Stochastic Life-Cycle Cost Analysis of Wind Parks

8192019 Stochastic Life-Cycle Cost Analysis of Wind Parks

httpslidepdfcomreaderfullstochastic-life-cycle-cost-analysis-of-wind-parks 127

Authors Accepted Manuscript

Stochastic Life-Cycle Cost Analysis of WindParks

Nikos D Lagaros Matthew G Karlaftis Maria KPaida

PII S0951-8320(15)00220-3DOI httpdxdoiorg101016jress201507016

Reference RESS5369

To appear in Reliability Engineering and System Safety

Received date 5 April 2014Revised date 27 June 2015Accepted date 17 July 2015

Cite this article as Nikos D Lagaros Matthew G Karlaftis Maria K PaidaStochastic Life-Cycle Cost Analysis of Wind Parks Reliability Engineering and

System Safety httpdxdoiorg101016jress201507016

This is a PDF file of an unedited manuscript that has been accepted forpublication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting typesetting andreview of the resulting galley proof before it is published in its final citable formPlease note that during the production process errors may be discovered whichcould affect the content and all legal disclaimers that apply to the journalpertain

wwwelseviercomlocateress



1

Stochastic Life-Cycle Cost Analysis of Wind Parks

Nikos D Lagaros1+

Matthew G Karlaftis Maria K Paida

+

+ Institute of Structural Analysis amp Antiseismic ResearchDepartment of Structural Engineering

School of Civil EngineeringNational Technical University of Athens

9 Iroon Polytechniou Str Zografou CampusGR-15780 Athens Greece

e-mail nlagaroscentralntuagrmarypaidagmailcom

Department of Transportation Planning andEngineering



e-mail mgkcentralntuagr

Abstract We develop a life-cycle cost model for assessing wind parks implementing the model

requires calculation of cost components that are related to wind tower structural performance formultiple wind hazard levels We compute the structural capacity of the wind towers by means ofnonlinear static structural analysis for three wind hazard levels then the limit state dependentand life-cycle costs for the wind park are calculated based on the proposed model The wind loadfor each wind hazard level is based on actual collected data and is generated probabilisticallyApplication of the proposed life-cycle cost analysis model is tested for a wind park with knowncharacteristics (number and location of wind towers wind potential and so on)

Keywords Life-cycle cost assessment stochastic wind load nonlinear static structural analysiswind parks

1

INTRODUCTION

Renewable energy sources (RES) or mild forms of energy offer exploitable energy derived from

natural processes such as solar radiation movement of water wind and geothermal energy 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 Wind turbines are used to produce electricity windmills are used to produce

mechanical power and wind pumps are used for water pumping Since the early 1980rsquos many

wind parks have been constructed and significant work has been published related to theassessment of wind park efficiency

The principles of life-cycle cost analysis (LCCA) are based on economic theories and are

related to the monetary estimation of possible losses due to events with random occurrence and

intensity during a projectrsquos service life In the past several studies concerning such analyses of

1 Corresponding author



2

wind energy related systems have been published The work by Schleisner [1] concentrated on

the assessment of energy and emissions related to the production and manufacture of materials

for offshore and onshore wind parks based on a life-cycle analysis model Granovskiiet al [2]

extended life-cycle assessment procedures and used to evaluate the efficiency economicity and

impact environmentally of producing hydrogen using wind and solar energy instead of fossilfuels Celik [3] analysed statistically the wind energy potential of a region in Turkey based on

five year hourly time series wind speed data for two different wind turbine heights and carried

out life-cycle cost analysis over a 20-year system The results of life-cycle assessment performed

in the work by Wagner and Epe [4] depicted that wind energy converters also in comparison

with other renewable energy systems have a short energy payback time and low emissions

Dismukes et al [5] presented an assessment procedure that addresses the factors that have

influenced the lengthy industrial technology life-cycle of wind electrical power generation since

its inception in late 19th Century They applied also the accelerated radical model that provides a

general approach to understanding the dynamics of the industrial technology life-cycle for a wide

variety of radical innovations as well as wind electrical power Lee et al [6] aimed to identify the

environmental and economic aspects of the wind-hydrogen system using life-cycle assessment

and costing methodologies and they demonstrated the future competitiveness of the wind energy

pathway in both environmental and economic aspects Aziz et al [7] analysed the current

information systems in order to assess them within wind energy organizations towards achieving

causal understanding of problems requirements capabilities and characteristics and to provide

the ability to enhance current practices Weaver [8] investigated the commercial attractiveness of

the offshore wind energy industry using two commercial scale operational projects as illustrative

cases and applied a methodology aiming to justify the financial value over life-cycle of plant

based on historical data and future market scenarios Schaumman et al [9] conducted life-cycle

assessment and provided the results on the sustainability of steel substructures of offshore wind

turbines in the framework of the German research project NaStafEE The aim of the study by

