Life-cycle cost structural design optimization of steel wind towers

Computers and Structures xxx (2015) xxx–xxx

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Life-cycle cost structural design optimization of steel wind towers Nikos D. Lagaros a,⇑, Matthew G. Karlaftis b a Institute of Structural Analysis & Antiseismic Research, Department of Structural Engineering, School of Civil Engineering, National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece b Department of Transportation Planning and Engineering, School of Civil Engineering, National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece

a r t i c l e

i n f o

Article history: Accepted 22 September 2015 Available online xxxx Keywords: Structural design optimization Life-cycle cost Wind towers Metaheuristics Eurocodes Vestas

a b s t r a c t The objective of this work is to propose a design procedure, formulated as a structural design optimization problem, for designing steel wind towers subject to constraints imposed by the Eurocode. For this purpose five test examples are considered, in particular steel wind towers varying on their height are optimally designed with minimum initial cost, while the wind load is calculated according to Eurocode 1 Part 2–4 and design constraints imposed by Eurocode 3 on shell buckling and local buckling of flat ring-stiffeners are implemented. Furthermore, two formulations are examined differing on the shape of the wind tower along the height. While, for the solution of the optimization problem the Optimization Computing Platform (OCP) was used. Ó 2015 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

1. Introduction Renewable energy sources (RES) are traditionally inefficient providing low rate of return which suggests high implementation cost; however, much work in the areas of technology and proper power source location is tackling these issues. Since early 1980s, many wind parks have been constructed and significant work has been published related to the assessment of wind park efficiency. So far several hundred individual wind turbines, which are connected to the electric power transmission network, form large offshore or onshore wind parks have been in-stalled. Although, offshore wind farms can take advantage of more frequent and powerful winds than are available to land-based installations and they have less visual impact on the landscape, their construction cost is considerably higher. On the other hand, onshore wind facilities harness winds at higher altitudes, which are stronger and more consistent, through the appropriate tower height, while small wind parks are used to provide electricity to isolated locations. During the last three decades many numerical methods have been developed to meet the demands of structural design optimization. These methods can be classified in two categories, the deterministic and probabilistic ones. Mathematical programming methods are the most popular methods of the first category and ⇑ Corresponding author. E-mail addresses: [email protected] (N.D. Lagaros), [email protected] (M.G. Karlaftis).

in particular the gradient-based optimizers. These methods make use of local curvature information, derived from linearization of the objective and constraint functions by using their derivatives with respect to the design variables at points obtained in the process of optimization, to construct an approximate model of the initial problem. Heuristic and metaheuristic algorithms are nature-inspired or bio-inspired and belong to the probabilistic category of methods. Metaheuristic algorithms for engineering optimization among others include genetic algorithms (GA), simulated annealing (SA), particle swarm optimization (PSO), ant colony algorithm (ACO), artificial bee colony algorithm (ABC), harmony search (HS), firefly algorithm (FA), and many others [1]. A rather limited number of studies concerning design optimization of wind tower systems has been published so far. In particular, Negm and Maalawi [2] presented optimization models for the design of wind tower structures where the main tower body was composed by uniform segments while cross-section area, radius of gyration and height of each segment were the design variables. Harte and Van Zijl [3] presented some aspects of the classical wind energy turbines and discussed topics related to the solar chimney concept (like the wind action, eigen frequencies, stiffening and shape optimization). Uys et al. [4] studied the problem of minimum cost design of steel wind towers where the optimum shell thickness, number of stiffeners and dimensions of the stiffeners were defined. Maalawi [5] presented a design optimization model for engaging wind tower structure frequencies (tower/nacelle/ rotor) in free yawing motion. The objective of the work by Silva et al. [6] was to find optimized designs of reinforced concrete

http://dx.doi.org/10.1016/j.compstruc.2015.09.013 0045-7949/Ó 2015 Civil-Comp Ltd and Elsevier Ltd. All rights reserved.

