the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Sep 2023
Dynamic wind farm flow control using free-vortex wake models
Maarten J. van den Broek
Marcus Becker
Benjamin Sanderse
Jan-Willem van Wingerden
Abstract. A novel dynamic economic model-predictive control strategy is presented that improves wind farm power production and reduces the additional demands of wake steering on yaw actuation when compared to an industry state-of-the-art reference controller. The novel controller takes a distributed approach to yaw control optimisation using a free-vortex wake model. An actuator-disc representation of the wind turbine is employed and adapted to the wind-farm scale by modelling secondary effects of wake steering and connecting individual turbines through a directed graph network. The economic model-predictive control problem is solved on a receding horizon using gradient-based optimisation, demonstrating sufficient performance for realising real-time control. The novel controller is tested in a large-eddy simulation environment and compared against a state-of-the-art look-up table approach based on steady-state model optimisation. Under realistic variations in wind direction and wind speed, the novel controller yields additional gains in power production during transients as well as a reduction in yaw actuator usage.
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Maarten J. van den Broek et al.
Status: final response (author comments only)
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RC1: 'Comment on wes-2023-119', Jaime Liew, 11 Oct 2023
Dear authors,
Thank you for the opportunity to review your paper. The work presents some exciting new capabilities for model-based wind farm control using free-vortex methods. The results demonstrate potential for improved wind farm performance through anticipating transients and accounting for propagation dynamics and secondary wake steering effects. I enjoyed reading through the details and believe the paper will be a strong contribution once the following points are addressed:
- Table 1: There have been some recent advancements in quantifying yaw control costs that indicate the parameters in Table 1 may need updating. In particular, the literature suggests beta_p may be lower than 3 (see Howland, Michael F., et al. "Influence of atmospheric conditions on the power production of utility-scale wind turbines in yaw misalignment." _Journal of Renewable and Sustainable Energy_ 12.6 (2020).) and beta_t closer to 2 in reality (see see Li, Zhaobin, and Xiaolei Yang. "Large-eddy simulation on the similarity between wakes of wind turbines with different yaw angles." _Journal of Fluid Mechanics_ 921 (2021): A11.). Additionally, beta_p and beta_t are not constants and depend on the thrust force (See Heck, Kirby S., Hannah M. Johlas, and Michael F. Howland. "Modelling the induction, thrust and power of a yaw-misaligned actuator disk." _Journal of Fluid Mechanics_ 959 (2023): A9.) and the heterogeneity of the wind field (See Liew, Jaime, Albert M. Urbán, and Søren Juhl Andersen. "Analytical model for the power–yaw sensitivity of wind turbines operating in full wake." _Wind Energy Science_ 5.1 (2020): 427-437.) It would strengthen the work to acknowledge these recent findings and discuss any implications on your modeled results.
- Line 110: The assumption of optimal induction is reasonable for your below-rated cases. For completeness, considering control actions above rated conditions would further generalize the approach and should be discussed or, ideally, implemented. This could be as simple as using a C_P-C_T look-up table.Â
-Figure 4: This figure shows multiple turbines with free-vortices, but you mention you are only modelling 1 vortex wake at a time. Some clarification on how you are modeling secondary wake effects with only a single wake in the free vortex model would help interpretation of the results. Are any simplifying assumptions made? How is $u_r$ determined for downstream turbines when you are only simulating a single wake in your free-vortex model?Â
- Line 130: you mention a simulation study used to determine a method for induced yaw angle but do not provide details of the results of this study. Consider showing some results to justify this method.
- Additional details on the graph network topology selection (e.g. how are the sector angle and length determined?) would improve reproducibility and clarity of the control design.
- Â Section 3: Additional details on the controller iterative scheme would improve reproducibility and clarity of the control approach. In particular, what measurements are used as inputs to the controller, and how (and when) are these measurements connected to the control algorithm? Are the measurements received by the controller incrementally as new data is received? What information exactly is shared between the independent optimisations? A more concise controller definition, and perhaps a pseudocode of the algorithm, would help clarify these points.Â
- Line 198: Just out of curiosity, what do you think causes the set point at the horizon to approach greedy control? I would expect it to approach a steady-state optimal for the wind farm.
- Section 4.5 There appears to be turbulence in the LES simulations but there is no mention of turbulence in the manuscript. Furthermore, realistic low-frequency wind variations are mentioned, but again, a mention of turbulence is absent. How does the controller account for and respond to turbulence? Expanding the analysis to include effects of turbulence would provide valuable insights into robustness. Even simple benchmarking studies on a few turbulence levels could indicate where the controller succeeds or struggles.
- In connection with the previous point, how are measurements incorporated, and how are measurement uncertainties dealt with?