Abbes and Belhadj [10] was to investigate the prospects of wind energy development in the El-

Kef governorate in Tunisia especially the annual energy production of the wind park wascalculated and the economic feasibility was evaluated based on a life-cycle costing methodology

in order to estimate wind projects viability and profitability in the region In order to assess more

effectively the safety risk of a wind park project the life-cycle indicators can be divided into

quality cost time security and sustainable development indices in this direction Yang and

Zheng [11] presented a principal component analysis model of the life-cycle indicators aiming to



3

define the safety status of a wind park project Wind turbines incorporate prognostics and health

management to reduce life-cycle costs and increase availability Haddad et al [12] introduced

maintenance options in order to quantify the decisions significance and were implemented on a

US land-based wind park

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 and industrial

level [1314] The main factors that affect the total cost include installation operation-

maintenance (OampM) 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 operation-maintenance costs are about 95 of the

total cost [15] 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 tower by improving the turbine performance and increasing the

power generation efficiency costs of wind power production are decreased

In this paper a LCCA framework - explicitly based on nonlinear structural analysis - is used

to assess wind park projects Our goal is to present a new life-cycle cost model that assesses the

efficiency of wind park projects and is explicitly based on structural engineering principles

Following the assessment of the structural capacity for each type of wind tower in a park (and for

three wind hazard levels) limit state dependent and life-cycle costs for the wind park are

calculated The wind load for each wind hazard level is considered stochastically and is generated

based on collected data The implementation of the life-cycle cost analysis model proposed in this

study is presented for a wind park located in Cyprus with known characteristics (number andposition of wind towers wind potential etc)



4

2 LIFE-CYCLE COST ANALYSIS FRAMEWORK

The life-cycle cost C LC of a structural system refers either to the remaining life period of an

existing or retrofitted system or to the design life period of a new one The calculation formula of

the life-cycle cost is defined as follows [1617]

( ) ( ) ( ) LC IN LS C t C C t = + s s s (1)

where C IN and C LS are the initial and limit state dependent costs both expressed in present value s

designates the design vector that corresponds to resistances material properties and design loads

all affecting the design procedure of the structural system while t is the lifespan period (design

life period or remaining life one) Material and labour construction costs are taken into account in

the initial cost while limit state dependent cost denotes to the potential damage cost due to

hazards (wind earthquake) that may occur during the lifespan of the system It accounts for the

repair cost the rental-income losses the loss of contents cost along with other direct or indirect

losses due to natural hazards

It should be noted that there are other not necessarily hazard driven factors that might

influence the behaviour of the wind towers and consequently the performance of the wind park

However based also on the works of others wind hazard is considered as a major factor that

affect the wind tower life-cycle cost In particular as mentioned in the work by Lavassas et al

[18] during the design procedure wind load is among the most critical ones and is mainly used for

dimensioning of the steel structural part of the wind tower Such factors not hazard driven ones

are the corrosion fatigue mechanical failure and others that affect the reliability of the systemSuch factor can be considered into a life cycle model as suggested indicatively in the work by

Akiyama et al [19] and in particular into the LCC model used herein

Several engineering demand parameters (EDPs) have been proposed aiming to quantify

structural damage whose values can be related to specific structural damage states The idea of

describing the state of structural damage by specific parameter value that is defined based on a

distinct scale is attractive because of its simplicity Many researchers have studied so far various

EDPs applicable to reinforced concrete or steel structures a detailed survey can be found in the

work by Ghobarah et al [20] The maximum roof drift (θ ) is considered as the response

parameter which best characterises the structural damage and is associated with all types of losses

and is suggested by the American Society of Civil Engineers and American Wind Energy

Association [21] that can be used for assessing proper turbine operation and performance The

correlation between drift and damage states employed in this study is provided in Table 1 these