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(RC) towers taking into consideration the cost, computational time, construction techniques and precision of structural models, when subjected to dynamic wind loads while considering the vibration effects of the wind energy generator’s components. The objective of the study by Yıldırım and Özkol [7] was to optimize the mass of a 1.5 MW steel wind tower using GA in accordance to ASCE 7-98, AISC-89 and IEC61400-1 design codes. Nicholson [8] used Microsoft Excel in order to design wind towers along with their foundation taking into account the effect of deflections. In the chapter by Maalawi [9] various optimization approaches for wind tower design were presented aiming to develop topology optimization methods for designing smart wind turbine structures. Zwick et al. [10] implemented an iterative optimization approach and have found that structural analyses of the wind tower systems, keeping constant the dimension of the members over the tower height, provide indications about the required dimensions for optimized designs. Katyal et al. [11] performed parametric optimization of wind turbine tubular towers using the wind speed NWP model data for a complex hilly terrain at Idukki District of Kerala. In this paper, a design procedure formulated as a structural design optimization problem is proposed, for designing steel wind towers subject to constraints imposed by the Eurocodes. For this purpose five test examples are considered, in particular steel wind towers varying on their height are optimally designed with minimum initial cost, while the wind load is calculated according to Eurocode 1 Part 2–4 [12] and design constraints imposed by Eurocode 3 [13] on shell buckling and local buckling of flat ringstiffeners are implemented. Furthermore, two formulations are examined differing on the shape of the wind tower along the height resulted from the optimization procedure, while three material properties for the structural steel are considered. For the solution of the optimization problem the Optimization Computing Platform (OCP) [14] is used, while in [15] OCP was further developed to be able to handle steel wind towers in terms of optimization and structural design. In the current study, we have added advanced features to the variant of OCP presented in [15], in particular: (1) various problem formulations of the optimization problem are now allowed, varying with reference to the type of constraint functions and the design variables used for the formulation, (2) the cascade optimization concept can now be used for this type of structures while (3) various techno-economic studies have been performed on the optimum designs achieved.

structure or the stress concentrations around the door opening which must be thoroughly examined. Their design is governed by the extreme wind loading, while earthquake loading should also be taken into account when designing the turbine tower on seismic hazardous areas. The evaluation of the shell thicknesses of the tower is performed by means of the plastic limit state design (LS1-plastic), buckling limit state design (LS3-buckling) and fatigue limit state design (LS4-fatigue) as described in Eurocode 3 [13]. More details can be found in the work by Baniotopoulos et al. in [16]. The generic overall cross section design, defined by the outside diameter and wall thickness should take into account the restriction on transportation and therefore may not exceed 4.5 m. Additionally, due to limitations on carbon steel that can be processed in a standardized cylindrical form, the wall thickness shall not exceed 40 mm. Regarding the allowable maximum displacement of the top of the wind turbine tower, this is set conservatively to 1.00% of the total height of the tower in order to avoid excessive movement that would oppose the efficient operation of the wind turbine. Critical for the wind turbine tower design is the avoidance of the resonance phenomenon. Specifically, the natural frequency of the whole structure of the tower should be at a safe distance from the excitation frequencies in the operation phase of the wind turbine rotor. Common values for operating frequencies for small wind turbines are between 0.23 Hz and 0.52 Hz and for large wind turbines in the range of 0.10–0.30 Hz. The natural frequency of the tower structure should remain above the highest operating frequency of the wind turbine, multiplied by a safety factor, typically between 1.1 and 2.0, in order to avoid resonance at any point during the operation. 3. Solving the optimization problem The solution of the optimization part of the problems presented herein, is dealt with the optimizer component of the optimization computing platform (OCP) presented recently by the authors in [14]. This component is equipped with the eight metaheuristic optimization algorithms (MOA) which have been successfully applied in various challenging problems [17,18]. Furthermore, the cascade optimization concept [19] also implemented into the computing platform was used in the current study. 3.1. Differential evolution

2. Wind tower design optimization – Generic problem formulation An optimization problem can be formulated in standard mathematical terms as a non-linear programming problem that in general form can be written as follows:

opt ðmin = maxÞ FðsÞ subject to

g j ðsÞ 6 0 j ¼ 1; . . . ; m

ð1Þ

sLi 6 si 6 sUi i ¼ 1; . . . ; n where s is the vector of design variables, FðsÞ is the objective function to be optimized, g j ðsÞ are the behavioral constraints imposed by the design codes and/or the design engineer while sLi and sUi are lower and upper bounds of the ith design variable. In order to ensure an efficient design of the wind turbine tower structural system, several limit state design checks based on static or dynamic analyses should be satisfied along with limitations on the maximum allowable tower top displacement and natural frequencies. Wind turbine towers are mainly simulated numerically as simple cantilever beams; however, their section forms a thinwalled cylindrical shell and therefore, several issues are identified through structural analysis such as the local buckling of the shell

