Set-point optimization in wind farms to mitigate effects of flow blockage induced by atmospheric gravity waves

Abstract

Abstract. Recently, it has been shown that flow blockage in large wind farms may lift up the top of the boundary layer, thereby triggering atmospheric gravity waves in the inversion layer and in the free atmosphere. These waves impose significant pressure gradients in the boundary layer, causing detrimental consequences in terms of a farm's efficiency. In the current study, we investigate the idea of controlling the wind farm in order to mitigate the efficiency drop due to wind-farm-induced gravity waves and blockage. The analysis is performed using a fast boundary layer model which divides the vertical structure of the atmosphere into three layers. The wind-farm drag force is applied over the whole wind-farm area in the lowest layer and is directly proportional to the wind-farm thrust set-point distribution. We implement an optimization model in order to derive the thrust-coefficient distribution, which maximizes the wind-farm energy extraction. We use a continuous adjoint method to efficiently compute gradients for the optimization algorithm, which is based on a quasi-Newton method. Power gains are evaluated with respect to a reference thrust-coefficient distribution based on the Betz–Joukowsky set point. We consider thrust coefficients that can change in space, as well as in time, i.e. considering time-periodic signals. However, in all our optimization results, we find that optimal thrust-coefficient distributions are steady; any time-periodic distribution is less optimal. The (steady) optimal thrust-coefficient distribution is inversely related to the vertical displacement of the boundary layer. Hence, it assumes a sinusoidal behaviour in the streamwise direction in subcritical flow conditions, whereas it becomes a U-shaped curve when the flow is supercritical. The sensitivity of the power gain to the atmospheric state is studied using the developed optimization tool for almost 2000 different atmospheric states. Overall, power gains above 4 % were observed for 77 % of the cases with peaks up to 14 % for weakly stratified atmospheres in critical flow regimes.

An improved fringe-region technique for the representation of gravity waves in large-eddy simulation with application to wind farms

“…In both cases, the reflectivity is rather constant along lines of low and constant P N values while it shows a higher variability for higher P N . This behaviour was also noted by Smith (2010) and Lanzilao and Meyers (2021), who observed that the flow response is less sensitive to changes in Froude number when P N is low. The median of the reflectivity distribution obtained with π L = 15 in the stdFR-RDL and newFR-RDL cases differs of several percentage points, going from 4.1% to 1.6%.…”

“…numbers can be written as Fr = 1/ √ π g π ∆θ and P N = 1/ √ π g π Γ , meaning that we will effectively vary the two non-dimensional quantities that characterize the impact of gravity waves on the flow dynamics (Smith, 2010;Allaerts andMeyers, 2018, 2019;Lanzilao and Meyers, 2021).…”

“…Wind-farm induced gravity waves absorb energy within the ABL, transporting it at higher altitudes (Smith, 2010(Smith, , 2022Wu and Porté-Agel, 2017;Allaerts and Meyers, 2017Lanzilao and Meyers, 2021;Devesse et al, 2022;Maas and Raasch, 2022). The energy is then released when the waves break down (Nappo, 2002;Sutherland, 2010).…”

Effects of the thrust force induced by wind turbine rotors on the incoming wind field: A wind LiDAR experiment

“…Understanding pressure-induced effects on wind farm performance is also important in light of the local speed enhancement for turbine arrays, as reported in theoretical [8], numerical [9,10], and wind tunnel studies [11] , which clashed with the power losses observed for real wind farms [6,11]. Further, the role of atmospheric stratification on rotor-induced effects of the incoming wind is still not clearly understood, yet recent studies showed more severe flow modifications occurring under stable atmospheric conditions as a consequence of the vertical flow confinement due to a smaller boundary layer height and gravity waves [12,13].…”

Including realistic upper atmospheres in a wind-farm gravity-wave model

“…Such displacements can propagate through the overlying inversion layer and free atmosphere as waves in stably stratified atmospheres, conditions which frequently occur at sea. As offshore wind farms in Europe increase in size and installed capacity (WindEurope, 2018), improving the understanding and simulation of gravity-wave-wind-farm interaction becomes crucial to optimizing turbine control and windfarm layout (see, e.g., Lanzilao and Meyers, 2021b).…”