A novel spatial distribution method for wind farm parameterizations based on the Gaussian function

Abstract

Wind farm parameterizations are crucial for quantifying the wind-farm atmosphere interaction, where wind turbines are typically modeled as elevated momentum sinks and sources of turbulence kinetic energy (TKE). These quantities must be properly distributed to the mesoscale grid. Existing parameterizations use the single-column method. However, this method can easily lead to the errors of the spatial distribution of the sink and source, thereby impacting the accuracy of mesoscale flow simulations. To this end, we propose a multi-column spatial distribution method based on the Gaussian function. This method distributes the sink and source to multiple vertical grid columns based on the grid weights, which are analytically determined by integrating the two-dimensional Gaussian function over the mesoscale grid. We have applied this method to the classic Fitch model, proposed the improved Fitch-Gaussian model, and integrated it into the mesoscale Weather Research and Forecasting model. Using high-fidelity large eddy simulation as a benchmark, we compared the performance of the proposed method with the single-column method. The results show that the proposed method captures the spatial distribution of the sink and source more accurately, with a higher correlation coefficient and lower normalized root mean square error. Furthermore, the Fitch-Gaussian model better captures the overall spatial distribution patterns of velocity deficit and added TKE. Therefore, the proposed method is recommended for future mesoscale wind farm simulations, especially when the influence of the wind turbine rotor spans multiple mesoscale grid columns.

Figures (19)

• Figure 1: (a) Schematic of calculating the weight $W_{ij,n}$ of the mesoscale vertical grid column $(X_i,Y_j)$ based on the two-dimensional Gaussian function for wind turbine $T_n$, (b) the spatial distribution of $W_{ij,n}$ determined based on the proposed Gaussian-based multi-column method, and (c) the spatial distribution of $W_{ij,n}$ determined based on the conventional single-column method.
• Figure 2: Schematic of the wind farm layouts and their relative locations with the mesoscale grid ($\Delta$=1km). (a) Single-A, (b) Single-B, (c) Gap, and (d) Jump wind farms. (The wind turbine enclosed with a skyblue box is the reference wind turbine.)
• Figure 3: Schematic of the WRF-PBL and WRF-LES inner domain with the Gap wind farm. ($(x_{PBL}^r, y_{PBL}^r)$ and $(x_{LES}^r, y_{LES}^r)$ are the coordinates of the reference wind turbine in the WRF-PBL and WRF-LES inner domain, whose values can be found in table \ref{['tab2']}.)
• Figure 4: Comparison of the last-hour and spatial averaged vertical inflow profiles corresponding to the fully developed region of the WRF-LES framework and the inner domain of the WRF-PBL framework. (a) Potential temperature, (b) horizontal wind speed, (c) wind direction, and (d) turbulent kinetic energy. The top, bottom, and hub height of the wind turbine rotor are depicted with horizontal black lines for reference.
• Figure 5: Comparison of the hub-height spatial distribution of the total momentum sink $T$ for the Single-A and Single-B cases obtained from the WRF-LES and WRF-PBL frameworks