Multi-blade Coordinate Transformation Research Articles

Comprehensive analysis of rotor edgewise whirling mode interaction with rotor speed harmonics

This paper presents a model that investigates the interaction between rotor edgewise whirling modes and rotor speed harmonics in wind turbines. The model is based on kinematic and dynamic principles, with a focus on the multi-blade coordinate transformation, which is critical for simulating the behaviour of the rotor whirling modes in wind turbines. The research has two objectives: to investigate the interaction between the rotor edgewise whirling modes and the rotor speed harmonics, and to provide clearer graphs that explain the complex nature of this non-intuitive rotor dynamics. The paper concludes by highlighting the practical implications of the research findings, in particular the effectiveness of visualisation techniques in identifying and explaining unexpected interactions.

Optimization with genetic algorithms of individual pitch control design with and without azimuth offset for wind turbines in the full load region

The reduction of fatigue loadings in wind turbines to increase their lifetime has become of special interest from a control viewpoint. Individual Pitch Control (IPC) is a well-known approach used to mainly mitigate periodic blade loads, and it is usually implemented with the assistance of the multi-blade coordinate (MBC) transformation, which transforms and decouples the measured blade load signals from a rotating frame into a non-rotating tilt-axis and yaw-axis. Nevertheless, these axes still show coupling between them in practical scenarios adversely affecting the system performance. Previous studies have demonstrated the benefits of including an extra tuning parameter in the MBC, the azimuth offset, in improving the performance achieved by the IPC. However, the tuning of this parameter and its real improvements that can be obtained compared to the IPC without this offset require more research. Here, two 1P+2P IPC, with and without additional azimuth offset, are designed and applied to the 5 MW reference turbine model developed by NREL using the FAST software as a simulation platform. The controller parameter tuning is formulated as an optimization problem that minimizes the blade fatigue load according to the Dirlik index and that is resolved through genetic algorithms. To fairly analyze the improvement entailed by the addition of the azimuth offset, both optimized IPC schemes, with and without azimuth offset, are compared qualitatively and quantitatively using a classical controller as the baseline case. From the simulation results, it can be stated that the optimal IPC scheme with azimuth offset compared with the IPC scheme without offset achieves improvements of around 11% in load reduction and pitch signal effort.

Simplified complex-valued modal model for operating wind turbines through aerodynamic decoupling and multi-blade coordinate transformation

An efficient modelling methodology for steady-state operating wind turbines is proposed, combining aerodynamic decoupling, multi-blade coordinate transformation, and modal reduction. This leads to a complex-valued, reduced-order modal model for prediction of dynamic responses of the tower-rotor-blade wind turbine system, considering rotor rotation, blade flexibility, and vibration coupling between rotor and tower. A fully coupled finite element model was first developed, with aerodynamic forces linearised and expressed as a viscous aerodynamic damping matrix. A time-invariant state space model is formed using multi-blade coordinate transformation, allowing standard modal analysis. The complex-valued eigenvalues and mode shapes were obtained, and it is shown that the operating wind turbine modes exhibit a combination of tower and blade vibrations. Various degrees of modal reduction are applied to the state space model to obtain a modal model with fewer degrees of freedom, whose performance was evaluated in the time and frequency domain for operating wind turbines in normal condition. The displacement and stress responses are in close agreement with those of the fully coupled model with the first 21 modes included. The model already performs well with 8 modes considered to capture relevant fundamental frequency peaks. This allows significantly reduced computational effort and can be particularly beneficial for fatigue prediction, reliability analysis, and structural identification.

Along-the-path exponential integration for Floquet stability analysis of wind turbines

Traditionally, stability assessment of wind turbines has been performed by eigenanalysis of the azimuthally-averaged linearized system after applying the Multi-Blade Coordinate (MBC) transformation. However, due to internal or external anisotropy, the MBC transform does not produce an exact Linear Time-Invariant (LTI) system, and a Floquet analysis is required to capture the influence of all periodic terms, leading to a more accurate stability analysis. In this paper exponential integration methods that use system linearizations at different blade azimuth positions are used to integrate the perturbed system state and compute the Floquet monodromy matrix. The proposed procedure is assessed for a simple 6 DOF wind turbine model and a more complex aeroelastic model of a 5MW onshore wind turbine. The defined along-the-path or moving-point exponential integrator is found to be the suitable in order to perform a Floquet stability analysis even using a coarse linearization grid.

