structural design optimization of wind turbine towers · structural design optimization of wind...

Structural design optimization of wind turbine towers

Hani M. Negma, Karam Y. Maalawib,*aAerospace Engineering Department, Cairo University, Cairo, Egypt

bMechanical Engineering Department, National Research Center, Cairo, Egypt

Received 22 May 1998; accepted 18 February 1999

Abstract

This paper describes several optimization models for the design of a typical wind turbine tower structure. Themain tower body is considered to be built from uniform segments where the e�ective design variables are chosen tobe the cross-sectional area, radius of gyration and height of each segment. The nacelle/rotor combination is

regarded as a rigid non-rotating mass attached at the top of the tower. Five optimization strategies are developedand tested. The last one concerning reduction of vibration level by direct maximization of the system naturalfrequencies works very well and has shown excellent results for both tower alone and the combined tower/rotor

model. Extensive computer experimentation has shown that global optimality is attainable from the proposeddiscretized model and a new mathematical concept is given for the exact placement of the system frequencies. Thenormal mode method is applied to obtain forced response for di�erent types of excitations. The optimization

problem is formulated as a nonlinear mathematical programming problem solved by the interior penalty functiontechnique. Finally, the model is applied to the design of a 100-kW horizontal axis wind turbine (ERDA-NASAMOD-0). It has succeeded in arriving at the optimum solutions showing signi®cant improvements in the overallsystem performance as compared with a reference or baseline design. # 2000 Elsevier Science Ltd. All rights

reserved.

Keywords: Renewable energy; Wind turbine; Vibration; Tower structures; Dynamics; Optimization

1. Introduction

Many wind energy research and development pro-grams have been initiated since the 1973 oil crisis, and

di�erent con®gurations of wind turbines have beeninstalled in many countries. These clean energy sourcescan make a substantial and economically competitivecontribution to the future energy needs. Concerning

system design optimization, most of the publishedresearch work has mainly been focused on power-out-put economics and cost optimization [1,2]. Little may

be found that deals with the problem of structural de-

sign optimization. It is the main intent of the present

study to consider the basic aspects of design optimiz-

ation of the combined tower/rotor structure of a wind

turbine. A good survey of the various theories and nu-

merical methods of structural design optimization can

be found in Refs. [3,4] which are usually classi®ed into

two groups: the mathematical programming approach,

and the optimality criteria approach. The latter was

employed by Takewaki [5] who considered optimiz-

ation of a concrete tower structure with round tubular

cross section. An approximate concept was applied for

®nding the optimal bending sti�ness distribution which

minimizes the total structural weight for a speci®ed

Computers and Structures 74 (2000) 649±666

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* Corresponding author.

fundamental natural frequency. Later on, Takewakisolved the same problem taking into consideration the

¯exibility e�ect of the tower base connection [6]. Heused ®nite element method and a piecewise-linear func-tion to approximate the sti�ness distribution along the

tower height.In the present paper, several optimization strategies

for the design of a wind turbine tower are developed

and tested, and di�erent forms of the objective func-tion are examined. The tower is considered to be builtfrom uniform round tubular segments (modules),

where the e�ective design variables are the cross-sec-tional area, radius of gyration and height of each seg-ment. Isolated tower dynamics including the completekinematical analysis and formulation of the applied

loads are treated in detail. A simpli®ed set of the gov-erning dynamical equations of motion is given in anappropriate non-dimensional form, and an exact

method for the determination of the dynamic charac-teristics and forced response is presented. The ®nalmodel formulation identi®es the optimum values of the

design variables which make the tower structure ful®lits design objectives in the best possible manner whilesatisfying all design constraints. The problem is formu-

lated as a nonlinear mathematical programming pro-blem. Optimum design solutions are obtained by theinterior penalty function method coupled with Powell'smulti-dimensional and quadratic-interpolation one-

dimensional search techniques.Implementation of the ®nal model form to an exist-

ing wind turbine; the ERDA-NASA MOD-0 machine,

is given and the optimum design trends are examinedand discussed. It is demonstrated that global optimal-ity can be attained from the proposed discretized

models, and exact placement of the tower frequenciesis also achievable.

2. Optimization problem formulation

In formulating an optimization problem, three prin-cipal phases must be considered:

. De®nition and measure of the system design objec-tives.

. Selection of the design variables and preassigned

parameters.. De®nition of the design constraints.

2.1. Basic assumptions

. The basic structural model of the tower is rep-

resented by an equivalent long, slender cantileverbeam built from segments (modules) having di�erentbut uniform cross-sectional properties. The tower is

cantilevered to the ground, and is carrying a concen-trated mass at its free end approximating the inertia

properties of the nacelle/rotor unit. This mass isassumed to be rigidly attached to the tower top.

