Wind turbines are being used to generate electricity as
an alternative energy source to conventional fossil fu-
els, and it is well known that wind towers must to sus-
tain continuous vibration forces throughout their opera-
tional life. In this paper, a stability analysis of bending
deflection of a wind turbine steel tower is presented.
The wind turbine is modelled as the structure of a sim-
plified beam-column by a switched system. It is mod-
elled by using a Hamiltonian system, which simplifies
the system under study and allows to analyze the sta-
bility dynamics of the system. An eigenvalue analysis
have been done in order to analyze the stability of the
system; finally, also, some transient simulations of the
system are presented to verify the results obtainedPostprint (published version

Domínguez García, José Luis

García Planas, María Isabel

Mediano Valiente, Begoña

English

Wind turbines are being used to generate electricity as

an alternative energy source to conventional fossil fu-

els, and it is well known that wind towers must to sus-

tain continuous vibration forces throughout their opera-

tional life. In this paper, a stability analysis of bending

deflection of a wind turbine steel tower is presented.

The wind turbine is modelled as the structure of a sim-

plified beam-column by a switched system. It is mod-

elled by using a Hamiltonian system, which simplifies

the system under study and allows to analyze the sta-

bility dynamics of the system. An eigenvalue analysis

have been done in order to analyze the stability of the

system; finally, also, some transient simulations of the

system are presented to verify the results obtaine

UPCommons

Analysis of support structure of wind turbine tower

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PHYSCON 2013, San Luis Potosı´, Me´xico, 26–29th August, 2013ANALYSIS OF SUPPORT STRUCTURE OF WIND TURBINETOWERBegon˜a Mediano-ValienteDepartament de Matema`tica Aplicada IUniversitat Polite`cnica de CatalunyaSpainbego med@yahoo.esJ.L. Domı´nguez-Garcı´aIRECSpainjldominguez@irec.catM. Isabel Garcı´a-PlanasDepartament de Matema`tica Aplicada IUniversitat Polite`cnica de CatalunyaSpainmaria.isabel.garcia@upc.eduAbstractWind turbines are being used to generate electricity asan alternative energy source to conventional fossil fu-els, and it is well known that wind towers must to sus-tain continuous vibration forces throughout their opera-tional life. In this paper, a stability analysis of bendingdeflection of a wind turbine steel tower is presented.The wind turbine is modelled as the structure of a sim-plified beam-column by a switched system. It is mod-elled by using a Hamiltonian system, which simplifiesthe system under study and allows to analyze the sta-bility dynamics of the system. An eigenvalue analysishave been done in order to analyze the stability of thesystem; finally, also, some transient simulations of thesystem are presented to verify the results obtained.Key wordsWind turbine tower, switched linear systems, reacha-bility.1 IntroductionOne of the major forms to produce renewable energyis to use the wind and one of the most efficient waysof converting the kinetic energy in wind into mechani-cal power is extracting energy from wind, using Windturbines [1; 2].It is known that many wind turbine damage occurs dueto the structural failure of wind turbine towers. Themost of these failures are caused by the strong windstriking the structure or wind induced vibrations. Theaim of this paper is to analyze of bending deflection ofa wind turbine steel towerThe steel tower of wind-turbine structure can be sim-plified as a beam-column with varying diameter involvein bending and torsion.The Finite Elements theory is the most popularmethod used in strength and buckling analysis of wind-turbine structures. In this paper a new method to modelFigure 1. Structure of Wind Turbine Towerthe structure of a simplified beam-column based onswitched system is presented.Roughly speaking, a switched system is a family ofcontinuous-time (or discrete-time) dynamical