An improved theory in the determination of aerodynamic damping for a horizontal axis wind turbine (HAWT)

https://doi.org/10.1016/j.jweia.2021.104619Get rights and content

Highlights

  • An aerodynamic damping theory including blade flexibility, shaft tilt, yaw angle and tower shape effects is developed.

  • Qualitative trend of aerodynamic damping related with influential factors is determined.

  • Quantitative range of related factors within which the aerodynamic damping is highly affected.

  • A potential application of the improved theory in wind turbine dynamic analysis is depicted.

Abstract

The aerodynamic damping reflects the aero-structure interaction and is intrinsically involved in a fully coupled turbine. However, it is still of great importance to theoretically quantify the aerodynamic damping of a HAWT in some cases. The existing theory can reasonably characterize the aerodynamic damping level of a HAWT with rigid blades, minute shaft tilt and yaw angles, while certain discrepancies were observed when compared with either experimental or numerical damping results of a more realistic turbine. This study aims to provide an improved theory to incorporate more realistic conditions (i.e., blade flexibility, shaft tilt and yaw angle) in aerodynamic damping estimation. Good agreements are found between the proposed theory and numerical results with varied influential factors, upon which modification factors against the original theory are created and discussed. Finally, the aerodynamic damping of tower is determined and included in a decoupled fatigue analysis framework to demonstrate the potential application of this improved aerodynamic damping theory.

Introduction

The wind industry is booming to cope with the exhaustion of fossil fuel and expansion of power demand. A report revealed that the total installed wind capacity was over 651 ​GW by 2019, and an increase of over 355 ​GW newly installed capacity was expected in the next 5 years, though the full impact of Covid-19 to the wind industry is still unknown (Lee and Zhao, 2020). Besides the increase of total capacity, the rated power for a single wind turbine is increasing with an enlarged rotor diameter to harvest more energy (Hand and Cashman, 2018; Bangga et al., 2018; Ju et al., 2020). To ensure that each of the designed turbine can work till its expected decommissioning, a feasible dynamic analysis approach is required in terms of both structural strength and fatigue life design (Liu et al., 2018, 2021; Wen et al., 2020). However, in a preliminary design stage where the conceptual models have not been realized in turbine analysis software with aero-hydro-structure interaction involved, an alternative way need to be resorted to. One of the applicable methods is to replace the rotor nacelle assembly (RNA) with a lumped mass plus a dashpot damping to reflect the aero-structure interaction, where the magnitude of damping is regarded as the most crucial factor to be determined in structural dynamic analysis, and is called aerodynamic damping. In addition to preliminary design, such practice is also widely adopted in final design such as in earthquake analysis and fatigue analysis of a wind turbine where potential soil-structure interaction can be achieved (Alati et al., 2014; Santangelo et al., 2016; Rezaei et al., 2018; Ju and Huang, 2019). To this end, an accurate prediction for the aerodynamic damping is of crucial importance in the structure design of a wind turbine under different operational conditions. It is noteworthy that the focus of this study is majorly on the aerodynamics of a HAWT. Relevant research for vertical axis wind turbines (VAWT) can be found in Chen et al. (2019), Nguyen et al. (2020), Posa (2020), Hau et al. (2020) and Hohman et al. (2020), and the aerodynamic damping of a VAWT is to be investigated for interested scholars.

From a comprehensive review of past work, the determinations of aerodynamic damping were majorly conducted in four ways: in-situ experiment, laboratory experiment, theoretical deduction and numerical simulation, in which the numerical simulation was often presented with other determination approaches as a reference and was not introduced separately in the following paragraphs.

The experiment on a full scale offshore or onshore wind turbine in terms of aeroelastic damping determination majorly relied on two different approaches which were systematically described by Hansen et al. (2006). One was forced excitation method, the other was ambient excitation method (Hansen et al., 2006). The overall damping and possible aerodynamic damping in different directions estimated from both methods applied on onshore or offshore wind turbines under different working status are listed in Table 1.

The determination of wind turbine aerodynamic damping through laboratory experiments was relatively limited and immature. Three representative studies were listed in Table 2 with summarized advantages and deficiencies.

Besides what presented in Table 2, the work by Fontecha et al. (2019) also revealed that the measured aerodynamic damping was found 5–40% larger than the theoretical estimations from Kühn (2001) and Valamanesh and Myers (2014), even though the lift coefficient derivative in the experiment model was of similar magnitude to the prototype. Such results suggested that the analytical approach proposed by Kühn (2001) and Valamanesh and Myers (2014) may omit some factors that are also crucial in aerodynamic damping determination.

