MOTION SIMULATION AND STRESS AND STRAIN ANALYSIS OF ELASTIC WIND POWER GENERATORS *

Transcription

1 th 11 National Congress on Theoretical and Applied Mechanics, 2-5 Sept. 2009, Borovets, Bulgaria MOTION SIMULATION AND STRESS AND STRAIN ANALYSIS OF ELASTIC WIND POWER GENERATORS * EVTIM ZAHARIEV, EMIL MANOACH Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. bl. 4, Sofa 1113, Bulgaria evtimvz@bas.bg, e.manoach@imbm.bas.bg MIHAIL TODODROV, VALENTIN ILIEV Technical University Sofia, Department of Aeronautics, bl. 10,8 Kliment Ohridski St. Sofia 1000, Bulgaria michael.todorov@tu-sofia.bg, viliev@aero.tu-sofia.bg ABSTRACT. In the paper motion simulation and stress and strain analysis of horizontal axis wind turbines is presented. Three blade design scheme is considered. Finite elements in relative coordinates are applied for modeling of large flexible deflections superimposed by the global motion of the blades. Spatial beam elements are used. Generalized Newton-Euler dynamic equations are applied for deriving the dynamic equations of motion. Motion simulation of the deflections and vibrations, stress and strain analysis of the tower and the blades is implemented. KEY WORDS: wind generators, dynamics, flexible systems 1. Introduction The main purpose of the structural model of a wind turbine is to be able to determine temporal variation of the loads in various components in order to estimate fatigue damage. To calculate the deflections and velocities of various components in the wind turbine in the time domain a structural model including the inertia terms is needed. Then the dynamic structural response of the entire construction can be calculated subject to the time dependent load found using an aerodynamic model, such as the beam element model. The wind turbines are large flexible structures which blades implement complex motion in space with high velocities and rapidly changing accelerations of the blade deflections. The vibrations cause unstable working conditions, noise and random loads of the units. Because of the changing * The financial support of National Fund Scientific Investigations, Ministry of Education and Science of Bulgaria, Contract MI-1507/05 is acknowledged.

2 Evtim Zahariev, Emil Manoach, Mihail Tododrov, Valentin Vasilev external loading over the blades the turbine shaft is imposed on bending and torsion including also impacts in the bearings. Modeling wind turbines for predicting of their power quality is reported in the literature. Models of wind turbines of varying complexity are presented. Electrical engineers, for example, tend to simplify the aerodynamic and mechanic parts of the system and usually emphasize on the generator description. In contrast, mechanical engineers often overlook generator performance details. Simplified aerodynamic modeling of wind turbines has been presented in [1]. The main idea in these articles is to adjust wind speed data at one point (hub level) by the use of various filters in order to represent the interaction of turbine blades with wind speed distribution over the rotor. The resulting wind data are applied to determine the driving torque. In contrast to this, an advanced approach to aerodynamic modeling, that uses a professional software package, has been presented in [2]. The complexity of the reported drive-train models varies considerably however rather simplified descriptions that often incorporate a soft shaft representation dominate completely in the literature. The soft shaft representation is studied, for example, in [3]. A drivetrain model representation that might suffer from the unavailability of system parameters is presented in [4] and [5]. Wind turbines are being designed larger and more flexible than ever before. The consequence is blade deflections that require special design and computational considerations regarding loads, displacements, blade tower interaction, etc. The aeroelastic tools available in the design process are generally based on modal analysis and a finite element (FE) representation. The general approach is to assume small displacements and to apply the aerodynamic loads in the undeflected position [6]. However, as the blades in some situations deflect considerably, the validity of the small-displacement assumption needs to be investigated. Examples of codes that take large deformations into account are ADAMS/WT and RCAS [7, 8]. The wind power generator (WPG) is a typical multibody system compiled of large flexible bodies (the blades) connected to a hub and shaft (rigid or flexible) that transmits the torque to a gear with elastic clutch and electric generator machine. The system is mounted on a tower and a base that also could be considered as a compilation of rigid and flexible bodies. And finally, the ground has elastic nature with damping properties that are to be taken into account especially during earthquake simulation. The methodology of the Multibody System Dynamics is able to solve many complicated problems related to nonlinear effects of large flexible structures including these in the WPGs. Multibody system dynamics is a rapidly developing discipline worldwide [9]. The methods and algorithms for deriving accurate dynamic equations of rigid and flexible multibody systems have its recent development. The classical methods of the finite element theory [10] have been constantly developing to solve successfully the-up-to date problems. Almost in every book of finite elements that discusses the dynamics of flexible systems the dynamic equations are presented in the following form Q e = M Δ&& e + C Δ& e + K e Δ, where Δ & is matrix-column compiled of linear and angular accelerations of the element