subsys-tems and a rule that determines the switching betweenthem.Given an initial time t0, a switching path is a functionof time σ : [t0, T ) −→M = {1, 2, . . . , `}, T > t0.Definition 1.1. A switched system is a system whichconsists of several subsystems and a rule that orches-trates the switching between them.x˙(t) = Aσx(t) +Bσu(t), (1)where Aσ ∈ Mn(R), Bσ ∈ Mn×m(R), and x˙ =dx/dt.he pairs of matrices (Ak, Bk) for k ∈ M are referredas the subsystems of (1) Given an initial state x(t0) =x0, input u, and a switching path σ : [t0, tf ] −→ M ,the solution of state equation (1) is given byx(t) = eAik (ttk) . . . eAi0 (t1t0)x0+eAik (ttk) . . . eAi1 (t2t1)× ∫ t1t0eAi0 (t1τ)Bi0u(τ)dτ + . . .+∫ ttkeAik (tτ)Biku(τ)dτ,(2)for tk < t ≤ tk+1, 1 ≤ k ≤ s where t0, t1, . . . , ts isthe switching time sequence of σ, ts+1 = tf , and i0 =σ(t0), . . . is = σ(ts) is the switching index sequenceof σ.Consider nowΨ(t, σ, x0) = eAik (t−tk)eAik−1 (t−tk−1) . . . eAit0 (t−t0),t ∈ [tk, tk+1], so the state transition matrix is given byΦ(t1, t2, σ, x0) = ψ(t1, σ, x0)(ψ(t2, σ, x0))−1 and thesolution can be rewritten asφ(t, t0, x0, u, σ) =Φ(t, t0, σ, x0)x0 +∫ tt0Φ(t, τ, σ, x0)u(τ)dτ.The structure of the paper is as follows. In section2 modeling assumptions are presented, in section 3the structure of wind turbine tower is modelled as aswitched system, in section 4 the reachability of thesystem is analyzed. In section 5 some simulations havebeen realized and eigenvalues have been calculated fora particular steel tower in order to analyze the stabilityof the system. Finally, some conclusions are summa-rized in section 6.2 Modeling AssumptionsFor static analysis of a wind turbine tower, take intoaccount the following series of forces. The first onedue to gravity consist of a concentrated force at the topof the tower, representing the weight nacelle, gear box,runner generator, etc., and the weight of the tower it-self distributed along of its height. The second series offorces due to the wind resistance, distributed wind pres-sure along the height of the tower q(z) and the aerody-namics loads at the elevation of the engine axis. Thewind forces on the rotors are calculated and transferredto the top of the tower by equivalent static method con-sisting of the following three forces and moments1. Fy wind force2. Fx yaw force3. Fz vertical force4. My blade bending moment5. Mx tower bending moment6. Mz yaw momentThe direction of the y axis is horizontal in the winddirection, the z axis is vertical with upward direction.ZYXFigure 2. Wind Turbine tower seen as beam-column3 Switched system modelThe structure of wind turbine tower can be simplifiedas a beam column with linearly varying diameter andstepwise varying thickness subject to an axial force Pand it can be modeled by means the following differen-tial equation∂2∂z2(EI(z)∂2y∂z2)+∂∂z(P (z)∂y∂z)= q(z) (3)Where EI is the flexural rigidity, y(z) is the deflectionin the yz-plane and q is the transverse loadWhen the column is divided into a number of elementsit can be assumed that the axial force, wind loads, sec-tion modules are constant with respect z in each ele-ment.As a consequence the equation is reduced toEI∂4y∂z4+ P∂2y∂z2= q (4)corresponding to the linear dynamical system0 1 0 00 0 1 00 0 0 10 0 − PEIi 0yy′y′′y′′′+000qEIi =y′y′′y′′′yiv (5)Notice that y is the deflection, y′ = θ the slope angle,EIy′′ = M the moment and EIy′′′ = Q the shearforce at the corresponding section.Remark 3.1. A solution of the equation (5) is X(z) =Xp(z) + Xh(z) where Xp(z) is a particular solutionand Xh(z) is the general solution of the homogeneousassociated linear systemProposing a second order polynomial vector as a par-ticular solution, we obtain Xp(z) =q2P z2qP zqP0.The characteristic polynomial of the homogeneoussystem is t4 + PEI t2 and the roots are t = 0 doubleand t = ±√− PEI = ±√PEI i. SoXh(z) =αz + β + γ sin√PEI z + δ cos√PEI zα+ γ√PEI cos√PEI z − δ√PEI sin√PEI z−γ PEI sin√PEI z − δ PEI cos√PEI z−γ PEI√PEI cos√PEI z + δPEI√PEI sin√PEI zand the general solution isX(z) =αz + β + γ sin√PEI z + δ cos√PEI z +q2pz2α+ γ√PEI cos√PEI z − δ√PEI sin√PEI z +qP z−γ PEI sin√PEI z − δ PEI cos√PEI z +qP−γ PEI√PEI cos√PEI z + δPEI√PEI sin√PEI z .