Wind turbine load reduction based on 2DoF robust individual pitch control

The individual pitch controller can mitigate asymmetric loads in wind turbines. However, wind turbine dynamics have strong nonlinearity and high uncertainty. In addition to target value tracking, suppression of wind disturbance is also a performance requirement of the individual pitch controller, while the traditional PI controller cannot take into account the tracking and disturbance rejection performance simultaneously. Therefore, the two-degree-of-freedom (2DoF) robust individual pitch controller is proposed to reduce loads in the above-rated region. Besides, parameter tuning is complicated in the design of the robust individual pitch controller. So the reference model method is proposed to preset the closed-loop system response. Firstly, the state-space model is established to describe the dynamics of the wind turbine. The multi-blade coordinate transformation is applied to transform the model into the fixed coordinate system. Subsequently, the μ-synthesis problem is solved by the D-K iterative algorithm to get the controller parameter. Finally, the control method is verified in GH Bladed. It is shown that rotor loads are suppressed without affecting the output power. Tower loads are also mitigated. Moreover, the relationship between the pitch actuator action and the load reduction capability is discussed by designing different bandwidths of the proposed method.

Efficient tuning of Individual Pitch Control: A Bayesian Optimization Machine Learning approach

With the trend of increasing wind turbine rotor diameters, the mitigation of blade fatigue loadings is of special interest to extend the turbine lifetime. Fatigue load reductions can be partly accomplished using individual pitch control (IPC), and is commonly facilitated by the so-called multiblade coordinate (MBC) transformation. This operation transforms and decouples the blade load signals in a non-rotating yaw-axis and tilt-axis. However, in practical scenarios, the resulting transformed system still shows coupling between the axes. To cope with this phenomenon, earlier research has shown that the introduction of an additional MBC tuning variable – the azimuth offset – decouples the multivariable system. However, the introduction of this extra variable complicates the controller design process, and requires expert knowledge and specialized analysis software. To provide an efficient method for the optimization of fixed-structure IPC controllers, based on black box and computationally costly objective functions, this paper considers a Bayesian optimization controller tuning framework. Results show the efficiency of the framework to tune a combined 1P + 2P IPC implementation, without prior knowledge, and based on high-fidelity simulation results using a computationally expensive objective function.

Modal Analysis of a Quad-Rotor Wind Turbine

Unlike Single-Rotor wind turbines, stability analysis of Multi-Rotor wind turbines is still in its initial stages. This paper presents the modal analysis of a Quad-Rotor wind turbine and identifies the new modes or possible instability modes that are otherwise not present on a Single-Rotor wind turbine. Multi-Blade Coordinate transformation scheme is adapted to a Quad-Rotor wind turbine to write the system’s equation of motion in fixed-reference frame followed by Eigenvalue analysis to determine the natural frequencies and mode shapes of the Quad-Rotor wind turbine. A Campbell diagram of the Quad-Rotor wind turbine is presented. Results indicate that the Quad-Rotor turbine is soft-soft to the first tower modes (fore-aft, side-side, and torsion). Furthermore, the modes with low natural frequency other than the tower modes are a combination of tower, boom, and blade modes. Therefore, due to the presence of blade modes, the modal frequency of these modes increases or decreases with increasing rotor speed due to centrifugal stiffening.

Development of semi-active vibration control strategy for horizontal axis wind turbine tower using multiple magneto-rheological tuned liquid column dampers

Vibration control is an essential component of safe and sustained operation of modern large wind turbine towers. As the blade-tower assembly offers time-dependent system matrices that periodically evolves with the rotational speed of the turbine, control theories (e.g. linear quadratic Gaussian) can not be directly adopted in this case. With this in view, the present study aims to develop a semi-active strategy using multiple magneto-rheological tuned liquid column dampers (MR-TLCDs) to mitigate excessive vibration of the horizontal axis wind turbine towers. The design of the controller is achieved by the multi-blade coordinate transformation that converts the system matrices into a non-rotating framework followed by time averaging to enable the application of linear control law. Parameters of the non-linear MR-TLCDs are optimally tuned to dissipate maximum energy. Aerodynamic loads are evaluated using modified blade element momentum theory. The performance of the proposed control strategy is assessed against different wind speeds. Clipping laws are employed to keep the semi-active control force in the feasible range. Two different clipping laws are compared in this paper. Finally, sensitivity analysis is carried out to demonstrate the performance envelope and robustness of the proposed controller.

