the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Modelling of cup anemometry and dynamic overspeeding in average wind speed measurements
Abstract. Cup anemometers measure average wind speed in the atmosphere, and has been used for one and a half century by meteorologists. Within the last half century cup anemometers has been used extensively in wind energy to measure wind resources and performance of wind turbines. Meteorologists researched on cup anemometer behaviour and found dynamic overspeeding to be of an inherent and significant systematic error. The wind energy community has strong accuracy requirements for power performance measurements on wind turbines and this lead in the last two decades to new research on cup anemometer characteristics, which was taken to a new level with development of improved calibration procedures, cup anemometer calculation models and classification methods.
Research projects in wind energy demonstrated by field and wind tunnel measurements, that angular response was a significant contributor to uncertainty, and that dynamic overspeeding was a significant but less important contributor. Earlier research was mainly made on cup anemometers with hemispherical cups on long arms, and dynamic overspeeding was considered an inherent and high uncertainty on cup anemometers. Newer research on conical cups on short arms showed that zero and low overspeeding at low to medium turbulence intensities is present. Different cup anemometer calculation models were investigated in order to find derived overspeeding characteristics. The general and often used parabolic torque coefficient model show that zero overspeeding is present when the speed ratio roots of the torque coefficient curve go through the equilibrium speed ratio and zero. The twocup drag model is a special case of the parabolic torque coefficient model, but with the second root being reciprocal to the equilibrium speed ratio. The drag model always results in a positive maximum overspeeding in the order of 1.1 times the turbulence intensity squared. A linear torque coefficient results in maximum overspeeding levels equal to the turbulence intensity squared. Torque characteristics of a cup anemometer with hemispherical cups fits slightly well to the drag model, but a cup anemometer with conical cups do not fit to neither the drag model nor the parabolic model, but better to a partial linear model, and even better to an optimized torque model. Most accurate modelling of cup anemometer characteristics is at present made with the ACCUWIND model. This model uses tabulated torque coefficient and angular response data measured in wind tunnel. The ACCUWIND model is found in IEC wind turbine power performance standards, where it is used in a classification system for estimation of operational uncertainties. For an actual comparison of two cup anemometers, with hemispherical and conical cups respectively, the influence of dynamic overspeeding was found to be relatively low compared to angular response, but for conical cups it was specifically low.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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 Final revised paper
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Interactive discussion
Status: closed
 AC1: 'Comment on egusphere20231291', Troels Friis Pedersen, 06 Sep 2023

RC1: 'Comment on egusphere20231291', Anonymous Referee #2, 06 Sep 2023
Excelent work on cup anemometer overspeeding.
Some minor comments:
 Line 45. SanzAndres et al 2014. (Within References: SanzAndrés, A.; Pindado, S.; Sorribes, F. Mathematical analysis of the effect of the rotor geometry on cup
anemometer response. Sci. World J. 2014, Article ID 537813, DOI: 10.1155/2014/537813.) Line 679. "...mathematical formulae".
Citation: https://doi.org/10.5194/egusphere20231291RC1 
AC2: 'Reply on RC1', Troels Friis Pedersen, 19 Sep 2023
Reply on comments from Referee #2:
Ref#2: Excelent work on cup anemometer overspeeding.
Reply: Grateful and humble
Ref#2: Some minor comments:
 Line 45. SanzAndres et al 2014. (Within References: SanzAndrés, A.; Pindado, S.; Sorribes, F. Mathematical analysis of the effect of the rotor geometry on cup
anemometer response. Sci. World J. 2014, Article ID 537813, DOI: 10.1155/2014/537813.) Line 679. "...mathematical formulae".
Reply: Forgot coauthors. This is corrected according to referee. Misspelling is corrected
Citation: https://doi.org/10.5194/egusphere20231291AC2

