\r\nfor a three-bladed horizontal axis wind turbine (HAWT) suited

\r\nfor control applications. It is based on a 8-dofs, lumped

\r\nparameters structural dynamics coupled with a quasi-steady sectional

\r\naerodynamics. In particular, using the Euler-Lagrange Equation

\r\n(Energetic Variation approach), the authors derive, and successively

\r\nvalidate, such model. For the derivation of the aerodynamic model,

\r\nthe Greenbergs theory, an extension of the theory proposed by

\r\nTheodorsen to the case of thin airfoils undergoing pulsating flows,

\r\nis used. Specifically, in this work, the authors restricted that theory

\r\nunder the hypothesis of low perturbation reduced frequency k,

\r\nwhich causes the lift deficiency function C(k) to be real and equal

\r\nto 1. Furthermore, the expressions of the aerodynamic loads are

\r\nobtained using the quasi-steady strip theory (Hodges and Ormiston),

\r\nas a function of the chordwise and normal components of relative

\r\nvelocity between flow and airfoil Ut, Up, their derivatives, and

\r\nsection angular velocity ε\u02d9. For the validation of the proposed model,

\r\nthe authors carried out open and closed-loop simulations of a 5

\r\nMW HAWT, characterized by radius R =61.5 m and by mean chord

\r\nc = 3 m, with a nominal angular velocity Ωn = 1.266rad\/sec.

\r\nThe first analysis performed is the steady state solution, where

\r\na uniform wind Vw = 11.4 m\/s is considered and a collective

\r\npitch angle θ = 0.88\u25e6 is imposed. During this step, the authors

\r\nnoticed that the proposed model is intrinsically periodic due to

\r\nthe effect of the wind and of the gravitational force. In order

\r\nto reject this periodic trend in the model dynamics, the authors

\r\npropose a collective repetitive control algorithm coupled with a PD

\r\ncontroller. In particular, when the reference command to be tracked

\r\nand\/or the disturbance to be rejected are periodic signals with a

\r\nfixed period, the repetitive control strategies can be applied due to

\r\ntheir high precision, simple implementation and little performance

\r\ndependency on system parameters. The functional scheme of a

\r\nrepetitive controller is quite simple and, given a periodic reference

\r\ncommand, is composed of a control block Crc(s) usually added

\r\nto an existing feedback control system. The control block contains

\r\nand a free time-delay system eτs in a positive feedback loop, and a

\r\nlow-pass filter q(s). It should be noticed that, while the time delay

\r\nterm reduces the stability margin, on the other hand the low pass

\r\nfilter is added to ensure stability. It is worth noting that, in this

\r\nwork, the authors propose a phase shifting for the controller and

\r\nthe delay system has been modified as e^(−(T−γk)), where T is the

\r\nperiod of the signal and γk is a phase shifting of k samples of the

\r\nsame periodic signal. It should be noticed that, the phase shifting

\r\ntechnique is particularly useful in non-minimum phase systems, such

\r\nas flexible structures. In fact, using the phase shifting, the iterative

\r\nalgorithm could reach the convergence also at high frequencies.

\r\nNotice that, in our case study, the shifting of k samples depends

\r\nboth on the rotor angular velocity Ω and on the rotor azimuth

\r\nangle Ψ: we refer to this controller as a spatial repetitive controller.

\r\nThe collective repetitive controller has also been coupled with a C(s) = PD(s), in order to dampen oscillations of the blades.

\r\nThe performance of the spatial repetitive controller is compared

\r\nwith an industrial PI controller. In particular, starting from wind

\r\nspeed velocity Vw = 11.4 m\/s the controller is asked to maintain the

\r\nnominal angular velocity Ωn = 1.266rad\/s after an instantaneous

\r\nincrease of wind speed (Vw = 15 m\/s). Then, a purely periodic

\r\nexternal disturbance is introduced in order to stress the capabilities

\r\nof the repetitive controller. The results of the simulations show that,

\r\ncontrary to a simple PI controller, the spatial repetitive-PD controller

\r\nhas the capability to reject both external disturbances and periodic

\r\ntrend in the model dynamics. Finally, the nominal value of the

\r\nangular velocity is reached, in accordance with results obtained with

\r\ncommercial software for a turbine of the same type.","references":null,"publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 117, 2016"}