the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
On the power and control of a misaligned rotor – Beyond the cosine law
Abstract. We present a new model to estimate the performance of a wind turbine operating in misaligned conditions. The model is based on the classic momentum and liftingline theories, considering a misaligned rotor as a lifting wing of finite span, and accounts for the combined effects of both yaw and uptilt angles.
Improving on the classical empirical cosine law in widespread use, the new model reveals the dependency of power not only on the misalignment angle, but also on some rotor design parameters and – crucially – on the way a rotor is governed when it is yawed out of the wind. Additionally, the model also shows that a sheared inflow is responsible for the observed lack of symmetry for positive and negative misalignment angles. Notwithstanding its simplicity and insignificant computational cost, the new proposed approach is in excellent agreement with large eddy simulations (LES) and wind tunnel experiments.
Building on the new model, we derive the optimal control strategy for maximizing power on a misaligned rotor. Additionally, we maximize the total power of a cluster of two turbines by wake steering, improving on the solution based on the cosine law.
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Status: closed (peer review stopped)

RC1: 'Comment on wes2023133', Majid Bastankhah, 01 Dec 2023
"Review of “On the Power and Control of a Misaligned Rotor – Beyond the Cosine Law”
The paper presents a new analytical relationship to predict the variation of thrust and power coefficients with the yaw angle. This is an important and timely topic, as an accurate estimation of these two dimensionless quantities is of great importance in implementing wake steering. The power coefficient directly determines the amount of power loss on the turbine that is yawed, and the thrust coefficient is an important parameter in engineering wake models used to estimate the amount of wake deflection in yawed conditions. Two assumptions are made to develop the model: (i) the relative flow angle is small, which is valid for hightip speed ratios, and (ii) the induction factor is uniformly distributed across the rotor disk. The latter assumption is supported by a discussion on the significance of the nonuniformity for misaligned rotors. Model predictions are compared with both LES and wind tunnel data. Overall, the paper is wellwritten. The work is timely and of interest to the wind energy community. An interesting coordinate transformation was performed to group both yaw and tilt effects into one combined angle (called the misalignment angle) in a bespoke coordinate system. It is great to see that all coordinate systems and parameters are very carefully and clearly defined. Detailed discussions were made on the impact of inflow shear and its effect on the asymmetrical distribution of thrust and power. I believe this work will have a high impact, especially in the field of wake steering. However, before publication, I would like to ask the authors to address the following comments (sorted by the line number) to improve the clarity and completeness of their work:
Line 44: The paragraph is too short. It can be merged with the following one.
Line 84: Verb missing in “… and Sect. 3 for its validation…”
Line 112 and 117: Line 112 says that Mu is always positive, but in line 117, it has a negative value.
Line 116 and Figure 2: I agree with the argument that different distributions of yaw and tilt angles produce a similar wake in the II plane if their misalignment angle is the same. However, this is only valid if we neglect ground effect. Cases with a large tilt angle may have a strong interaction with the ground. This can be clarified here.
Line 128: The radial and azimuthal coordinates can be shown in Figure 1d.
Line 125: A brief discussion on the relevance of linear shear in comparison to widely used power law or logarithmic profiles can be mentioned here. In other words, you can better justify why linear shear was used instead of those profiles.
Line 150: “r” is first defined in 128 and later in 150 with apparently two different definitions.
Line 199: I am not sure if I understand what 0P means here. Some clarification in the text would be useful.
Figure 4: I think the angle shown in the figure should be mu instead of gamma.
Line 207: Authors can better justify the validity of the small inflow angle approximation. For instance, the value of phi for the blade tip of a turbine with a TSR of 8 can be reported here.
Line 210: Typo: “we” and “list” should be replaced with “be” and “lift”.
Line 210: I believe the assumption of C_L=C_L,alpha*alpha is only valid for a symmetrical aerofoil. This can be clarified in the text.
