26 Oct 2021
26 Oct 2021
FLOWERS: An integral approach to engineering wake models
 ^{1}National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO 80401, USA
 ^{2}Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
 ^{3}Department of Engineering, Durham University, Durham DH1 3LE, UK
 ^{1}National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO 80401, USA
 ^{2}Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
 ^{3}Department of Engineering, Durham University, Durham DH1 3LE, UK
Abstract. Annual energy production (AEP) is often the objective function in wind plant layout optimization studies. The conventional method to compute AEP for a wind farm is to first evaluate power production for each wind direction and speed using either computational fluid dynamics simulations or engineering wake models. The AEP is then calculated by weightedaveraging (based on the wind rose at the wind farm site) the power produced across all wind directions. We propose a novel formulation for timeaveraged wake velocity that incorporates an analytical integral of a wake deficit model across every wind direction. This approach computes the average flow field more efficiently, and layout optimization is an obvious application to exploit this benefit. The clear advantage of this new approach is that the layout optimization produces solutions with comparable AEP performance yet is completed about 700 times faster. The analytical integral and the use of a Fourier expansion to express the wind speed and wind direction frequency create a more smooth solution space for the gradientbased optimizer to excel compared with the discrete nature of the existing weightedaveraging power calculation.
Michael LoCascio et al.
Status: final response (author comments only)

CC1: 'Comment on wes2021116', Mads Pedersen, 28 Oct 2021
Thank you for a very interesting paper. I really like the idea of fitting the wind rose with a fourier series to avoid the wake rays which is clearly seen when applying the tophat model for a limited number of wind directions.
I have a few comments, questions and corrections:
 Is it possible to include local flow conditions without a timeconsuming iterative approach?
 As I understand it, you will try to fix the power integral and allow a windspeed dependent cp.
Will this solution capture the fact that a annual wind distribution normally also includes wind speeds above rated with lower cp and is it possible to also allow a windspeed dependent ct  I assume the codes are implemented in python using numpy. The problem is that a python loop is so much slower than a numpy vector operation.
The time comparison is therefore only fair if the none of the codes contain extra python loops compared to the other, which could be vectorized.  line 48: 180 > 270
 section 3.1: You indicate that you are using local waked wind speed instead of freestream wind speed when scaling the Jensen deficit, is that correct?
 line 177: "FLOWERS is only computing the velocity through each turbine at a single point instead of an array of points on the rotor area.". How many points are evaluated with the Jensen model
 line 209: speed > direction?
 line 245: "One reason why the Jensen optimizer favors this type of solution is that wakeadded turbulence is included in the modeling framework, so maximizing the streamwise spacing of the turbines improves the wake recovery for downstream turbines."
You claim to have identical wake expansion factor, which I assume means a constant factor. This conflicts with improved wake recovery due to wakeadded turbulence.
As far as I can see, the Jensen simulation gives a interference mesh of wake "fingers", which results in many local optima, where the optimizer gets stuck as you also write in section 4.1  line 253: "corresponds to Case 3". It looks more as Case 2
 Is it correct understod that all 4x10 runs in figure 8, 9, 11 and 12 would end up with the same AEP gain (same optimized layout) if the optimizations were perfect?
Maybe the same 10 cases could be compared in the one figure for the five different approaches (Jensen_72, Jensen_360, FLOWERS, Gaus_360, Gaus_9) if it does not spoil your nice flow of the story

RC1: 'Comment on wes2021116', Anonymous Referee #1, 26 Nov 2021
GENERAL COMMENTS
This is a generally well written paper and presents an apparently new and interesting method for wind farm AEP calculations. I think that some of the content could be improved and my comments are below.SPECIFIC COMMENTS
Abstract.
What does gradientbased mean? This doesn't seem to be explained anywhere within the article.
Perhaps consider replacing the sentence:
"The analytical integral and the use of a Fourier expansion to express the wind speed and wind direction frequency create a more smooth solution space for the gradientbased optimizer to excel compared with the discrete nature of the existing weightedaveraging power calculation."
with somethng like
"The analytical integral and the use of a Fourier expansion to express the wind speed and wind direction frequency create a relatively smooth solution space for the gradientbased optimizer in comparison to the existing weightedaveraging power calculation."