. Material of construction is linearly elastic, isotropic

and homogeneous. The tower has a thin-walled cir-cular cross-section.

. The Euler±Bernoulli beam theory is used for predict-

ing de¯ections. Secondary e�ects such as axial andshear deformations, and rotary inertia are neglected.

. Distributed aerodynamic loads are restricted to pro-

®le drag forces. A two-dimensional (2D) steady ¯owmodel is assumed.

. Nonstructural mass will not be optimized in the de-sign process. Its distribution along the tower height

will be taken equal to some fraction of the structuralmass distribution.

. Structural analysis is con®ned only to the case of

¯apping motion (i.e. bending perpendicular to theplane of rotor disk).

2.2. Tower design objectives

A wind turbine tower is the main structure which

supports rotor, power transmission and control sys-tems, and elevates the rotating blades above the earthboundary layer. A successful structural design of the

tower should ensure e�cient, safe and economic designof the whole wind turbine system. It should provideeasy access for maintenance of the rotor componentsand sub-components, and easy transportation and

erection. Good designs ought to incorporate aestheticfeatures of the overall machine shape.In fact, there are no simple criteria for measuring

the above set of objectives. However, it should berecognized that the success of tower structural designis judged by the extent to which the wind turbine main

function is achieved. The di�erent optimization strat-egies considered herein are discussed below.

2.2.1. Light weight designA minimum weight structural design is of para-

mount importance for successful and economic oper-

ation of a wind turbine. The reduction in structuralweight is advantageous from the production and costpoints of view. For a tower composed of Ns segmentsthe weight (mass) function, to be minimized, can be

expressed in the non-dimensional form

minimize; M �XNs

k�1DktkHk �1�

For the de®nition of the various parameters, refer toTable 1 and Table 2

H.M. Negm, K.Y. Maalawi / Computers and Structures 74 (2000) 649±666650

2.2.2. High sti�nessThe main tower structure must possess an adequate

sti�ness level. Maximization of the sti�ness is essentialto enhance the overall structural stability and decreasethe possibility of fatigue failure. For a cantilevered

tower, sti�ness can reasonably be measured by themagnitude of a horizontal force applied at the free endand producing a maximum de¯ection of unity. This

can be expressed mathematically as in Ref. [7]:

maximize; S � 1=XNs

k�1

Hk

Ik

�1ÿ �xk�1 � xk �

� 1

3

�x 2k�1 � xkxk�1 � x 2

k

�� 2�

2.2.3. High (sti�ness/mass)-ratioPerhaps, maximization of the sti�ness-to-mass ratio,

which is directly related to the physical realities of thedesign, is a better and more straight forward design

criterion than maximization of the sti�ness alone orminimization of the structural mass alone. The related

mathematical expression can be obtained by dividingEq. (2) by Eq. (1).

2.2.4. Design for minimum vibrationMinimization of the overall vibration level is one

of the most cost-e�ective solutions for a successfulwind turbine design. It fosters other important design

goals, such as long fatigue life, high stability andlow noise level. Two di�erent criteria for measuringvibration reduction are stated in the following

sections [7].

2.2.4.1. Frequency-placement criterion. Reduction of vi-

bration can be achieved by separating the natural fre-quencies of the structure from the exciting frequenciesto avoid large amplitudes caused by resonance [8].

This may be measured by minimizing the performanceindex

Table 1

De®nition of reference beam parameters

Parameter Notation Description

Height H0 Tower reference height

Diameter D0 Mean value of the inner and outer diameters

Thickness t0 Wall thickness

Cross-sectional area (structurally e�ective) A0 A0 � pD0t0Second moment of inertia of the cross section I0 I0 � pD3

0t0=8:0Radius of gyration r0 � ��

I0=A0

p � D0=2��2p

Mass density r0 � rTNon-structural mass per unit length mns0 Not structurally e�ective in carrying stresses (e.g. stairs)

Structural mass/length ms0 r0A0

Mass/unit length m0 � ms0 �mns0

Young's modulus E ET

H.M. Negm, K.Y. Maalawi / Computers and Structures 74 (2000) 649±666 651

minXi

Wfi

ÿoi ÿ o�i

�2 �3�

where Wfi are weighting factors, oi are the actual fre-quencies and o�i are the corresponding desired (target)

values chosen to be within close ranges (called fre-quency-windows) of those of a reference or baselinetower design, which are well separated from the critical

exciting frequencies.