5

seven limit states are considered in the procedure described below and the two test examples

studied

For the i-th limit state limit state dependent cost is expressed as follows [22]

i EDP i i i i

LS dam con ren incC C C C C = + + + (2)

where i

damC corresponds to the damage repair cost (that refers to the wind tower) i

conC is related to

the loss of contents cost (that refers to the wind turbine) i

renC designates the loss of rental cost

(that refers to the rental cost of the land) and i

incC refers to the income loss cost (that refers to the

income attained from the sale of electricity) In particular i

damC and i

conC refer to the repair cost of

the steel wind tower that is calculated as the replacement cost times the mean damage index and

that of the wind turbine that is calculated as the unit contents (blades nacelle gear box generator

etc) cost times the mean damage index respectively Additionally i

renC and i

incC refer to the

rental cost that is calculated as the rental rate times the gross leasable area and loss of function

and income cost that is calculated as income rate times down time respectively A more detailed

description of the four cost components given in Eq (2) and in particular for the case of wind

towers can be found in Table 2 while step by step implementation can be found in previous work

of the authors in [17] The basic costs provided in Table 2 are expressed in monetary units (MU

corresponding to Dollars or Euros) while they refer to the first component of the calculation

formulas on the other hand the income loss cost is defined based on the sale price of 005

MUKWh The values of mean damage index loss of function and down time used in this study

are based on the values given in [1623] Table 3 provides the ATC-13 [24] and FEMA-227 [25]

limit state dependent damage consequence severities which have been implemented for the case

of buildings bridges underground structures and general structural systems [1726-28]

Based on the assumption that after major damage systems are immediately retrofitted to their

original intact conditions and that wind occurrences are modelled with a Poisson process model

Wen and Kang [16] proposed the following limit state dependent cost formula considering N

limit states

( )

1

( ) 1 N

t i EDP EDP

LS LS i

i

C t e C Pλ ν

λ

minus

=

= minus sdotsum s

(3)

where

1( ) ( ) EDP

i i iP P EDP EDP P EDP EDP

minus= gt minus gt (4)

and

1( ) ln[1 ( )]

i iP EDP EDP P EDP EDPt ν

minusgt = sdot minus gt (5)



6

wherei EDP

LS C is the limit state dependent cost value if the i-th limit state is violated EDP

iP denotes

the corresponding exceedance probability given the wind load occurrence ( )iP EDP EDPgt is

the exceedance probability given wind load occurrence over the period [0t ] for the engineering

demand parameter considered EDPi-1 and EDPi denote the lower and upper bounds of the i-th

limit state (the maximum roof drift is used in this study) for the i-th

damage level( )

iP EDP EDPgt denotes the annual exceedance probability based on the Poisson process model

ν denotes the annual occurrence rate of wind load and t is the service life period of a new system

or the remaining life of a retrofitted one The calculation steps of the life-cycle cost analysis

procedure implemented in this study for wind towers are given in Figure 1 Where the

exponential term of Eq (3) is used in order to express C LS in present values while λ is the annual

monetary discount rate defined in the range of 3 to 6 as proposed in previous studies [23] In

this work the annual monetary discount rate λ is taken to be constant and equal to 5 since a

constant discount rate is considered a common practice [2930]

The formulation given in Eqs (2) and (3) was applied initially to buildings [1622] however

the applicability of the model is not limited to such structural systems only This formulation is

not structural type dependent since with the proper reformulation of the counter parts (as

described in Table 2) the specific LCC model can be applied to wind towers too Furthermore

the specific model among others was used for the assessment of bridges [3132] in order to apply

the specific LCC model into wind tower systems the modifications indicated in the description of

the cost components were implemented

3 STOCHASTIC MODELLING OF WIND LOAD

Two approaches are basically used for modelling uncertain properties random variables or

random fields In the case of random variables an uncertain property has constant value across a

specific domain this constant value is then assumed to follow some probability distribution

which describes the random characteristics of the property In the case of random fields the value

of the property varies across the domain according to a correlation pattern

The basic assumption in stochastic formulations is that the variation of an uncertain propertyP

across a two-dimensional domain can be represented at each location of the domain x y as

( ) [ ]0 1 ( )P x y P f x y= + (6)

where P0 is the mean value of P and f ( xy) is a zero-mean 2D Gaussian homogeneous stochastic

field which describes the variation of P about P0 Stochastically modelled wind load properties



7

are utilized in the present work and in this respect the spectral representation method is used to

generate Gaussian stochastic field samples

31 Spectral representation of stochastic fields

According to the spectral representation method the sample required for the i-th simulation for a

two-dimensional univariate homogeneous Gaussian stochastic field which in the present study

describes the uncertain wind velocity can be generated by the following series of cosines formula

as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn

i x Α x f φ κ (7)

where )(inφ (n=01hellip N -1) denotes the i-th realization of the random phase angle

nφ which is

uniformly distributed over the range [0 2π] while An is given by

κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)

