Two methods for estimating limits to large-scale wind power generation

To evaluate the limits to wind power generation, we use a reference climatology of Central Kansas for the time period of May 15 to September 30, 2001 using the WRF-ARW v3.3.1 regional weather forecasting model (15, 16), forced with North American Regional Reanalysis data (17). This particular time period is climatologically representative for this region: a near-neutral El Niño southern oscillation phase, a climatologically standard position and strength of the Great Plains low-level jet, and an average summer soil moisture content (18). The simulation uses a single domain with a horizontal grid spacing of 12 km and 31 vertical levels, and the first 15 d of the simulation are excluded from the analysis to avoid spin-up effects. This WRF simulation represents our control simulation, which is used as input to the VKE method and as a reference for various WRF simulations with different densities of installed wind turbines to obtain the limit for wind power generation using the WRF method.

As shown in Fig. 1, the WRF simulations show that a greater installed capacity within the wind farm region increases the total electricity generation rate. This increase is almost linear at the lower installed capacities (0.3 MW_i⋅km⁻² $\approx$ 0.13 W_e⋅m⁻², 0.6 MW_i⋅km⁻² $\approx$ 0.24 W_e⋅m⁻²; subscripts i and e refer to the installed capacity and electricity generation, respectively). With further increases in the installed capacity, the marginal return of electricity generation predominantly occurs during higher wind speed periods. Such greater generation rates during windy periods can be seen in the differences between the simulations with 5.0 and 10 MW_i⋅km⁻² during the high wind speeds of June, whereas the difference is smaller during the lower wind speeds of August and September. Because the greater generation rates occur during periods that are less frequent, the increase in generation is no longer linear. This is reflected by comparing the generated electricity of the 5.0 MW_i⋅km⁻² to the 0.3 MW_i⋅km⁻² simulation, which generates seven times more electricity with 16 times as many wind turbines. Stated differently, each wind turbine at 5.0 MW_i⋅km⁻² generates electricity at half the rate as wind turbines with the same technical specifications but installed at 0.3 MW_i⋅km⁻².

This difference in the relationship between generation rate and installed capacity is reflected in a change in the capacity factor. First, we use the hub-height wind speeds of the control simulation and the turbine power curve for the Vestas V112 turbine (SI Appendix, Fig. 6) to calculate the generation rate of a single isolated wind turbine deployed to each location and time. This yields a capacity factor of 47%, which represents the upper bound value for the case of no interactions between the wind turbines and the atmospheric flow. This estimate compares well to the capacity factors of 22–36% (1, 7) derived from installed capacity and operational generation data from Kansas during 2006–2012, even though this estimate includes turbines of various technical specifications taken over a much longer timescale than this study. Using the 2012 installed capacity of 2,713 MW_i (7) and the area of 213,000 km² for Kansas yields a state-scale installed capacity of 0.013 MW_i⋅km⁻², which falls below the lowest installed capacity that we used. Our simulation with the lowest installed capacity of 0.3 MW_i⋅km⁻² corresponds to a slightly reduced capacity factor of 42%, and 39% with 0.6 MW_i⋅km⁻² (SI Appendix, Table 2). These capacity factors compare well with the previously used values for this region of 37% (6) and 40–47% used by the National Renewable Energy Laboratory (7). However, the estimates of refs. 6 and 7 used an installed capacity of 5.0 MW_i⋅km⁻², which in our simulations yield a much lower capacity factor of 19%, which should thus result in much lower estimates for wind power generation.

The reduction in capacity factor with greater installed capacity results from an enhanced interaction of wind turbines with the atmospheric flow. Because a greater installed capacity of wind turbines removes more kinetic energy from the atmosphere and converts it into electric energy, this causes a decrease in the hub-height wind speed downwind (26), which decreases the mean per-turbine electricity generation rate of the wind farm. This reduction in wind speeds within the wind farm and its effects on the per-turbine electricity generation rate is shown in Fig. 2 in relation to the power curve of the turbine and the wind speed histogram (Fig. 2A) as well as the mean wind speed and mean per-turbine generation rate (Fig. 2B). The point spread around the 3.0 MW_i turbine power curve in Fig. 2A, with some values below the 3.0 m⋅s⁻¹ cut-in wind speed, is due to the use of mean hourly hub-height wind speed and electricity generation rate for the entire wind farm region. Additionally, the variability in hub-height wind speed decreases with greater installed capacity (Fig. 2B), which also decreases the variability of per-turbine electricity generation. This reduction in wind speeds has also been observed in previous modeling studies (9–12, 27, 28).

