the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Feb 2026
Virtual sensing for strain estimation in wind turbine support structures based on a single accelerometer
Abstract. This paper introduces a novel model-based approach for virtual sensing of wind turbine support structures for full-field strain estimation using a single DC-capable accelerometer. It enables displacement and strain estimation in the quasi-static frequency band, which accounts for a large proportion of accumulated fatigue damage in offshore wind turbine support structures, while saving costs by relying solely on accelerations from a single accelerometer as input. The introduced method extends the modal decomposition and expansion by using displacement estimations based on tilt-error compensated acceleration time series, utilising the tower's static bending line. It is applied here in two validation case studies: a small-scale laboratory experiment and a full-scale offshore wind turbine. In both cases, the estimated strain is validated against strain measurements conducted at various locations along the structure. The results show excellent agreement between the estimated and measured strains for both case studies. In the laboratory experiment, both displacements and strains are estimated accurately with errors below 2.2 % and 1.2 %, respectively. For the offshore wind turbine, the damage equivalent loads at the tower can be estimated with a maximum error of 21 % in the worst case and 6 % in the best case. The presented approach offers an improvement over established methods for strain estimation, achieving similar accuracy with fewer sensors, resulting in a low-maintenance load monitoring.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: open (until 20 Mar 2026)
- CC1: 'Comment on wes-2026-5', Xin Feng, 09 Mar 2026 reply
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RC1: 'Comment on wes-2026-5', Anonymous Referee #1, 12 Mar 2026
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In their manuscript, the authors present a method for virtual sensing in wind turbine support structures that relies on a single biaxial DC capable accelerometer. The approach allows for displacement and strain estimation in the low frequence range that includes the quasi-static contribution as well as the one from the lower structural modes. The key idea in the method is to compensate the measured acceleration for the tilt error. The method is validated first using data from a laboratory test with excitation at a known location and next actual data from an actual off-shore wind turbine. The performance of the method is quantified through the error in damage equivalent strains as well as the normalized root mean square error.
The manuscript addresses a highly relevant problem and proposes a novel method with potential for industrial application. The method is clearly described, the assumptions properly formulated and the validation by lab and field data convincing. I would recommend a minor revision of the manuscript where the authors address the following comments:
- The method requires a single biaxial and DC capable accelerometer. I would suggest to also state it in this way in the title of the paper.
- Line 79. Please specify that the accelerometer is capable of measuring down to DC.
- Lines 95-98. Please reformulate, the structure of these two sentences is grammatically incorrect.
- Lines 114-115. The present formulation suggests the term Ritz vector is proposed in the quoted reference. Please adapt.
- Equation (3). Please indicate whether m would depend on the considered frequency band.
- Table 2. Are the MAC values determined based on the accelerometer data only? Please specify.
- Line 403. Why is the conventional MAC value not used to compare the measured and predicted mode shape? Please explain the S2MAC as this is not a conventionally used metric and indicate why it has been used here.
- Caption of table 8 contains a repeated article “the” in front of eigenfrequencies.
- Line 440. Please specify that the accelerometer needs to be biaxial and DC capable.
Citation: https://doi.org/10.5194/wes-2026-5-RC1 -
RC2: 'Comment on wes-2026-5', Anonymous Referee #2, 16 Mar 2026
reply
The introduction to the work, and the state of the art is complete and without clear error. I appreciate the authors’ effort and care.
I also appreciate the effort to make the code and data open access.
Overall the work is clear, the ambitions and methods of the authors are clear and in my opinion fairly novel. While there is still much to do to truly validate this technology, the authors are aware of this and do reflect it in their conclusions. It is honest and fair, in its shortcomings and objectives.
I’ve written down my comments as I read the document, some comments were later addressed by the authors, I’ll use (UPDATE) to highlight comments that were updated as I continued reading the document.
Bigger comments:
- It is maybe nitpicking but I would argue that the authors are using ‘transmissibility’ functions T(omega) to compute the virtual sensor, rather than transfer functions H(omega). A transfer function is a system property and independent of the load, it provides the relation between inputs and outputs, respectively forces and responses.
Opposingly, Transmissibility functions relate responses to other responses (even if they are different physical quantities, like strain and displacement) AND transmissibility functions depend on the localization of the forces.
In my reading, I would pose that the authors are using transmissibility functions. Fundamentally this doesn’t change anything about the methods, but it might be relevant to consider for clarity to not use the term transfer function and the typical symbol H. - Interesting the authors do not introduce a yaw transformation at any point. This is somewhat surprising as the thrust load is aligned with the nacelle. Meaning that one could consider using a different transfer/transmissibility function in FA and SS. Don’t get me wrong, there is beauty in simplicity not doing a yaw transformation.