- Figure 9: Why does FVW respond earlier than LUT? Is the preview you mention earlier implying that FVW receives the measurements earlier than LUT? What happens if LUT also receives (and responds to) the measurements with preview? Would this perhaps be a more fair comparison?
- Section 5.5 The transition from formulated closed-loop to open-loop implementation seemed unclear. The formulation in Section 3 and the illustration in Figure 6 appears to describe a closed-loop, but it is revealed here in Section 5.5 that the presented work is open-loop. Some clarification on this its implications would be helpful.
- Section 5.6 The computational speed results are promising, but more details of how the method scales would be more useful to the reader to gauge what scale of problem this method is suitable for. Further benchmarks on how the method scales with problem size would indicate limits and suitable conditions for real-time application. For example, a plot showing control horizon or wake length versus computational time.
Minor suggestions:
- Line 82: the subscripts 1, 2, and 3 have been used to refer to the three spatial dimensions, however you use x_1 and x_2 to refer to the start and end point of a vortex filament. To avoid confusion, perhaps use different subscripts for the start and end points, for example, x_s, and x_e.
- Line 137: change "we a method" to "we present a method"
- Figure 8: Please make the shaded regions more defined as they are quite faint on a black and white print out.
- Figure 10: Please define "E", and perhaps the other terms if they have not already.Â
I believe these suggestions can significantly strengthen the work and hope they are received in the constructive manner intended. I look forward to seeing an updated version addressing these points as the core ideas show exciting potential.Sincerely,Â
Jaime LiewCitation: https://doi.org/10.5194/wes-2023-119-RC1 -
RC2: 'Comment on wes-2023-119', Anonymous Referee #2, 12 Oct 2023
Good work, clearly presented. I concur with many of the comments made by Jaime Liew, and would just add a few more points:
Some readers would appreciate an explanation of the term 'economic' as applied to model predictive control.
Results for TTWF: FVW shows higher power gain than LUT, but with higher yaw activity too. If you make the LUT controller a bit more responsive (by reducing the dead-band for example), so that it matches the yaw travel of FVW, the power gain would presumably be improved, so this should be tested to see if the power gain remains lower than with FVW. When comparing a complicated new controller to an existing simple one, it's always important to be sure that simple tweaks to the existing controller can't do the job just as well. The results for HKN do suggest an improvement in both power gain and yaw travel with FVW, which is promising, although this is clearly sensitive to the low-frequency wind variations as the benefits vary significantly between the different cases, so I would say that a lot more simulations over different randomly selected conditions, or over a much longer period of historical site data, are required in order to give confidence that the additional complexity of the proposed method is really worthwhile.
I think it would be worth further investigation of the practicality of the proposed approach when applied to a large wind farm, both in terms of computation time, and also considering that (especially in low turbulence conditions), wakes may persist far enough downstream to affect a number of turbines, in a way which the free vortex wake method may not be able to capture accurately.
The paper makes no mention of turbine loading, although this is known to be an important issue for the practicality of wake steering. Some comments should be made to indicate how the proposed method could be extended to take loading into account.
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Citation: https://doi.org/10.5194/wes-2023-119-RC2 -
RC3: 'Comment on wes-2023-119', Michael Sinner, 19 Oct 2023
This paper presents a method for implementing model predictive control for farm-wide power maximization. Overall, I found the paper well presented, and I particularly appreciate the detailed analysis and insights on the results obtained. I have one larger comment and a handful of smaller comments to improve the paper.
My main comment pertains to the use of the lookup table using the current wind conditions. The FVW received preview wind direction information, which allows it to "preactuate" in anticipation of wind direction changes, whereas the reference LUT approach (as well as the greedy control) receive only the current wind condition. As the authors point out, the preview gives an advantage to FVW over the LUT approach when there are changing wind conditions. In Fig. 9, it appears that the main effect is that the FVW yaw signal leads the LUT signal during changing conditions. However, if the preview of wind direction is assumed available, couldn't this also be fed to the LUT (or greedy yaw controller, for that matter) instead of the "current" wind direction to overcome the latency in the yaw system? The precise amount to advance the wind direction signal to the LUT would likely be outside of the scope of this work, but looking at Fig. 9, about 100s of advance would likely help the LUT approach significantly (probably you can find a more precise number with the data you have from Fig. 9). I would be interested to see this case added as another "reference" case, to see whether the advantages for the FVW still persist over this "preview-enabled LUT" approach.
My smaller comments are as follows:
1. I feel that some of the terms used to describe the MPC problem are misleading. First, what makes this an "economic" MPC? This term is used several times, but to me, the MPC cost function (eq. 18) doesn't look especially "economic". Second, as I understand the method presented, there is no communication between the turbines. Each solves their own MPC problem without knowledge of the solutions of the other turbines. If that is correct, I would not describe this as a "distributed" MPC, as "distributed" usually implies communication between agents over a network. I'm not actually sure what the correct terminology for this set-up is (if I'm understanding it correctly); perhaps "isolated" or similar?