Fig. 2C shows the increasing importance of considering the reduction in wind speed for the mean generation rate of the wind farm with greater installed capacity. The dashed line in Fig. 2C is derived by applying the turbine power curve to the control hub-height wind speeds for a mean per-turbine capacity factor of 47% (slope = 0.47). The WRF simulations with installed capacities of less than about 1 MW_i⋅km⁻² yield similar estimates because the capacity factors remain high (see also SI Appendix, Table 2). At greater installed capacities, the WRF simulations resulted in proportionally lower estimates. For example, at an installed capacity of 2.5 MW_i⋅km⁻² the “no interactions” estimate would yield a generation rate per unit area of the wind farm of 1.18 W_e⋅m⁻², but this was simulated to be 0.68 W_e⋅m⁻². This discrepancy continues with greater installed capacities, so at 5.0 MW_i⋅km⁻² the estimate without interactions overestimates the average electricity generation rate by more than a factor of two (2.4 W_e⋅m⁻² for no interactions, 0.95 W_e⋅m⁻² with interactions). The maximum electricity generation rate of 1.1 W_e⋅m⁻² is obtained with an installed capacity of 10 MW_i⋅km⁻², at which the associated hub-height wind speed decreased by 42% and the capacity factor is reduced to 12%. Our WRF simulations suggest that previous estimates of mean wind energy generation potentials for Kansas of 1.9 W_e⋅m⁻² (6), 2.0–2.4 W_e⋅m⁻² (7), and 2.5 W_e⋅m⁻² (4) are likely to be too high because the effects of reduced wind speeds were not considered. To place this reduction into the context of present-day wind power deployment, note that such installed capacities are several orders of magnitude larger than presently operational Kansas wind farms. Our simulations thus suggest that an equidistant deployment of 50 times more installed wind power in Kansas than is presently operational ($\approx$ 0.013–0.6 MW_i⋅km⁻²) would maintain the presently high per-turbine capacity factors and thus increase the generation rate 50-fold.

The VKE method captures the magnitude of wind power generation as well as its temporal variations. In our Kansas scenario, we estimate a maximum 4-mo mean generation rate from WRF at 10 MW_i⋅km⁻² as 1.1 W_e⋅m⁻² and VKE as 0.64 W_e⋅m⁻². Based on the linear correlation, the daily mean estimates of the two methods are highly correlated: r² = 0.98, with a slope of $m=1.76$, an rmse of 0.60, and a mean absolute error (MAE) of 0.47. The WRF estimate from the 5.0 MW_i⋅km⁻² simulation, an installed capacity often used for wind power planning and policy analysis (6), also compares very well, with daily mean estimates being highly correlated with VKE with r² = 0.98, $m=1.47$, rmse = 0.39, and MAE = 0.32. The mean generation rate of this WRF simulation was 0.95 W_e⋅m⁻², nearly the same rate as the 10 MW_i⋅km⁻² simulation, but from half the number of turbines. When hourly estimates of WRF and VKE are compared (Fig. 3), we note that correlations are very high during day and night, but the slope is much better captured by VKE during the day, whereas at night VKE underestimates the magnitude of electricity generation by almost 45% in this simulation.