It is not clear why the "weightedaveraging power calculation" is discrete.1. Introduction.
Should 'tophat' be 'tophat'?
Line 30. What are gradientfree algorithms?
Line 46. The term "nonzero" before wind direction seems inappropriate. Perhaps remove the term altogether or replace with 'discrete'.2. Mathematical formulation
Line 69. To "the streamwise and spanwise position" I think you should add "with respect to the wind direction theta', where theta' is the wind direction in the X,Y frame"
It seems that the following should be defined before equation (2)
x = r cos(thetatheta')
y = r sin(thetatheta')
Then y/x = tan(thetatheta')
However, on line 73 it is stated that y/x = tan(theta) which doesn't seem correct.
It would be worth putting this in a diagram for clarity, as in the attached figure.
Line 83. It should be clarified that the equation in this line comes directly from the equation in line 72, and represents the boundary of the wake velocity deficit. Again, this would benefit from a diagram.
Line 86 to 88. I don't think this sentence is really true. Surely the wake deficit is defined as the difference between the freestream velocity and the wake velocity.
Line 92 onwards at bottom of page 4. Does U_infinity(theta') imply there is only one inflow wind velocity for each direction? Where does the Weibull distribution fit into this scheme? Is there no integration over wind velocity in the integral?
Line 96. says "the product is a vector with length equal to the number of wind direction bins", however, this will have units of m/s so how can it be a number? Please clarify.
Line 100. for a given g(theta') and N.
Line 102. Taylor series to second order?
2.2 Annual Energy Production. It is not clear to me why the integral is intractable (line 121).3. AEP Comparison
Could the authors briefly state how AEP is calculated? Typically this would be done with a wind rose/Weibull distribution and a wind turbine power curve? Is this the case here?
I think the various approaches need to be defined clearly and consistent terminology used throughout the paper. For example
Conventional Jensen approach = Jensen wake + numerical integration
FLOWERS = analytical formula
Conventional Gaussian approach = Gaussian wake + numerical integration
but it seems sometimes the first is referred to as 'Jensen integration' (line 183) or 'conventional numerical integration method' (line 192) or 'numerical integration' (line 198).
3.2 Generalised case. Line 170. Is the wind rose specified by f(theta') ?
Figure 2. caption says AEP comparison, but it seems that it is the wind velocity that is shown.
It is stated that "wind direction bins B used in (c) is B = 72", but this is for a single wind direction as shown in subfigure (a), is that right? Does this mean the single wind direction is split into 72?
Lines 183, 187, 192, 198. Again use consistent terminology instead of "Jensen integration", "conventional method" ... etc.
3.3 Improving computational efficiency
Can you give an indication of the absolute computational times? Minutes, hours, days?
Figure 5. Please state what they values are normalised with respect to. Is it with respect to the cases with maximum resolution?
Line 225. Perhaps replace "There is no reason to use the extended Fourier series if it only increases the computational cost of the FLOWERS solution"
with
"There is no reason to use the extended Fourier series if it increases the computational cost of the FLOWERS solution with no associated accuracy benefit."4. Optimization Comparison
Lines 229 to 226. Again, please be consistent when stating optimizer names.
4.1 FLOWERS and Jensen
Figure 7. Are labels (a), (b) necessary?
Figure 7. It is a bit concerning that the two results are vastly different, but possibly due to fact that a tophat wake is used instead of the more realistic gaussian wake. It is more reassuring that the layouts in Figure 10 are more similar.
Also, the initial positions are marked. Surely the final optimised layout should be independent of the initial positions?
Line 264. Please define AEP gain.
Figures 8 onwards. Are labels (a), (b) necessary if these labels are not referred to? Perhaps put "B=, N=" in captions instead.Conclusions. As future work it would be of interest to validate the AEPs of the various approaches against a real wind farm AEP.
Figures 8, 9, 11, 12 don't actually indicate which method is actually closest to the true AEP of a real wind farm. 
RC2: 'Review of wes2021116', Paul van der Laan, 02 Dec 2021
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes2021116/wes2021116RC2supplement.pdf
Michael LoCascio et al.
Michael LoCascio et al.
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