2.2.4.2. Maximum frequency criterion. Another alterna-

tive for reducing vibrations is the direct maximizationof the system natural frequencies. Higher natural fre-quencies are favorable for reducing both of the steady-

state and transient responses of the tower. Szyszkowski[9] considered maximization of the fundamental fre-quency for given structural weight via a multi-modaloptimality criterion. Another work dealing with opti-

mum shapes of beams for maximum fundamental fre-quency has been studied by Masad [10], who presentedan e�cient numerical method for computing the rate

of change of the associated eigenvalues. The present

work considers a di�erent optimization strategy bymaximizing a weighted-sum of the system natural fre-

quencies. This is described mathematically by the fol-lowing linear composite form of the objective function[7]:

maxXi

WfiGbi �4�

where Gbi � ��oip

are the non-dimensional frequency

parameters of bending vibrations (Eq. (14)).

2.3. The preassigned parameters

The following tower variables will be given preas-

signed ®xed values in order to decrease the dimension-ality of the optimization problem:

1. tubular single-pole con®guration

2. total height, H3. type of cross section is thin-walled hollow circular

cross section

4. material of construction is chosen to be steel

Table 2

De®nition of non-dimensional quantitiesa

Quantity Notation Non-dimensional multiplier factor

Spatial coordinates (X, Y, Z )

Height H H0

De¯ections (U, V, W )

Time t��m0H

40�=�EI0�

qSpatial derivatives @=@X � � �,x 1=H0

Time derivatives @=@ t� � � � �,t��EI0�=�m0H

40�

qElastic rotations jx,jy,jz 1.0

Cross-sectional properties of the kth segment

Diameter Dk D0

Thickness tk t0Area Ak A0

Area moment of inertia Ik I0Mass/unit length mk m0

Mass moment of inertia/unit length Im m0H20

Total mass M,MR m0H0

Mass density r m0=H20

Mass moment of inertia IM,IR m0H30

Concentrated forces Fx,Fy,Fz EI0=H20

Concentrated moments Mx,My,Mz EI0=H0

Distributed forces Px,Py,Pz EI0=H30

Distributed moments qx,qy,qz EI0=H20

Acceleration of gravity g �EI0�=�m0H30�

Natural frequency o��EI0�=�m0H

40�

qDamping factor gz m0

��EI0�=�m0H

40�

qa E.g. Dim. (Fz)=Non-dim.(Fz)� (EI0/H0

2) or Fz3 Fz� (EI0/H02).


2.4. Design variables

The design variables which are subject to changein the optimization process are chosen to be theradius of gyration, cross-sectional area and height

of each module composing the main towerstructure. For a round thin-walled tubular con-®guration, these correspond to the mean diameter,

wall-thickness and height of each module. Inother words, the design-variable vector Xd is de®nedas

Xd � ��D1,t1,H1 �,�D2,t2,H2 �, . . . ,�Dk,tk,Hk �, . . .,

k � 1,2, . . . ,Ns

�5�

which is of order �3Ns � 1).

2.5. Design constraints

For the present model formulation, design con-straints are stated in the following section.

2.5.1. Behavioral constraintsStrength requirements

se,k

sal

ÿ 1:0R0:0, k � 1,2, . . . ,Ns �6�

Maximum de¯ection

Wmax

Wal

ÿ 1:0R0:0 �7�

Resonance avoidance�1:0ÿ DiR

oi

o�iR1:0� Di

��8�

Mass limitation

�M=Mal � ÿ 1:0R0:0 �9�The various parameters and symbols of the above

inequalities are de®ned in the following:

se,k maximum e�ective axial stress within the kthmodule

sal allowable tensile or compressive stress

Wmax maximum tip de¯ectionWal an allowable value of de¯ectiono�i target (or desired) frequencies that are well

separated from the exciting ones

Di allowable tolerances of frequenciesMal allowable upper limiting value of the structural

mass

Regarding buckling stresses, the present simpli®edanalysis only imposes lower bounds on the wall thick-

nesses along the tower height in order to avoid localinstability.

2.5.2. Side constraints

DklRDkRDku �10a�

tklRtkRtku, k � 1,2, . . . ,Ns �10b�where Dkl and Dku are the lower and upper limitingvalues of the kth diameter respectively, and tkl is thelower bounding value determined either from the mini-

mum available sheet thicknesses or from considerationsof local wall instability. The upper limiting value tku isimposed in order that the thickness tk may not violate

the assumption of thin-walled tubular con®guration(e.g. tku � 0:1Dk). The total tower height is assumed tobe equal to that of a reference design, i.e.,

PHk � 1:

In addition, the non-negativity constraints Hkr0 mustalso be included to avoid having unrealistic negativeheights.