Effect of tilt angle: It is good to consider the uptilt angle effect, but it would be useful to show how big the effect is on C_T and C_P. Arguably, the effect on turbine performance and its wake should not be very significant. With that in mind, I suggest mentioning that the developed model will also be very useful for tilt angle control, which could become popular in the future generation of wind turbines, especially in floating turbines.
Line 315: It is useful to report the difference between the values of C_D and C_L,alpha obtained from the optimization study with the typical values reported for the blade aerofoil in 2D studies.
Section 4.1: The discussion in this section is interesting. One thing that I however struggled to understand (perhaps I’m missing something here) is that Figure 15d shows that the power loss for the first turbine is more significant when the optimal model is used. However, this contrasts with what is mentioned in lines 439 and 488.
Section 4.1 C_T effect: The manuscript points out that the thrust coefficient for the optimal model used in section 4.1 is higher for yawed cases compared with the standard approach. It is mentioned that C_T has an effect on the wake felt downstream. I agree with the authors, but I suggest elaborating on this more. As we know, C_T has two effects on the wake: (i) it increases the streamwise velocity deficit and (ii) also increases the amount of wake deflection. These two have counteracting effects on the downwind turbine. It is interesting to understand this effect in greater detail, at least discussing it in the paper.
Missing references: I appreciate that the authors did a thorough literature review, but some relevant references are missing. For instance, the streamtube model in section 2.5 and finding the wake spanwise velocity in the farm wake based on lifting theory are highly relevant to Shapiro et al. (2018). For instance, eq 14 in this manuscript could be compared with equation 2.13 in that paper.
Heck et al. (2022): It is true that your developed model improving the stateoftheart by including tilt and shear effects in greater details. However, I believe it is still informative for the reader to show whether your model, compared with the one proposed in Heck et al., provides similar predictions if they are used for similar operating conditions.
References:
Shapiro, Carl R., Dennice F. Gayme, and Charles Meneveau. "Modelling yawed wind turbine wakes: a lifting line approach." Journal of Fluid Mechanics 841 (2018): R1.
Citation: https://doi.org/10.5194/wes2023133RC1  RC2: 'Comment on wes2023133', Anonymous Referee #2, 07 Dec 2023
Status: closed (peer review stopped)

RC1: 'Comment on wes2023133', Majid Bastankhah, 01 Dec 2023
"Review of “On the Power and Control of a Misaligned Rotor – Beyond the Cosine Law”
The paper presents a new analytical relationship to predict the variation of thrust and power coefficients with the yaw angle. This is an important and timely topic, as an accurate estimation of these two dimensionless quantities is of great importance in implementing wake steering. The power coefficient directly determines the amount of power loss on the turbine that is yawed, and the thrust coefficient is an important parameter in engineering wake models used to estimate the amount of wake deflection in yawed conditions. Two assumptions are made to develop the model: (i) the relative flow angle is small, which is valid for hightip speed ratios, and (ii) the induction factor is uniformly distributed across the rotor disk. The latter assumption is supported by a discussion on the significance of the nonuniformity for misaligned rotors. Model predictions are compared with both LES and wind tunnel data. Overall, the paper is wellwritten. The work is timely and of interest to the wind energy community. An interesting coordinate transformation was performed to group both yaw and tilt effects into one combined angle (called the misalignment angle) in a bespoke coordinate system. It is great to see that all coordinate systems and parameters are very carefully and clearly defined. Detailed discussions were made on the impact of inflow shear and its effect on the asymmetrical distribution of thrust and power. I believe this work will have a high impact, especially in the field of wake steering. However, before publication, I would like to ask the authors to address the following comments (sorted by the line number) to improve the clarity and completeness of their work:
Line 44: The paragraph is too short. It can be merged with the following one.
Line 84: Verb missing in “… and Sect. 3 for its validation…”
Line 112 and 117: Line 112 says that Mu is always positive, but in line 117, it has a negative value.
