This paper presents a parametric wind direction rose model based on an ellipse and demonstrates how the model, which includes 3 parameters, can be fit to measured wind direction data for a variety of sites. There is little published on parametric wind direction rose models, so this is a novel contribution to the literature. When combined with wind speed distributions, the model could potentially serve as a standard wind rose definition for computing wake losses, lifetime loads from wakes, and wind farm control benefits for a wind farm, similar to how the Weibull distribution is used to model wind speed probabilities for annual energy production and fatigue load calculations for individual turbines. The parameters of the elliptical wind direction rose model could be used to standardize the characterization of wind direction distributions in the wind industry. They could also be varied to explore the sensitivity of wind plant performance, loads, and wind farm control strategies to different wind direction distributions.

The main comment I have is about how the elliptical parameterized wind rose can be made more useful for sites with more complex wind direction distributions. The presented parameterization works well for sites with unimodal wind roses or bimodal wind roses where the prevailing wind directions are in opposite directions, as shown in Fig. 3. However, similar to what is shown in Fig. 3e, there are many sites where the most common wind directions are not 180 degrees apart. To provide another example, in Fig. 3 of Bensason et al. 2021 (https://pubs.aip.org/aip/jrse/article/13/3/033303/285076/Evaluation-of-the-potential-for-wake-steering-for), the most common wind direction sectors are from the northwest and south. It would greatly strengthen the paper to discuss possible extensions of the elliptical wind rose model that could more accurately describe these types of wind direction distributions. For example, could you consider a linear combination of elliptical wind roses with different prevailing wind directions such that the sum of the distributions integrates to 1? This could be a nice solution if the different prevailing wind directions could be included as optimization parameters, rather than being identified manually. Of course the idea behind a parameterized wind rose model is to keep it relatively simple, but considering only wind roses with prevailing wind directions 180 degrees apart might be too much of a simplification for many sites. Among other issues, this could be important when estimating wake losses at a site with long distances between rows of turbines but close spacing in the perpendicular direction. Underrepresenting the likelihood of common wind directions aligned with the close turbine spacing in order to fit the wind rose model to the prevailing direction might cause wake losses to be significantly underestimated.

Comments:

1. Pg. 3, ln. 60: Is this equation only supposed to be valid when theta^tilde_1 and theta^tilde_2 are less than or equal to pi/2? If so, please clarify. Also, looking at Fig. 1b, how is y^tilde_+(x) defined for x_2 < x <= x_1? The segment area is no longer bounded by two lines like it is for x < x_2.
2. Pg. 4, ln. 82: "not be equally" -> "not equally"
3. Section 2.4: Scaling the wind direction probability by 1-f for pi/2 < theta_c,i < 3pi/2 and 1+f for 0 <= theta_c,i < pi/2 or 3pi/2 < theta_c,i <= 2pi causes a sharp discontinuity at theta_c,i = pi/2 and 3pi/2, which doesn’t seem very realistic. Would a smooth (e.g., linear) transition from 1-f to 1+f be more appropriate? One example would be scaling P_el by (1 + f - (2*f/pi)*theta_c,i) for 0 <= theta_c,i < pi. This way you would still get 1 - f for theta_c,i = pi, 1 for theta_c,i = pi/2 and 1 + f for theta_c,i = 0.
4. Section 2.5: Could you also optimize the prevailing wind direction theta_prev when fitting a wind rose to empirical data?
5. Pg. 6, equation after line 110: in the first two lines, it would clarify the equation if "i" were added as a subscript for P^dagger_g because this represents the probability of the specific bin "i".
6. Pg. 6, ln. 115: "The partial derivative del P^dagger_g / del a is readily obtained using del A_theta_1,theta_2 / del a…": To help the reader, it would be good to refer to the specific equations earlier in the text that show how these two partial derivatives are linked. This might require more equations to be numbered.
7. Section 3: It would be helpful to discuss the choice of bin sizes shown. What are typical wind direction bin widths for wind roses in the wind industry, for example for energy yield assessments or controls analysis?
8. Pg. 7, ln. 130: "the RMSE-scale is dependent on the number of wind direction bins." Couldn't the RMSE be normalized to account for the number of wind direction bins so it can be used to compare the goodness-of-fit for roses with different bin sizes?
9. Pg. 7, ln. 131: "Limitations of R^2 should be kept in mind" Please briefly discuss these limitations here.

GENERAL COMMENTS

The manuscript proposes a parametric model for the probability of observed wind directions, and it might be used for any circular probability distribution. The aim is to provide a smooth wind direction rose, suitable for optimization of wind farm layouts or advanced wind farm control.

The basic model takes the shape of an ellipse, and, to allow more flexibility, it folds part of the probability mass in half of the ellipse upon the opposite half. An expression for the area of an ellipse sector is presented and used to fit the parametric model to observed wind sector frequencies. The model-fitting principle is the minimization of the sum of squared errors. For this purpose, the author presents equations for derivatives of the objective function with respect to model parameters. The model does not yet include a directional variation of the wind speed distribution.

Gradient-based layout optimization algorithms will accept larger wake effects in sectors with low frequency of occurrence and thereby smaller contributions to annual energy production. If the input wind rose is too detailed, the algorithm's convergence may be slow, and the solution will be sensitive to random variations. Thus, models with smoother directional variation are needed for some purposes. On the other hand, the wind-rose simplification should not significantly alter the predicted energy production with or without wake correction. At most sites, the wind speed distribution depends on direction, so we risk that the energy production estimate changes after modifications of the wind rose.

An ellipse is symmetrical over both major and minor axes, so we might fold over either or both of them. Just remember that the rotation angle should be included as an optimization parameter if we choose to fold over both axes. Unfortunately, the fold-over procedure introduces discontinuities in the dictations along the minor axis, which might reintroduce the disadvantages of the raw wind rose.

The model is fitted by a raw wind rose with discrete sector statistics, but it might be more accurate to fit directly by data. The result seems to be a new sector-based distribution, but working with the underlying continuous distribution in optimization algorithms might be better.

The von Mises (vM) distribution is the classic model for circular statistics. Due to its unimodal distribution, it is rarely used in wind engineering, but the generalized von Mises distribution (GvM) supports an arbitrary number of modes. GvM models are not easy to fit to data, but Kim and SenGupta present a promising numerical method based on the maximum likelihood principle, see https://doi.org/10.1080/02664763.2020.1796938. The book "Directional Statistics" by Mardia and Jupp discusses more options, see https://onlinelibrary.wiley.com/doi/book/10.1002/9780470316979.

I once used a more straightforward approach, fitting Fourier splines to the observed sector frequencies and directional variations of the mean and cube of the wind speed. A low-pass filter in wave number domain provided flexible directional smoothing, and Weibull distributions for wind speed from different directions were derived by statistical moments of the wind speed.

I suspect that the new elliptical model offers too little flexibility for accurate wind farm production estimates. However, it might be useful for special purposes like the development of wind farm control strategies or fast approximate layout optimization.

SPECIFIC COMMENTS:

P1, line 15: I was puzzled by the expression “energy uplift obtained for a single candidate wind rose”. Try to reformulate for clarity.  

P2, line 36-48: The explanation of the eccentricity is not used in model formulation, so it might be left out.

Section 2.3: The multi-case equations in this paragraph are complex to read. Maybe you could simplify by using the Arg or Atan2 functions.