# Investigating the Effect of Large Wind Farms on Energy in the Atmosphere

^{*}

## Abstract

**:**

_{2}. Also, the net heat added to the environment due to wind dissipation is much less than that added by thermal plants that the turbines displace.

## 1. Introduction

## 2. Blade Element Momentum Model

## 3. Power Curve Comparisons

_{0}, for all three turbines. The model-generated power curves were compared with the expected power output of these turbines.

Turbine type | No. of blades | Blade radius (m) | Rotation speed (rpm) | Rated Power (kW) | Cut-in wind speed (m/s) |
---|---|---|---|---|---|

NREL UAE | 2 | 5 | 71.63 | 19.8 | 6 |

LMG | 3 | 29 | 19 | 1,500 | 5 |

Tjaereborg | 3 | 30 | 22 | 2,000 | 5 |

**Figure 2.**NREL power comparison as a function of incoming wind speed, V

_{0}. The gray lines indicate the range of wind speeds that are relevant for wind energy production.

**Figure 6.**Previous power comparisons weighted with the Rayleigh probability distribution: (A) NREL vs. model, (B) LMG vs model, and (C) Tjaereborg vs model.

## 4. Energy Reduction Estimate

_{0}, and the velocity at the wake, V

_{wake}:

_{wake}is not the velocity throughout the whole wake, but an estimate of the velocity in the region of the wake where the static pressure is equal to the static pressure outside the stream tube. This distance is typically 2D to 3D, where D is the diameter of the rotor [27]. The thrust, T, due to the pressure difference across the rotor can be computed as the momentum change across the disc:

_{P}, which is the ratio of the generated to available power in the wind, is then calculated as:

_{wake}:

_{0}is an input, so V

_{wake}is easily solved for. V

_{wake}is a measure of the axial velocity downstream of the disc and rotation in the wake is not taken into consideration. The rotation in the wake adds to the pressure drop across the disc and further reduces the kinetic energy of the flow, so V

_{wake}calculated here is an overestimate. Including wake rotation should lead to lower axial velocities downstream (e.g., ~50% lower), but typically only near the wake edge.

_{wake}changes with V

_{0}for a GE 1.5 MW turbine with a 77 m rotor diameter [28]. This turbine was chosen because it is typical of the type of turbines being erected in many wind farms today. It can be seen from the figure that the ratio of the two velocities increases with V

_{0}implying that less energy is taken out of the flow as the incoming wind speed increases. This makes sense as the efficiency of this specific turbine is maximum at lower wind speeds. The efficiency, C

_{p}, is plotted together (dashed line) with the velocity to show that V

_{wake}is small when the efficiency is high.

**Figure 8.**Change in wake velocity as a function of incoming velocity overlapped with turbine efficiency.

_{wake}/V

_{0}) to be about 50% at 2.5D downstream. Although the wind speed at which the ratio was measured was not specified, it is still possible to compare this with the minimum ratio shown in Figure 8 since this minimum is found in the 8–9 m/s wind speed range, which is the most common wind speed at many wind farms. Another measure for comparison is the relative velocity deficit, computed as:

_{f}is the height of the frustum (or, in this case, depth, since the frustum is on its side), and r

_{1}and r

_{2}are the radii of the top and base circles. Previous studies have found that the wind speed in the wake of a turbine has been recovered by more than 90% of the free stream wind speed at a distance of 10D, where D is the rotor diameter [30] and that at that distance (10D), the average wake width was about 3D. Translating these lengths into the dimensions of a frustum results in h

_{f}= 10D, r

_{1}= 1D/2, and r

_{2}= 3D/2 (see Figure 9). The volume of the wake, S

_{wake}, is then:

**Figure 10.**Energy loss (%) calculated using Equation (11) (solid line) and this same loss after factoring in the fraction of atmosphere actually affected by the wake (dashed line). In the case of a wind farm with 3D × 10D spacing, this fraction is 13πD/36h (see Equation (15)).

_{2}from all fossil fuel energy sources (both electric and non-electric), and scenario B, which assumes that all onroad vehicles are replaced by wind-powered battery electric vehicles. For scenario A, two cases were analyzed. The first case assumes that wind displaces all fossil fuel energy worldwide that produces carbon. The second case assumes that wind displaces all fossil fuel carbon in the United States. Scenario B also examines two cases: replacing onroad vehicles throughout the whole U.S. and replacing onroad vehicles in California. California is of interest because it is the U.S. state with the largest vehicle density. Table 2 summarizes the cases. The energy losses from all these cases are examined both as the loss from the lower 1 km of the atmosphere (heretofore referred to as L1)—over global land and oceans—and the loss from only sections of the L1 layer that are above the land area of interest, e.g., over U.S. land.

