Limits to the power density of very large wind farms

TL;DR

This paper presents an analysis establishing an upper limit on the power density of very large wind farms, indicating that maximum extractable power is constrained by natural shear stress and undisturbed wind speed, not just farm size or height.

Contribution

It introduces a theoretical upper limit on wind farm power density based on boundary layer physics, highlighting fundamental constraints on large-scale wind energy extraction.

Findings

Maximum power density is about 0.38 times shear stress times undisturbed wind speed.

Maximum extractable power is proportional to undisturbed wind speed, not farm height.

Power density does not increase indefinitely with larger wind farms.

Abstract

A simple analysis is presented concerning an upper limit of the power density (power per unit land area) of a very large wind farm located at the bottom of a fully developed boundary layer. The analysis suggests that the limit of the power density is about 0.38 times $\tau_{w0}U_{F0}$, where $\tau_{w0}$ is the natural shear stress on the ground (that is observed before constructing the wind farm) and $U_{F0}$ is the natural or undisturbed wind speed averaged across the height of the farm to be constructed. Importantly, this implies that the maximum extractable power from such a very large wind farm will not be proportional to the cubic of the wind speed at the farm height, or even the farm height itself, but be proportional to $U_{F0}$.