Simulating Feedback Linearization control of wind turbines using high‐order models

As wind turbines are becoming larger, wind turbine control must now encompass load control objectives as well as power and speed control to achieve a low cost of energy. Due to the inherent non‐linearities in a wind turbine system, the use of non‐linear model‐based controllers has the potential to increase control performance. A non‐linear feedback linearization controller with an Extended Kalman Filter is successfully used to control a FAST model of the controls advanced research turbine with active blade, tower and drive‐train dynamics in above rated wind conditions. The controller exhibits reductions in low speed shaft fatigue damage equivalent loads, power regulation and speed regulation when compared to a Gain Scheduled Proportional Integral controller, designed for speed regulation alone. The feedback linearization controller shows better rotor speed regulation than a Linear Quadratic Regulator (LQR) at close to rated wind speeds, but poorer rotor speed regulation at higher wind speeds. This is due to modeling inaccuracies and the addition of unmodeled dynamics during simulation. Similar performance between the feedback linearization controller and the LQR in reducing drive‐train fatigue damage and power regulation is observed. Improvements in control performance may be achieved through increasing the accuracy of the non‐linear model used for controller design. Copyright © 2009 John Wiley & Sons, Ltd.     

Equations of motion for a rotor blade,including gravity,pitch action and rotor speed variations

This paper extends Hodges–Dowell's partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade interaction, and the model will be extended to include an aerodynamic model and be used for analysis of basic properties of pitch blade interaction. Copyright © 2007 John Wiley & Sons, Ltd.     

Wind turbine extreme gust control

This paper focuses on the problem of extreme wind gust and direction change recognition (EG&DR) and control (EEC). An extreme wind gust with direction change can lead to large loads on the turbine (causing fatigue) and unnecessary turbine shutdowns by the supervisory system caused by rotor overspeed. The proposed EG&DR algorithm is based on a non‐linear observer (extended Kalman filter) that estimates the oblique wind inflow angle and the blade effective wind speed signals, which are then used by a detection algorithm (cumulative sum test) to recognize extreme events. The non‐linear observer requires that blade root bending moments measurements (in‐plane and out‐of‐plane) are available. Once an extreme event is detected, an EEC algorithm is activated that: (i) tries to prevent the rotor speed from exceeding the overspeed limit by fast collective blade pitching; and (ii) reduces 1p blade loads by means of individual pitch control algorithm, designed in an ?_∞ optimal control setting. The method is demonstrated on a complex non‐linear test turbine model. Copyright © 2009 John Wiley & Sons, Ltd.     

Flap/lead–lag aeroelastic stability of wind turbine blade sections

The scope of this article is to investigate the aeroelastic stability of wind turbine blade sections subjected to combined flap/lead–lag motion. The work is motivated by recent concern about destructive ‘edgewise' vibrations of modern, half‐megawatt‐scale, blades. The aeroelastic governing equations derive from the combination of a spring–mass–damper equivalent of the structure and a ‘non‐stationary' aerodynamic model. The aerodynamic model used in the present context is the differential dynamic stall model developed at ONERA. The resulting equations of motion are linearized and their stability characteristics are investigated in terms of the system entries, expressed through suitable, non‐dimensional, structural and aerodynamic parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

Flap/lead–lag aeroelastic stability of wind turbine blades

A numerical tool for investigating the aeroelastic stability of a single wind turbine blade subjected to combined flap/lead–lag motion is presented. Its development is motivated by recent concern about destructive edgewise vibrations of modern stall‐controlled blades. The stability tool employs a finite element formulation to discretize in space the structural and aerodynamic governing equations. Unsteady aerodynamics is considered by means of the extended ONERA lift and drag models. The mathematical form of these models allows for a combined treatment of dynamics and aerodynamics through the introduction of a so‐called ‘aeroelastic beam element’. This is an extended two‐node beam element having both deformation and aerodynamic degrees of freedom. Several linear and non‐linear versions of the stability tool are available, differing in the way that instantaneous lift and/or drag is treated. In the linear case, stability is investigated through eigenvalue analysis. Time domain integration is employed for non‐linear stability analysis. Results are presented and discussed for a 17 m stall‐controlled blade. Copyright © 2001 John Wiley & Sons, Ltd.     