2.6. Choice of the ®nal model form: functional behavior

of the objective function

Perhaps the most important part of the mathemat-

ical model is the objective function. Care must be exer-cised in constructing such a function because thedesign recommended by the model will directly depend

on its form. In this section, we shall attempt to test thedi�erent alternatives of the objective function throughcomprehensive computer implementation.

2.6.1. Mass and sti�ness optimization analysisUsing Eqs. 1 and 2, the level curves of the mass,

sti�ness and sti�ness-to-mass ratio can be generated

and plotted. Fig. 1 shows those of the sti�ness/massratio of a two-segment cantilever model. In order tominimize the mass alone below its reference value of

one the sti�ness is substantially degraded as it sinks toits lowest levels. The opposite trend is true, in order toraise the sti�ness level above the reference value of 3.0,

unacceptable high values for the mass can be reachedand, hence, unrealistic cost of fabrication and pro-duction. Examining the well-behaved sti�ness/massfunction, it is seen that the middle region of the design

space encompasses the global optimum solutions wherebalanced improvement in both mass and sti�ness isattained. Several other con®gurations including two,

three and four segments have been implemented andtested. It has been con®rmed that maximization of thesti�ness-to-mass ratio is a much better and straight

forward measure of the system performance. Minimiz-ation of the mass alone or maximization of sti�nessalone requires imposition of lower limits on sti�ness or


upper limits on mass, respectively. The choice of thevalues of such limits is not easy.

2.6.2. Exact frequency optimization analysisVibration reduction may be achieved either by sep-

arating the natural frequencies from the exciting onesor by directly maximizing these frequencies. Whateveris the approach taken, it is useful to begin with study-

ing the behavior of the system natural frequencies andsee how they are changed with the selected design vari-ables.

2.6.2.1. Frequency behavior of tower alone: new math-ematical concept. Consider the simplest case of a uni-form one-segment cantilever beam with non-

dimensional height H1 and radius of gyration r1. Theassociated transcendental equation of bending frequen-cies can be shown to have the form

cos

�GbH1��

r1p

�cosh

�GbH1��

r1p

�� ÿ1 �11�

Fig. 2 shows variation of the fundamental frequencyparameter Gb1 � ��

o1p

with r1 � D1 and H1 � 1 for theround thin-walled tubular con®guration. The mass-to-thickness ratio, M=t1, is also shown in the ®gure. It is

seen that both of the frequency and mass functions are

well behaved and continuous in the variable D1. Now

it is possible to choose prescribed values for Gbi andeither one of the variables r1�� D1� or H1 and solve

for the remaining variable. Extensive computer analy-

sis for numerical frequency solutions has proved thevalidity of such a mathematical concept. It is thus

possible to freely place the natural frequencies at any

desired values, but of course within prescribed masslimitations. This useful conclusion has been con®rmed

by consideration of tower models built of more thanone segment [7]. Typical results showing the associated

mass and frequency isomert curves are depicted in Fig.

3 and Fig. 4.

2.6.2.2. Frequency optimization of the combined tower/

rotor structure. Two major factors should be includedwhen considering frequency analysis of a wind turbine

tower [11]. The ®rst factor is the tip mass approximat-

ing the inertia and mass properties of the nacelle/rotorunit at the top of the tower. The computed results for

a selected case is shown in Fig. 5. It is remarked that

the frequency behaves in a manner di�erent from thatof the case of the tower alone because the tip mass has

considerable contribution to the total system mass.The second factor is the axial compressive gravity

forces which may destabilize the tower in case of large

wind generator systems. Fig. 6 shows variation of the

Fig. 1. Sti�ness/mass isomert curves for a two-segment tower model (H1 = H2 = 0.5, D1 = t1 = 1.0).


fundamental frequency with the non-dimensional grav-ity parameter, g, for di�erent values of the tip mass,

MR: Variation of slenderness ratio, H0=r0, is alsoshown in the ®gure for di�erent tower heights. It isseen that a quick reduction in the frequency occurs

when the axial compressive stress becomes close to thecritical buckling stress of the tower. The calculated nu-merical values of the corresponding critical gravity fac-

tor, gcr, are approximately equal to 1.1, 2.0, 3.35, and7.8 for MR � 2:0, 1.0, 0.5, and 0.0, respectively.