AC2: 'Reply on RC1', Troels Friis Pedersen, 19 Sep 2023

RC2: 'Comment on egusphere20231291', Anonymous Referee #1, 08 Sep 2023
General comments:
The work is a significant analysis on different possible models to evaluate cup anemometers' overspeeding. The models are well explained and their performance when analyzing the maximum overspeeding or for a wind speed spectrum are correctly presented. Results, together with discussion about the step response and distance constant, provide helpful information for researchers who need to model cup anemometers.
Specific comments:
In chapter 8, it becomes slightly confusing to have the data calculated with the RIS∅ cup anemometer dimensions and rotor inertia, but then have results also compared with Thies. A discussion about the implications of this fact in the analysis would be appropiate.
Technical corrections:
146: English correction. "could not be explained alone on distant constant values" (could not be explained by the ditance constant values alone).
Figure captions should state with more detail the anemometer models.
191: Reference missing about the Pedersen research
The paper needs a through correction on equation referencing (many equation references appear as ()). See 349 and 491 as examples.
683: experienced. Maximum overspeeding (dot missing).
Citation: https://doi.org/10.5194/egusphere20231291RC2 
AC3: 'Reply on RC2', Troels Friis Pedersen, 20 Sep 2023
Reply to Referee #1
Ref#1: General comments:
The work is a significant analysis on different possible models to evaluate cup anemometers' overspeeding. The models are well explained and their performance when analyzing the maximum overspeeding or for a wind speed spectrum are correctly presented. Results, together with discussion about the step response and distance constant, provide helpful information for researchers who need to model cup anemometers.
Reply: Takes note of the positive comments
Ref#1: Specific comments:
In chapter 8, it becomes slightly confusing to have the data calculated with the RIS∅ cup anemometer dimensions and rotor inertia, but then have results also compared with Thies. A discussion about the implications of this fact in the analysis would be appropiate.
Reply: Admit that it might be sligtly confusing, and that a discussion on this will improve on understanding! Adding sentence: Risø and Thies are actual cup anemometers with individual dimensions and rotor inertia, but the Risø cup anemometer is the one that is interesting to optimize incrementally, and why the Risø properties is used.
Ref#1: Technical corrections:
146: English correction. "could not be explained alone on distant constant values" (could not be explained by the distance constant values alone).
Reply: Corrected as proposed
Ref#1: Figure captions should state with more detail the anemometer models.
Reply: Fig 6 added Thies and Risø cup anemometers, Fig 10 changed sentence from “Parabolic torque coefficient curves for …” to “Torque coefficient curves for parabolic model with equilibrium speed ratio , and slope at equilibrium speed ratio . Various values of _{ }as shown in the legend. Linear model in black and drag model in orange”, Fig 11 changed to “… Risø (red) and Thies (blue) are added”, Fig 12 changed to “Torque coefficient curves for partial linear model with various ratios. Linear model in black”, Fig 13 changed “coefficients” to “coefficient model”, Fig 17 changed from “…, Parabolic ranges, 0.280.32 light pink, 0.260.34 medium pink, and 0.240.36 pink. Risø red, Thies blue, data from Figure 6.” to “…, and parabolic ranges, 0.280.32 (beige light), 0.260.34 (beige medium), and 0.240.36 (beige dark). Added Risø (red), Thies (blue), data from Figure 6.”, Fig 18 changed to “Maximum overspeeding at sinusoidal wind of 8 m/s average wind speed for optimized torque coefficient curves from Figure 17, calculated with same dimensions and inertia as Risø. Added Risø (red) and Thies (blue, and with Thies properties).”, Fig 19 changed from “Overspeeding of Kaimal wind spectrum (u/v/w=1/0.8/0.5) at 8 m/s average wind speed. Curves include three parabolic model curves, two partial model curves, two optimized torque model curves, and Risø and Thies. ” to “Overspeeding for Kaimal wind spectrum (u/v/w=1/0.8/0.5) at 8 m/s average wind speed. Curves include: three parabolic model curves, drag (orange), linear (black), low Os (green), two partial model curves, ratio 1.2 (long dashed purple), ratio 0.8 (short dashed purple), two optimized torque model curves, 0.280.32 (beige light), 0.260.34 (beige medium), and Risø (red) and Thies (blue).”, Fig 20 changed from “Differences between Thies and Risø cup anemometers in field comparison and ACCUWIND calculation, with all influence parameters and with only torque” to “Differences between Thies and Risø cup anemometers from the field comparison in Figure 1, and with two ACCUWIND calculations, one with all influence parameters, and one where only torque is considered.”, Fig 21 changed from “ Speed ratio normalized (to torque coefficient curves for optimized torque with constants , , and three parabolic ranges (0.240.36, 0.260.34 and 0.280.32), and Risø and Thies torque coefficient curves normalized as well” to “Torque coefficient curves for optimized torque with constants , , and three parabolic ranges 0.240.36 (light beige), 0.260.34 (medium beige), 0.280.32 (dark beige), and added Risø (red) and Thies (blue) torque coefficient curves, normalized to speed ratio ”
Ref#1: 191: Reference missing about the Pedersen research
Reply: added “, (Dablberg et al. 2006),”
Ref#1: The paper needs a through correction on equation referencing (many equation references appear as ()). See 349 and 491 as examples.
Reply: Equation references are changed according to comments from Axel Albers:
349 equation() to equation(19)
442 () to equation(36)
489 equation () to equation(45)
491 equation() to equation(51)
493 equation() to equation(52)
497 equation() to equation(52)
In line 657 the sentence with missing equation: "The maximum overspeeding would be reduced with the drag ratio according to equation (), but this would cause a reduction in sensitivity (calibration gain), equation (22) and (10), which not is an advantage." is changed to: "Increasing the low drag coefficient by 10% would increase the drag ratio k by 10%, and reduce the equilibrium speed ratio by 8%, equation (23). The calibration gain would be increased by 8%, equation (10), because the rotor would run slower, and the maximum overspeeding would be reduced by less than 2%, equation (23) and (24). The maximum overspeeding would for further increase of the low drag coefficient converge towards the linear maximum overspeeding, turbulence intensity squared, and the drag model cannot provide a lower value for any drag ratio.
always has a radius of curvature of the torque coefficient curve to the wrong side, and will never become better in overspeeding than the linear model."
Ref#1: 683: experienced. Maximum overspeeding (dot missing).
Reply: dot inserted
Citation: https://doi.org/10.5194/egusphere20231291AC3

