These authors contributed equally to this work.

We propose a modification to the Fitch wind farm parameterization implemented in the Weather Research and Forecasting (WRF) model. This modification, derived from 1D momentum theory, employs a wind-speed-dependent induction factor to correct the local grid wind speed back to freestream before computing the turbine's power and thrust. While the original implementation underestimates power, the modified version shows good agreement with the power curve. We strongly recommend employing the modification for all studies that model at maximum one turbine per WRF grid cell. For simulations with more turbines per grid cell, additional inner-cell wake losses have to be considered.

As offshore wind energy is developing quickly and in relatively concentrated regions along the coastline, models that correctly represent large-scale wake effects and their interactions with the atmosphere are needed. Several attempts towards the realistic modeling of those effects have been made by employing fast engineering models

The most commonly used mesoscale model, especially when studying large-scale wake effects, is the Weather Research and Forecasting model

The Fitch parameterization does not consider the effect induction has on the local wind speed at the grid cell of the turbine. In this publication we show that, as a consequence, the turbines' power and thrust are underestimated.

The effect of axial induction increases in relevance with increasing ratios between turbine dimensions and grid sizes. It is therefore desirable to have a method that does not rely on precomputed correction factors but rather one that is directly generalizable. This brief communication proposes a physics-derived modification based on 1D momentum theory. It considers the induction factor of the wind turbine to correct the local grid wind speed back to a freestream wind speed, as is standard in actuator disk modeling. The validity of the proposed modification is directly verified by comparing the model's power estimations to power curve calculations using the wind speed from a reference simulation without a turbine.

WRF version 4.2.1 was employed for this study. Note that this version already includes the bug fix to the Fitch parameterization reported by

We conducted a simulation of 5 d (21–25 January 2020), preceded by 24 h that was omitted as spin-up. A single turbine was placed in the domain, which was centered around the German Bight. To analyze the sensitivity of the power calculations to the turbine dimensions, two turbine types were used: the NREL 5 MW wind turbine

The Fitch parameterization calculates each turbine's power output

For

Comparison of the 22 MW

Figure

To demonstrate the scalability of the modification with high capacity densities and multiple turbines, five 22 MW wind turbines were placed within a single grid cell in WRF. While this scenario is rather unrealistic, it tests whether the proposed modification holds in extreme cases. Under the assumption that all five turbines operate in free wind conditions, the average power production per turbine calculated with the modification for multiple turbines (Fitch-mAIF) was compared with Fitch-o, Fitch-AIF and the reference (Fig.

Same as in Fig.

This exercise reveals that the underestimation of Fitch-o for this extreme dense case of turbines is about 23 %. This reduces to 18 % for Fitch-AIF but is almost eliminated when considering the number of turbines in the grid cell (Fitch-mAIF). The slightly positive

The Fitch wind farm parameterization implemented in WRF version 4.2.1 does not consider local induction effects and consequently underestimates power production of a single wind turbine in the dynamic region of the power curve. This issue is amplified for large turbines or when there are multiple turbines in one grid cell. To correct this underestimation, we propose a modification derived from 1D momentum theory. Instead of using the local wind speed of the grid cell, we use the wind-speed-dependent induction factor to estimate the freestream wind speed. Results from a simple analysis verify that the turbine's power curve is reproduced when including this modification, thus improving the power estimation of the wind turbine. Compared to measurement data, the power of a turbine will likely still differ from the simulated power by WRF, for example due to biases in the modeled wind speeds and power curves. A full-scale validation with measurement data is therefore considered important for future work. With the proposed induction correction, however, the model's negligence of the reduction in wind speed inside the grid cell due to the turbine's presence is not responsible for the difference anymore.

It is important to note that downstream wind speeds from wind farms modeled with WRF-Fitch have shown good agreement with measurements

Solving the induction correction becomes challenging when turbines in a grid cell have different dimensions. In such cases, the non-dimensional thrust coefficient needs to be converted to dimensional form, and variations in hub heights need to be considered. However, if precise yield calculations for individual turbines are desired, the mesoscale model may not be the most suitable choice due to its low spatial resolution. Regardless, it is worth considering whether increasing the model resolution and allocating more computing resources are worthwhile to mitigate unresolved inner-grid effects.

In short, for scenario calculations of wind farm yields with wind farms of not more than one single turbine per cell (e.g., calculations involving future turbine dimensions), we strongly recommend using the proposed modification as presented in this paper for more accurate yield and wind resource assessments.

The WRF code with the induction correction for the Fitch parameterization is available for download at

LV conceptualized the idea, BAMS implemented the correction in WRF and performed the simulations. MD initiated the associated research project and was thus involved in the funding acquisition and discussions. All authors contributed significantly to the writing and reviewing of the paper.

The contact author has declared that none of the authors has any competing interests.

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The results presented in this paper were derived within the framework of the X-Wakes (grant no. FKZ03EE3008) project. The X-Wakes project is funded by the German Federal Ministry for Economic Affairs and Climate Action (Bundesministerium für Wirtschaft und Klimaschutz – BMWK) due to a decision of the German Bundestag. The simulations were partly performed at the HPC cluster EDDY, located at the University of Oldenburg (Germany), and funded by the BMWK (grant no. FKZ0324005).

This paper was edited by Sukanta Basu and reviewed by Patrick Hawbecker and one anonymous referee.