**Table 2.**Different cases and scenarios used in the analysis of the energy loss from large wind farms.

Scenario | Description |
---|---|

A1 | Replace all fossil fuel energy globally with wind |

A2 | Replace all fossil fuel energy in the US with wind |

B1 | Replace all onroad vehicles in the US with wind-powered battery electric vehicles (BEVs) |

B2 | Replace all onroad vehicles in California with wind-powered BEVs |

_{8}, f

_{8.6}and f

_{4.8}are the Rayleigh distributions centered around 8 m/s, 8.6 m/s and 4.8 m/s, respectively. The variable A

_{land}represents the fraction of land included in the analysis. In analyzing the energy loss only over land, A

_{land}= 1. For the analysis involving the entire ABL, A

_{land}= 0.29, as the globe is 29% land and 71% ocean. From Equation (11), $\Delta E={V}_{0}^{2}-{V}_{3}^{2}$ and ${E}_{0}={V}_{0}^{2}$. The remaining variables are the number of turbines, N, the volume of the wake, S

_{wake}, and the volume of the atmosphere being analyzed, S

_{atm}. This latter volume is analogous to the control volume described earlier in the single turbine case. Here, the chosen control volume will be depend on the energy loss scenario being analyzed. It is computed as S

_{atm}= A

_{sft}× h where h = 1 km, and A

_{sfc}is the surface area for each case. For example, in analyzing case A1 for L1 over the whole globe, A

_{sfc}is the surface area of global land and oceans, but if only the loss in L1 above global land is desired, A

_{sfc}is only the global land surface area.

_{s}, by the energy produced by a single turbine, E

_{t}:

_{t}(kWh/yr), is calculated as:

_{r}is the turbine’s rated power (kW), CF is the turbine’s capacity factor, and γ is an efficiency factor that takes into account the energy loss due to heat, mechanical, and transmission issues. In this analysis, we assume γ = 0.85, or a loss of 15%. The capacity factor, CF, is a measure of how much energy the turbine produces over its maximum rated energy. An empirical equation for CF is:

_{r}is the turbine’s rated power (kW), and D is its rotor diameter (m) [26]. Note that since this equation is empirical, its units do not equate.

**Table 3.**Number of 1.5 MW turbines required for each scenario (see Table 2 for scenario description).

MEAN WIND SPEED (M/S) | NUMBER OF TURBINES | |||
---|---|---|---|---|

Scenario A1 | Scenario A2 | Scenario B1 | Scenario B2 | |

7 | 10 million | 1.8 million | 400,000 | 40,000 |

8 | 8 million | 1.4 million | 320,000 | 33,000 |

9 | 6.7 million | 1.2 million | 270,000 | 27,000 |

10 | 5.7 million | 1 million | 230,000 | 23,000 |

**Figure 11.**Energy loss (%) for Scenario A1, where wind is used to power global energy. The first set of bars indicates the relative energy loss in the entire boundary layer, while the second set indicates the losses in the boundary layer over just the global land area. The different bars represent the different mean wind speeds, 7, 8, 9, and 10 m/s, with the number of turbines needed for each mean wind speed given in Table 3.

**Figure 12.**Energy loss (%) for Scenario A2, where wind farms supply U.S. energy. Similar to the previous figure, the first set of bars represents the relative loss in the entire boundary layer, and the second set represents the relative loss in the boundary layer just above the U.S. The number of turbines for each wind speed is given in Table 3.

**Figure 13.**Energy loss (%) for Scenario B1, where all U.S. onroad vehicles are replaced with WBEV. The number of turbines for each mean wind speed case is given in Table 3.

**Figure 14.**Energy loss (%) for Scenario B2, where all California onroad vehicles are replaced with WBEV. Here, the second set of bars represents the relative loss in the boundary layer just above California. The number of turbines for the different mean wind speeds are just roughly an order of magnitude less than that needed for the Scenario B1 and shown in Table 3.

^{2}) and compared with radiative forcing due to increased carbon dioxide (CO

_{2}) in the atmosphere. The maximum energy loss in Scenario A1 is largest at a mean wind speed of 7 m/s. The equation used to calculate the power density loss, R (W/m

^{2}), is:

_{sfc}= 5.1 × 10

^{8}km

^{2}, the surface area of the Earth. The energy loss, ∆E (J), is calculated as

_{a}is the mass (kg) of the lower 1 km of the atmosphere. In contrast to the power density loss from Equation (21), the radiative forcing due to CO

_{2}in the atmosphere is 1.6 W/m

^{2}[34]. A doubling of CO

_{2}in the absence of replacing current energy generation with wind, therefore, will have a several-order-of-magnitude larger effect on the overall climate than the atmospheric momentum loss from supplying the world’s energy demands with wind.

## 5. Limitations and Context

## 6. Conclusions

_{2}in the atmosphere. Also, any heating effects of this energy loss is outweighed by the thermal pollution that it will avert when wind farms displace the thermal power plants driven by fossil fuels.

## Acknowledgements

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**MDPI and ACS Style**

Sta. Maria, M.R.V.; Jacobson, M.Z.
Investigating the Effect of Large Wind Farms on Energy in the Atmosphere. *Energies* **2009**, *2*, 816-838.
https://doi.org/10.3390/en20400816

**AMA Style**

Sta. Maria MRV, Jacobson MZ.
Investigating the Effect of Large Wind Farms on Energy in the Atmosphere. *Energies*. 2009; 2(4):816-838.
https://doi.org/10.3390/en20400816

**Chicago/Turabian Style**

Sta. Maria, Magdalena R. V., and Mark Z. Jacobson.
2009. "Investigating the Effect of Large Wind Farms on Energy in the Atmosphere" *Energies* 2, no. 4: 816-838.
https://doi.org/10.3390/en20400816