AC3: 'Reply on RC2', Troels Friis Pedersen, 20 Sep 2023
Interactive discussion
Status: closed
 AC1: 'Comment on egusphere20231291', Troels Friis Pedersen, 06 Sep 2023

RC1: 'Comment on egusphere20231291', Anonymous Referee #2, 06 Sep 2023
Excelent work on cup anemometer overspeeding.
Some minor comments:
 Line 45. SanzAndres et al 2014. (Within References: SanzAndrés, A.; Pindado, S.; Sorribes, F. Mathematical analysis of the effect of the rotor geometry on cup
anemometer response. Sci. World J. 2014, Article ID 537813, DOI: 10.1155/2014/537813.) Line 679. "...mathematical formulae".
Citation: https://doi.org/10.5194/egusphere20231291RC1 
AC2: 'Reply on RC1', Troels Friis Pedersen, 19 Sep 2023
Reply on comments from Referee #2:
Ref#2: Excelent work on cup anemometer overspeeding.
Reply: Grateful and humble
Ref#2: Some minor comments:
 Line 45. SanzAndres et al 2014. (Within References: SanzAndrés, A.; Pindado, S.; Sorribes, F. Mathematical analysis of the effect of the rotor geometry on cup
anemometer response. Sci. World J. 2014, Article ID 537813, DOI: 10.1155/2014/537813.) Line 679. "...mathematical formulae".
Reply: Forgot coauthors. This is corrected according to referee. Misspelling is corrected
Citation: https://doi.org/10.5194/egusphere20231291AC2