2.6.3. Final model form

Di�erent formulations of the optimization modelhave been tested on the computer using the actual dataof the ERDA-NASA-MOD-0 wind turbine [7]. It was

found that the most signi®cant and promising optimiz-ation strategy is the direct maximization of the systemnatural frequencies (Eq. (4)). Frequency-windows con-

straints (Eq. (8)) were discarded from the model for-mulation because of the di�culty in ®nding a feasiblestarting point in the selected design space. If it hap-

pened that one of the critical forcing frequencies wereclose to any one of the natural frequencies of the opti-mized tower structure, another value for the naturalfrequency can be chosen near its optimum value and

the associated eigenvalue problem is resolved for anyone of the design variables instead. This is one of the

advantages of the present mathematical formulation.In addition, computer experimentation has shown that

minimization of the frequency-placement index of Eq.(3) results in a more time consuming optimization pro-cess and a slow rate of convergence towards the opti-

mum solution. Besides, the attained optimum solutionswere found to be strongly dependent on the chosenvalues of the target frequencies, which are rather arbi-

trarily chosen. Finally, it was decided to discard theconstraints on the module's diameters because it wasfound that optimization favors increasing diameters

for maximizing frequencies.

2.6.4. Determination of the weighting factors

There are many procedures reported in the literaturethat help extract the values of the weighting factors,Wfi of Eq. (4). For instance, one may give the funda-

mental frequency the largest weight, while ignoring theimportance of other higher frequencies. A morereliable procedure of adjusting the values of Wfi is

given in Ref. [7] which is summarized as follows:

. Initialize Wfi by the reciprocal of the values of thefrequencies of a reference uniform tower,

Wfi � 1=�Gbi �reference

. Normalize the resulting values by dividing each one

Fig. 2. Variation of frequency and mass with cross-sectional diameter (case of uniform one-segment model).


Fig. 3. Mass and frequency level curves of two-segment towers. (a) Gb1-contours, height vs diameter (D1=1, t1=1, t2=0.5).

(b) Mass contours, height vs diameter (D1=1, t1=1, t2=0.5). (c) Gb2-contours, thickness vs height (D1=1, D2=0.5, t1=1).


by their sum in order to make the ®nal sum equal to

one, i.e.

Wfi3Wfi=X

Wfi

Therefore, the ®nal appropriate values of the weight-ing factors are given as:* Tower alone: 61.0, 24.4, and 14.6%.* Combined tower/rotor: 61.2, 23.6, and 15.2%.

3. Tower analysis

This section describes brie¯y the method of analysis

of the bending motion of an isolated tower/rotor struc-tural model. Fig. 7 shows the intermediate coordinatesand degrees of freedom required to de®ne motion. Theassociated transformation matrices and the complete

kinematical analyses can be found in reference [7]where the more general case of bending-bending-tor-sional motion is given in detail. Distributed-load vec-

tors along tower height are expressed in theundeformed inertial coordinate system �x,y,z� andinclude aerodynamic, inertial, gravitational as well as

damping contributions. The ®nal approximate govern-

ing partial di�erential equation for the bending displa-cement W is cast in the following non-dimensionalform:

�IWxx �xxÿ�FxWx �xÿPz � 0 �12�which must be satis®ed over the tower height, i.e.0RxR1: The non-dimensional equivalent distributed

load Pz can be calculated from [7]:

Pz � ÿmTwtt ÿ gzwt � Paero � FZNBd�xÿ 1�

�MYNBZ�xÿ 1� �13�

where the concentrated tip force FZNB and momentMYNB transmitted from the NB-rotating blades aretransformed to distributed loads by the use of the

Dirac-delta, d, and the unit doublet, Z, functions. Thedistributed aerodynamic force is given by [12]

Paero � 4rairV20DCD�x�2ax �H0=D0 �f�t� �14�

where f �t� is a time-dependent function accounts forthe dynamic and gusty nature of the wind, ax is called

the wind shear exponent, CD is the drag coe�cient and

Fig. 3 (continued)


V0 is the wind speed at hub height H0 non-dimensio-nalized by the term

��E=r�Tp

: The associated boundary

conditions are:

At x=0 (cantilevered end)

W � 0 and Wx � 0:

At x=1 (free end)

� ÿ IWxx �x�FxWx �MRWtt � 0

� ÿ IWxx � ÿ IR�Wx �tt� 0 �15�

3.1. Natural vibration analysis: the eigenvalue problem

The ®rst step in the solution procedure is the deter-mination of the free natural vibration characteristics

by removing all of the forcing functions and consider-ing only the homogeneous equation. The resultingeigenvalue problem is described by the 4th order ordin-

ary di�erential equation:

g 0000i� �x� ÿ ÿfxkg 0i � �x�� 0ÿo2

kgi� �x� � 0 �16�

which must be satis®ed over any module (k ) extendingfrom x � xk to x � xk�1 or 0R �xRHk, where �x �xÿ xk is a local coordinate system. fxk is a non-dimen-sional quantity representing compressive force due togravity and is given by

fxk � gk�xÿ fk0

�, fk0 � 1

Ak

MR �

XNs

i�kAiHi

!,

gk � g=r2k and ok � oi=rk

�17�

oi and gi are the non-dimensional eigenvalues (naturalfrequencies) and eigenfunctions (mode shapes), respect-ively. For thin-hollow circular cross section, the non-

dimensional properties are:

Ak � Dktk, Ik � D3ktk, and rk � Dk: �18�

The above di�erential equation has been shown tohave an exact solution by the method of Frobenius inwhich the spatial functions gi�x� is represented by a

four-term power series [7]:

Fig. 4. Fundamental frequency for towers built of four segments.


gi�x� �X4n�1

Cnln�x�,

ln�x� �X1m�n

anm�x�mÿ1, mrn

�19�

where ln are four linearly independent solutions and

anm are the unknown coe�cients given by

anm � ÿ gkfk0an,mÿ2�mÿ 1��mÿ 2� �

gk�mÿ 4�an,mÿ3�mÿ 1��mÿ 2��mÿ 3�

� o2kan,mÿ4

�mÿ 1��mÿ 2��mÿ 3��mÿ 4� mrn �20�

Fig. 5. Fundamental frequency and mass isomert curves for the combined tower/rotor structural models. (a) One-segment model.

(b) Three-segment model.


Since an exact solution is available for one uniformsegment, then a non-uniform tower structure built

from Ns-uniform segments must have an exact solutionfor its natural frequencies and mode shapes. The trans-fer matrix method has been found to be suitable for

the present discretized model formulation. Applicationof the associated boundary conditions and consider-ation of the non-trivial solution yields the following

exact frequency equation:

a1o4i � a2o2

i � a3 � 0 �21�

where the coe�cients ai are de®ned as

a1 � IRMR�T23T14 ÿ T24T13 �,

a2 � IR�T24T43 ÿ T23T44 � �MR�T34T13 ÿ T33T14 �,

a3 � �T33T44 ÿ T34T43 � �22�Tij are the elements of an overall transfer matrix

formed by taking the products of all the intermediatetransfer matrices of the di�erent segments. Detailedderivations can be found in Ref. [7]. Once the natural

frequencies oi, i � 1,2, . . . ,N, have been determined,the associated mode shapes gi�x� can be obtained fromEqs. (19) and (20).

3.2. Forced response

Application of the normal mode method to the gov-

erning equation of motion and the associated bound-ary conditions (Eq. (12)) results in N-independent set

of uncoupled di�erential equations of the form

�xi�t� � 2Zioi_xi�t� � o2

i xi�t� � Qi=Mi,

i � 1,2, . . . ,N�23�

where Qi is called the ith generalized load given by

Qi�t� ��10

gi�x� Paero�x,t� dx� gi�1�FZNB

ÿ g 0i �1�MYNB

�24�

Assuming zero initial conditions, the response is givenby:

xi�t� � �1=Mioid ��t0

Qi�t�exp�ÿ Zioi�tÿ t��

� sin oid�tÿ t� dt: �25�

where oid � oi

��1ÿ Z2i

pis the ith damped natural fre-

quency and Mi is the ith generalized mass de®ned as:

Mi ��10

mTg2i dx�MRg2i �1� � IRg 0i2�1� �26�

3.3. Internal loads and stress analysis

After the response has been found, displacements,

Fig. 6. Variation of fundamental frequency with gravity (case of one-segment tower with tip mass).


velocities and accelerations can be substituted for thecalculation of the applied distributed loads. The in-

ternal forces and moments can then be determined atany location x along the tower height by integratingthe equilibrium equations of motion. Stresses resulting

from axial, bending, torsion and shearing e�ects canbe systematically calculated at any point within thestructure. For the present structural model of round

tubular con®guration the most signi®cant stress com-ponents are sxx and txy: Other components are usuallyneglected. Stresses are calculated within the limitations

of the engineering beam theory of bending and torsion,and the combined stress is determined on the basis ofVon Mises equivalent stress theory.

3.4. Other critical loading conditions

The most important critical loading conditions

which should be taken into consideration in the design

phase of the supporting tower structure were investi-gated in detail in Ref. [7]. They include the cases of

suddenly applied severe wind, transient loads inducedby tower base motion and vortex-shedding induced vi-brations.

