With the current expansion of wind power as a renewable energy source, wind turbines increasingly extract kinetic energy from the atmosphere, thus impacting its energy resource. Here, we present a simple, physics-based model (the Kinetic Energy Budget of the Atmosphere; KEBA) to estimate wind energy resource potentials that explicitly account for this removal effect. The model is based on the regional kinetic energy budget of the atmospheric boundary layer that encloses the wind farms of a region. This budget is shaped by horizontal and vertical influx of kinetic energy from upwind regions and the free atmosphere above, as well as the energy removal by the turbines, dissipative losses due to surface friction and wakes, and downwind outflux. These terms can be formulated in a simple yet physical way, yielding analytic expressions for how wind speeds and energy yields are reduced with increasing deployment of wind turbines within a region. We show that KEBA estimates compare very well to the modelling results of a previously published study in which wind farms of different sizes and in different regions were simulated interactively with the Weather Research and Forecasting (WRF) atmospheric model. Compared to a reference case without the effect of reduced wind speeds, yields can drop by more than 50 % at scales greater than 100 km, depending on turbine spacing and the wind conditions of the region. KEBA is able to reproduce these reductions in energy yield compared to the simulated climatological means in WRF (

The use of wind energy as a renewable energy resource has substantially increased over the last decades in the attempt to decarbonize the energy system. Particularly wind over the sea is seen as a tremendous yet under-utilized energy resource. In Europe alone, the current installed capacity of 22 GW in offshore wind power has increased by 3.5 GW in 2019

There is, however, a substantial discrepancy in how efficient wind turbines are in generating electricity, depending on the scale of deployment. An isolated turbine in an offshore environment with high, continuous wind speeds may generate electricity highly efficiently, with a capacity factor (i.e. the ratio of generated electricity to the capacity of the turbine) above 50 % and more than 4300 full load hours per year. These high efficiencies are typically used in assessments of offshore wind resource potentials

However, the more wind turbines that are deployed within a region, the more these remove kinetic energy from the atmosphere, leaving less behind, resulting in lower wind speeds and lower efficiencies of turbines downwind. Idealized climate model simulations at the planetary scale showed that this wind depletion effect results in much lower large-scale limits to wind power

Illustration of the Kinetic Energy Budget of the Atmosphere (KEBA) approach to estimate regional wind energy potentials, which considers the fluxes of kinetic energy in and out of a virtual volume that encloses a given region of dimensions

Here, we describe a modelling approach to estimate regional-scale wind energy resource potentials that explicitly accounts for the wind speed reductions and lower yields. The goal of this modelling approach is to provide simple and transparent first-order estimates based on physical concepts. To do so, we focus on the Kinetic Energy Budget of the Atmosphere (KEBA), as shown in Fig.

Naturally, our KEBA approach needs to be tested to see whether it can reasonably reproduce the effects and magnitudes simulated by far more complex simulation models. To do this, we use the published results of numerical simulations performed by

In the following, we first describe the mathematical formulation of KEBA in Sect.

The goal of the KEBA model is to provide a simple and transparent yet physically based approach to estimate wind energy potentials for a given region across scales that can reproduce the wind speed reductions found in much more complex numerical simulation models. It uses an observed record of wind speeds, dimensions of the region, and turbine characteristics as well as the number of turbines as an input. It predicts the reduction in wind speeds as well as the generated yields as output. The simplicity of the approach allows for it to be implemented in a spreadsheet, which is provided in the Supplement.

KEBA derives an effective wind speed within a region of wind turbines from the different fluxes that add, remove, or dissipate kinetic energy within the associated atmospheric air volume that encloses the region (Fig.

The effective wind speed

Overview of KEBA variables and how these are specified or computed.

The total influx of kinetic energy,

The loss terms of kinetic energy are described with respect to an effective wind speed,

For the electricity generation, or yield,

The outflux of kinetic energy,

Dissipation of kinetic energy by wake turbulence,

The Eqs. (

The yield of the wind farm,

In the case in which the wind farm operates at its rated capacity (

To link and visualize reductions in yield and wind speeds, it is instructive to explicitly look at the fluxes that shape the kinetic energy balance. The influxes of kinetic energy,

These two equations are a reformulation of Eqs. (

Equations (

To estimate wind energy yields, KEBA needs meteorological input in form of wind speeds,

The implementation of KEBA as well as the evaluations shown in the following section is provided in the Supplement as an Excel spreadsheet.

We evaluated KEBA with a set of sensitivity simulations with the WRF regional weather model published by

Wind forcing and turbine power curve used for the evaluation of KEBA. Panels

The wind speed histograms of the three regions considered are shown in Fig.