- Along with comment 2 I think the authors should give a way forward when wave loads become more present, they are a bit trickier than thrust load. So it would be appreciated if the authors spent a little bit of time discussing expected challenges with those. A similar comment applies to the accuracy of this method when looking at submerged sensors. Sensors below the ‘impact point’ of the waves might struggle more.
(UPDATE: this is later mentioned in the Benefits and limitations, it is ok for me like this, but I also wouldn’t mind a more extensive discussion)
Minor comments:
1) “By design, this approach does not allow for compensation of alignment errors in the yaw direction.”
- The use of “By design” implies that this was a deliberate choice to not compensate the yaw misalignment. While in reality it simply is not possible with the given setup, please remove the ‘by design’.
2 ) Eq 15 – 17: In these equations it is IMO important to flag that the mode shape matrix is truncated (i.e. not all infinite modes are preserved). Moreover, it is also important to flag this is a Real mode shape matrix, else you would have complex time domain signals, while in theory the mode shape matrix is Complex.
These are subtle things but do impose some (albeit minor) restrictions/limitations to the method and it is relevant to flag these ‘shortcuts’.
3) The authors ambition a virtual sensing strategy, and particularly focus their attention on the quasi static content. This is a fair ‘focus’ but one should be attentive that (l391) filtering everything beyond 1Hz is a bit harsh, especially for smaller turbines I would not recommend to employ such a stringent lowpass filter if a full fatigue picture is to be preserved. It is ok within the context of this paper, but it should be put to the readers’ attention that this is not offering the full picture. I’m particularly concerned about how this handles very sudden events, it might ‘clip’ the peaks of the strains and thus underestimate fatigue loads.
4) Table 8 Can the authors motivate the use of the S2MAC over the classic MAC (as was done in Table 2)? Is this motivated by the fact that MAC values tend to be very high for ‘sparse’ setups?
5) (l. 411) “It is assumed for this proof of concept that the dominant load in the quasi-static frequencyrange is the thrust load on the rotor.”
- > to avoid ambiguity (dominant =\= only), do the authors mean it is the ONLY load? -> Or were waves considered?
6) (l 419) The use of the high-pass filter implies that very low frequency loads, including the static loads, are ignored. How do the authors plan to handle the very low frequency variations in load (eg. Diurnal cycles) . Something that can be picked up readily with a strain gauge. Arguably you could say the method is no longer ‘DC-capable’
(UPDATE; the authors raise this sufficiently in their conclusion, OK for me)
7) Figure 17, I understand the authors have some requirement to normalize the scale, but it would be clear if the first mode is clearly indicated (not by number but just as a line). What are the spikes to the right of the black box? And how does the black box relate to the first mode (e.g. 25%?)
Can the orientation of the turbine at that time (nacelle position/yaw) and wind speed be indicated for that record?
8) Figure 18, is the settling of the error in the beginning of the timeseries linked to the high pass frequency?
9) Table 10, is nice but is only shown for m=3, potentially the results are better/worse for m=5, it would be interesting to also see the results for this other slope that focuses more on the few large cycles in the record.
Citation: https://doi.org/10.5194/wes-2026-5-RC2 - It is maybe nitpicking but I would argue that the authors are using ‘transmissibility’ functions T(omega) to compute the virtual sensor, rather than transfer functions H(omega). A transfer function is a system property and independent of the load, it provides the relation between inputs and outputs, respectively forces and responses.
Data sets
A benchmark structure for virtual sensing on tower structures Jonathan Thurn, Clemens Jonscher, Raimund Rolfes https://doi.org/10.25835/lyav246d
Model code and software
VirtualSensingTowerStructures Jonathan Thurn, Clemens Jonscher, Benedikt Hofmeister https://github.com/isd-luh/VirtualSensingTowerStructures
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Comment on “Virtual sensing for strain estimation in wind turbine support structures based on a single accelerometer” by Thurn et al. (2026)
Xin Feng
School of Infrastructure Engineering, Dalian University of Technology, Dalian Chian
E-mail: fengxin@dlut.edu.cn
Thurn et al. (2026) present a model-based approach for virtual sensing of wind turbine support structures for full-field strain estimation using a single DC-capable accelerometer. The method is explicitly aimed at the quasi-static frequency range and is presented as enabling strain estimation while saving costs by relying solely on accelerations from a single accelerometer as input. The concept is technically interesting, but this central claim is stronger than the method formulation and validation evidence can support.
1. The proposed single-accelerometer virtual sensing is theoretically conditional on an accurate structural model, rather than being established solely from one acceleration input.