Evaluation of a wind speed estimator for effective hub‐height and shear components

Estimates of the effective wind speed disturbances acting on a wind turbine are useful in a variety of control applications. With some simplifications, it is shown that for zero yaw error, any wind field interacting with a turbine can be equivalently described using a hub‐height (uniform) component as well as linear horizontal and vertical shear components. A Kalman filter‐based wind speed estimator is presented for estimation of these effective hub‐height and shear components. The wind speed estimator is evaluated in the frequency domain using the FAST aeroelastic simulator with the National Renewable Energy Laboratory's 5 MW reference wind turbine model and realistic hub‐height and shear disturbances. In addition, the impact of the inflow model, used to simulate the rotor aerodynamics, on the Kalman filter performance is investigated. It is found that the estimator accuracy strongly depends on the inflow model used. In general, the estimator performs well up to a bandwidth of 1 Hz when the inflow model used for simulation matches the model used to create the linear Kalman filter model and blade pitch angle remains close to the linearization operating point. However, inaccuracies in the linear model of the turbine when dynamic inflow is used for simulation as well as nonlinearities in the turbine dynamics due to blade pitch actuation cause performance to degrade. Finally, the improvement gained by employing a non‐causal wind speed estimator is assessed, showing a minor increase in performance. Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling,control, and simulation of dual rotor wind turbine generator system

A new wind turbine generator system (WTGS) is introduced, and its mathematical model, blade pitch control scheme, and nonlinear simulation software for the performance prediction are presented. The notable feature of WTGS is that it consists of two rotor systems positioned horizontally at upwind and downwind locations, and a generator installed vertically inside the tower. In this paper, this new WTGS is treated as a constrained multi-body system, and the equations of motion are obtained by using the multi-body dynamics approach. Aerodynamic forces and torques generated from each of rotor blades are calculated using the blade element theory. Various pitch control schemes depending on the wind speed and the main rotor's rotational speed are implemented. A relatively simple model for the load torque is obtained by using the test data of the doubly fed induction generator adopted in the new WTGS. Finally, FORTRAN and Matlab/Simulink-based hybrid simulation software is developed and used to predict and analyze the performance of the WTGS.     

Numerical study of turbulent flow around a wind turbine nacelle

This article presents a numerical method for the simulation of turbulent flow around the nacelle of a horizontal axis wind turbine. The flow field around the turbine and nacelle is described by the Reynolds‐averaged Navier–Stokes equations. The k–? model has been chosen for closure of the time‐averaged turbulent flow equations. The rotor disc is modelled using the actuator disc concept. The main purpose of this article is to assess the impacts of the variation of some operational parameters (e.g. blade pitch angle changes) and atmospheric turbulence upon the relationship between wind speed measured near the nacelle and freestream wind speed established for an isolated turbine. Simulation results were compared with experimental data (from a typical stall‐controlled, commercially available wind turbine rated higher than 600 kW). In general, good qualitative agreements have been found that validate the proposed method. It has been shown that a level of accuracy sufficient for use in power performance testing can be obtained only when a proper aerodynamic analysis of the inboard non‐lifting cylindrical sections of the blade is included. Furthermore, the numerical method has proven to be a useful tool for locating nacelle anemometers. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability analysis of pitch‐regulated,variable‐speed wind turbines in closed loop operation using a linear eigenvalue approach

A multi‐body aeroelastic design code based on the implementation of the combined aeroelastic beam element is extended to cover closed loop operation conditions of wind turbines. The equations of a controller for variable generator speed and pitch‐controlled operation in high wind speeds are combined with the aeroelastic equations of motion for the complete wind turbine, in order to provide a compound aeroservoelastic system of equations. The control equations comprise linear differential equations for the pitch and generator torque actuators, the control feedback elements (proportional–integral control) and the various filters acting on the feedback signals. In its non‐linear form, the dynamic equations are integrated in time to provide the reference state, while upon linearization of the system and transformation in the non‐rotating frame, the linear stability equations are derived. Stability results for a multi‐MW wind turbine show that the coupling of the controller dynamics with the aeroelastic dynamics of the machine is important and must be taken into account in view of defining the controller parameters. Copyright © 2008 John Wiley & Sons, Ltd.     