Each scenario considered a set of 2 MW Vestas V-80 turbines (

The scenarios consider four different sizes of wind farms arranged in a square, ranging from about 5 to 337 km, with three turbine spacings of

We compare the KEBA estimates also to an estimate in which no wind speed reductions are considered (“isolated” case), so that each turbine operates as if it were an isolated wind turbine. This case is represented in KEBA by a reduction factor of

Scenarios used to evaluate KEBA, as defined in

KEBA parameters used to evaluate the scenarios of

The comparison of KEBA estimates to the estimates by

Mean yields estimated for the case of isolated wind turbines (without wind speed reductions) and yield estimates with wind speed reductions by KEBA and by

The key variable that describes the effect of the kinetic energy removal by the wind turbines is the reduction factor,

Figure

The value of the reduction factor

Diagnosed terms of the kinetic energy budget, with the relative contributions to the influxes of kinetic energy (KE) shown in the left column and the conversion to electricity, dissipation, and outflux of kinetic energy shown in the three right columns (for three different turbine spacings: W: wide; I: intermediate; N: narrow). The horizontal and vertical contributions to the KE influx are shown by dark and light blue, respectively. The yield (conversion to electricity) is shown in yellow and the dissipation by surface friction (red) and wake turbulence (orange) and outflux of kinetic energy (purple). The plots show the KE budgets for the scenarios of wind farms of four different areas (S, M, L, and XL) for region A (central US, land,

Since KEBA is explicitly based on the budgeting of kinetic energy, we can further analyse these scenarios in terms of changes in the energy fluxes within this budget. These terms are approximated here using the mean kinetic energy fluxes (with the densities given by

Estimated kinetic energy influxes for the different scenarios and wind farm sizes in comparison to the estimated yields. The ranges for the “KEBA” and “Volker et al. (2017)” cases refer to the different turbine spacings (narrow, intermediate, wide) with lowest CFs and yields corresponding to the narrowest spacing and highest turbine densities.

The analysis of the kinetic energy budget illustrates how an increasing share of the kinetic energy influx is taken by the turbines and converted into electricity, resulting in less kinetic energy outflux and reduced wind speeds. For the scenarios of small wind farms, essentially all of the kinetic energy supply is provided by the horizontal influx, and the wind farm removes an insignificant fraction of it. The larger the wind farm region, the more the vertical influx of kinetic energy contributes to the supply of kinetic energy, and wind farms remove an increasingly significant fraction of this supply (yellow bars in Fig.

It is hence this constrained nature of the fluxes that feed the kinetic energy budget of the regional lower atmosphere that encloses the wind farm that results in the diagnosed magnitude of yield reductions. As the estimates by KEBA match those derived from much more complex model simulations by

We next evaluated the sensitivity of the KEBA estimates to the height of the boundary layer

KEBA sensitivity to downwind length

The sensitivity to downwind length

To evaluate this sensitivity, we use the meteorological forcing of region B (North Sea) and evaluate how the KEBA reduction factor,

As

For very long downwind lengths (

The KEBA approach is, clearly, extremely simple and neglects many complicating factors, such as the role of stability, different drag coefficients, variations in boundary layer height, or the wind direction. KEBA can nevertheless reproduce the wind energy yields and their reductions in large wind farms simulated by the much more complex WRF simulations of

Our KEBA approach also only crudely describes the wakes that develop directly behind wind turbines through the wake dissipation term,

In its present form, KEBA can adequately capture wind energy resource estimates at the regional scale and, as such, can inform the planning and policy development regarding the future expansions of wind energy. By being implemented in a spreadsheet, it can quickly estimate the yields of different scenarios in a transparent and reproducible way, given the prescribed wind conditions of the region.
What are the likely cases where KEBA would provide useful insights? Current wind farms on land typically have installed capacity densities of less than 3 MW km

We presented a model to estimate wind energy resource potentials at the regional scale that explicitly accounts for wind speed reductions caused by the wind turbines. This formulation yielded analytical solutions to estimate these wind speed reductions and the associated mean yields of the wind farms. We compared this formulation to a set of sensitivity simulations with the WRF regional weather forecasting model by

The modelling of the kinetic energy budget thus provides valuable insights for estimating wind speed reductions and wind energy resource potentials at the regional scale but also at a more general level in terms of understanding the impacts that large-scale wind energy use has on the atmosphere. While our approach can be extended in future work to address some of the shortcomings, it seems that an explicit analysis of kinetic energy fluxes would be informative and provides valuable information. In its present form, KEBA seems well suited to provide first-order estimates of wind energy resource potentials at the regional scale that are based on atmospheric physics.

The KEBA implementation is provided in an Excel spreadsheet available in the Supplement. All data used to evaluate KEBA are contained in the Excel spreadsheet.

The supplement related to this article is available online at:

AK and LEM designed the study, performed the analysis, and wrote the paper; AK performed the KEBA simulations.

The authors declare that they have no conflict of interest.

The authors thank one anonymous reviewer and Dan Kirk-Davidoff for their helpful comments.

The article processing charges for this open-access publication were covered by the Max Planck Society.

This paper was edited by Simon Unterstrasser and reviewed by Daniel Kirk-Davidoff and one anonymous referee.