4. Optimization analysis by mathematical programming

The tower optimization model developed in the pre-vious sections is a non-linear mathematical program-

ming model. The model uses the interior penaltyfunction technique [13] in ®nding the constrained opti-mum solution. In contrast to the exterior technique,the interior technique presents a series of improving

feasible designs which gradually approach the ®nal op-timum design. For multi-dimensional search the modeluses Powell's technique. The quadratic interpolation

method has been chosen for the determination of the

Fig. 7. Isolated tower structural model.


step size along the search direction. A computer pro-gram was developed to automate the tower analysisand design procedures. Detailed description of the pro-

gram can be found in Ref. [7].

5. Model implementation: a case study

As a case study, the proposed optimization modelwas implemented on an experimental, horizontal axis

wind turbine; namely, the ERDA-NASA MOD-0 [11].Its rotor has two blades with 38.1 m diameter locateddownwind from the tower at 30 m height above the

ground. The original tower con®guration is a steel±lat-tice truss structure having four legs connected togetherby chord members.

5.1. The reference cantilever beam and baseline design

data

Table 3 presents the pertinent data of the reference

cantilever beam which has a round tubular cross-sec-tion type. Its total mass is chosen to be the same asthat of the original MOD-0 con®guration. Table 4

gives the associated values of the non-dimensional mul-tiplier factors of the various quantities. The equivalentdiscretized model of the baseline tower design is deter-

mined on the basis of having the same total mass andsti�ness and mass distributions as that of the actualMOD-0 design. It is selected to be composed of sixsegments with the non-dimensional design variables

given in Table 5.

5.2. Results of structural analysis

Fig. 8 depicts the calculated frequencies and modeshapes of the reference cantilever beam and baseline

designs. It is seen that the ®rst and second frequenciesof the baseline design are higher than those of thereference cantilever indicating a good tower structural

con®guration. The third frequency, however, is slightlyless than its reference value. The fundamental fre-quency of the combined tower/rotor structure is about

Table 3

Parameters of the reference cantilever beam (thin-walled

round tubular con®guration)

Parameter Notation (units) Numerical values

Height H0 (m) 30.0

Mean diameter D0 (m) 2.421

Wall-thickness t0 (m) 0.01

Cross-sectional area A0 (m2) 76:058� 10ÿ3

Second moment of inertia I0 (m4) 557:243� 10ÿ4

Radius of gyration r0 (m) 0.856

Material of construction

Type: Steel

Mass density r0 (kg/m3) 8:7� 103

Young's modulus E (N/m2) 21:0� 1010

Allowable normal stress sa (N/m2) 1:6� 108

Allowable shear stress ta (N/m2) 0:8� 108

Poisson's ratio n 0.3

Structural mass/length ms0 (kg/m) 661.667

Non-structural mass/length mns0 (kg/m) 181.667

Total mass/length m0 (kg/m) 843.333

Total structural mass Ms0 (kg) 19850.0

Total non-structural mass Mns0 (kg) 5450.0

Total mass M0 (kg) 25300.0

Table 4

Non-dimensional multiplier factors

Non-dimensional quantity Notation Multiplier factor Units

Concentrated moments Mx,My,Mz 3:90� 108 Nm

Concentrated forces Fx,Fy,Fz

Distributed moments qx,qy,qz 1:30� 107 N

Distributed forces px,py,pz 4:333� 105 N/m

Natural frequency o 4.1389 sÿ1

Table 5

Baseline design variables. (Tubular construction)

Segment �Dk,tk,Hk�, k � 1,2, . . .Ns

1 (1.4460, 1.25, 0.233)

2 (1.1360, 1.05, 0.183)

3 (0.9290, 1.00, 0.167)

4 (0.8260, 0.85, 0.150)

5 (0.6196, 0.75, 0.133)

6 (0.5163, 0.55, 0.134)

Rotor mass, MR 0.5

Rotor inertia, IR 0.01

Tower mass, M 1.0

Gravity, g 0.019


1.82 Hz �� 4:1389� 1:660782=2p), which is higherthan the primary forcing frequency 2/rev showing a

sti� baseline design. The peak values of the concen-trated loads transmitted from the rotor at the top ofthe tower were calculated at a rated wind speed of 8.0

m/s.The associated maximum stress at the tower base

and de¯ection at the top were found to be 0:268� 108

N/m2 and 0.02844 m, respectively, which are much lessthan the allowable values. On the other hand, consid-ering critical loading conditions discussed above in

Section 3.4, the maximum stress at tower base hasreached a value of 0:6523� 108 N/m2 for the overloaddesign condition with wind velocity V0 � 27 m/s. Thetip de¯ection increased to 0.0658 m.