3 Motion simulation. of wind power generator nodes and these accelerations depend on the generalized velocities and accelerations, i.e.: Δ & = Δ&& ( q &, q& ). The matrices Me, Ce and Ke are the mass, damping and stiffness matrices, respectively; Q e is the matrix-column of the generalized forces. Although the flexible elements are much more complex than the rigid bodies, no velocity dependent term (similar to that of the inertia moments in the Newton Euler equations for rigid body) is assumed that leads to inaccurate dynamic equations of flexible elements. Zahariev [11] proposed a novel method and generalized Newton- Euler dynamic equations for rigid and flexible multibody systems. The equations are valid for every rigid and flexible body which kinetic energy is quadratic form of the quasi-velocities including finite element discretization. They are successfully applied for deriving precise dynamic equations. Mainly two wind turbine concepts with their control strategies have been applied in practice, namely: active stall control wind turbine with induction generator; variable speed, variable pitch wind turbine with doubly-fed induction generator. In the paper the precise dynamic models of active stall control and variable speed wind turbines are proposed. Motion and deflection analysis as a result of wind loading is achieved. The elasticity of blades and the tower, as well as, of the transmission and the gear is taken into account. Reonomic constraints that represent the external disturbances and compulsory motion affected by earthquakes and waves are included in the dynamic equations. Examples of simulation of elastic deflections and vibrations due to wind loading are presented. 2. Dynamics of a wind power generator In Fig. 1 a simplified presentation of a WPG design scheme is presented. In Fig. 2 its finite element discretization of the flexible part is depicted. The blades transform the wind power into a torque M w transmitted via the shaft to a gear that increases the blade angular velocity to an optimal value for the electric power generator. The blades and the shaft a firmly fixed to a hub presented as a rigid body. In the figure M e and ωe are the electric power generator torque and angular velocity, respectively. The reducer is assumed a rigid block with stiffness k r and damping coefficient c r presented by the manufacturer. The power unit is adjusted to an elastic tower and rigid body foundation. The elastic and damping properties of the ground are presented by elastic kg and damping cg coefficients Topology and kinematics A major step in the dynamics simulation of a multibody system compiled of rigid and flexible bodies is an algorithm for topology description. For numerical simulation of flexible system discretization of the continuum should be implemented and mass and stiffness properties of the flexible bodies are to be reduced to a finite number of points called nodes. Most often these nodes are treated just like points but actually they are coordinate systems. The node of a flexible element is a free object that, in the three dimensional space, has six degrees of freedom. The node motion is restricted by elastic forces acting between the neighbor nodes and defined by the

4 Evtim Zahariev, Emil Manoach, Mihail Tododrov, Valentin Vasilev stiffness properties of the material. Detailed explanations of the method and the procedures of discretization of flexible bodies into moving coordinate systems of elements and nodes are presented in [11]. The definition of the nodes as coordinate systems allows the flexible particles of the multibody systems, similarly to the systems of rigid bodies, to be decomposed to systems of moving coordinate systems connected by joints. These blades reducer shaft hub electric power generator tower foundation ground Fig. 1. A simplified presentation of a wind power generator design scheme electric power generator reducer c r M e ω e k r M w flexible nodes k g Fig. 2. Discretization of the flexible part of wind power generator c g