Analysis and optimal individual pitch control decoupling by inclusion of an azimuth offset in the multiblade coordinate transformation

AbstractWith the trend of increasing wind turbine rotor diameters, the mitigation of blade fatigue loadings is of special interest to extend the turbine lifetime. Fatigue load reductions can be partly accomplished using individual pitch control (IPC) facilitated by the so‐called multiblade coordinate (MBC) transformation. This operation transforms and decouples the blade load signals in a yaw‐axis and tilt‐axis. However, in practical scenarios, the resulting transformed system still shows coupling between the axes, posing a need for more advanced multiple input multiple output (MIMO) control architectures. This paper presents a novel analysis and design framework for decoupling of the nonrotating axes by the inclusion of an azimuth offset in the reverse MBC transformation, enabling the application of simple single‐input single‐output (SISO) controllers. A thorough analysis is given by including the azimuth offset in a frequency‐domain representation. The result is evaluated on simplified blade models, as well as linearizations obtained from the NREL 5–MW reference wind turbine. A sensitivity and decoupling assessment justify the application of decentralized SISO control loops for IPC. Furthermore, closed‐loop high‐fidelity simulations show beneficial effects on pitch actuation and blade fatigue load reductions.

Aeroelastic modeling and dynamic analysis of a wind turbine rotor by considering geometric nonlinearities

Due to the increased flexibility of modern multi-megawatt wind turbine structures, more advanced analyses are needed to investigate the effects of geometric nonlinearities originating from large blade deformations under operational loads. The main objective of this paper is to study the related dynamics and the aeroelastic effects of these nonlinearities by using a multi-flexible-body aeroelastic model of an entire three-bladed wind turbine assembly instead of a more conventional single-blade model. A geometrically-exact beam formulation is employed to model the rotating blades connected to the wind turbine tower tip via hub and nacelle components; and the revolute joint constraint is applied to model the rotor rotation. The derived governing equations are analyzed by applying a reduced-order approach based on the Galerkin method. After verifying the model by comparing it with an FEM model, it is used in several aeroelastic and dynamic studies. All the performed simulations are based on the specifications of the state-of-the-art 5 MW-NREL wind turbine. Moreover, the time-invariant forms of the governing equations are derived by applying the multi-blade coordinate transformation method. The obtained equations are used to study the whirling characteristics of wind turbine rotor by different models including the model of unloaded blades as well as the linear and nonlinear aeroelastic models. The eigenfrequencies indicate the splitting characteristics according to the gyroscopic effects that originate from the whirling motion in the forward and backward directions. The simulation results reveal that single-blade models are helpful in demonstrating the overall trends of rotor aeroelasticity. However, because single-blade models overestimate the instability limits, an analysis based on a full wind turbine model is crucial. Besides, the geometric nonlinearities significantly alter the eigenfrequency results (especially for the modes dominantly incorporated into flutter instability) as well as the dynamic responses of the system, notably around the flutter speed.

Optimal design of semiactive MR-TLCD for along-wind vibration control of horizontal axis wind turbine tower

Present study aims to address the design of smart vibration control scheme for horizontal axis wind turbine tower using magneto-rheological tuned liquid column damper. With this in view, a reduced order model of the blade-tower system is used, considering centrifugal stiffening and gravitational effects that lead to time-dependent dynamic stiffness matrix. Aerodynamic load on the blades is modeled using blade element momentum theory. Semiactive control law in linear quadratic regulator framework is developed to mitigate the along-wind vibration of the tower. To implement the control law, multiblade coordinate transformation is adopted that converts the system matrices in the nonrotating framework to tackle its time dependency. The performance of the proposed control algorithm is demonstrated using numerical simulations with and without controller. Clipped optimality of the control force is imposed to keep the parameters of magneto-rheological tuned liquid column damper in the feasible range. Finally, sensitivity analysis is carried out to demonstrate the performance envelope of the proposed control algorithm for different operational scenario. Results presented in this paper clearly demonstrate that the proposed algorithm can be employed for effective along-wind vibration control of large HWAT tower.