AC2: 'Reply on RC1', Troels Friis Pedersen, 19 Sep 2023

RC2: 'Comment on egusphere20231291', Anonymous Referee #1, 08 Sep 2023
General comments:
The work is a significant analysis on different possible models to evaluate cup anemometers' overspeeding. The models are well explained and their performance when analyzing the maximum overspeeding or for a wind speed spectrum are correctly presented. Results, together with discussion about the step response and distance constant, provide helpful information for researchers who need to model cup anemometers.
Specific comments:
In chapter 8, it becomes slightly confusing to have the data calculated with the RIS∅ cup anemometer dimensions and rotor inertia, but then have results also compared with Thies. A discussion about the implications of this fact in the analysis would be appropiate.
Technical corrections:
146: English correction. "could not be explained alone on distant constant values" (could not be explained by the ditance constant values alone).
Figure captions should state with more detail the anemometer models.
191: Reference missing about the Pedersen research
The paper needs a through correction on equation referencing (many equation references appear as ()). See 349 and 491 as examples.
683: experienced. Maximum overspeeding (dot missing).
Citation: https://doi.org/10.5194/egusphere20231291RC2 
AC3: 'Reply on RC2', Troels Friis Pedersen, 20 Sep 2023
Reply to Referee #1
Ref#1: General comments:
The work is a significant analysis on different possible models to evaluate cup anemometers' overspeeding. The models are well explained and their performance when analyzing the maximum overspeeding or for a wind speed spectrum are correctly presented. Results, together with discussion about the step response and distance constant, provide helpful information for researchers who need to model cup anemometers.
Reply: Takes note of the positive comments
Ref#1: Specific comments:
In chapter 8, it becomes slightly confusing to have the data calculated with the RIS∅ cup anemometer dimensions and rotor inertia, but then have results also compared with Thies. A discussion about the implications of this fact in the analysis would be appropiate.
Reply: Admit that it might be sligtly confusing, and that a discussion on this will improve on understanding! Adding sentence: Risø and Thies are actual cup anemometers with individual dimensions and rotor inertia, but the Risø cup anemometer is the one that is interesting to optimize incrementally, and why the Risø properties is used.
Ref#1: Technical corrections:
146: English correction. "could not be explained alone on distant constant values" (could not be explained by the distance constant values alone).
Reply: Corrected as proposed
Ref#1: Figure captions should state with more detail the anemometer models.
Reply: Fig 6 added Thies and Risø cup anemometers, Fig 10 changed sentence from “Parabolic torque coefficient curves for …” to “Torque coefficient curves for parabolic model with equilibrium speed ratio , and slope at equilibrium speed ratio . Various values of _{ }as shown in the legend. Linear model in black and drag model in orange”, Fig 11 changed to “… Risø (red) and Thies (blue) are added”, Fig 12 changed to “Torque coefficient curves for partial linear model with various ratios. Linear model in black”, Fig 13 changed “coefficients” to “coefficient model”, Fig 17 changed from “…, Parabolic ranges, 0.280.32 light pink, 0.260.34 medium pink, and 0.240.36 pink. Risø red, Thies blue, data from Figure 6.” to “…, and parabolic ranges, 0.280.32 (beige light), 0.260.34 (beige medium), and 0.240.36 (beige dark). Added Risø (red), Thies (blue), data from Figure 6.”, Fig 18 changed to “Maximum overspeeding at sinusoidal wind of 8 m/s average wind speed for optimized torque coefficient curves from Figure 17, calculated with same dimensions and inertia as Risø. Added Risø (red) and Thies (blue, and with Thies properties).”, Fig 19 changed from “Overspeeding of Kaimal wind spectrum (u/v/w=1/0.8/0.5) at 8 m/s average wind speed. Curves include three parabolic model curves, two partial model curves, two optimized torque model curves, and Risø and Thies. ” to “Overspeeding for Kaimal wind spectrum (u/v/w=1/0.8/0.5) at 8 m/s average wind speed. Curves include: three parabolic model curves, drag (orange), linear (black), low Os (green), two partial model curves, ratio 1.2 (long dashed purple), ratio 0.8 (short dashed purple), two optimized torque model curves, 0.280.32 (beige light), 0.260.34 (beige medium), and Risø (red) and Thies (blue).”, Fig 20 changed from “Differences between Thies and Risø cup anemometers in field comparison and ACCUWIND calculation, with all influence parameters and with only torque” to “Differences between Thies and Risø cup anemometers from the field comparison in Figure 1, and with two ACCUWIND calculations, one with all influence parameters, and one where only torque is considered.”, Fig 21 changed from “ Speed ratio normalized (to torque coefficient curves for optimized torque with constants , , and three parabolic ranges (0.240.36, 0.260.34 and 0.280.32), and Risø and Thies torque coefficient curves normalized as well” to “Torque coefficient curves for optimized torque with constants , , and three parabolic ranges 0.240.36 (light beige), 0.260.34 (medium beige), 0.280.32 (dark beige), and added Risø (red) and Thies (blue) torque coefficient curves, normalized to speed ratio ”
Ref#1: 191: Reference missing about the Pedersen research
Reply: added “, (Dablberg et al. 2006),”
Ref#1: The paper needs a through correction on equation referencing (many equation references appear as ()). See 349 and 491 as examples.
Reply: Equation references are changed according to comments from Axel Albers:
349 equation() to equation(19)
442 () to equation(36)
489 equation () to equation(45)
491 equation() to equation(51)
493 equation() to equation(52)
497 equation() to equation(52)
In line 657 the sentence with missing equation: "The maximum overspeeding would be reduced with the drag ratio according to equation (), but this would cause a reduction in sensitivity (calibration gain), equation (22) and (10), which not is an advantage." is changed to: "Increasing the low drag coefficient by 10% would increase the drag ratio k by 10%, and reduce the equilibrium speed ratio by 8%, equation (23). The calibration gain would be increased by 8%, equation (10), because the rotor would run slower, and the maximum overspeeding would be reduced by less than 2%, equation (23) and (24). The maximum overspeeding would for further increase of the low drag coefficient converge towards the linear maximum overspeeding, turbulence intensity squared, and the drag model cannot provide a lower value for any drag ratio.
always has a radius of curvature of the torque coefficient curve to the wrong side, and will never become better in overspeeding than the linear model."
Ref#1: 683: experienced. Maximum overspeeding (dot missing).
Reply: dot inserted
Citation: https://doi.org/10.5194/egusphere20231291AC3

AC3: 'Reply on RC2', Troels Friis Pedersen, 20 Sep 2023
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Troels Friis Pedersen
JanÅke Dahlberg
The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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