Storn and Price in [20] proposed a floating point evolutionary algorithm for global optimization and named it differential evolution (DE). Several variants of DE have been proposed so far [21], according to the variant implemented in OCP a donor vector vi,g+1 is generated first:

v i;gþ1 ¼ sr ;g þ F  ðsr ;g  sr ;g Þ 1

2

3

ð2Þ

Integers r1, r2 and r3 are chosen randomly from the interval [1, NP] while i – r1, r2 and r3. NP is the population size, F is a real constant value, called the mutation factor. In the next step the crossover operator is applied by generating the trial vector ui,g+1 which is defined from the elements of si,g or vi,g+1 with probability CR. The last step of the generation procedure corresponds to the implementation of selection operator where vector si,g, is compared to trial vector ui,g+1:




si;gþ1 ¼

ui;gþ1

if Fðui;gþ1 Þ 6 Fðsi;g Þ

si;g otherwise i ¼ 1; 2; . . . ; NP

ð3Þ

where FðsÞ is the objective function to be optimized (see Eq. (1)), while without loss of generality the implementation described in

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Eq. (3) corresponds to minimization. A more detailed description of the DE algorithm and its implementation can be found in [14,22] while the flowchart of DE is presented in Fig. 1.

3.2. The cascade optimization concept

influence of the corresponding disadvantages on the optimum design achieved. The selection of the optimizers to be included in a cascade procedure, their exact sequence and the number of cascade stages performed can be determined. 4. Wind energy

It is generally accepted that there is still no unique optimization algorithm capable of handling, with equal efficiency, all types of optimization problems. Cascade optimization attempts to alleviate this deficiency by applying a multi-stage procedure where optimization algorithms are implemented successively [19]. In the first stage of the cascade procedure the first optimizer starts from a user specified design known as ‘‘cold-start”. The cold-start is a userspecified or randomly selected design, which defines the starting solution d0 for the initial optimizer utilized in the first stage of the cascade procedure. The starting solution of the first stage is referred to as a cold-start, because, most often, it lies far from the region of the global optimum. After running the first-stage optimizer, the optimal solution reached is used as the starting solution d0 for the second cascade stage. The intermediate optimal solution reached in the first cascade stage, which may be perturbed using a pseudo-random technique, is called a ‘‘hot-start” and is used to initiate the second optimization stage. This new starting solution is called a hot-start, because during the execution of the initial optimizer the achieved optimal solution has moved toward the region of the global optimum. Accordingly, each optimization stage of the cascade procedure starts from the optimum design achieved in the previous stage (possibly perturbed). Thus, each cascade stage initiates from a hot-start and produces a new hot-start for the next stage. Each autonomous optimization stage of the cascade procedure starts from an initial design d0, which is either a ‘cold-start’ or a ‘hot-start’. This way the autonomous computations of successive optimization stages are coupled. The flowchart of Fig. 2 graphically illustrates the cascade optimization procedure. In general, the optimization algorithm implemented at each stage of a cascade process may or may not be the same. Cascade optimization has been implemented using different deterministic and/or probabilistic optimizers in the cascade stages [19]. The main motivation in combining different optimizers into a successive manner has to do with the exploitation of the advantages offered by different optimization algorithms and reducing the Step 1: Initialize Parameters F(s): objective function si: design variable n: number of design variables NP: size of the population K: control variable F: mutation factor CR: mutation probability TC: termination criterion

Wind energy stands for the kinetic energy generated by blowing air streams. The wind energy produced by a wind turbine can be calculated through the following relationship:

E¼

1 1 1 m  u2 ¼ ðA  u  t  qÞu2 ¼ A  t  q  u3 2 2 2

ð4Þ

where A is the surface of the wind generator (rotor area) perpendicular to the wind direction, t is the time duration, q is the air density and u is the wind velocity. The product ðA  u  tÞ is the volume of air passing while the product ðA  u  t  qÞ is the mass m passing through the rotor area per unit time. The wind power is the quotient of energy per unit time:

P¼

E 1 ¼ ðA  qÞu3 t 2

ð5Þ

The wind characteristics like velocity and direction, vary in time and location, making difficult to predict the amount of wind energy that a wind turbine could exploit at a particular place. A considerably high volume of statistical data should be taken into account in order to determine these magnitudes, which in many cases is difficult to obtain. In these cases, probability distribution functions are often fit to the observed data in order to predict the wind characteristics for the installation area of the wind park. Weibull distribution is considered to be the proper one that best fits to the distribution of hourly wind velocities [23]. It can been shown theoretically that any windmill can only possibly extract a maximum of 59.3% of the power from the wind (this is known as the Betz limit). Thus, the formula of Eq. (5) is modified as follows:

Pt ¼

1 ðA  q  C p Þu3 2

ð6Þ

that denotes the power extracted by the wind turbine relative to the energy available in the wind stream while Cp is the coefficient of performance of the wind machine. Step 2: Initialize Population For i=1 to NP Random generation of the solution vector si,g=1 Calculate F(si,g=1)

Step 3: Generate population g+1 For i=1 to NP Generate donor vector vi,g+1 according to Eq. (2) Generate trial vector ui,g+1 according to Eq. (3) Select the solution vector si,g+1 Calculate F(si,g+1)

Step 4: Check of convergence Yes Satisfied

Terminate Computations

No

Repeat Step 3

Fig. 1. Flowchart of the differential evolution algorithm.

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Set initial design d0 (cold-start)

i =1

Design optimization

Attained optimum design: di

STOP ?

YES

Finally achieved optimum design: di

NO i = i +1

Set initial design d0=di (hot-start)

Fig. 2. Flowchart of the cascade structural optimization procedure.

Wind parks may consist of several hundred wind towers and cover hundreds of square miles, while wind towers should be located in certain distances between each other, which usually should be greater than 50 m or four times the diameter of the rotor. The wind potential is the most critical factor for choosing the site to install a wind park, while another important factor is accessibility to local demand or transmission capacity. The wind potential of a region is characterized by the mean value of wind velocity, wind orientation, frequency of wind data, etc. These characteristics are usually obtained from wind atlases and are validated using measurements. Investing on a wind park project requires determining the site potential based on historical meteorological wind data and more accurate measurements concerning the wind velocity, orientation, etc. Wind data are usually recorded for more than one year using radars placed on the top of guyed towers and local wind maps are designed after having collected detailed measurements of the wind speed, orientation, etc. Usually, wind speed is measured at a relative height of 10 m from the ground surface. However, as the influence of drag reduces at higher altitudes, the wind velocity increases significantly. This increase depends on the topography, the upwind barriers (e.g. buildings, trees) and the surface roughness. Consequently, as the altitude of wind tower increases, the expected wind velocity increases too, resulting in increased produced power. Typically,

the increase of wind speeds with increasing height follows a wind profile model, which indicates that wind velocity rises incrementally to the seventh root of altitude. The construction of wind parks requires designing and installing a collection of systems (towers, substation, etc.). The wind generators are connected via a communication network through power collection systems. Then, this medium-voltage electrical network increases its voltage at the substation with a transformer in order to be connected to the high voltage transmission system. From financial point of view, design of wind parks results from the balance between two conflicting economic factors: reduction of installation and operation costs and increase of the profit derived from the sale of the produced energy. The total cost of the construction and operation of wind parks is of particular concern in both academic level as well as in industry [24,25]. The main factors that affect the total cost include: installation, operationmaintenance (O&M) and electricity production costs. The installation cost includes: purchase cost of the wind turbine, design, construction and connection of the wind towers to the electricity grid costs; it was observed that installation and operationmaintenance costs are about 95% of the total cost. Generally, production of wind power requires a high capital investment, while it has low ongoing costs. As the wind generator technology develops by using more sustainable materials for the blades and

Fig. 3. Wind tower: (a) structural configuration (indicatively for H = 120 m) and (b) side view and typical cross-section.

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tower, by improving the turbine performance and increasing the power generation efficiency, costs of wind power production are decreased. 5. Case studies 5.1. Description and structural analysis of wind towers For the purposes of this study five wind towers were examined in order to investigate, among others, how the tower height affects the structural design. In particular, it is studied whether the increased wind speed can be exploited better and how the height influences the life-cycle cost assessment of the different wind towers and consequently that of the wind park. The first type has tower height equal to 80 m and rotor diameter equal to 90 m, while the tower is conical tube with gradually decreasing diameter in relation to the height increase. Structural steel of class with nominal yield stress of 235, 275 or 355 MPa and modulus of elasticity equal to 200 GPa was considered. The wind turbine used is the Vestas V90-3 MW which has a rotor diameter of 90 m with swept area of 6362 m2, whereas it has three rotor blades, each of which has length equal to 44 m. Fig. 3 depicts the wind tower structural configuration, while the finite element (FE) numerical model along with the loading conditions implemented are shown in Fig. 4. The FE model was developed in the framework of the design software SAP2000 [26] that is integrated to the OCP used for solving the optimization problem at hand; whereas the power curve of the Vestas V90-3 MW turbine is shown in Fig. 5. The other types of wind tower examined, have the same technical characteristics with respect to the wind turbine, except from the