Fault diagnosis and condition monitoring of wind turbines

SummaryThis paper describes a model‐free method for the fault diagnosis and condition monitoring of rotor systems in wind turbines. Both fault diagnosis and monitoring can be achieved without using a model for the wind turbine, applied controller, or wind profiles. The method is based on measurements from standard sensors on modern wind turbines, including moment sensors and rotor angle sensors. This approach will allow the method to be applied to existing wind turbines without any modifications.The method is based on the detection of asymmetries in the rotor system caused by changes or faults in the rotor system. A multiblade coordinate transformation is used directly on the measured flap‐wise and edge‐wise moments followed by signal modulation. Changes or faults in the rotor system will result in unique signatures in the set of modulation signals. These signatures are described through the amplitudes and phase information of the modulation signals. It is possible to detect and isolate which blade is faulty or has been changed based on these signatures. Furthermore, the faulty component can be isolated, ie, the actuator, sensor or blade, and the type of fault can be determined. The method can be used both on‐ and off‐line.

Passive suppression of helicopter ground resonance using nonlinear energy sinks attached on the helicopter blades

This paper investigates the passive control of a rotor instability named helicopter Ground Resonance (GR). The passive device consists of a set of essential cubic nonlinear absorbers named Nonlinear Energy Sinks (NES) each of them positioned on a blade. A dynamic model reproducing helicopter GR instability is presented and transformed to a time-invariant nonlinear system using a multi-blade coordinate transformation based on Fourier transform mapping the dynamic state variables into a non-rotating reference frame. Combining complexification, slow/fast partition of the dynamics and averaging procedure, a reduced model is obtained which allowed us to use the so-called geometric singular perturbation analysis to characterize the steady state response regimes. As in the case of a NES attached to the fuselage, it is shown that under suitable conditions, GR instability can be completely suppressed, partially suppressed through periodic response or strongly modulated response. Relevant analytical results are compared, for validation purposes, to direct integration of the reference and reduced models.

Iterative tuning of feedforward IPC for two-bladed wind turbines

At present, the cost of offshore wind energy does not meet the level of onshore wind and fossil-based energy sources. One way to extend the turbine lifetime, and thus reduce cost, is by reduction of the fatigue loads of blades and other turbine parts using Individual Pitch Control (IPC). This type of control, which is generally implemented by feedback control using the MultiBlade Coordinate transformation on blade load measurement signals, is capable of mitigating the most dominant periodic loads. The main goal of this article is to develop a self-optimizing feedforward IPC strategy for a two-bladed wind turbine to reduce actuator duty cycle and reduce the dependency on blade load measurement signals. The approach uses blade load measurement data only initially for tuning of the feedforward controller, which is scheduled on the rotor azimuth angle and wind speed. The feedforward strategy will be compared to the feedback implementation in terms of load alleviation capabilities and actuator duty cycle. Results show that the implementation is capable of learning the optimal feedforward IPC controller in constant and turbulent wind conditions, to alleviate the pitch actuator duty cycle, and to considerably reduce harmonic fatigue loads without the need for blade load measurement signals after tuning.

Whirl speeds of mistuned bladed rotors supported by isotropic stator

As an initial step toward understanding the fully coupled dynamics of mistuned bladed disk-shaft systems, this paper investigates the frequency characteristics of natural whirl speeds associated with the in-plain vibration of a rotating mistuned bladed disk mounted on an isotropic support. Through complex multi-blade coordinate transformation and modulation, a simplistic analysis model describing the essential in-plain whirling behavior of mistuned bladed rotor is derived in a typical form of linear differential equations with time-constant coefficients. By applying ordinary eigenvalue analysis for linear time-invariant systems, the whirl speeds of mistuned bladed rotor are examined for cases of weak and strong inter-blade coupling conditions. The mistuning effect on the whirl speeds of the bladed rotor is then explained by classifying the whirling modes into three types according to their cause of manifestation and the frequency relationship: namely, original, coupled multi-blade, and conjugate whirling modes.

Individual blade pitch for yaw control

Individual pitch control (IPC) for reducing blade loads has been investigated and proven successful in recent literature. For IPC, the multi-blade co-ordinate (MBC) transformation is used to process the blade load signals from the rotating to a stationary frame of reference. In the stationary frame of reference, the yaw error of a turbine can be appended to generate IPC actions that are able to achieve turbine yaw control for a turbine in free yaw. In this paper, IPC for yaw control is tested on a high-fidelity numerical model of a commercially produced wind turbine in free yaw. The tests show that yaw control using IPC has the distinct advantage that the yaw system loads and support structure loading are substantially reduced. However, IPC for yaw control also shows a reduction in IPC blade load reduction potential and causes a slight increase in pitch activity. Thus, the key contribution of this paper is the concept demonstration of IPC for yaw control. Further, using IPC for yaw as a tuning parameter, it is shown how the best trade-off between blade loading, pitch activity and support structure loading can be achieved for wind turbine design.