5 Motion simulation. of wind power generator coordinate systems are one and the same as for the rigid bodies so for the flexible elements and nodes, and their kinematic schemes are presented by a consequence of adjacent coordinate systems connected by joints. So, from the kinematic point of view it does not matter if the coordinate systems correspond to rigid or flexible bodies. An example of rigid and flexible bodies and their connectivity in kinematic chains are shown in Fig. 3 a. In the figure a rotating beam composed of rigid and flexible parts is depicted. The beam presented by a coordinate system i is compiled of two rigid parts part 1 (a hub) and part 2 (a rigid body beam). Their coordinate systems are also numbered by 1 and 2 (Figure 3, b). The two coordinate systems are firmly fixed to coordinate system i and are part of it. The flexible part of the beam is compiled of flexible element 1 and 2, as well as, from nodes 1, 2 and 3. Node 1 is presented by coordinate system 3, node 2 by 4, and node 3 by 5. The mass objects, rigid bodies and nodes are firmly fixed to the moving coordinate systems as follows: rigid part 1, rigid part 2 and node 1 to coordinate system i; node 2 to 4; node 3 to 5. In Figure 3 (c) the topological graph of the discretized beam is presented. So, the coordinate systems i, 4 and 5 compile an open kinematic chain. In the figure the numbers of the black circles present the moving coordinate systems. The numbers enclosed by circles point out the relative degrees of freedom of the adjacent coordinate systems. Z i body i node 1 node 2 node 3 Z Z i Z 1 2 Z 3 Z Z 4 5 part 1 Y i X 1 X i element 1 element 2 Y 5 part 2 X i X 2 X3 X X 4 5 (a) base 0 i 4 5 (b) (c) Fig. 3. Discretization of a rotating rigid and flexible beam Following this consequence the kinematic scheme of a discretized structure of the WPG (Fig. 2) is shown in Fig. 4. With arrows the flexible nodes are pointed out. With dense circles the rigid bodies are pointed out, which number and position depends on the WPG design scheme. Each flexible node adds six dof. Some of them could be neglected (for example the longitudinal deflections) that depends on the loading and the comparison between the relative deflections in the elements. The translational joints numbered with 1 and 2 present the possible two dof displacements of the ground. With m f the mass of the foundation is denoted. The flexible blades are connected to a common hub presented as a rigid body with mass m h and inertia

6 Evtim Zahariev, Emil Manoach, Mihail Tododrov, Valentin Vasilev m e J e 5 k e c e ω e Jr mr 4 k r c r ωr J b m b 3 ωw J h m h q 2 2 m f 1 q 1 c g2 k g2 c g 1 k g1 Figure 4: Kinematic scheme of the discretized structure of the wind power generator tensor J h. The end nodes and the blades coincide with the hub. Three nodes, in the hub, joints 3 and 4, represent the flexible shaft and the elastic clutch between the gear and the turbine. The gear (mass properties m r and J r ) is located in joint 4. The stiff and damping properties of the gear ( k r, c r ) are provided from the manufacturer. The nodes in joints 4 and 5 represent the flexible shaft and clutch (stiff and damping k e, c e ) between the electric generator and the gear. The dof of both flexible elements presenting the shafts are 1 (the other 5 are neglected) since the axial rotational deflections are dominant. The coordinate systems of the rigid bodies and the nodes in some cases (hub and blades) could coincide. Further details for discretization of a flexible structure could be found in [11]. The definition of the nodes as coordinate systems allows the flexible particles of a multibody system, similarly to the systems of rigid bodies, to be decomposed to systems of moving coordinate systems connected by joints. These coordinate systems are one and the same as for the rigid bodies so for the flexible elements and nodes, and the kinematic scheme (open or closed) is presented by consequence of adjacent coordinate systems connected by joints. So, from the kinematic point of view it does not matter if these coordinate systems correspond to rigid or flexible bodies and a generalized algorithm for dynamic analysis presented in [11] is applied.