5.3. Optimum tower solutions

In this section we shall consider, as a case study,tower optimization when subjected to a suddenlyapplied wind with a velocity equal to the survival

value of 70.0 m/s. Two di�erent optimum designs wereobtained by considering two types of the objective

function. The ®rst type, type-1, considered maximiza-tion of the ®rst three bending frequencies, Gbi, for thecombined tower/rotor structure, whereas the second

type, type-2, considered maximization of the frequen-cies of the tower alone. Fig. 9 presents the attained op-timum tower designs for the ERDA-NASA MOD-0

wind turbine by implementing each of the above twoformulations. As seen, the optimum tower con®gur-ation produced from the ®rst type consists of six-mod-

ules while that from the second type consists of eight-modules, although the starting design was the same forboth cases. It was found that the optimization algor-ithm treated the number of modules as an additional

implicit variable. Sometimes the computer discardedone or more segments by letting their heights sinktowards their lower limits (i.e. zeros), and sometimes

made two consecutive modules identical, i.e. havingthe same diameter and thickness. It is also remarkedthat type-1 formulation results in a higher frequency

Fig. 8. Natural frequencies and mode shapes of reference and baseline designs.


level than type-2, with the penalty of increasing struc-

tural mass. Both types produce substantial improve-

ments in mass and sti�ness as compared with those of

the baseline design described in Table 5. Mass saving

achieved from type-1 is shown to be about 7.6% and

the calculated maximum de¯ection at tower top is

0.0393 m, which is much less than the corresponding

value of 0.097 m of the baseline design. On the other

hand, type-2 formulation produces a slightly softer de-

sign where mass saving has reached a value of 12.4%

but with increased tip de¯ection to 0.0563 m.

The optimum values of the second mode frequencies

obtained from type-1 were too close to 29/rev and 49/

rev for the combined system and tower alone, respect-

ively. Even though they are far away from the primary

forcing frequencies (1/rev, 2/rev, 3/rev), they have been

Fig. 9. Optimum tower con®gurations of MOD-0 wind turbine (round tubular sections).


separated by applying the previously suggested math-ematical concept. That is, instead of solving the eigen-value problem in the natural frequencies, we choosethe desired frequency and solve the eigenvalue problem

in one of the design variables. The resulting modi®edoptimum frequencies were given by:

Combined system Gbi � 2:14344,5:44208,9:17620:

n=rev � 4:53961,29:26349,83:19975:

Tower alone Gbi � 3:43703,7:01730,11:08690:

n=rev � 11:67249,48:65001,121:45533:

which are seen to be well separated from the exciting

frequencies.Comparison between the present optimum solution

and the original design of MOD-0 tower is given in

Table 6. The frequency interference diagram (Camp-bell-diagram) is presented in Fig. 10, in which both the

natural and exciting frequencies are normalized withrespect to the rotor design speed, O0 � 4:2 rad/s.

6. Conclusions

Several optimization strategies for the structural de-sign of wind turbine towers are developed and investi-gated through extensive computer implementations.The developed models have been successfully applied

to an existing 100 kW wind turbine (ERDA-NASA,MOD-0), where optimum design trends were obtainedthrough the use of the interior penalty function tech-

nique. It has been proved that maximization of aweighted-sum of the system natural frequencies is themost representative objective function which directly

re¯ects the major design goals and ensure a balancedimprovement in both mass and sti�ness. Extensivecomputer experimentation has proved that the natural

frequencies, even though implicit functions in the de-sign variables, are well-behaved, monotonic and

Table 6

Comparisons of MOD-0 tower designs

Tower design Fundamental frequency (Hz) Mass saving Maximum de¯ection m, (V0 = 70 m/s)

Tower alone Combined system

Original space truss 5.1 1.59, 1.76, 2.1 (di�erent rotor positions) ± ±

Baseline cantilever 5.06 1.82 0.0 0.097

Optimum cantilever 7.92 3.06 7.60% 0.039

Fig. 10. Campbell frequency-diagram of combined tower/rotor. ERDA-NASA MOD-0 wind turbine �O0 � 40 rpm).


de®ned every where in the design space. Global optim-ality can be attained from the proposed models and

exact placement of the frequencies can be obtained byfreely selecting their desired values. Appropriate non-dimensionalization of the various parameters and vari-

ables throughout the problem formulation has led to anaturally scaled optimization model, which eliminatesthe need for scaling the design variables as usually

suggested by similar optimization procedures. Anotheruseful conclusion is the possibility of selection of themodule height as a main design variable. This import-

ant variable is always missed by most of the previousresearch work dealing with structural optimizationwhere the cross-sectional parameters are considered theonly e�ective design variables. Finally, the present

exact optimization analysis saves much of the compu-ter time required by the ®nite-element and other discre-tized approximate methods.

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