2. As I understand the conclusions of the paper, the key area of performance benefits for FVW over the LUT approach is in dynamic conditions. How often do these dynamic wind direction conditions actually occur, compared to steady conditions? Do they happen often enough to justify using FVW, especially given the possible underperformance (due to the finite prediction horizon length) during steady conditions?
3. I found Section 2.1 a bit difficult to follow. Various physics-based equations are given, then a generic model form (eq. 9), followed by more physical equations. Consider reordering this section for clarity.
4. I find the notation u_i(x_0, x_1, x_2) in equation 1. confusing. I understand that u_i is the velocity at point x_0, but it depends on x_1 and x_2 (via the r_j); is that correct? Although the notation "u_i(x_0, x_1, x_2)" may be correct, I would suggest using simply "u_i(x_0)", as I would find it clearer. Feel free to disagree with me here.
5. Eq. 8---is this also based on van Kuik? Are there any assumptions made there?
6. Lines 197--202. The authors mention that c_5, c_6, and c_7 are matched to the wind direction, which forces greedy (i.e. suboptimal in the infinite horizon) behavior at the end of the horizon. Why is this suboptimal behavior enforced? Instead, couldn't one enforce that c_5 = c_6 = c_7 = c_4, so as to reduce the spline dimension while allowing the turbine to maintain the offset at c_4? This may also not be optimal for the infinite horizon, but it seems to me a better approximation of optimality (without haven proven anything, admittedly).Â
7. Line 214: The use of N_c as a "control horizon" referring to the number of actions that will be taken before the MPC problem is solved again is not the usual definition of "control horizon" in MPC literature (usually, "control horizon" refers to a subset of the prediction horizon where the control actions are allowed to change, while the MPC problem is still resolved at every "real" time step). It may be worth clarifying this.
8. Throughout, the authors highlight the capability of modeling secondary steering. However, I don't see any clear evidence that this modeling is needed; in fact, during steady-state periods, it seems that the MPC approach is essentially equivalent to the LUT in terms of power produced. Is the point here that waked turbines can achieve a satisfactory wake deflection with a smaller offset (due to the effective offset produced by the upstream wake), so loads may be kept lower? Perhaps this could be clarified.
9. Fig. 14 and paragraph starting at 355: The authors point out the differences in yaw offsets compared to the LUT approach due to the length of the horizon in the steady-state segments. Are these leading to a noticeable loss in power using the FVW controller?
10. Table 2: The different simulations seem to have quite different results in terms of yaw travel comparing LUT to FVW. Is this simply due to random variation? Why are results so different in different cases? Could you run more simulations or a longer simulation of the yawing behavior without LES, just to provide a clearer idea of the yaw activity? This is perhaps outside of the scope of the present work, but the yaw results are a little too varied to be convincing as it stands.
11. Line 392: The authors mention that the FVW is occasionally leading to oscillating yaw offsets. Shouldn't the R term in the MPC cost function limit this, or couldn't it be tuned to limit this behavior? As I understand it, \psi(k_0 - 1) is the true previous offset applied.I noted the following typos and very minor corrections:
-- Line 81: Please provide a reference for the Biot-Savart law
-- Line 105: Please provide the full form of q_k, as it is not clear to the reader exactly what makes up the model state
-- I didn't see a wake deflection model---perhaps I missed this?
-- Line 137: "we a method" seems to be missing a word
-- Line 195: Are the splines turbine-specific? Or the same for all turbines? I do not quite understand the use of n_b. Is n_b the number of degrees of freedom in a single spline? Are there multiple "splines" for one turbine? I am not very familiar with the terminology around splines.
-- Lines 226--227. The statement about the meaning of Q and R is repeated in this paragraph.
-- Line 250. Why is the more strict yaw controller needed to facilitate a fair comparison? Aren't all controllers using the same yaw logic regardless?
-- Line 293. I believe "LES" is used before the acronym is defined.
-- Fig. 12 caption: "turbine" --> "turbines"
-- Line 421: "more optimal" is not very clearly defined, as formally, the value is either optimal or it is not. The authors could consider rephrasing this sentence.
-- Lines 450--455: I appreciate the discussion of computational speed, however, I found these sentences difficult to follow. Please consider rephrasing.Â
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Citation: https://doi.org/10.5194/wes-2023-119-RC3 - AC1: 'Comment on wes-2023-119', Maarten van den Broek, 21 Nov 2023
Maarten J. van den Broek et al.
Data sets
Simulation data and code accompanying the publication: Dynamic wind farm flow control using free-vortex wake models Maarten J. van den Broek https://doi.org/10.4121/50138917-cf01-4780-9d1d-443593b7e974
Maarten J. van den Broek et al.
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