tower height, which is equal to 100 m, 120 m, 140 m and 160 m, respectively. The structural analysis part required into an optimization framework in order to assess the structural performance of the wind towers were carried out with the software SAP2000 [26]. For the structural analyses of the towers, beam elements with hollow circular section were used, whose length is equal to 5 m and their diameter reduces as the height increases. Each beam element has a different diameter depending on the distance from the ground (z = 0). In the structural analysis, two types of loads are applied: the static load of the rotor’s self-weight and the wind load. It was assumed that the self-weight of the rotor is 1100 kN and it is placed with an eccentricity of 0.75 m with reference to the axis of the tower resulting into a moment of 825 kN m, while the wind load was simulated with a static horizontal load at the top of the pillar. 5.2. Calculation of wind velocity, atmospheric air density and wind load The calculation of the wind speed at the top of the wind tower is required in order to define the wind load at the top of the pillar. The wind velocity at the desirable height is calculated through the following formula:

  ln zho uðhÞ ¼ ug h  ln zrefo

ð7Þ

where uðhÞ is the wind speed at height h, ug is the average wind speed at the reference height (href), the reference height is usually taken equal to 10.00 m while z0 is the roughness length. For the

Fig. 4. Wind tower (a) finite element model and (b) loading conditions.

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where C(s) represents the construction cost, H is the height of the wind tower, gEC3 represents the constraints imposed by Eurocode 3 [13], fn is the fundamental eigen frequency of the wind tower while fr is the largest frequency of the wind turbine (60 Hz for Vestas V90-3 MW turbine) and utop is the displacement at the top of the tower. According to Eurocodes, design checks should be satisfied for specific load combinations, like the following ones:

8 X > Gkj 1:35  > > > > j > > > > j > > X > > > Gkj \ þ "Ed > :

Fig. 5. Power curve of the Vestas V90-3 MW turbine.

ð12Þ

j

wind park at hand, the construction region is a countryside area, without tall buildings and dense vegetation; thus, the roughness length z0 was taken to be equal to 0.01 [27]. The density of the atmospheric air affects the wind load developed at the top of each wind tower. Air density is a function of the atmospheric pressure and temperature that decreases in relation to the height increase. The density do of the air at the sea surface is equal to 1.225 kg/m3, when the temperature is equal to 20 °C while the corresponding pressure is equal to 101.6 kPa [28]. The relation of air pressure with respect to height is defined by [28]:

 gM L  h RL p ¼ p0  1  T0

ð8Þ

where p is the air pressure (Pa), p0 is the sea level standard atmospheric pressure (101,325 Pa), L is the temperature lapse (0.0065 K/m), h is the height from the sea surface, T0 is the sea level standard temperature (288.15 K), g is the earth-surface gravitational acceleration (9.80665 m/s2), M is the molar mass of dry air (0.0289644 kg/mol) and R is the universal gas constant (8.31447 J/ (mol K)), while the air density d is defined according to the following expression [28]:

d¼

pM RT

ð9Þ

where T is the absolute temperature in Kelvin (K = 273+°C). Having calculated the wind speed at the height of the wind turbine’s rotor, the operating wind load can be defined according to the following formula:

Fw ¼

1  C d  d  A  u2 ðhÞ 2

ð10Þ

where Cd is drag coefficient (Cd = 0.9) [12], d is the air density at the wind tower’s height and A is the surface of the wind generator (rotor area). 5.3. Description of the optimization problems In this study two formulations of the optimization problem are examined which rely on the general description of Eq. (1) where the objective function F(s) represents the construction cost while the constraints considered are those imposed by Eurocode 3 [13] along with frequency and displacement constraints. The basic formulation of the problem is expressed as follows:

min

CðsÞ

subject to g EC3 ðsÞ 6 0 f n ðsÞ P 1:1  f r utop ðsÞ 6 0:0125  H s ¼ ½DðzÞ; tðzÞ; 0 6 z 6 H