(6)In our particular setup the boundary conditions of thei section areyz=0 = yi−1, θz=0 = θi−1,Mz=0 = Mi−1, Qz=0 = Qi−1so, substituting the above conditions in the general so-lution of the linear system we obtain the values of theparameters for this case:α = y′i−1 +EIi−1P y′′′i−1,β = yi−1 +EIi−1P y′′i−1 − EIi−1P 2 q,γ = −EIi−1P 1√ PEIi−1y′′′i−1,δ = qP 2EIi−1 − EIi−1P y′′i−1.Substituting the value of these parameters to (6) wecan obtain the deflection, slope angle, moment andshear force status of the ends of the ithe beam section.In order to obtain the solution for any section from ini-tial conditions we consider the switched linear systemAσX +Bσ = X˙, σ = {0, 1, . . . , n}whereAi =0 1 0 00 0 1 00 0 0 10 0 − PEIi 0 , Bi =000qEIifor each i, z ∈ [z1, zi+1] [0, L) = [0, z1) ∪ [z1, z2) ∪. . . ∪ [zn, L).It is possible to obtain the state vector at any sectionof the wind turbine tower from initial state vector in thefollowing manner:Xi = Ai . . . A1X0+Ai . . . A2B1+. . .+AiBi−1+Bi.4 Reachability of the systemDefinition 4.1. A switched singular linear system issaid to be reachable when for any t0 ∈ R, xf andw ∈ Rn, there exists a real number tf > t0, a switch-ing well-defined path σ : [t0, tf ] −→ M and an in-put u : [t0, tf ] −→ Rm s.t.: φ(tf , t0, x0, u, σ) = xf ,φ˙(tf , t0, x0, u, σ) = w.Let us consider the following matrix sequence.N0 = (B1 ... B` )Nk = ( In A1 ... An−11 ... A` ... An−1` ) ·diag(Nk−1, . . . ,Nk−1),for k > 0Remark 4.1. Note thatrkN0 ≤ rkN1 ≤ · · · ≤ rkNn = rkNn+1 = . . .Proposition 4.1. A necessary and sufficient conditionfor reachability of a switched linear system (1) isrkNn = nProposition 4.2. If there exists j ∈ {1, . . . , n} suchthat Nj has full rank, then the switched linear system(1) is reachable.Remark 4.2. Obviously, when one of the subsystems(Ak, Bk) of system (1), is reachable, the system is alsoreachable.In our particular setup we analyze the reachability ma-trix of any subsystem (Ai, Bi) corresponding to the i-section of the beam0 0 0 qEIi0 0 qEIi 00 qEIi 0 0qEIi0 0 0λ1 λ2 λ3 λ40 0 j· 0.4849 · 10−3 -j· 0.4849 · 10−3Table 1. Eigenvalues of the first section−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5−4−3−2−1012345x 10−4Figure 3. Eigenvalues corresponding to the first sectionwhich rank is 4. So the subsystem (Ai, Bi) is reach-able, then the switched system is reachable.5 Application to a steel towerAs an application case of study, a steel tower for windturbine generation is considered. For the analysis, thefollowing parameters are considered P = 11000 · 9.81which represents the nacelle weight supported by thetower, E = 210000Nmm2corresponding to the steelmaterial, and I1 = 2.185·106, and I2 = 2.195·106 andI3 = 2.197 · 106, as the inertia value for the analyzedtower sections. They are different since the diameter ofthe tower is not constant.5.1 First SectionThe first section (lower inertia constant) represents asection of the wind turbine tower top part. Its inertia islower due to the fact that its diameter is smaller.From Figure 3 and Table 1, it can be observed that thesystem under study is marginally stable, since all theeigenvalues of the system exist on the negative or nullreal part of the complex plain. It is worth to say thatin Hamiltonian systems it can be understood as a stablesystem.In order to verify, the results obtained in the steadystate analysis (eigenvalues), some transient simulationsare presented. In Figure 4, it can be seen that all thevariable states are keeping almost constant since theaxis values are really low. Such results reveal that thesystem is keeping stable as predicted by the previousanalysis.024x 10−11To: Out(1)024x 10−11To: Out(2)012x 10−11To: Out(3)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52.17932.17942.17942.1794x 10−12To: Out(4)Impulse ResponseTime (sec)AmplitudeFigure 4. Impulse-response graphics corresponding to the first sec-tionλ1 λ2 λ3 λ40 0 j· 0.4838 · 10−3 -j· 0.4838 · 10−3Table 