The proposed approach requires a structural analysis model as its basis. The static bending line, modal shapes, and strain-transfer relations used by the method are all extracted from the finite-element (FE) model. The reliability of the estimated strain therefore depends fundamentally on whether the FE model can represent the true displacement-to-strain transfer from the measurement point to the target point. This is the critical issue. The paper itself acknowledges that, in the offshore application, the FE model is constructed from design assumptions and neglects manufacturing and construction deviations, uncertainty in soil-stiffness distribution, mass and stiffness asymmetry, and the influence of varying environmental and operational conditions on both the tilt constant and the frequency-dependent transfer function . Some discrepancy between FE prediction and true structural response is therefore unavoidable. Under a single- accelerometer configuration, the measurement does not contain sufficient independent information to identify, separate, or correct these model-induced errors. The method thus strongly relies on a prior FE-predicted response mapping without providing a theoretical mechanism to handle modeling error, parameter and loading uncertainty, or transfer-relation mismatch. In this sense, the claim of “saving costs by relying solely on accelerations from a single accelerometer as input” is theoretically questionable, because the method is not supported by one accelerometer alone, but by one acceleration measurement together with a sufficiently accurate structural model. Although the paper describes the framework as “a novel model-based approach”, its application to practical offshore wind turbine support structures lacks a firm theoretical foundation unless modeling error is explicitly addressed.
2. The low-frequency displacement estimation is governed by a prior kinematic assumption rather than by independent identification from measurement.
In the proposed method, the measured acceleration is decomposed into structural inertial acceleration and a gravity-induced component due to tilt, and the tilt constant is defined as the ratio between tilt and lateral displacement in the quasi-static bending line extracted from the FE model. This quantity is then embedded directly in the combined tilt-error compensation and double integration used for displacement estimation. The method therefore does not identify the actual low-frequency rotation-displacement relation from measurement, but prescribes it through the quasi-static bending line of the prior model. In the offshore proof-of-concept study presented in the paper, this modelling assumption is further specified by assuming that rotor thrust is the dominant load in the quasi-static frequency range, so that the quasi-static values of are determined from the static bending line resulting from a horizontal force applied at the tower top. The low-frequency displacement estimation is therefore explicitly dependent on load modelling and on the assumed structural representation. If the true quasi-static response is influenced by soil-structure interaction, varying environmental and operational conditions, structural asymmetry, or other load components, systematic bias should be expected. The paper also states that low-frequency noise can still cause long-term drift, and that, in the offshore case, both measured and estimated strains had to be high-pass filtered at Hz, which impeded the determination of static strains. Accordingly, the kinematic assumption underlying the low-frequency estimation cannot be regarded as guaranteed in practice, because it is affected by both modelling error and measurement technology, and a single acceleration measurement provides no mechanism to identify or correct this deficiency.
3. The validation evidence is too narrow to support a full-field reconstruction.
The laboratory study is a controlled beam experiment with simple geometry, a prescribed top displacement boundary condition, and an updated FE model with near-perfect mode-shape agreement. Accordingly, the paper concludes only that strains can be reliably estimated from one accelerometer for a known excitation. That result cannot be directly extended to an operating offshore wind turbine under uncertain support conditions and variable environmental loading. The offshore validation is similarly limited. It uses 9.5 h of part-load data, evaluates signals after low-pass filtering at 1 Hz, and validates strains only near the transition piece at one elevation. It provides no independent evidence at the fatigue-critical locations around the mudline or below it, even though the paper itself notes that such regions are of primary interest in offshore wind turbine support structures. The reported field deviations in damage-equivalent strain range from about 6 % to 21 %. The field study therefore supports, at most, a local proof of concept near the transition piece, not full-field strain reconstruction throughout the support structure.
In summary, the paper is more appropriately interpreted as a model-assisted proof of concept for quasi-static response estimation under strong prior assumptions. In practical offshore applications, however, obtaining a truly predictive finite-element model is intrinsically difficult. The response of a monopile wind turbine is governed by uncertain support conditions, soil-structure interaction, nonlinear damping and loading effects, and time-varying environmental and operational conditions. Moreover, the structural state itself evolves over time through processes such as scour, marine growth, and damage accumulation, all of which can alter the actual static and dynamic behaviour of the system as well as the associated response transfer. Under such conditions, a sufficiently accurate FE model cannot be assumed a priori. Nor can a single acceleration measurement realistically provide a basis for model updating capable of resolving such multi-source uncertainties and improving the predictive accuracy of the FE model to a practically sufficient level. One sensor therefore does not provide enough information to compensate for these modelling deficiencies. The paper therefore does not demonstrate that the spatial observability limit of single-point monitoring has been overcome, nor that robust full-field strain estimation from one accelerometer is generally available for offshore wind turbine support structures.
Reference:
Thurn, J., Jonscher, C., Hofmeister, B., Zorzi, G., and Rolfes, R.: Virtual sensing for strain estimation in wind turbine support structures based on a single accelerometer, Wind Energ. Sci. Discuss. [preprint], https://doi.org/10.5194/wes-2026-5, in review, 2026.