We attribute this underestimation of wind power generation by VKE at night to its use of the preturbine downward kinetic energy flux of the control. The atmospheric flow in this region typically decouples from the stable surface conditions at night in the summer, which leads to the formation of the low-level jet (LLJ) near the surface (29). The typical nighttime structure of the LLJ (Fig. 4B) with a mean stable boundary layer height of 40 m (12–124 m, 5th–95th percentile, respectively) from June–September 2001 in the WRF control mean is consistent with height observations of about 50–350 m in southeastern Kansas during October 1999 (30). Observed LLJ maxima at about 100 m after sunset with an increase in height to about 225 m over the course of the night were also observed for this region on October 25, 1999 (30). The rotors of the wind turbines extend from 28 to 140 m in height and thus reside above, within, or at the upper boundary of the stable boundary layer. The wind turbines in the WRF simulations can thus sometimes directly use the kinetic energy from above the constant stress layer and the LLJ at night. This increased utilization of kinetic energy of the LLJ and the flow of the free atmosphere results in an increased downward kinetic energy and thus a greater maximum generation rate in WRF compared with the VKE method, which does not account for this effect. Based on the nighttime hourly mean values for the wind farm region, a hub-height speed of 9.5 m⋅s⁻¹ and a surface momentum flux of 0.15 kg⋅m⁻¹⋅s⁻² yields a downward kinetic energy flux of 1.39 W⋅m⁻² with an associated maximum generation rate of 0.36 W_e⋅m⁻² by VKE. Daytime atmospheric conditions are different. The daytime mean convective boundary layer height in the WRF control simulation is 1,268 m. Of this total height, the constant stress layer, the vertical depth over which the downward kinetic energy flux is considered negligible, typically constitutes the lowest 10% of the convective boundary layer (31, 32). Therefore, during the daytime, the upper extent of the turbine rotors is likely to be within the constant stress layer. Based on mean daytime values, a hub-height speed of 6.9 m⋅s⁻¹ and a surface momentum flux of 0.37 kg⋅m⁻¹⋅s⁻² yields a downward kinetic energy flux of 2.55 W⋅m⁻² with an associated maximum generation rate of 0.65 W_e⋅m⁻² by VKE. Note how the daytime VKE estimate is about double the nighttime estimate, even though the wind speed during the daytime is lower. These differences between the nighttime and daytime downward kinetic energy fluxes also help explain the similarities and discrepancies between the daytime and nighttime VKE and WRF estimates (Fig. 3).

One last point to note is that the maximum mean electricity generation rate of 1.1 W_e⋅m⁻² achieved in WRF has notable effects on the atmosphere and would likely induce considerable differences in climate. Although several recent studies evaluated how wind power generation caused climatic differences in measurements (33, 34) and modeling (10, 12, 13, 27, 35–37), the reduction of wind speeds is relevant here, because this reduction sets the large-scale limit to wind power generation. The mean hub-height wind speed in the 10 MW_i⋅km⁻² decreased by 42% compared with the control (Fig. 2B). This decrease is consistent with VKE, which provides an analytic expression for the decrease in wind speed at maximum generation of $\left(1-\sqrt{3}/3\right)\cdot v_0=42\text{\%}$. As described above, it is this decrease in wind speed with greater kinetic energy extraction by more wind turbines that limits the wind power generation at large scales. That VKE reproduces the decrease in $v_0$ very well is likely the reason why it captures the magnitude and temporal dynamics of limits to large-scale wind power generation of the WRF simulation.

Our estimates from both methods are compared with several other recent studies in Fig. 5. There is a clear discrepancy between estimates based on climatological wind speeds (black symbols) from estimates derived with atmospheric models (colored symbols), which are generally lower. We attribute these discrepancies to the inclusion of turbine–atmosphere interactions in the case of the atmospheric models that result in the reduction of wind speeds in the wind farm. However, one study included in Fig. 5 was derived from existing operational wind farms and observed generation rates, which calls for a more detailed explanation of the discrepancy between those and our estimates. Numerous footprints of operational wind farms in the United Kingdom were digitized (38) and compared with their documented generation rate, thereby inherently including turbine–atmosphere interactions. With the majority of the wind farms used in ref. 38 covering relatively small areas of about 2.4 km² (0.1–13 km²) of “footprint area” in hilltop or offshore locations, the wind farms have a mean generation rate of about 2.9 W_e⋅m⁻² (0.8–6.6 W_e⋅m⁻²) from a mean installed capacity of about 11 MW_i⋅km⁻² (3.5–24 MW_i⋅km⁻²). These generation rates are substantially higher than our 1.1 W⋅m⁻² limit of large-scale wind power generation in Kansas, although the size of the wind farms is also notably smaller.

