A Linear Theory of Wind Farm Efficiency and Interaction

Abstract

Abstract We investigate the role of gravity waves (GW), farm shape, and wind direction on the efficiency and interaction of wind farms using a two-layer linearized dynamical model with Rayleigh friction. Five integrated diagnostic quantities are used: total wind deficit, the first moment of vorticity, turbine work, disturbance kinetic energy, and vertical energy flux. The work done on the atmosphere by turbine drag is balanced by dissipation of disturbance kinetic energy. A new definition of wind farm efficiency is proposed based on “turbine work.” While GWs do not change the total wind deficit or the vorticity pattern, they alter the spatial pattern of wind deficit and typically make a wind farm less efficient. GWs slow the winds upwind and reduce the wake influence on nearby downstream wind farms. GWs also propagate part of the disturbance energy upward into the upper atmosphere. We applied these ideas to the proposed 45 km × 15 km wind energy areas off the coast of New England. The proximity of these farms allows GWs to play a significant role in farm interaction, especially in winter with northwesterly winds. The governing equations are solved directly and by using fast Fourier transforms (FFT). The computational speed of the linear FFT model suggests its future use in optimizing the design and day-by-day operation of these and other wind farms. Significance Statement When a wind farm is generating electricity, the drag of the wind turbines slows the regional winds. As wind farms grow larger and more closely spaced, this wind reduction will limit the efficiency of wind farms and their economic return. In this paper we analyze an idealized mathematical model of the atmospheric response to wind farm drag including nonlocal gravity wave effects. We propose a new definition of farm efficiency based on the atmospheric disturbance that a farm creates. We also propose a fast Fourier transform (FFT) method for carrying out these estimates.