Wind farm models

Introduction

ERF supports models for wind farm parametrization in which the effects of wind turbines are represented by imposing a momentum sink on the mean flow and/or turbulent kinetic energy (TKE). Currently only the Fitch model (Fitch et al. 2012) is supported.

Fitch model

The Fitch model for wind farms introduced in Fitch et al. 2012 models the effect of wind farms (See Fig. 1) as source terms in the governing equations for the horizontal components of momentum (i.e., \(x\) and \(y\) momentum) and the turbulent kinetic energy (TKE). The wind turbine is discretized only in the vertical (ie. z direction). At a given cell \((i,j,k)\), the source terms in the governing equations are

\[\begin{split}\frac{\partial u_{ijk}}{\partial t} &= \frac{u_{ijk}}{|V|_{ijk}}\frac{\partial |V|_{ijk}}{\partial t} \\ \frac{\partial v_{ijk}}{\partial t} &= \frac{v_{ijk}}{|V|_{ijk}}\frac{\partial |V|_{ijk}}{\partial t} \\ \frac{\partial \text{TKE}_{ijk}}{\partial t} &= \frac{0.5N^{ij}C_{TKE}(|V|_{ijk})|V|_{ijk}^3A_{ijk}}{z_{k+1}-z_k}\end{split}\]

where

\[\frac{\partial |V|_{ijk}}{\partial t} = \frac{0.5N^{ij}C_T(|V|_{ijk})|V|_{ijk}^2A_{ijk}}{z_{k+1}-z_k}\]

where u and v are horizontal components of velocity, |V| is the velocity magnitude, \(N^{ij}\) is the number of turbines in cell \((i,j)\), \(C_T\) is the coefficient of thrust of the turbines, \(C_{TKE}\) is the fraction of energy converted to turbulent kinetic energy – both of which are functions of the velocity magnitude, and \(A_{ijk}\) is the area intersected by the swept area of the turbine between levels \(z=z_k\) and \(z= z_{k+1}\), and is explained in the next section.

Intersected area \(A_{ijk}\)

Consider \(A_k^{k+1}\) – the area intersected by the swept area of the wind turbine between \(z=z_k\) and \(z = z_{k+1}\). We have (see Figs. 2 and 3 below)

\[A_k = \frac{\pi R^2}{2} - A_{ks}\]

where \(A_{ks}\) is the area of the segment of the circle as shown in Fig. 3. We have from geometry, \(d_k = \min(|z_k - z_c|,R)\) is the perpendicular distance of the center of the turbine to \(z = z_k\), where \(z_c\) is the height of the center of the turbine from the ground. The area of the segment is

\[A_{ks} = R^2\theta_k - d_k\sqrt{R^2 - d_k^2}\]

where \(\theta_k = \cos^{-1}\left(\frac{d_k}{R}\right)\).

Hence, we have the intersected area \(A_{ijk}\equiv A_k^{k+1}\) as

\[\begin{split}A_k^{k+1} = \begin{cases} |A_k - A_{k+1}| & \text{if } (z_k - z_c)(z_{k+1}-z_c) > 0 \\ |A_k + A_{k+1}| & \text{if } (z_k - z_c)(z_{k+1}-z_c) \le 0 \\ \end{cases}\end{split}\]

An example of the Fitch model is in Exec/Fitch

1 Horizontal view of the wind farm in the Fitch model – shows a wind farm in cell (i,j) with 5 wind turbines. The turbines are discretized only in the vertical direction.

2 The vertical discretization of the wind turbine in the Fitch model.

3 The area terminology in the Fitch model. The circle represents the area swept by the turbine blades.