This difference in wind power generation rates can be understood by relating the kinetic energy used by the wind turbines to their sources. For this, we distinguish between the import of kinetic energy by horizontal and vertical fluxes into the wind farm region. These two contributions change as the spatial scale of the wind farm increases. This change can be illustrated by using the mean values of the wind farm region from the WRF control simulation over the 4-mo period. The mean horizontal flux of kinetic energy is given by $K{E}_{in,h}=\left(1/2\right)\rho v_0^3\cdot x\cdot h$, where $\rho =1.1$ kg⋅m⁻³ is the air density at hub height, $v_0=8.0$ m⋅s⁻¹ is the hub-height wind speed at 84 m, $x=\mathrm{360,000}$ m is the east–west extent of the wind farm that is perpendicular to the mean wind direction, and $h=112$ m is the height of the wind farm, assumed here to be equivalent to the rotor diameter of the 3.0 MW_i turbine. This yields a mean horizontal kinetic energy flux of $K{E}_{in,h}=11$ GW (or 282 W⋅m⁻² per unit cross-sectional area) into the upwind vertical cross-section of the wind farm region. The mean vertical kinetic energy flux is given by $K{E}_{in,v}=\rho u_{*}^2\cdot v_0\cdot x\cdot y$, where the mean (spatial and temporal) friction velocity at the surface $u_{*}=0.45$ m⋅s⁻¹ and $y=\mathrm{312,000}$ m is the north–south extent of the wind farm that describes the downwind length of the wind farm. This yields a mean vertical kinetic energy flux downward into the entire wind farm region of $K{E}_{in,v}=200$ GW or 1.8 W⋅m⁻² per unit surface area of the wind farm region, so that in the Kansas setup, $K{E}_{in,v}$ provides about 20 times as much kinetic energy as the horizontal influx. Note that this vertical flux of kinetic energy, derived from the WRF control simulation, served as the input to the VKE estimate. When the wind farm increases in downwind length with a greater value of y, the contribution by the vertical kinetic energy flux into the wind farm region increases linearly whereas the horizontal contribution remains relatively unchanged. WRF simulations with an installed capacity of 1 MW_i⋅km⁻² or greater (>110 GW_i) represent wind farms in which the installed capacity is of the order of the mean kinetic energy flux into the wind farm region (about 211 GW), which is when the reductions of wind speed start to play a role in shaping the generation rate.

In the context of the Kansas wind farm region, we can use these considerations to estimate the downwind depth at which the horizontal kinetic energy flux is fully consumed by electricity generation and turbulence. Assuming a conservative 33% loss to turbulence during the extraction process (25), the 11-GW mean horizontal kinetic energy flux would result in a maximum electricity generation rate of 7.4 GW_e. This generation rate is equivalent to about 5,800 wind turbines of 3.0 MW_i capacity with a 42% capacity factor, which is close to our WRF simulation at the lowest installed capacity of 0.3 MW_i⋅km⁻². When considering the much greater installed capacity of 5.0 MW_i⋅km⁻², the 11 GW of horizontal kinetic energy flux would be fully consumed within a downwind depth of about 10 km (see also ref. 22). Therefore, as the downwind extent of the wind farm grows, electricity generation rates of successive downwind turbines are derived progressively less from the horizontal flux and more from the vertical flux. This results in an edge effect of higher generation rates at the upwind border of the wind farm compared with lower generation rates in the interior of the wind farm region (see also SI Appendix, Fig. 9). This edge effect does not exist for the VKE estimate (SI Appendix, Fig. 9), because it neglects the horizontal kinetic energy flux as an energy source. This can in part explain the lower estimates of the VKE method. However, when considering wind farms of greater sizes, the influence of this edge effect on the mean generation rate becomes progressively less important to consider.

Generation rates above those estimated by VKE could be achieved if the incoming horizontal kinetic energy flux is available to the wind farm because it was not extracted by upwind turbines, or relate to an increase in the vertical kinetic energy flux by the wind turbines, as shown to particularly occur in the WRF simulations at night. The spatial extent over which this enhanced vertical kinetic energy flux can be maintained, how much it alters the LLJ, and possibly how this results in a regional redistribution in this flux remain as open questions.

An overall increase in the downward kinetic energy flux at larger deployment scales seems unlikely to occur, because climate model simulations performed at continental and global scales do not predict such an increase for present-day radiative forcing conditions (10, 13). Although these studies did not include a full analysis of the energetics, their predictions broadly agree with the predictions of the VKE method in terms of a maximum of 25–27% of the natural dissipation rate that could be used for electricity generation (10) and a slowdown of hub-height wind velocities by 51% globally, 50% over land, and 51% over the ocean (13). Despite its lack of considering changes in the downward kinetic energy flux, it would nevertheless seem that the VKE method is suitable to provide first-order estimates of the magnitude of wind power generation by large wind farms, but this would require further confirmation.