Damping of edgewise vibration in wind turbine blades by means of circular liquid dampers

This paper proposes a new type of passive vibration control damper for controlling edgewise vibrations of wind turbine blades. The damper is a variant of the liquid column damper and is termed as a circular liquid column damper (CLCD). Rotating wind turbine blades generally experience a large centrifugal acceleration. This centrifugal acceleration makes the use of this kind of oscillatory liquid damper feasible with a small mass ratio to effectively suppress edgewise vibrations. A reduced 2‐DOF non‐linear model is used for tuning the CLCD attached to a rotating wind turbine blade, ignoring the coupling between the blade and the tower. The performance of the damper is evaluated under various rotational speeds of the rotor. A special case in which the rotational speed is so small that the gravity dominates the motion of the liquid is also investigated. Further, the legitimacy of the decoupled optimization is verified by incorporating the optimized damper into a more sophisticated 13‐DOF aeroelastic wind turbine model with due consideration to the coupled blade‐tower‐drivetrain vibrations of the wind turbine as well as a pitch controller. The numerical results from the illustrations on a 5 and a 10MW wind turbine machine indicate that the CLCD at an optimal tuning can effectively suppress the dynamic response of wind turbine blades. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐fidelity linear time‐invariant model of a smart rotor with adaptive trailing edge flaps

A high‐fidelity linear time‐invariant model of the aero‐servo‐elastic response of a wind turbine with trailing‐edge flaps is presented and used for systematic tuning of an individual flap controller. The model includes the quasi‐steady aerodynamic effects of trailing‐edge flaps on wind turbine blades and is integrated in the linear aeroelastic code HAWCStab2. The dynamic response predicted by the linear model is validated against non‐linear simulations, and the quasi‐steady assumption does not cause any significant response bias for flap deflection with frequencies up to 2–3 Hz. The linear aero‐servo‐elastic model support the design, systematic tuning and model synthesis of smart rotor control systems. As an example application, the gains of an individual flap controller are tuned using the Ziegler–Nichols method for the full‐order poles. The flap controller is based on feedback of inverse Coleman transformed and low‐pass filtered flapwise blade root moments to the cyclic flap angles through two proportional‐integral controllers. The load alleviation potential of the active flap control, anticipated by the frequency response of the linear closed‐loop model, is also confirmed by non‐linear time simulations. The simulations report reductions of lifetime fatigue damage up to 17% at the blade root and up to 4% at the tower bottom. Copyright © 2016 John Wiley & Sons, Ltd.     

Individual blade pitch control of floating offshore wind turbines

Floating wind turbines offer a feasible solution for going further offshore into deeper waters. However, using a floating platform introduces additional motions that must be taken into account in the design stage. Therefore, the control system becomes an important component in controlling these motions. Several controllers have been developed specifically for floating wind turbines. Some controllers were designed to avoid structural resonance, while others were used to regulate rotor speed and platform pitching. The development of a periodic state space controller that utilizes individual blade pitching to improve power output and reduce platform motions in above rated wind speed region is presented. Individual blade pitching creates asymmetric aerodynamic loads in addition to the symmetric loads created by collective blade pitching to increase the platform restoring moments. Simulation results using a high‐fidelity non‐linear turbine model show that the individual blade pitch controller reduces power fluctuations, platform rolling rate and platform pitching rate by 44%, 39% and 43%, respectively, relative to a baseline controller (gain scheduled proportional–integral blade pitch controller) developed specifically for floating wind turbine systems. Turbine fatigue loads were also reduced; tower side–side fatigue loads were reduced by 39%. Copyright © 2009 John Wiley & Sons, Ltd.     

A torque matched aerodynamic performance analysis method for the horizontal axis wind turbines

An analysis method is developed to test the operational performance of a horizontal axis wind turbines. The rotor is constrained to the torque–speed characteristic of the coupled generator. Therefore, the operational conditions are realized by matching the torque generated by the turbine over a selected range of incoming wind velocity to that needed to rotate the generator. The backbone of the analysis method is a combination of Schmitz' and blade element momentum (BEM) theories. The torque matching is achieved by gradient‐based optimization method, which finds correct wind speed at a given rotational speed of the rotor. The combination of Schmitz and BEM serves to exclude the BEM iterations for the calculation of interference factors. Instead, the relative angle is found iteratively along the span. The profile and tip losses, which are empirical, are included in the analysis. Hence, the torque at a given wind speed and rotational speed can be calculated by integrating semi‐analytical equations along the blade span. The torque calculation method is computationally cheap and therefore allows many iterations needed during torque matching. The developed analysis method is verified experimentally by testing the output power and rotational speed of an existing wind turbine model in the wind tunnel. The generator's torque rotational speed characteristic is found by a separate experimental set‐up. Comparison of experiments with the results of the analysis method shows a good agreement. Copyright © 2013 John Wiley & Sons, Ltd.     

