  The Reynolds-averaged Navier-Stokes approach coupled with the standard
$k-\varepsilon$ model is widely utilized for wind-energy applications. However,
it has been shown that the standard $k-\varepsilon$ model overestimates the
turbulence intensity in the wake region and, consequently, overpredicts the
power output of the waked turbines. This study focuses on the development of an
extended $k-\varepsilon$ model by incorporating an additional term in the
turbulent kinetic energy equation. This term accounts for the influence of
turbine-induced forces, and its formulation is derived through an analytical
approach. To assess the effectiveness of the proposed model, we begin by
analyzing the evolution of normalized velocity deficit and turbulence intensity
in the wake region, and the normalized power of the waked turbines. This
investigation involves a comparison of the predictions against results from
large-eddy simulations in three validation cases with different layouts. We
then simulate a wind farm consisting of 30 wind turbines and conduct a
comparative analysis between the model-predicted normalized streamwise velocity
and wind-tunnel measurements. Finally, to conclude our assessment of the
proposed model, we apply it to the operational wind farm of Horns Rev 1 and
evaluate the obtained normalized power with the results from large-eddy
simulations. The comparisons and validations conducted in this study prove the
superior performance of the extended $k-\varepsilon$ model compared to the
standard version.
