In this paper, an analytical wake model with a double-Gaussian velocity
distribution is presented, improving on a similar formulation by
Analytical engineering wind farm models are low-fidelity approximations used
to simulate the performance of wind power systems. A wind farm model includes
both a model of the wind turbines and a model of the modifications to the
ambient flow induced by their wakes, together with their mutual interactions.
Analytical wake models, as opposed to high-fidelity computational fluid
dynamics (CFD) models, are simple, easy to implement, and computationally
inexpensive. In fact, they only simulate macroscopic average effects of wakes
and not their small scales or turbulent fluctuations. Engineering wake
models find applicability in all those cases that do not need to resolve
small spatial and fast temporal scales, such as the calculation of the power
production of a wind plant over a sufficiently long time horizon. Such models
are also extremely useful in optimization problems, where a large number of
simulations might be required before a solution is reached or where
calculations need to be performed on the fly in real time. Analytical wake
models are thus often utilized in wind farm layout planning and in the
emerging field of wind farm and wake control
Because of their indisputable usefulness, engineering wake models have been
extensively studied in the literature. The Jensen (PARK) formulation is one
of the most widely used wake models, to the extent that it is sometimes
considered the industry standard
All such models have been designed to faithfully represent the average flow
properties of the far-wake region. However, in the near wake (which is
usually defined as the region up to about 4 diameters (4 D) downstream of the
rotor disk), the models seem to lack accuracy. Nowadays, onshore wind farms
tend to be closely packed, and turbine spacing often reaches values close to
or even below 3 D
In this short note, a double-Gaussian wake model, based on Keane's model and with emphasis on near-wake flow behavior, is derived, calibrated, and
validated. The present formulation addresses and resolves some issues found
in the original implementation of
This paper is organized as follows. The derivation of the double-Gaussian
wake model is detailed in Sect.
The double-Gaussian wake model is derived in a similar
way to the Frandsen
At the downstream distance
Stream tube with nomenclature:
The conservation of momentum principle is now applied on the ansatz velocity
deficit distribution, using the amplitude function
If the wake velocity, defined in Eqs. (
The derived expressions for
In the previous section, following the conservation
of momentum, the shape of the double-Gaussian wake deficit has been defined
as a function of the Gaussian parameter
In previous work by
In the present work, the stream tube outlet is not assumed to be located at
the turbine rotor (
To derive
At the Betz stream tube outlet (
Visualization of the width of the double-Gaussian function
The remaining parameters,
To calibrate and validate the double-Gaussian wake
model, time-averaged flow measurements from an LES numerical solution have
been used. The CFD simulation replicates an experiment conducted with the
scaled
A complete digital copy of the experiment was developed with the LES
simulation framework developed by
The double-Gaussian model proposed in this work has
three tunable parameters:
In the original formulation by Keane,
The goal of model calibration is to ensure that the wind velocity profiles
match the reference data set as closely as possible. To this end, the squared
error between the wake model and CFD-computed wake profiles is minimized with
respect to the free parameters. This estimation problem was solved using the
Nelder–Mead simplex algorithm implemented in the MATLAB function
The identified parameters are presented in Table
Identified model parameters.
Figure
Normalized wind velocity profiles of the double-Gaussian model (solid blue line) compared to experimental measurements (black circles) and CFD simulations (red
The model exhibits good generality, as demonstrated by the good matching of the profiles at distances that were not used for model estimation. Especially in the near-wake region up to about 3 D, the placement of the Gaussian extrema appears to be in good agreement with the measured one.
The performance of the model clearly depends on the data used for its
calibration. Using reference data close to the turbine rotor is important for
accurately gauging the positions of the velocity profile extrema, while a
wider rage of distances leads to an improved expansion behavior. In the
present case, more data from the near-wake region (1.7 and 3 D) were
considered in the tuning process than in the far wake (6 D). This leads to a
slight overestimation of the velocity deficit at 9 D, which could be
attributed to an underestimation of the expansion slope
Figure
As a comparison, Fig.
Root mean square error between the reference CFD velocity deficit
data and the engineering wake models. Double-Gaussian (DG) wake model
identified using measurements at 1.7, 3, and 6 D: solid blue line with
This short paper presented an analytical double-Gaussian wake model. The proposed formulation corrects and improves a previously published model proposed by
The model was calibrated and validated using a set of time-averaged CFD
simulation results, which replicate wind tunnel experiments performed with a
scaled wind turbine in a boundary layer wind tunnel. Results show that the
model fits the reference data with good accuracy, especially in the near-wake
region where a single-Gaussian wake is unable to describe the typically
observed bimodal velocity profiles. In the far wake, a slight overestimation
of the wake deficit could be observed. It is speculated that this might be
due to the wake expansion gradient being slightly different in the near- and
far-wake regions. This claim, however, would need additional work to be
substantiated. The different shape of the wake in the near- and far-wake regions also suggests stitching the two models together, the double Gaussian being used in the near-wake region and the single Gaussian further downstream. This would avoid the need for a single tuning that has to cover such a long distance and different behaviors. Additional future work could extend the wake model
to include wake deflection, which could be done in a rather straightforward
manner by following
Equation (
Term
Finally,
A MATLAB implementation of the wake model and the data contained in this article can be obtained by contacting the authors.
JS conducted the main research work, AB implemented the correct model and CLB closely supervised the whole research. All three authors provided important input to this research work through discussions, feedback and by writing the paper.
The authors declare that they have no conflict of interest.
The authors express their gratitude to Jesse Wang and Filippo Campagnolo of the Technical University of Munich, who respectively provided the numerical and experimental wake measurements.
This research has been partially supported by the European Commission, H2020 Research Infrastructures (CL-Windcon (grant no. 727477)).This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.
This paper was edited by Gerard J. W. van Bussel and reviewed by Matthew J. Churchfield and one anonymous referee.