Aside from experimental study, theoretical estimation of aerodynamic damping of a HAWT was developed coordinately. The aerodynamic damping of a blade in different directions (flapwise or edgewise) and under different wake assumptions (steady or unsteady) were theoretically studied by several scholars (Petersen et al., 1998; Thomsen et al., 2000; Hansen 2003; Liu et al., 2010). These aerodynamic damping theories were associated to a single blade, which can be useful in characterizing the dynamic performance of a blade under aero-structure interaction consideration. However, dynamic description of the entire structure requires an appropriate estimation of aerodynamic damping in the aspect of rotor and tower. Garrad (1990) investigated the aerodynamic damping of a HAWT with flexible tower and rigid blade elements. With the modification of Garrad’s theory, Kühn (2001) proposed an aerodynamic damping expression of a wind turbine rotor, which was related to the lift coefficient derivative and blade geometric properties. The theory developed by Garrad (1990) and Kühn (2001) was only applicable for a turbine with constant wind speed, and an advanced theory for a variable-speed wind turbine was developed by Salzmann and Van der Tempel (2005). They also pointed out that, with the consideration of drag coefficient, the aerodynamic damping could be augmented by 16% under high wind speed, but the tangential induction factor was not involved in the derivation of aerodynamic damping. Along this line, Liu et al. (2017) obtained a transient aerodynamic damping for a variable-speed wind turbine by introducing a time-dependent correction factor. However, the aerodynamic damping before correction was based on a series of assumptions (e.g., small inflow angle and no drag term) that may undermine the final damping accuracy (Liu et al., 2017). One of the most influential aerodynamic damping estimation methods currently available was provided by Valamanesh and Myers (2014), where the BEM theory was combined with the single degree of freedom (SDOF) structural dynamic theory to analyze the variation of aerodynamic thrust under unit rotor centre translation. Nonetheless, compared with the numerical simulation results from a fully coupled model, the theory proposed by Valamanesh and Myers (2014) tends to somewhat underestimate the aerodynamic damping due to a few assumptions made through the theory deduction, such as rigid blade, zero shaft tilt and yaw angle. Most recently, Xi et al. (2020) included the tower top rotational degree of freedom in aerodynamic damping determination, whereas such theory was still based on a rigid rotor assumption and the effects of shaft tilt and yaw were not explored.

In conclusion of the aerodynamic damping determination approaches, the in-situ experiment method can be conducive in post-construction structural health monitoring, but is relatively lack of instructive significance in initial design of a wind turbine due to the unclear contribution of aerodynamic damping in the overall damping and the case-dependent damping levels. The laboratory experiment method is relatively immature due to the difficulties in uncertainty control and similarity criterion satisfaction. The numerical simulation method based on a fully coupled turbine model can provide desirable results. However, it is sometimes unavailable in the initial design stage and the influential factors for aerodynamic damping are hard to determine, thus hampering potential structural optimization. Finally, the theoretical method provides a clear aerodynamic damping mechanism, but is based on a few idealized conditions so that there is still discrepancies when compared with either experimental or numerical damping results of a more realistic turbine. In view of this, this study makes an attempt to improve the existing damping theory by including several realistic conditions so that the corresponding aerodynamic damping mechanism can be elucidated and more accurate structural analysis result can be obtained. The effect of blade flexibility, shaft tilt and yaw angles on aerodynamic damping were theoretically derived and then numerically verified in Section 2 Blade flexibility consideration in aerodynamic damping determination, 3 Shaft tilt consideration in aerodynamic damping determination, 4 Yaw angle consideration in aerodynamic damping determination respectively; the aerodynamic damping of a turbine tower is estimated in Section 5; a potential application of the improved theory is presented in Section 6; followed by the final conclusion made in Section 7.

Section snippets

Blade flexibility consideration in aerodynamic damping determination

The aerodynamic damping theory in previous studies assumed a rigid rotor translating under upstream wind load, so the entire RNA was replaced by a lumped mass, and the structure was simplified to a SDOF model in the fore-aft direction. Nonetheless, in reality, the rotor should be treated as a multi degree of freedom (MDOF) system due to the blade flexibility.

Shaft tilt consideration in aerodynamic damping determination

The effect of shaft tilt angles are often dismissed in previous studies (Garrad, 1990; Kühn, 2001; Salzmann and Van der Tempel, 2005; Valamanesh and Myers, 2014; Liu et al., 2017; Xi et al., 2020), and the corresponding aerodynamic damping are assumed to be equivalent to a turbine with aclinic shaft. This is comprehensible because the common tilt angle for a wind turbine is around 5° (upwind and upward), which is designed to avoid potential collision of turbine blade and tower due to blade

Yaw angle consideration in aerodynamic damping determination

According to International Electrotechnical Commission (2019), the yaw error is suggested to be in the range of ±15° in most scenarios except for an extreme yaw misalignment up to ±30° considered in DLC 6.3 with a 1-year return period. Nevertheless, analogous to what explained in Section 3, many recent studies demonstrated that yaw misalignment could be used as an active control strategy in wake induction to promote the overall output power of a wind farm (Fleming et al., 2015; Porté-Agel

Turbine tower consideration in aerodynamic damping determination

The turbine tower aerodynamic load and corresponding aerodynamic damping are often neglected or ignored in the structural dynamic analysis (Alati et al., 2014; Rezaei et al., 2018), which is comprehensible because the upstream wind speed Vw accepted by a tubular tower is much lower than the relative wind speed Vrel accepted by a rotating turbine blade. However, when a turbine is under parked condition, simple disregard of the tower aerodynamics may incur certain inaccuracy. This section is

Potential application

In Section 2 to Section 5, the effect of blade flexibility, shaft tilt, yaw angle and tower shape in terms of aerodynamic damping determination are systematically studied and verified (the tower aerodynamic damping is intended to be verified in this section). An example of potential application of the improved damping theory is demonstrated in this section in the aspect of fatigue evaluation of a parked-unfeathered NREL 5 ​MW wind turbine under extreme wind conditions such as cyclones or

Conclusion

This article presents an improved theory with the consideration of blade flexibility, shaft tilt, yaw angle and tower shape in aerodynamic damping determination of a HAWT. The theory is proposed with detailed verification through the parametric analysis, and is intended to provide a qualitative and quantitative guide for potential aerodynamic damping estimation of a wind turbine when the effects of those factors turn to be prominent. One example of potential application in decoupled analysis is

CRediT authorship contribution statement

Yisu Chen: formalism, Writing – original draft, wrote the article. Di Wu: formalism, Writing – original draft, wrote the article. Yuguo Yu: performed the computations and verified the computational. Wei Gao: Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The work presented in this paper has been supported by the Australian Research Council projects IH150100006 and IH200100010.

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