General Comments

The use of engineering wake models, and hence their development, is still of key interest to the wind industry. Over recent years, the individual building blocks of such models have been rightfully challenged and revised. This paper presents a rigorous exploration of one key aspect of such models that has received comparatively little attention, and in so doing derives a analytic solution to the Rotor Averaged Wind Speed problem that may significantly improve the computational speed of such models.

The mathematical derivation is clear, well linked to the physics of the problem, and relies on deep mathematical insights.

Specific Comments

• The wake model of Bastankhah and Porté-Agel is used extensively to demonstrate the solution, however the solution would be applicable to a wide range of other wake models in which the Gaussian profile is used. The wider applicability of this result should be more clearly stated in section 2.1.
• Rather than “Rotor Average Windspeed”, many wake models use Root Mean Squared (RMS) speed or “Root Mean Cubed” (RMS) in calculation of thrust or power respectively. Assuming this method cannot be readily extended to RMS and RMC, a note to this restriction should be made in the text.
• Many wake models only use the “nacelle wind speed” (i.e. no rotor averaging) in order to reduce computational cost. I would recommend:
  1. In the abstract “These interactions are typically quantified for each turbine by evaluating its rotor-averaged wind speed” be amended,
  2. The nacelle point wind speed be included on the axes in figure 2 to highlight the benefit or rotor average wind speed over single point wind speed.

The following 2 comments relate to all “rotor average wind speed” methods, but should also be considered in the text:

• The impact of rotor induction perturbing the inflow profile (i.e. the rotor average speed the rotor experiences could be different from that calculated here)
• The impact of the blade geometry (i.e. in a real turbine, the wind-speed at the nacelle is much less important than the wind speed at ~2/3 of the blade length).

Technical Points:

• Line 23: Jensen 1983 also proposed a “Cosine-bell” profile.
• Figure 1: this figure is not that clear given the number of measurements that must be shown. Perhaps a set of orthographic views would be clearer?
• Line 121: “solution” (end of line) should be “approximation” or “approximate solution”.
• Line 136: This is the first introduction of Kappa in the text and its importance and meaning are lost. Please define kappa after Eqn. 8 or 9 (as a numbered display equation), and include a short description of it’s physical meaning (i.e. “equation 8 is valid for low values of kappa. Kappa is high if…”).
• Line 247 to end of page: “the number of turbines with non-negligible deficits”… In large windfarms, the sum of a large number of upstream “negligible” wakes becomes extremely significant. It is not safe to “neglect” the large number of up-stream turbines just because each one has a small impact, as this results in significant deltas in total windfarm power.

Overall Thoughts

The authors derive an analytical expression for the rotor-averaged wake velocity deficit downstream of a wind turbine, building on their previous work to now include non-axisymmetric wakes due to yaw misalignment or wind veer. The wake model predictions from this method are compared to those of the standard approach, which numerically integrates the velocity deficit at a set of discrete points across the rotor, and show excellent agreement.

The presented work is interesting and demonstrates mathematical rigor, but my main concern is demonstrating whether this model improves on the numerical averaging technique. I think the scope of this journal paper needs to be expanded to provide evidence that the work is advantageous over current methods and a convincing contribution to the literature.

Major Comments

There seem to be two ideas that motivate the derivation of this analytical model. First, according to the introduction (line 36), “uncertainties can arise from the number, distribution, and averaging weights of the control points” used in the numerical integration process. Second, according to the discussion section (line 282), there is a use case for a differentiable wake velocity deficit model to obtain gradients for rotor-averaged wind speed with respect to turbine positions and operating parameters, which can be applied to optimization problems such as yaw control or plant layout.

Regarding the first point, I do not think this paper adequately explains how the proposed method addresses these uncertainties. Section 3 is titled “Validation,” but this is really a comparison between two low-fidelity models, which I would argue is more of a verification or benchmarking process than a validation process. Without a comparison to, say, large eddy simulation results or wind tunnel experiments, how can we conclude that this model is an accurate prediction of the rotor-averaged velocity? The paper does establish good agreement between the proposed analytical method and the standard numerical approach, but it cannot make the case that it improves on the predictions of the numerical integration method.

I think the authors could argue that this proposed analytical method agrees very well with current practices, and it is a more practical/easy to use model in design applications. One reason could be reduced computational cost, as Referee #1 states, but unfortunately the authors do not discuss how this compares between their method and the numerical approach. I suspect from the complexity of the final expressions that this cost is not trivial.

Regarding the second point, the authors do argue that because the model is an analytical expression of the rotor-averaged velocity, it is differentiable and can be applied to these optimization problems more easily than the numerical method. However, this paper does not derive those expressions, and I wonder how manageable the analytical formulation of that gradient would be when you consider the derivative of the proposed model, plus the wake deflection model used to obtain the wake centerline deflection, and then the expression for turbine power production. Other numerical approaches exist (i.e., algorithmic differentiation) that can calculate these derivatives for the numerical approach, so the presence of the analytical derivatives of the wake model does not automatically translate to superior optimization performance.

So, I think the authors need to clearly establish the contribution of this analytical formulation. If the objective is to reduce the error of the rotor-averaging process by evaluating an analytical integral instead of a numerical integral, then I think we need to see the velocity predictions compared against proper validation data. Or, if the objective is to improve performance in an optimization application, then I’d like to see the proposed method applied to a case study where the benefit of the differentiable expression is clear, or at least a comparison of computation time between the methods.

Minor Comments

• Line 93: The important of accounting for wind veer is discussed in this paper, but what about shear? Both have a significant impact on power production (https://onlinelibrary.wiley.com/doi/full/10.1002/we.2917) and contribute to more complex wakes (https://iopscience.iop.org/article/10.1088/1742-6596/753/5/052004).
• Line 205: Can all three of these superposition methods be defined for the reader?
• Figures 2 and 3: A legend here indicating that the lines are the analytical model and the markers are the numerical model would help with readability.
• Also Figure 3: I don’t think the “no yaw” curves here are necessary since no comparisons between the models are done with this data and it adds clutter to the image.
• Section 3.2: Can power production be defined here? It is not explicitly stated to the reader how power depends on wind speed (or yaw angle).
• Line 272: It is discussed here that the simplification in the derivation that the considered turbine is normal to the free-stream flow is negligible. I am wondering how the calculations for the numerical method are performed in this case—is the rotor plane taken at the yaw-misaligned angle, or is it making the same simplification as the derived analytical expression?
• Line 284: Not to be nitpicky, but I think this is an oversimplification of these control/design optimization problems. The analytical formulation of the gradient of the rotor-averaged wind speed with respect to some design variables of the upstream turbine could be used in a gradient-based approach to solve the optimization problem. However, I don’t think it’s fair to say that it would reduce the problem to a simple root-finding problem, or that it would be able to find the global optimum.