In the above expressions the power spectrum S ff of the stochastic field is a real non-negative

function of the wave number κ while ∆κ is the wave number increment The upper cut-off wave

number κ u defines the active region of the power spectrum since S ff (κ ) is assumed to be zero

beyond the value of κ u The power spectral density function used in the present work is expressed

as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)

where σ represents the coefficient of variation of the stochastic field while b is the correlation

length parameter The correlation length b measures the distance of two different stochastic field

locations over which the correlation between the respective random variables approaches zero

Therefore brarrinfin indicates a perfectly correlated stochastic field with all its variables being

linearly dependent while brarr0 corresponds to white noise generated by an uncorrelated random

field

32 Stochastic wind load

The implementation of LCCA procedure requires determining the wind load for at least three

hazard levels Therefore the structural performance of the wind park is assessed for three hazard

levels corresponding to the frequent rare and extreme wind hazard levels For each hazard level



8

the distribution of the wind speed is taken into account in order to estimate its effect on the

structural performance of the wind towers for the wind park considered The first hazard level is

determined based on statistical data that are available for the specific region For the second

hazard level it was assumed that the mean value of wind velocity is equal to 25 ms which is

considered rare for the specific region while for the last hazard level the average wind speeddeveloped in the wind park is considered to be equal to 36 ms which is an extremely infrequent

velocity value In order to take into account the random distribution of the wind speed for the

above three hazard levels over the landscape of the wind park location 700 stochastic fields are

generated 600 of them correspond to the frequent hazard level representing the wind velocity for

twelve different wind directions varying by 30o while the rest 100 correspond to the other two

hazard levels (25 ms and 36 ms) Based on the available data for the case of the first (frequent)

wind hazard level stochastic fields were sampled with reference to wind speed and its direction

while for the case of the rare and extreme hazard levels the corresponding stochastic fields were

sampled with reference to wind speed only Furthermore it should be stated that in this study a

correlation length b equal to 5000 m was considered according to the work by Tompson et al

[34] Indicatively fourteen stochastic fields are shown in Figures 2 and 3 corresponding to the

three hazard levels considered In particular Figure 2 depicts twelve stochastic fields for the

different wind directions considered while Figure 3 provides one stochastic filed for the rare and

extreme hazard levels respectively All fourteen sample functions were generated using the

aforementioned correlation parameter b=5000 m The 50 wind field simulations (generated for

wind hazard levels) times (12 each wind direction for the frequent hazard level plus 1 direction for

rare and extreme hazard levels) are such that they encompass the measured wind time series (for

the frequent wind hazard level) and the average wind speed for the rest ones

4 STRUCTURAL ANALYSIS METHOD

41 Description

An onshore wind park located in Pafos [35] Cyprus was chosen to implement the life-cycle cost

analysis procedure 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 2MW each Figure 4 depicts the landscape of the location



9

42 Description and structural analysis of wind towers

For the purposes of this study two types of wind towers were examined in order to investigate

how the tower height affects the structural performance whether the increased wind speed can be

better exploitable and how the height influences the life-cycle cost assessment of the wind park

The first type has tower height equal to 80 m and rotor diameter equal to 90 m while the tower isconical tube with gradually decreasing diameter in relation to the height increase The bottom

diameter is equal to 43 m and the top diameter is equal to 295 m when H=80 m and equal to

260 m when H=100 m while the thickness of the hollow circular cross section is equal to 007

m Structural steel of class with nominal yield stress of 355 MPa and modulus of elasticity equal

to 200 GPa was considered The wind turbine 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 The second

type of wind tower examined has the same technical characteristics with the first one except

from the tower height which is equal to 100 m

The structural analyses of the wind towers were carried out with the software Seismostruct

[36] 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) Specifically

the diameter of each beam element is determined through the relationship D( z)= D0-0008middot z

where base diameter D0 is equal to 43m while z is the height In the structural analysis two

types of loads are applied the static load of the rotorrsquos self-weight and the wind load It was

assumed that the self-weight of the rotor is 800 kN and it is placed with an eccentricity of 10 m

with reference to the axis of the tower while the wind load was simulated with a static horizontal

load at the top of the pillar During the nonlinear static analysis the structural model is pushed

using the static lateral load at the top of the pillar which is applied incrementally The nonlinear

static analysis resulted in the capacity curves for the two types of wind towers which are

depicted in Figures 5(a) and 5(b)

43 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 [37]