Implicit Floquet analysis of wind turbines using tangent matrices of a non‐linear aeroelastic code

The aeroelastic code BHawC for calculation of the dynamic response of a wind turbine uses a non‐linear finite element formulation. Most wind turbine stability tools for calculation of the aeroelastic modes are, however, based on separate linearized models. This paper presents an approach to modal analysis where the linear structural model is extracted directly from BHawC using the tangent system matrices when the turbine is in a steady state. A purely structural modal analysis of the periodic system for an isotropic rotor operating at a stationary steady state was performed by eigenvalue analysis after describing the rotor degrees of freedom in the inertial frame with the Coleman transformation. For general anisotropic systems, implicit Floquet analysis, which is less computationally intensive than classical Floquet analysis, was used to extract the least damped modes. Both methods were applied to a model of a three‐bladed 2.3 MW Siemens wind turbine model. Frequencies matched individually and with a modal identification on time simulations with the non‐linear model. The implicit Floquet analysis performed for an anisotropic system in a periodic steady state showed that the response of a single mode contains multiple harmonic components differing in frequency by the rotor speed. Copyright © 2011 John Wiley & Sons, Ltd.     

Open‐loop frequency response analysis of a wind turbine using a high‐order linear aeroelastic model

Wind turbine controllers are commonly designed on the basis of low‐order linear models to capture the aeroelastic wind turbine response due to control actions and disturbances. This paper characterizes the aeroelastic wind turbine dynamics that influence the open‐loop frequency response from generator torque and collective pitch control actions of a modern non‐floating wind turbine based on a high‐order linear model. The model is a linearization of a geometrically non‐linear finite beam element model coupled with an unsteady blade element momentum model of aerodynamic forces including effects of shed vorticity and dynamic stall. The main findings are that the lowest collective flap modes have limited influence on the response from generator torque to generator speed, due to large aerodynamic damping. The transfer function from collective pitch to generator speed is affected by two non‐minimum phase zeros below the frequency of the first drivetrain mode. To correctly predict the non‐minimum phase zeros, it is essential to include lateral tower and blade flap degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

Preview predictive control layer design based upon known wind turbine blade‐pitch controllers

The use of upstream wind measurements has motivated the development of blade‐pitch preview controllers for improving rotor speed tracking and structural load reduction beyond that achievable via conventional feedback control. Such preview controllers, typically based upon model predictive control (MPC) for its constraint handling properties, alter the closed‐loop dynamics of the existing blade‐pitch feedback control system. This can result in a deterioration of the robustness properties and performance of the existing feedback control system. Furthermore, performance gains from utilising the upcoming real‐time measurements cannot be easily distinguished from the feedback control, making it difficult to formulate a clear business case for the use of preview control. Therefore, the aim of this work is to formulate a modular MPC layer on top of a given output‐feedback blade‐pitch controller, with a view to retaining the closed‐loop robustness and frequency‐domain performance of the latter. The separate nature of the proposed controller structure enables clear and transparent quantification of the benefits gained by using preview control, beyond that of the underlying feedback controller. This is illustrated by results obtained from high‐fidelity closed‐loop turbine simulations, showing the proposed control scheme incorporating knowledge of the oncoming wind and constraints achieved significant 43% and 30% reductions in the rotor speed and flap‐wise blade moment standard deviations, respectively. Additionally, the chance of constraint violations on the rotor speed decreased remarkably from 2.15% to 0.01%, compared to the nominal controller. Copyright © 2017 John Wiley & Sons, Ltd.     

