Capacity Factor (CF), which represents the fraction of the year a turbine can operate at peak power. Cf depends on both the turbine and site characteristics. A CF of .3 or above is considered good and economically viable. [5]
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| Fig. 2: Simulated power curve as given by Eqs. (1) and (2) with parameters ρ = 1.225 kg m-3, A = 3925 m2 and Cp = 0.59. The cut-in speed is taken to be 3 m/sec, the optimal speed is taken to be 10 m/sec, and the cut-off speed is taken to be 30 m/sec. (Image Source: A. Sheoran) |
The air density (ρ) is 1.225 kg m-3. Let the radius of the blade be 50 meters. Using the rule of thumb the effective cross-section of a turbine is about half the physical cross-section of the blade disc at the optimal wind speed. [6] The area of the cross-section can be assumed to be
| A | = | 1 2 |
π × ( 50 m )2 | = | 3925 m2 |
Let the cut-in speed be 3 m/s (at which the turbine starts to produce power), optimal wind speed be 10 m/s, and cut-off speed be 30 m/s (beyond which the turbine stops functioning. Assuming a CP of .59, the total power output (rated output) at the optimal speed is
| PT | = | 0.59 2 |
× 1.225 kg m-3 × 3925 m2 × ( 10 m/sec )3 |
| = | 1.4 × 106 watts | ||
A simulated power curve for this turbine is shown in Fig. 2.
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| Fig. 3: Mineral requirement (kg per MW). (Image Source: A. Sheoran, after the EIA. [7]) |
Sufficient space must be there to avoid upstream wakes and dissipation of kinetic energy by previous turbines. As a rule of thumb, to avoid interaction among wind energy extraction machines, a spacing of 5 to 15 times the rotor diameter is recommended between the turbines. The parameter of interest for this analysis is the ratio of the rotor-swept area to the land surface area (Λ). Λ value above .001 leads to a significant fall in energy produced per machine. [4]
The intermittency of power generation is a major issue. The power generated is not dispatchable, and many times, power produced by thermal power plants is adjusted to take care of the intermittency issue. Battery storage, pumped hydroelectric energy storage (PHES), and compressed air energy storage (CAES) are other ways to tackle the problem. Friedman provides a good overview of costs associated with intermittency and ways to handle it. [6]
Onshore wind plant requires nine times more mineral resources than a gas-fired power plant. Rare earth elements are essential for permanent magnets that are vital for wind turbines (Fig. 3). [7] The concentration of critical mineral supply chains is a major challenge. Mineral Security Partnership is one way to ensure that countries come together and explore ways to ensure supply chain resilience of such minerals.
The calculations above show that the available wind energy is multiple times the existing energy consumption rate. During recent years, wind energy installed capacity has increased significantly. Resolving challenges such as intermittency, storage, and securing supplies of critical minerals can ensure a sustained growth of the installed capacity. This will be critical for the world to secure the temperature goal of the Paris Agreement and the goals countries agreed to as part of the first Global Stock Take (GST) outcome.
© Amit Sheoran. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] "World Energy Outlook," International Energy Agency, 2024, p.157.
[2] "Decision 1/CMA.5: Outcome of the First Global Stocktake", in Decisions Adopted by the Conference of the Parties Serving as the Meeting of the Parties to the Paris Agreement, United Nations, FCCC/PA/CMA/2023/16/Add.1, March 2024.
[3] "BP Statistical Review of World Energy", British Petroleum, June 2022, p.47.
[4] M.R. Gustavson, "Limits to Wind Power Utilisation", Science 204, 13 (1979).
[5] T. M. Letcher, ed., Wind Energy Engineering: A Handbook for Onshore and Offshore Wind Turbines, 2nd Ed, (Academic Press, 2023), Chs. 3 and 4.
[6] O. Friedman, "Wind Turbines," Physics 240, Stanford University, Fall 2021.
[7] "The Role of Critical Minerals in Clean Energy Transition," International Energy Agency, March 2022, p.26.