ð11Þ

where ‘‘+” implies ‘‘to be combined with”, the summation symbol ‘‘R” implies ‘‘the combined effect of”, Gkj denotes the characteristic value ‘‘k” of the permanent action j while Wd and Ed are the design values of the wind and seismic actions, respectively. Three allowable analysis methods are allowed for performing the design checks for the three limit states considered herein (LS1, LS3 and LS4): including linear elastic (LA), material nonlinear (MNA), and geometric and material nonlinear analysis with imperfections included (GMNIA). In this study the linear elastic (stress design) analysis method was adopted. Specifically, in order to perform the verification checks of LS1, the design stresses should satisfy the following condition:

req;Ed 6 f eq;Ed

ð13Þ

where req;Ed is the von Mises equivalent design stress and f eq;Ed is von Mises design strength defined based on EC3 [13]. Eurocode’s linear elastic buckling analysis is also used for LS3, where interaction checks for the combined membrane stress state are carried out [13]. The LA method is also employed for assessing fatigue strength for all structural steels. In our study five different heights have been considered for the wind tower ranging from 80 m to 160 m high. For the type of the wind tower that we have studied herein, the main load bearing structure of the steel tower consists of thin-wall cylindrical and conical parts, of varying diameters and wall thicknesses, which are linked together by bolted circular rings. Therefore, in the formulation described previously in Eq. (11) the design variables used are the diameter of the wind tower D(z) and the thicknesses t(z) along the weight z (starting from the base where z = 0 to the top of the wind tower where z = H). Since the wind tower is composed by conical parts the two types of design variables D and t correspond to the top diameter of each conical part and the corresponding thickness. In this study the conical parts (beam finite elements) were chosen to be equal to 5 m long thus the 80 m long wind tower is simulated with 16 beams, the 100 m long wind tower is simulated with 20 beams, the 120 m long wind tower is simulated with 24 beams, the 140 m long wind tower is simulated with 28 beams and the 160 m long wind tower is simulated with 32 beams. Hence the number of design variables used for such a formulation are equal to two times the number of the beams plus one that denotes the diameter D0 at the base of the wind tower. However, such a formulation might result to nonfeasible designs in terms of constructability, since it might lead to undesirable variability of the diameter along the height. In order to avoid such design configurations, two formulations have been implemented, where the values of the design variables (diameter and thickness) are defined through to the following expressions:

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Fig. 6. Steel 235 (a) FORM1 and (b) FORM2.

DðzÞ ¼ a  z þ D0

)

tðzÞ ¼ b  DðzÞ or DðzÞ ¼ a  z0:5 þ D0 tðzÞ ¼ b  DðzÞ

min CðsÞ subject to g EC3 ðsÞ 6 0

FORM1 )

ð14Þ

FORM2

The first one is denoted as FORM1 and the second one as FORM2. According to the first formulation the dimensions of the conical parts (diameter and thickness) are reduced linearly as the tower height increases, while based on the second one the dimensions are reduced based on the power expression of Eq. (14). Aiming to guide the shape of the wind tower plan view along its height and mainly to avoid non-feasible designs in terms of constructability are the two reasons that justifies the use of these two variations for the problem formulation. Consequently, based on these two formulations instead of considering diameters D(z) and thicknesses t(z) along the weight z as independent design variables, the formulation of Eq. (11) is modified as follows:

f n ðsÞ P 1:1  f r utop ðsÞ 6 0:0125  H s ¼ ½D0 ; t 0 ; a; b

0:05 6 a 6 0:15

ð15Þ

0:5% 6 b 6 1:0% 0:1 m 6 Dðz ¼ 0Þ 6 4:5 m 0:1 m 6 Dðz ¼ HÞ 6 3:4 m 1:0 mm 6 tðzÞ 6 40:0 mm In addition to formulations FORM1 and FORM2 three material properties have been examined, in particular structural steel of class with nominal yield stress of 235 MPa, structural steel of class with nominal yield stress of 275 MPa and structural steel of class with nominal yield stress of 355 MPa, affecting both strength and material cost. Thus, for S235 the unit material cost was considered

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Fig. 7. Steel 275 (a) FORM1 and (b) FORM2.

to be equal to 1.7 €/kg, for S275 the unit material cost was considered to be equal to 2.0 €/kg and for S355 the unit material cost was considered to be equal to 2.3 €/kg.

6. Numerical results In this study five wind towers are considered for presenting the efficiency of the two problem formulations and that of the structural optimization computing platform. In the optimization problem formulation the construction cost was considered as the objective function to be minimized while the constraint functions considered are those imposed by the design codes (i.e. Eurocodes [12,13,29]). An onshore wind park located in Pafos [30], Cyprus, was chosen to implement the design optimization framework presented in this study. The wind park examined in this study was installed in 2010 and occupies area of 36 square kilometres while it consists of 41 Vestas wind turbines with nominal power of 3 MW each.