This agreement does not resolve the apparent discrepancy between our estimates and the observation-based estimates from small UK wind farms (38); note that these wind farms have downwind depths much less than 10 km, making their electricity generation rates almost exclusively dependent on the horizontal kinetic energy flux. Formulated differently, edge effects determine the generation rate of these small wind farms. To illustrate compatibility with WRF-simulated results, we apply the footprint area definition of ref. 38 for isolated 3.0 MW_i wind turbines (i.e., a circle with diameter five times the turbine diameter, or 0.25 km² per turbine) to our simulation of 0.3 MW_i⋅km⁻². This results in each 3.0 MW_i turbine being spaced 3.1 km apart and yields a comparable 5.1 W_e⋅m⁻² for the turbines. For progressively larger installed capacities, this estimate decreases to 4.7 W_e⋅m⁻² for an installed capacity of 0.6 MW_i⋅km⁻², to 4.0 W_e⋅m⁻² for 1.3 MW_i⋅km⁻², to 3.3 W_e⋅m⁻² for 2.5 MW_i⋅km⁻², to 2.3 W_e⋅m⁻² for 5.0 MW_i⋅km⁻², and to 1.3 W_e⋅m⁻² for 10 MW_i⋅km⁻².

In summary, these considerations illustrate the strong dependence of small-scale wind farms on a horizontal kinetic energy flux that is not influenced by other wind farms upwind. Our results suggest that expanding wind farms to large scales will limit generation rates by the vertical kinetic energy flux, thereby constraining mean large-scale generation rates to about 1 W_e⋅m⁻² even in windy regions. Large-scale estimates that exceed 1 W_e⋅m⁻² thus seem to be inconsistent with the physical limits of kinetic energy generation and transport within the Earth’s atmosphere.

We evaluated large-scale limits to wind power generation in a hypothetical scenario of a large wind farm in Kansas using two distinct methods. We first used the WRF regional atmospheric model in which the wind farm interacts with the atmospheric flow to derive the maximum wind power generation rate of about 1.1 W_e⋅m⁻². This maximum rate results from a trade-off by which a greater installed capacity resulted in a greater reduction of wind speeds within the wind farm. This reduction in wind speeds reflects the strong interaction of the wind farm with the atmospheric flow, with speeds reduced by 42% at the maximum generation rate. We then showed that these estimates can also be derived by the VKE method, which used the downward influx of kinetic energy of the control climatology and its partitioning into turbulent dissipation and wind-energy generation as a basis. The VKE method predicts that the maximum generation rate equals 26% of the instantaneous downward transport of kinetic energy through hub height. This method only required the information of wind speeds and friction velocity of the control climate to provide an estimate of a maximum wind power generation rate. With an estimate of 0.64 W_e⋅m⁻², the VKE method underestimates the maximum wind power generation rate, particularly during night, but it nevertheless captures the temporal dynamics as well as the reduction in wind speeds very well.

Both methods used here yield estimates for the limits to large-scale wind power generation that are energetically consistent. Although many current wind farms are still comparatively small and can therefore sustain greater generation rates, an energetically consistent approach becomes relevant when the installed capacity of the wind farm approaches the kinetic energy flux into the wind farm region. Although the VKE method assumes this influx to be fixed, it nevertheless demonstrates that an energetically consistent estimate can be done in a comparatively simple way, thus providing a useful means to derive a first-order estimate of large-scale wind power generation from preturbine climatologies. We conclude that large-scale wind power generation is thus limited to a maximum of about 1 W_e⋅m⁻² because of this inevitable reduction of wind speeds and the comparatively low vertical kinetic energy fluxes in the atmosphere.

We thank C. Dhara, M. Renner, N. Carvalhais, two anonymous reviewers, and the editor for their suggestions and constructive comments that helped improve the mansucript. This study was funded by the Max Planck Society through the Max Planck Research Group of A.K. A.J.M. was funded by the National Science Foundation.