7 Motion simulation. of wind power generator 2.2. External forces The main loading of the structure of the WPGs is excited by the wind. Modeling of the aerodynamics of the blades is the base for estimation of the forces that set turbine in motion and produce torque and electricity. The spanwise velocity of a cross-section of the turbine blades is much lower than the streamwise components, and it is therefore assumed in many aerodynamic models that the flow at a given radial position is 2-dimensional. Two dimensional flow realized in a plane is described with a coordinate system XY as shown in Fig. 5, the velocity components in the z-direction being zero. In order to realize 2-D flow it is necessary to neglect the velocity of the airfoil along the blade, i.e. perpendicular to the blade crosssection. This velocity is much more less than the airfoil tangential velocity because of the blade rotation. Never-the-less, some of the aerodynamic models consider 3D flow. The 2D aerodynamics is of practical interest. Fig. 5 shows the airflow around of the airfoil, the two zones of high an low pressure and of stagnation streamline and leading edge stagnation point. The relative wind velocity V r is the vector resulting of the sum of the blade cross-section tangential velocity ( ω. ρ ) and the wind velocity ( V w ) vectors The force F from the flow is decomposed into the perpendicular direction to the velocity V r and to direction parallel to it. The former component is known as the lift, L; the latter is called the drag, D (see Fig. 5). Their values depend on the relative velocity of the fluid and on the lift and drag coefficients Cl, Cd. To describe the forces completely it is also necessary to know the moment M about a point in the airfoil. This point is often located on the chord line at c 4 from the leading edge. In addition, for a precise estimation of the torque generation, it is important to consider the leakage at the tip of the blades, which can be well described with a 3-D aerodynamics representation. This leakage produces a vortices system that reduces the angle of attack α seen locally on the blades and consequently decreases the power extracted from the wind [6] L ω.r V r leading edge F low pressure zone stagnation point α M V w D stagnation streamline high pressure zone WPG horizontal axis c y x WPG vertical axis Fig. 5. Airfoil streamline and distribution of the wind lading to the blade nodes

8 Evtim Zahariev, Emil Manoach, Mihail Tododrov, Valentin Vasilev 2.3. Reonomic constraints Reonomic constraints for structures of WPGs are external disturbances and compulsory motion. Most often they are caused by ground shaking because of earthquakes and, for example of floating dock structures, the motion of the platforms. The reonomic constraints are included in the dynamic equations as algebraic constraints. For example, the simplest way the disturbances because of earthquakes to be taken into account it is restrictions of the motion of the ground to be imposed. In analytical form they could be functional dependencies of the ground motion with respect to time, i.e.: q1 = q1( t), q 2 = q 2 ( t), where q1 and q2 are the coordinates of two dimensional compulsory motion of the ground (Fig. 4). These dependences are estimated from experimental investigations and data collection from previous earthquakes and explosions and comprise the frequencies and amplitudes of the deflections. For the purposes of the simulation they are interpolated by polynomials and presented in analytical form. The reonomic constraints for floating docks are similar. The data for the characteristics of the disturbances are obtained from experimental investigation or solution of hydro-dynamical problems. The possible motion of the foundation is three dimensional. Particular reonomic constraints are to be taken into account for dynamic simulation of both types of WPGs the constant speed and variable speed (stall controlled) turbines. The constant speed turbines are directly coupled to the grid and the electric machine generate electricity with constant frequency and voltage. As a result the torque on the rotor varies. The reonomic constraint is expressed in analytical form as constant speed of the generator, i.e. ω e = A = const. The variable speed turbines do not directly couple the grid. Power electronic converter is used as intermediate unit to transform the electricity from the turbine constant frequency and voltage. The reonomic constraint for such turbines is expressed as a condition for constant torque of the electric generator Explicit form dynamic equations The dynamic equations of a multibody system are presented by a system of ordinary differential equations of the unconstrained systems with respect to all (of number n) coordinates subject to algebraic constraints (of number m). The algebraic constraints define the number (g) of the generalized coordinates q (g=n-m). In the paper the dynamic equations of a rigid and flexible multibody system are defined with respect to the generalized coordinates as follows: & & \ & (2.1) M q = P q B( q) q = Φ where M is g g mass matrix B is g g g is three dimensional matrix and P are the generalized external forces, superscript \ denotes matrix transpose. 3. Example The example presented in the paper consider constant speed wind turbine. This restriction for the turbine leads, because of the variable speed of the wind, to the