Line 116 and Figure 2: I agree with the argument that different distributions of yaw and tilt angles produce a similar wake in the II plane if their misalignment angle is the same. However, this is only valid if we neglect ground effect. Cases with a large tilt angle may have a strong interaction with the ground. This can be clarified here.
Line 128: The radial and azimuthal coordinates can be shown in Figure 1d.
Line 125: A brief discussion on the relevance of linear shear in comparison to widely used power law or logarithmic profiles can be mentioned here. In other words, you can better justify why linear shear was used instead of those profiles.
Line 150: “r” is first defined in 128 and later in 150 with apparently two different definitions.
Line 199: I am not sure if I understand what 0P means here. Some clarification in the text would be useful.
Figure 4: I think the angle shown in the figure should be mu instead of gamma.
Line 207: Authors can better justify the validity of the small inflow angle approximation. For instance, the value of phi for the blade tip of a turbine with a TSR of 8 can be reported here.
Line 210: Typo: “we” and “list” should be replaced with “be” and “lift”.
Line 210: I believe the assumption of C_L=C_L,alpha*alpha is only valid for a symmetrical aerofoil. This can be clarified in the text.
Effect of tilt angle: It is good to consider the uptilt angle effect, but it would be useful to show how big the effect is on C_T and C_P. Arguably, the effect on turbine performance and its wake should not be very significant. With that in mind, I suggest mentioning that the developed model will also be very useful for tilt angle control, which could become popular in the future generation of wind turbines, especially in floating turbines.
Line 315: It is useful to report the difference between the values of C_D and C_L,alpha obtained from the optimization study with the typical values reported for the blade aerofoil in 2D studies.
Section 4.1: The discussion in this section is interesting. One thing that I however struggled to understand (perhaps I’m missing something here) is that Figure 15d shows that the power loss for the first turbine is more significant when the optimal model is used. However, this contrasts with what is mentioned in lines 439 and 488.
Section 4.1 C_T effect: The manuscript points out that the thrust coefficient for the optimal model used in section 4.1 is higher for yawed cases compared with the standard approach. It is mentioned that C_T has an effect on the wake felt downstream. I agree with the authors, but I suggest elaborating on this more. As we know, C_T has two effects on the wake: (i) it increases the streamwise velocity deficit and (ii) also increases the amount of wake deflection. These two have counteracting effects on the downwind turbine. It is interesting to understand this effect in greater detail, at least discussing it in the paper.
Missing references: I appreciate that the authors did a thorough literature review, but some relevant references are missing. For instance, the streamtube model in section 2.5 and finding the wake spanwise velocity in the farm wake based on lifting theory are highly relevant to Shapiro et al. (2018). For instance, eq 14 in this manuscript could be compared with equation 2.13 in that paper.
Heck et al. (2022): It is true that your developed model improving the stateoftheart by including tilt and shear effects in greater details. However, I believe it is still informative for the reader to show whether your model, compared with the one proposed in Heck et al., provides similar predictions if they are used for similar operating conditions.
References:
Shapiro, Carl R., Dennice F. Gayme, and Charles Meneveau. "Modelling yawed wind turbine wakes: a lifting line approach." Journal of Fluid Mechanics 841 (2018): R1.
Citation: https://doi.org/10.5194/wes2023133RC1  RC2: 'Comment on wes2023133', Anonymous Referee #2, 07 Dec 2023
Data sets
On the power and control of a misaligned rotor – Beyond the cosine law S. Tamaro, F. Campagnolo, C. L. Bottasso https://zenodo.org/record/8388901
Model code and software
On the power and control of a misaligned rotor – Beyond the cosine law S. Tamaro, F. Campagnolo, C. L. Bottasso https://zenodo.org/record/8388901
Interactive computing environment
On the power and control of a misaligned rotor – Beyond the cosine law S. Tamaro, F. Campagnolo, C. L. Bottasso https://mybinder.org/v2/gh/sTamaroTum/Beyond_the_cosine_law/main
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