Response and instability prediction of helicopter dynamics on the ground

In helicopters with hinged blades an unstable dynamical phenomenon known as ground resonance may occur during take-off and landing and lead to the total destruction of the aircraft. Predicting the phenomenon is necessary to determine the stability of periodical equations of motion. The instability boundaries can be easily obtained for isotropic rotor configurations through multi-blade coordinate transformation once the periodic terms are eliminated. However, Floquet׳s theory is commonly used to treat the periodic motion equations when introducing the asymmetric effects of spring or damper aging or rotor rupture (anisotropic rotors). In additional, it is known that when treated as parametric excitations, periodic terms may lead to instability in dynamical systems under parametric resonances. In this paper a helicopter in contact with the ground is considered as a parametrically excited system and the equations are treated analytically by applying the method of multiple scales (MMS). A stability analysis verifies the existence of parametric instabilities by first order sets of equations for an isotropic rotor configuration. The results are compared and validated with those obtained by using Floquet׳s Theory. Moreover, the amplitude responses of the aircraft at equilibrium in the remaining resonant cases are studied. The results are then compared with those obtained from the time response analysis.

A novel individual pitch control algorithm based on μ-synthesis for wind turbines

In this paper, we present a novel individual pitch control algorithm based on μ-synthesis for blade load reduction for wind turbines. Our algorithm is different from the conventional individual pitch control in the sense that the multi-blade coordinate transformation is not used at all. Instead, the wind turbine model coefficients that depend on the azimuth angle are regarded as structured uncertainties. The control objective is formulated with weighting functions on the disturbance input and performance output in the framework of the μ-synthesis approach. Our control algorithm can be added on to the collective pitch control as an on-off mechanism with minimal effect on it. Simulation results show that the proposed controller achieves very good robust stability and performance and can significantly reduce blade bending moments.

Linear individual pitch control design for two‐bladed wind turbines

AbstractIn this article, the conventional individual pitch control (IPC) strategy for wind turbines is reviewed, and a linear IPC strategy for two‐bladed wind turbines is proposed. The typical approach of IPC for three‐bladed rotors involves a multi‐blade coordinate (MBC) transformation, which transforms measured blade load signals, i.e., signals measured in a rotating frame of reference, to signals in a fixed non‐rotating frame of reference. The fixed non‐rotating signals, in the so‐called yaw and tilt direction, are decoupled by the MBC transformation, such that single‐input single‐output (SISO) control design is possible. Then, SISO controllers designed for the yaw and tilt directions provide pitch signals in the non‐rotating frame of reference, which are then reverse transformed to the rotating frame of reference so as to obtain the desired pitch actuator signals. For three‐bladed rotors, the aforementioned method is a proven strategy to significantly reduce fatigue loadings on pitch controlled wind turbines. The same MBC transformation and approach can be applied to two‐bladed rotors, which also results in significant load reductions. However, for two‐bladed rotors, this MBC transformation is singular and therefore, not uniquely defined. For that reason, a linear non‐singular coordinate transformation is proposed for IPC of two‐bladed wind turbines. This transformation only requires a single control loop to reduce the once‐per‐revolution rotating blade loads (‘1P’ loads). Moreover, all harmonics (2P, 3P, etc.) in the rotating blade loads can be accounted for with only two control loops. As in the case of the MBC transformation, also the linear coordinate transformation decouples the control loops to allow for SISO control design. High fidelity simulation studies on a two‐bladed wind turbine without a teetering hub prove the effectiveness of the concept. The simulation study indicates that IPC based on the linear coordinate transformation provides similar load reductions and requires similar pitch actuation compared with the conventional IPC approach. Copyright © 2014 John Wiley & Sons, Ltd.

Monitoring of a Wind Turbine Rotor using a Multi-blade Coordinate Framework

In this paper a method to detect asymmetric faults in a wind turbine rotor is presented. The paper describes how fault diagnosis using an observer-based residual generator approach is able to distinguish between the nominal and faulty case by the injection of e.g. a sinusoidal excitation signal into the system. In the case of a wind turbine, an excitation signal is automatically generated by the rotation of the rotor in a turbulent wind field. Using the multi-blade coordinate transformation, the detection of asymmetries in the rotor of the wind turbine is greatly improved.