9 Motion simulation. of wind power generator exaggeration of variations of the turbine shaft torque and, respectively, in the structure. In the example the full nonlinear dynamic model of the WPG is regarded and the deflections of the structure are simulated. The example consider the following mass and inertia parameters: mass of the reducer and the electric power generator 300 kg; mass of the hub 100 kg; mass of the foundation kg; length of a blade 10 m; length of the tower 10 m; length of an flexible element m. The size and stiffness properties of the elements are as follows: data for the blade elements: E = ;G = ; L = 10; ρ = ρ = ; I x = I z = Ic = ;S = data for the tower elements: E = ;G = ; L = 10; ; I x = I z = Ic = ;S = In Fig. 6 the results of the analysis and simulation are presented. ω t Angular velocity of the turbine V w ω X Y Snapshot of the turbine at t=15 M Torque of the input shaft Relative deviations of the blade tips Deviations of the tower tip ± t Fig. 6. Analysis of constant speed WPG

10 Evtim Zahariev, Emil Manoach, Mihail Tododrov, Valentin Vasilev 3. Conclusion A methodology for analysis and motion simulation of WPG is proposed. Precise dynamic model based on generalized Newto-Euler dynamic equations is developed. The algorithm is experimentally tested on a WPG virtual design scheme. R E F E R E N C E S [1] WILKIE, J., W.E. LEITHEAD, C.ANDERSON, Modelling of Wind turbines by Simple Models. Wind Engineering, 13 (1990) No. 4. [2] BOSSANYI, E.A., P. GARDNER, L. CRAIG, Z. SAAD-SAOUD, N. JENKINS, J. MILLER, Design tool for prediction of flicker. European Wind Energy Conference, Dublin Castle, Ireland (1997). [3] AKHMATOV, V., H.KNUDSEN, A.H. NIELSEN, Advanced simulation of windmills in the electrical power supply. International Journal of Electrical Power and Energy Systems, 22 (2002), No. 6, [4] LEITHEAD, W.E., M.C.M. ROGERS, Drive-train Characteristics of Constant Speed HAWT's: Part I - Representation by Simple Dynamics Models. Wind Engineering, 20 (1996), No. 3. [5] LEITHEAD, W.E., M.C.M.ROGERS, Drive-train Characteristics of Constant Speed HAWT's: Part II - Simple Characterization of Dynamics. Wind Engineering, 20 (1996) No. 3. [6] RASMUSSEN, F., M.H. HANSEN, K. THOMSEN, T.J. LARSEN, F. BERTAGNOLIO, J. JOHANSEN, H.A. MADSEN, C. BAK, A.M. HANSEN, Present status of aeroelasticity of wind turbines. Wind Energy, (2003), No. 6: [7] HANSEN, A.C., D. J. LAINO, User s guide to the Wind Turbine Dynamics Computer Programs YawDyn and AeroDyn for ADAMS. Technical Report, Mechanical Engineering Department, University of Utah, Salt Lake City, UT (1998). [8] JONKMAN, J, J.,COTRELL, A demonstration of the ability of RCAS to model wind turbines. Technical Report NREL/TP , National Renewable Energy Laboratory, Golden, CO, (2003). [9] SCHIEHLEN, W., Multibody Dynamics Fundamentals and Applications, In: Multibody Dynamics: Monitoring and Simulation Techniques III (Rahnejat H. and Rothberg S., eds.), Professional Engineering Publishing, Longhborough, UK, (2004), [10] ZIENKIEWICZ, O. C., The Finite Element Method, McGraw-Hill, (1979). [11] ZAHARIEV E. V., Generalized Finite Element Approach to Dynamics Modeling of Rigid and Flexible Systems, Mechanics Based Design of Structures and Machines, Taylor & Francis Group, 34(2006), No. 1,