2. Eigenvalues of the second section−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5−4−3−2−1012345x 10−4Figure 5. Eigenvalues corresponding to the second section5.2 Second SectionThe second section (medium inertia constant) repre-sents a section of the wind turbine tower middle part.Its inertia is in between the maximum and minimumdue to the fact that its diameter is medium.From Figure 5 and Table 2, it can be seen that the sys-tem under study is marginally stable, as in the previouscase, since all the eigenvalues of the system do not be-long to the right hand plane.Again, some dynamic simulations are done to com-pare them to the results shown by the eigenvalues. InFigure 6, can be seen that the system is stable as ex-pected.012x 10−11To: Out(1)012x 10−11To: Out(2)2.17942.17942.1794x 10−12To: Out(3)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52.16942.16942.1694x 10−12To: Out(4)Impulse ResponseTime (sec)AmplitudeFigure 6. Impulse-response graphics corresponding to the secondsection5.3 Third SectionThe third section (bigger inertia constant) represents asection of the wind turbine tower lower part. Its inertiais higher since its tower diameter is bigger.As in all the previous cases, Figure 7 and Table 3 showthat the system under study is marginally stable, sinceall the eigenvalues of the system exist on the negativeor null real part of the complex plain.λ1 λ2 λ3 λ40 0 j· 0.4836 · 10−3 -j· 0.4836 · 10−3Table 3. Eigenvalues of the third section−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5−4−3−2−1012345x 10−4Figure 7. Eigenvalues corresponding to the third sectionFinally, the simulations are drawn in Figure 8, whereall the variable states are keeping almost constant, asoccur in all the previous cases.012x 10−11To: Out(1)2.17942.17942.1794x 10−12To: Out(2)2.16942.16942.1694x 10−12To: Out(3)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 52.16752.16752.1675x 10−12To: Out(4)Impulse ResponseTime (sec)AmplitudeFigure 8. Impulse-response graphics corresponding to the first sec-tion6 ConclusionTheoretical model of a wind turbine tower has beenintroduced, in this paper. An approximation of suchmodel to a hamiltonian system has been proposed. Itis worth to remark that wind turbine tower can be un-derstood as a switched system, which is an importantissue to consider. The proposed model is useful to pre-dict instability of the system depending on the parame-ters. Finally, in order to verify the results obtained fromthe model, dynamic responses and eigenvalue analysishave been developed considering real case parameters.ReferencesT. Burton, N. Jenkins, D. Sharpe, E. Bossanyi “WindEnergy Handbook” Second Edition, Whiley Ed. 2011.J. Fahad, To Study of Wind Resistant Stability ofTubular Wind Turbine Tower. PhD Thesis, 2012.L. Gurvits, R. Shorten, O. Mason, On the Stability ofSwitched Positive Linear Systems. IEEE Transactionson Automatic Control, 52-6, pp. 1099-1103, (2007).A.A. Mailybaev, H. Yabuno, and H. Kaneko. Optimalshapes of parametrically excited beams Struct Multi-disc Optim 27, pp. 435445 (2004).B. Mediano, Ana´lisis y simulacio´n del compor-tamiento de una tuberı´a mediante MEF. Master The-sis. UPC. (2011).B. Mediano, M.I. Garcı´a-Planas, Stability of apipeline hydraulic fluid with one end fixed. IPACSelectronic library, (2011).B. Mediano, M.I. Garcı´a-Planas, Stability Analy-sis of a Clamped-Pinned Pipeline Conveying Fluid.Preprint.B. Petersen, M. Pollack, B. Connell, D. Greeley, D.Davis, Ch. Slavik, B. Goldman, Evaluate the effect ofturbine period of vibration requirements on structuraldesign parameters: technical report of findingsw, Ap-plied Physical Sciences. (M10PC00066).Z. Sun, S.S. Ge, (2005) “Switched Linear Systems”.London England. Springer.M. Wang, Zh. Wang, H. Zhao, Analysis of wind-turbine steel tower by transfer matrix method. 2009Int. Conference on Energy and Environment Technol-ogy. DOI: 10.1109/ICEET.2009.133.

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Analysis of support structure of wind turbine tower

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