Damping of Wind Turbine Tower Oscillations through Rotor Speed ...

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model with concentrated parameters. This can be done using modal analysis that is very common tool in wind turbine analysis [1], [3]. It describes a complex oscillatory structure as a composition of several simple oscillatory systems each of them being described by means of mass, stiffness and damping. By this representation complex tower oscillations are seen as a sum of many simple oscillations characterized by their modal frequencies which are one of the most important structural properties of wind turbine. It has been shown in practice [5] that fairly good modeling of wind turbine tower nodding can be achieved using two modal frequencies (two modes). Since we are here primarily interested in building model suitable for controller design we use only the first modal frequency. The justification for this lies in the fact that for the turbine in scope second modal frequency is more than 6 times greater than the first modal frequency and therefore falls out of the controller frequency bandwidth. By using only one modal frequency tower dynamics can be described as: Mx �� + Dx� + Cx = F , (9) t t t where M, D, and C are modal mass damping and stiffness respectively and F is the generalized force that is originated by wind and that causes wind turbine tower oscillations. Tower modal properties in expression (9) are related to first tower modal frequency ω0t as follows [4]: D= 2 ζ ω ⋅M, t 0t 2 0t ( ω ) C = ⋅M, (10) where ζ t is structural damping. For steel structure structural damping is mostly set to 0.005 [4]. Modal mass M can be calculated as [1]: ht 0 ( ) ( ) 2 φ M = ∫ m h h dh, (11) where ht is the height of the tower, m(h) is the mass distribution along the tower and φ ( h) is the tower's first mode shape. Note that actual distribution of mass along tower has to be modified in order to include mass of the rotor and the nacelle which is assumed to be concentrated at the tower top. Driving force F is mostly the rotor thrust force Ft caused by wind. It can be shown [4] that thrust force, similar to aerodynamic torque, depends upon wind speed, rotor speed and pitch angle. So similarly to (6) it can be expressed as [4]: 1 2 2 Ft = ρairR πvwCt( λ, β) , (12) 2 where Ct is so called thrust coefficient. Expressions (6), (8), (9) and (12) form the simplified nonlinear model of wind turbine that is used in the following sections for controller design. Model is summarized below taking into account the fact that wind speed seen by the rotor is a sum of wind speed and tower nodding speed: J � tω= Mr −Mg −Ml, 1 3 2 Mr = ρairR πCQ( λ, β) ⋅( vw −x�t) , 2 . (13) 1 2 2 Ft = ρairR πCt( λ, β) ⋅( vw −x�t) , 2 F = Mx �� + Dx�+ Cx. t t t t Torque and thrust coefficients Cq and Ct are usually provided by wind turbine blade manufacturers or can be calculated using professional simulation tools. 4. PID Controller The PID controllers are still by far the most used controllers for wind turbine speed and power control. This is due to their simplicity and rather high robustness. The small number of parameters makes possible for designer to quickly arrive at satisfactory, although in many cases suboptimal, system behavior. As stated before wind turbine dynamics change in nonlinear fashion with change in wind speed. To control the wind turbine with linear controllers gain scheduling has to be used [3]. Although it seems straightforward to use wind speed as scheduling criterion this is not an appropriate solution since the wind speed is not measured fast and accurately enough. Therefore measured pitch angle is usually used as scheduling variable. For the design of a PID controller using analytical methods process model (13) has to be linearised around chosen operating point. After linearization of expressions (13) and
transition to Laplace domain simple algebraic manipulations yield transfer functions that are needed for controller design. Those transfer functions are: and ( ) G s β w ( ) G s Δω = Δβ w ( s) ( s) ( s) ( ) Δω = Δv s (14) . (15) These transfer functions are of the third order. A good insight in system properties of the wind turbine can be gained if we examine frequency characteristics of transfer function (14) shown in fig. 3. Figure 3: Frequency characteristics of G β . It can be observed that frequency characteristics of transfer function (14) at tower modal frequency exhibits magnitude and phase drop. Similar phenomena are present in frequency characteristics of (15) as well. This fact makes the pitch controller design very difficult. Physical explanation for observed effects becomes clear from the following analysis. Change in wind speed causes change in rotor speed what requires controller action and pitching of the blades in order to regulate the rotor speed to its rated value. Pitching the rotor blades, besides the aerodynamic torque, alters the thrust force significantly. Thrust force, according to (13), causes change in wind turbine tower top speed and thus the wind speed seen by the rotor is changed. This alters the aerodynamic conversion and in this way a feedback is formed. For this reason wind turbine can easily be driven into oscillatory behavior if the pitch controller is not designed properly. To prevent the pitch controller from driving the wind turbine into oscillatory behavior it must be assured that system frequency bandwidth is below the first tower modal frequency. Moreover, sufficiently small magnitude is required at the first modal frequency. As it can be seen from fig. 3. the first modal frequency of the turbine in scope is 3 rad/s so a bandwidth of 1 rad/s was chosen. PID controller was designed to assure phase margin of around 60 o what gives satisfactory behavior of the system. To fully explore the system behavior with chosen controller simulation tool GH Bladed was used. GH Bladed is professional simulation package designed for wind turbine simulations and load calculations [5]. It relies upon very complex mathematical model based on combined blade element and momentum theory [1]. Structural properties of the wind turbine are modeled in detail and inertial and gravitational loads are taken into account along with aerodynamic ones. Extensive testing showed that simulation results obtained in Bladed are in accordance with measurements taken on actual wind turbines what was recognized by major standardization and certification institutions (e.g. Germanischer Llyod). To model the structural properties of explored turbine many modes are used [5]. Tower nodding is modeled with two modes as well as tower naying (tower side-side motion). Rotor blades' motion in flapwise direction is modeled with 6 modal frequencies while blades' motion in edgewise direction (displacement of the rotor blades in the plane of rotation) is modeled with 5 modal frequencies. Wind shear and tower shadow are included in the model as well. PID controller designed based on linearised model (14) and (15) was implemented in C and included as external discrete time controller. In that way we could use Bladed for controller testing. In the following figures behavior of the system when PID controller is used for rotor speed control is shown for one representative operating point ( v w = 15 m/s). Wind that was used for simulation observed positive and negative stepwise change shown in fig. 4. Responses of rotor speed, pitch angle and tower top displacement are shown in figs. 5. 6. and 7. respectively. From these figures it can be seen that rotor speed is well regulated and quickly compensated for the influences of wind speed changes. The pitch control actions are moderate without any oscillations. Similar results were obtained for all operating points throughout wind turbine operating range.
Page 1 and 2: March 29 - April 1 International Co
Page 3: Pr = Pw⋅ CP. (3) The theoretical
Page 7 and 8: From (16) and (17) it follows that
Page 9 and 10: Pitch angle [deg] 16 15 14 13 12 11

Damping of Wind Turbine Tower Oscillations through Rotor Speed ...

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