10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)

where ( )u h is the wind speed at height hg

u is the average wind speed at the reference height

(href ) the reference height is usually taken equal to 1000 m while z0 is the roughness length For

the 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 001 [38] The stochastic

fields for the wind velocities are generated at the reference height of 10 m and they provide the

wind velocity at the reference height for twelve different directions of the wind (per 30o) and

three different hazard levels while the wind velocity at the top of each wind tower is calculated

using Eq (12)

The density of the atmospheric air affects the wind load developed at the top of each windtower Air density is a function of the atmospheric pressure and temperature that decreases in

relation to the height increase The density d o of the air at the sea surface is equal to 1225 kgm3

when the temperature is equal to 20οC while the corresponding pressure is equal to 1016 kPa

[39] The relation of air pressure with respect to height is defined by [39]

0

0

1

g M

R L L h p p

T

sdot

sdot sdot= sdot minus

(13)

where p is the air pressure (Pa) p0 is the sea level standard atmospheric pressure (101325 Pa) L

is the temperature lapse (00065 Km) h is the height from the sea surface T 0 is the sea level

standard temperature (28815 K) g is the earth-surface gravitational acceleration (980665 ms2)

M is the molar mass of dry air (00289644 kgmol) and R is the universal gas constant (831447

J(molmiddotK)) while the air density d is defined according to the following expression [39]

p M d

R T

sdot=

sdot (14)

where T is the absolute temperature in Kelvin (Κ =273+oC) Having calculated the wind speed at

the height of the wind turbinersquos rotor the operating wind load can be defined for the three wind

hazard levels considered according to the following formula

21( )

2w d

F C d A u h= sdot sdot sdot sdot (15)

where C d is drag coefficient (C d = 09) [40] d is the air density at the wind towerrsquos height and A

is the surface of the wind generator (rotor area)



11

5 NUMERICAL RESULTS

51 Implementation of life-cycle cost analysis

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 (eg 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

According to Eq (5) in order to calculate the probabilities ( )i

P EDP EDPgt it is necessary to

calculate first the t-year exceedance probabilities ( )t i

P EDP EDPgt for the seven limit states

described in Table 1 In this study the annual exceedance probabilities ( )iP EDP EDPgt are used

that are obtained as follows (i) The maximum values of the EDPs considered are calculated forat least three wind hazard levels based on nonlinear static analyses (ii) Using the pairs of

maximum EDPs - annual exceedance probabilities (for the at least three wind hazard levels

considered) a curve is fitted based on a relationship of the form

( ) ( ) k

i iP EDP EDP EDPγ minus

gt = (16)

where parameters γ and k are obtained by best fit of the known i iP EDPminus pairs Hence the annual

exceedance probability for the ith limit state ( )iP EDP EDPgt is directly read from the fitted

curve of Eq (16) ie for the seven limit states implemented in this study and that are given in

Table 1

Therefore the limit-state cost calculation procedure requires the definition of at least three

pairs of annual probability of limit-state exceedance ( )iP and the corresponding maximum value

of engineering demand parameter ( EDPi) developed at the wind tower Thus the assessment of

wind towerrsquos structural capacity for selected hazard levels of increased intensity has to be

performed first The abscissa values of the ( )i iP EDPminus pairs correspond to the maximum values



12

of the engineering demand parameter for at least three wind hazard levels and they are obtained

by means of nonlinear static analysis while the ordinate values denote the corresponding annual

probabilities of exceedance The capacity curves of the towers were used in order to calculate

EDP values for each hazard level In particular the wind load was defined at the top of each

tower using Eq (15) for every wind direction and hazard level and through the capacity curve thecorresponding EDP values were obtained In this work three different wind hazard levels were

considered corresponding to the frequent wind velocity level defined based on statistical

measurements for one year the rare wind velocity level defined with average velocity equal to 25

ms and the extreme wind velocity level defined with average velocity equal to 36 ms The wind

velocity corresponding to the frequent hazard level was defined for 12times50 stochastic fields

(twelve directions of the wind varying by 30o degrees and 50 different stochastic fields for each

direction) The corresponding maximum EDP value for the frequent hazard level was obtained

according to the following procedure the median value for the EDPs among the 12 directions

was defined first for each of the 50 simulations and amongst those the maximum EDP value was

chosen Similar steps were implemented for both rare (25 ms) and extreme (36 ms) hazard

levels and for each wind tower in particular 50 stochastic fields were generated for each hazard

level and among those the maximum EDP value was chosen for each hazard level

In the second part the annual exceedance probabilities of the three limit states were

calculated The annual exceedance probability for the frequent hazard level was obtained using

the Weibull distribution The probability density function of the Weibull distribution is

( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)

where k and λ are positive constants that define the shape and scale parameters respectively The

shape parameter k is taken equal to 20 and the scale parameter λ is obtained according to [41] as

defined by the statistical data for wind speed For the case of the rare and extreme wind hazard

level the annual exceedance probabilities were calculated assuming recurrence interval periods

of 2500 years and 6000 years respectively As a result the annual probability of exceedance forthe rare hazard level with mean wind speed potential of 25 ms is equal to 004 and the annual

probability of exceedance for the extreme hazard level with mean wind speed potential of 36 ms

is equal to 00167 In the current study the values used for the wind speed and return period for

the case of the rare and extreme hazard levels were taken from [4243]



13

52 Test case A

In the first test case it is assumed that the wind park consists of 41 wind towers of the same type

those with tower height H=80 m the positions of the wind towers are shown in Figure 6 As it

was described above the wind speed for each tower for the reference height of 10 m is obtained

through the wind velocity stochastic fields generated for each hazard level Subsequently these

values are adapted to the rotor height for each tower according to Eq (15) while the air pressure

and air density are derived through Eqs (14) and (15) respectively where d0 = 1225 kgm3 T =

293 K (20 oC + 273 = 293 K) Furthermore the wind load at the top of all wind towers for each

case was calculated using Eq (15) where Cd = 09 and A is equal to 6362 m2 Furthermore the

structural response corresponding to each hazard level was estimated through linear interpolation

in the capacity curve of the first type of wind towers given in Figure 5(a) Therefore in order to

calculate the limit state dependent cost of each wind tower the initial installation cost of this type

of wind towers should be defined this cost is assumed to be equal to 18middot106 MU [44]

( )i i i i i

LS dam con ren incC C C C C t s

θ = + + + (18)

The various cost components of Eq (18) are estimated assuming that they are proportional to the

initial installation cost of each wind tower according to corresponding limit state of the damage

index considered (roof drift) [23] The values provided in Table 3 indicate the relationship

between the roof drift value achieved for the wind tower and the percentage of the initial

installation cost The total limit state dependent cost of the wind park is the sum of the limit state

dependent costs of all wind towers

( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)

where N T is the number of the wind tower Consequently the limit state dependent of the wind

tower resulted to be equal to 454middot106 MU The initial cost of the wind park is the sum of the

initial installation costs of the wind towers 41times18middot106 = 738middot106 MU this installation cost

includes purchase cost of the wind turbine design construction and connection of the wind

towers to the electricity grid costs Finally the life-cycle cost of the wind park is the sum of the

initial installation cost and the limit state dependent cost

( ) ( ) ( ) 6 7834 10wind park wind park wind park

LC IN LS C s t C s C t s MU = + = sdot (20)

The contribution of the cost components to the limit state dependent cost for the 41 wind towers

is presented in Figure 7 As expected similar trend on the contribution of the limit state

dependent cost is observed among the 41 wind towers In particular C dam and C con are the

dominant contributors of the limit state dependent cost since they represent for all wind towers



14

of the park around 300 and 650 respectively of the limit state dependent cost while the

contribution to the limit state dependent cost of both C ren and C inc is less than 01 In addition

C dam corresponds to 125 - 260 of the initial cost while C con corresponds to 250 - 50

This is reasonable since the damages to the wind turbine (blades nacelle gear box generator

etc) that is associated with the contents cost is more sensitive to the damages to the steel windtower

53 Test case B

In the second test case it is examined the same wind park consists of the same number of wind

towers but of two different types of them In this case twenty one of them are 80 m high and the

remaining twenty are 100 m high The positions of the second type of wind generators were

chosen randomly and they are indicated with red colour in Figure 8 In order to estimate the limit

state dependent cost of the wind park the same procedure described for the test case A isimplemented for the two types of wind towers The wind velocity at the top of the wind towers

as well as the air density are calculated first Subsequently the wind load for each wind tower

and the corresponding roof drift values for each one of the three wind hazard levels considered

are obtained from the capacity curves of the 41 wind towers The three pairs (983120 minus 983108983113983081 required

for the calculations are determined according to the procedure described for the test case A

Based on the assumption for the test case A the initial installation cost of the first type of wind

towers (H=80 m) was taken equal to 18middot106 MU while the initial installation cost of the second

type of wind towers (H=100 m) was assumed to be equal to 20middot106 MU [44] The calculation

procedure resulted to the limit state dependent value for the wind park to be equal to 482middot106