Measurements and modelling of the power performance of a model floating wind turbine under controlled conditions

This paper describes power performance measurements undertaken on a model floating wind turbine installed on a tension‐leg platform (TLP) in a wind/wave generator facility. Initially, the surge of the platform was measured under different rotor and wave conditions. The surge behaviour depended considerably on the rotor tip speed ratio and the wave frequency and amplitude. High‐frequency data sampling techniques were then used to derive the instantaneous power coefficient and tip speed ratio directly from the measurements, together with the surge velocity of the floating system. The power measurements were compared with those predicted by three independent numerical models, two of which are based on the blade element momentum approach and the third involving a lifting‐line free‐wake vortex model. The fluctuations of the power coefficient with time predicted by the three models were in close agreement; however, these were all significantly larger than those derived from the rotor shaft torque measurements. This was found to be due to the limitations of the torque measurement technique. Although being accurate in measuring the time‐averaged torque, the sensor was incapable of measuring the high‐frequency low‐amplitude fluctuations in the rotor shaft torque induced by the TLP surge. This was confirmed using an alternative experimental technique involving hot‐wire near‐wake measurements. The study also investigated the influence of the platform surge motion on the time‐averaged power coefficients. Both the measurements and the free‐wake vortex model revealed marginal deviations in the time‐averaged power coefficients when compared with those obtained for a fixed, non‐surging rotor. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal blade pitch control with realistic preview wind measurements

In above‐rated wind speeds, the goal of a wind turbine blade pitch controller is to regulate rotor speed while minimizing structural loads and pitch actuation. This controller is typically feedback only, relying on a generator speed measurement, and sometimes strain gages and accelerometers. A preview measurement of the incoming wind speed (from a turbine‐mounted lidar, for example) allows the addition of feedforward control, which enables improved performance compared with feedback‐only control. The performance improvement depends both on the amount of preview time available in the wind speed measurement and the coherence between the wind measurement and the wind that is actually experienced by the turbine. We show how to design a collective‐pitch optimal controller that takes both of these factors into account. Simulation results show significant improvement compared with baseline controllers and are well correlated with linear model‐based results. Linear model‐based results show the benefit of preview measurements for various preview times and measurement coherences. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐linear ultimate strength and stability limit state analysis of a wind turbine blade

According to the design codes for wind turbine blades, it is sufficient to evaluate the blade's limit states using solely a linear analysis. This study, however, shows the need of non‐linear analyses in blade design. Therefore, a geometrically non‐linear structural response of a 34 m blade under flap‐wise loading has been compared with a linear response to determine the blade's resistance in the ultimate strength and stability limit states. The linear analysis revealed an unrealistic failure mechanism and failure mode. Further, it did not capture the highly non‐linear response of the blade that was measured in an ultimate full‐scale test to failure and determined by a geometrically non‐linear analysis. A design evaluation in accordance with the least stringent Germanischer Lloyd (GL) requirements has been compared with non‐linear approaches proposed by GL and Eurocode, which require the application of an imperfection. The more realistic non‐linear approaches yielded more optimistic results than the mandatory linear bifurcation analysis. Consequently, the investigated blade designed after the lesser requirements was sufficient. Using the non‐linear approaches, considering inter‐fibre failure as the critical failure mode, yielded still a significant safety margin for the designer (7–28%). The non‐linear response was significantly dependent on the scaling of the imperfection. Eurocode's method of applying an imperfection appeared more realistic than the GL method. Since the considered blade withstood 135% of the design load at a full‐scale test to failure and the blade has operated successfully in the field, GL's safety factors combined with the imperfection size may be too conservative. Copyright © 2015 John Wiley & Sons, Ltd.     

Model‐independent periodic stability analysis of wind turbines

In this work, a new method is proposed for the stability analysis of wind turbines. The method uses input–output time histories obtained by conducting virtual excitation experiments with a suitable wind turbine simulation model. Next, a single‐input/single‐output periodic reduced model is identified from the recorded response and used for a stability analysis conducted according to the Floquet theory. Since only input–output sequences are used, the approach is model independent in the sense that it is applicable to wind turbine simulation models of arbitrary complexity. The use of the Floquet theory reveals a much richer picture than the one obtained by widespread classical approaches based on the use of the multi‐blade coordinate transformation of Coleman. In fact, it is shown here that, for each principal mode computed by the classical approach, there are in reality infinite super‐harmonics of varying strength fanning out from the principal one at multiples of the rotor speed. The relative strength of each harmonic in a fan provides for a way of measuring how periodically one specific fan of modes behaves. The notion of super‐harmonics allows one to justify the presence of peaks in the response spectra, peaks that cannot be explained by the classical time‐invariant analysis. The Campbell diagram, i.e., the plot of system frequencies vs. rotor speed, is in this work enriched by the presence of the super‐harmonics, revealing a much more complex pattern of possible resonant conditions with the per‐rev excitations than normally assumed. Copyright © 2014 John Wiley & Sons, Ltd.