6.1. Algorithmic parameters For the solution of the various optimization problems the DE method is used, since it was found robust in previous numerical tests performed by the authors [31–33]. This should not be considered as an implication related to the efficiency of other algorithms, any algorithm available by OCP can be considered for the solution of the optimization problem based on user’s experience. The control parameters for DE are the population size (NP) defined in the range of [50, 200], probability (CR) and constant (F) are defined in the range of [0, 1]. The performance of metaheuristics is influenced by the values of their control parameters; however, their selection is not a straight forward procedure. Although it is not possible to define specific values for the algorithmic parameters that will be the proper ones for all test examples considered, the values given above were found to be the proper ones as a balance between robustness and computational efficiency for numerical tests examined in the past by the authors [31–33] and they are suggested by OCP.

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9

Fig. 8. Steel 355 (a) FORM1 and (b) FORM2.

6.2. Wind tower optimum design In this section, wind towers of varying height (80, 100, 120, 140 and 160 m) have been considered in order to apply the optimization computing platform. All towers have been optimally designed to meet the Eurocode requirements, in particular they were designed according to Eurocode1 1, 3 and 8 [12,13,29]. Whereas, steel of class with yield strength of 235, 275 and 355 MPa with modulus of elasticity equal to 200 GPa has been considered; and as denoted previously for the solution of the optimization problem the DE method was adopted. In order to calculate the construction cost of each wind tower, the material volume is multiplied by the weight coefficient that is proportional to the steel quality, including the cost of production and processing, transportation, assembly, components like connecting flanges and stairs. In addition, we have considered the total initial cost of the wind turbine by adding to the construction cost of each tower 3.0 million € that correspond to the cost of the electromechanical equipment model for the Vestas V90-3 MW turbine that also incorporates the cost of connecting to the network. Further to these costs, in order to incorporate foundation, road

construction, earthworks, rent of land costs, etc., we have taken into account an addition of 20% of the tower’s construction cost. Based on these costs we have calculated the net profit in terms of life-cycle of each wind turbine as follows: (i) First, the annual electrical power (kW per year) is calculated using Eq. (6) for a series of average annual wind speeds ranging from 3 to 12 m/s with step of 0.5 m/s. (ii) Then, the annual energy produced (kW h) is calculated for the corresponding wind speed taking into account the number of hours that wind blows per year. (iii) Finally, the gross profit out of the energy produced (kW h) for 20 years for the corresponding wind speed is calculated; while the net profit after subtracting the initial total cost of the wind turbine for each average annual rate of spectrum. 6.3. Life-cycle cost calculations with reference to wind velocity The produced energy by one wind tower per year is calculated by the following expression:

E ¼ hyear 

uX cut-out

½Pt ðuÞ  HðuÞ

ð16Þ

ucut-in

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where E is the wind energy produced annually (kW h), Pt is the electric power produced (kW), hyear is the number of hours that wind blows per year (usually it is taken equal to the 25–30% of the total hours of the year), H(u) is the distribution of the wind velocity (Weibull distribution), while ucut-in and ucut-out are the lower and upper operational limits for the wind turbine. Therefore, the total wind energy produced annually from a wind farm is equal to:

Etot ¼

N X Ei

ð17Þ

i¼1

where N is the number of wind towers constituting the farm, while the total profit obtained from the wind farm is defined as follows:

C ¼ Pt  cm  T  h  N  cgen  ce

ð18Þ

ce ¼ 0:25  N  cgen

ð19Þ

where C is the total profit obtained from the wind farm, Pt is the electric power generated by the wind turbine (W), N is the total number of wind towers, cm is the unit cost for electric energy generation (€/W h), T is the operational time of the wind farm (usually taken equal to 20–25 years for the wind towers, for example Southern California Edison has agreed to a 25-year power purchase agreement for the power produced as part of the power purchase agreements for up to 1500 MW [34]), h is the potential (capacity) of the wind farm (h/year), cgen is the cost of the wind tower, ce is the cost of the wind park (mechanical equipment, roads, design cost, rent of the land cost, etc.) calculated according to Eq. (19) [35]. As it was described above the wind speed considered for the design of each tower corresponds to the reference height of 10 m; subsequently, this reference value is adapted to the rotor height for each tower according to Eq. (7), while the air pressure and air density are derived through Eqs. (8) and (9), respectively; where d0 = 1.225 kg/m3, T = 293 K (20 °C + 273 = 293 K). Furthermore, the wind load at the top of all wind towers for each case was calculated using Eq. (10), where Cd = 0.9 and A is equal to 6362 m2. The comparative performance of the optimized designs for the two formulations tested here in (FORM1 and FORM2) with reference to life-cycle cost measurements are depicted in Figs. 6–8 corresponding indicatively to a single wind tower. As a remark obtained from this comparative study it can be stated that in order to gain profit, a reference velocity larger than 7 m/s is required. This observation can be stated for both formulations, all heights of wind towers and all material properties tested herein. Although, this cannot be generalized since this remark rely on characteristic values, such as the profit gained from the energy sold (that in the

Fig. 10. Reference velocity 9 m/s.

Fig. 11. Reference velocity 10 m/s.

current study was taken equal to 0.05 €/kW h), it proves that structural optimization represent a very significant component of techno economical analysis. Furthermore, for S235 and FORM1 it can be seen that 140 m wind tower outperforms the rest ones when reference velocity ranges between 7 and 8.5 m/s, while for reference velocity larger 9.5 m/s it is the 80 m wind tower that outperforms the rest ones. Similar observations are obtained for the second formulation FORM2, while it is worth mentioning that 140 and 160 m wind towers achieve the best profit when reference velocity is larger than 8 and 7.5 m/s, respectively. Similar remarks can be denoted for steel properties S275 and S355 as depicted in Figs. 7 and 8, respectively. For more elaborated comparison between the two formulations (FORM1 and FORM2), they are compared between each other in Figs. 9–11 for three reference velocities 8, 9 and 10 m/s, respectively. Out of this comparison it can be seen that the profit achieved by FORM2 is slightly higher compared to FORM1. In particular, the profit for FORM2 is between 1% and 5% higher compared to FORM1, except from the 160 meters high tower that for all three reference velocities and for S235 FORM2 outperforms FORM1 by almost 12% while for S275 FORM2 outperforms FORM1 by almost 26%. 7. Concluding remarks

Fig. 9. Reference velocity 8 m/s.

In this study the applicability of the structural optimization in real-world civil structural engineering was examined; in this

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N.D. Lagaros, M.G. Karlaftis / Computers and Structures xxx (2015) xxx–xxx

direction a special topic related to five wind tower design examples was employed in order to illustrate the capabilities of structural optimization. Furthermore, it was proved that structural optimization represent a very significant component of techno economical analysis in conjunction with life-cycle cost measurements. Therefore, two design optimization formulations were implemented, aiming to guide the shape of the wind tower plan view along its height and mainly to avoid non-feasible designs in terms of constructability. More specifically, according to the first formulation the dimensions of the conical parts (diameter and thickness) are reduced linearly as the tower height increases, while based on the second one the dimensions are reduced based on the power expression. Furthermore, wind towers of varying height (80, 100, 120, 140 and 160 m high) have been considered in order to apply the optimization computing platform. All towers have been optimally designed to meet the Eurocode requirements. Whereas, steel of class with yield strength of 235, 275 and 355 MPa have been considered. For the solution of the optimization problem the differential evolution algorithm was adopted. The comparative performance of the optimized designs for the two formulations tested was based on life-cycle cost measurements. As a remark obtained from this comparative study it can be seen that for both formulations, all heights of wind towers and all material properties tested is that in order to achieve profit it required reference velocity larger than 7 m/s, while the second formulation outperformed the first one in terms of profit gained by more than 5%. Although, these observations cannot be generalized since they are based on characteristic values, such as the profit gained from the energy sold, it was proved that structural optimization represent an important part of techno economical analysis.

[9]

[10]

[11] [12]

[13]

[14] [15]

[16]

[17]

[18]

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[22]

Acknowledgements

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This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program ‘‘Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) – Research Funding Program: THALES ‘‘Resolution of Complex Problems in the Analysis of ‘‘Next Generation” Wind Turbine Towers”, Investing in knowledge society through the European Social Fund. The authors would also like to acknowledge the help of M. Birdas to perform part of the techno economical study required in this study.

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Please cite this article in press as: Lagaros ND, Karlaftis MG. Life-cycle cost structural design optimization of steel wind towers. Comput Struct (2015), http://dx.doi.org/10.1016/j.compstruc.2015.09.013