MU The initial installation cost of the wind park is the sum of the initial installation costs of the

wind towers 21times18middot106 + 20times20middot106 = 778middot106 MU while according to the test case A the

installation cost includes purchase cost of the wind turbine design construction and connection

of the wind towers to the electricity grid costs Finally the life-cycle cost of the wind park is the

sum of the initial cost and the limit state dependent cost

( ) ( ) ( ) 6 8262 10wind park wind park wind par

C

k

IN LS LC s t C s C t s MU = + = sdot (21)


is presented in Figure 9 Similar to the test case A C dam and C con are the dominant contributors of

the limit state dependent cost since they represent for all wind towers of the park around 320

and 650 respectively while the contribution to the limit state dependent cost of both C ren and

C inc is less than 01 In addition C dam corresponds to 120 - 250 of the initial cost while

C con corresponds to 240 - 520



15

6 CONCLUDING REMARKS

This main objective of this study was to develop a tool based on structural life-cycle cost analysis

procedure for assessing the investment for the construction and operation of wind parks The

formulation of the analysis procedure focused on the evaluation of a wind park from the

structural engineering point of view The cost for renting the land financing construction and that

of the selling price from the produced energy were considered In particular the proposed

calculation procedure of the life-cycle cost for wind parks is based on the installation cost of each

wind tower the damage repair cost the loss of contents cost the loss of rental cost and the

income loss cost

The numerical tests were performed on two variants of one wind park located in Cyprus

(Orites Wind Park) Comparing the contributing cost components of the limit state dependent

cost it can be said that the damage and contents costs in both variants are the dominant

contributors for all wind towers In particular contents cost is the first dominant contributor

while damage cost represents the second dominant contributor It is also worth mentioning that

although the location of the 41 wind towers differ significantly income and rental costs represent

only a very small percentage of the limit state dependent cost

ACKNOWLEDGMENTS

This research has been co‐financed by the European Union (European Social Fund ndash ESF) and

Greek national funds through the Operational Program Education and Lifelong Learning of the

National Strategic Reference Framework (NSRF) ‐ Research Funding Program THALES

ldquoResolution of Complex Problems in the Analysis of ldquoNext Generationrdquo Wind Turbine Towersrdquo

Investing in knowledge society through the European Social Fund

REFERENCES

1 Schleisner L Life cycle assessment of a wind farm and related externalities Renewable Energy20(3) 279-288 2000

2 Granovskii M Dincer I Rosen MA Exergetic life cycle assessment of hydrogen production fromrenewables Journal of Power Sources 167(2) 461-471 2007

3 Celik AN A techno-economic analysis of wind energy in Southern Turkey International Journal

of Green Energy 4(3) 233-247 2007

4 Wagner H-J Epe A Energy from wind - Perspectives and research needs European Physical

Journal Special Topics 176(1) 107-114 2009

5 Dismukes JP Miller LK Bers JA The industrial life cycle of wind energy electrical powergeneration ARI methodology modeling of life cycle dynamics Technological Forecasting and

Social Change 76(1) 178-191 2009



16

6 Lee J-Y An S Cha K Hur T Life cycle environmental and economic analyses of a hydrogen stationwith wind energy International Journal of Hydrogen Energy 35(6) 2213-2225 2010

7 Aziz Y El-Thalji I Tahir M Bhatti F Operation and maintenance management information systemsfor wind energy organizations Review and assessment European Wind Energy Conference and

Exhibition (EWEC) pp 4050 Warsaw Poland 20-23 April 2010

8 Weaver T Financial appraisal of operational offshore wind energy projects Renewable and

Sustainable Energy Reviews 16(7) 5110-5120 2012

9 Schaumann P Bechtel A Wagner H-J Stranghoumlner N Indicators for sustainability assessment ofrenewables Life-Cycle and Sustainability of Civil Infrastructure Systems - Proceedings of the 3

rd

International Symposium on Life-Cycle Civil Engineering (IALCCE) pp 1745 Vienna Austria 3-6October 2012

10 Abbes M Belhadj J Wind resource estimation and wind park design in El-Kef region Tunisia Energy 40(1) 348-357 2012

11 Yang T-H Zheng Q-H Study on establishment of safety risk assessment system of wind farm projectlife cycle Asia-Pacific Power and Energy Engineering Conference (APPEEC) Shanghai China 27-29 March 2012

12 Haddad G Sandborn PA Pecht MG Using maintenance options to maximize the benefits ofprognostics for wind farms Wind Energy 17(5) 775-791 2014

13 Landry M Leclerc A Gagnon Y A methodology for the evaluation of the economic impacts of windenergy projects Energy and Environment 24(5) 735-748 2013

14 Oebels KB Pacca S Life cycle assessment of an onshore wind farm located at the north-easterncoast of Brazil Renewable Energy 53 60-70 2013

15 International Renewable Energy Agency (IRENA) Renewable Energy Technologies Cost AnalysisSeries Volume 1 Power Sector Issue 55 Wind Power Bonn Germany 2012

16 Wen YK Kang YJ Minimum building life-cycle cost design criteria I Methodology Journal of

Structural Engineering 127(3) 330-337 2001

17 Lagaros ND Life-cycle cost analysis of design practices for RC framed structures Bulletin of

Earthquake Engineering 5(3) 425-442 2007

18

Lavassas I Nikolaidis G Zervas P Efthimiou E Doudoumis IN Baniotopoulos CCAnalysis and design of the prototype of a steel 1-MW wind turbine tower Engineering Structures25(8) 1097-1106 2003

19 Akiyama M Frangopol DM Matsuzaki H Life-cycle reliability of RC bridge piers under seismicand airborne chloride hazards Earthquake Engineering and Structural Dynamics 40(15) 1671-16872011

20 Ghobarah A Abou-Elfath H Biddah A Response-based damage assessment of structures Earthquake Engineering and Structural Dynamics 28(1) 79-104 1999

21 ASCEAWEA Wind Turbine Structures Recommended Practice for Compliance of Large Land-based Wind Turbine Support Structures RP2011 by American Wind Energy Association (AWEA)and American Society of Civil Engineers (ASCE) Washington DC 2011

22 Wen YK Kang YJ Minimum building life-cycle cost design criteria II Applications Journal of


23 Ellingwood BR Wen Y-K Risk-benefit-based design decisions for low-probabilityhighconsequence earthquake events in mid-America Progress in Structural Engineering and Materials7(2) 56-70 2005

24 ATC-13 Earthquake Damage Evaluation Data for California Applied Technology CouncilRedwood City CA 1985

25 FEMA 227 A BenefitndashCost Model for the Seismic Rehabilitation of Buildings Federal EmergencyManagement Agency Building Seismic Safety Council Washington DC 1992



17

26 Venkittaraman A Banerjee S Enhancing resilience of highway bridges through seismic retrofit Earthquake Engineering and Structural Dynamics 43(8) 1173-1191 2014

27 Pitilakis K Tsinidis G Performance and seismic design of underground structures GeotechnicalGeological and Earthquake Engineering 28 279-340 2014

28 May PJ Making choices about earthquake performance Natural Hazards Review 5(2) 64-702004

29

Rackwitz R The effect of discounting different mortality reduction schemes and predictive cohortlife tables on risk acceptability criteria Reliability Engineering and System Safety 91(4) 469-4842006

30 Rackwitz R Lentz A Faber M Socio-economically sustainable civil engineering infrastructures byoptimization Structural Safety 27(3) 187-229 2005

31 Kumar R Gardoni P Renewal theory-based life-cycle analysis of deteriorating engineeringsystems Structural Safety 50 94-102 2014

32 Padgett JE Tapia C Sustainability of natural hazard risk mitigation Life cycle analysis ofenvironmental indicators for bridge infrastructure Journal of Infrastructure Systems 19(4) 395-4082013

33 Shinozuka M Deodatis G Simulation of multi-dimensional Gaussian stochastic fields by spectralrepresentation Appl Mech Rev ASME 49 29-53 1996

34 Tompson AFB Ababou R Gelhar LW Implementation of the three-dimensional turning bandsrandom field generator Water Resources Research 25(10) 2227-2243 1989

35 Fokaides PA Miltiadous I-C Neophytou MK-A Spyridou L-P Promotion of wind energy in isolatedenergy systems the case of the Orites wind farm Clean Technologies and Environmental Policy16(3) 477-488 2014

36 Antoniou S Pinho R Seismo Struct V6 - A computer program for finite element analysis 2012wwwseismosoftcom

37 IEC 61400-11 Wind turbines - Part 11 Acoustic noise measurement techniques InternationalElectrotechnical Commission Geneva Switzerland 2005

38 Dong Z Gao S Fryrear DW Drag coefficients roughness length and zero-plane displacement height

as disturbed by artificial standing vegetation Journal of Arid Environments 49(3) 485-505 200139 US Standard Atmosphere US Government Printing Office Washington DC 1976

40 Eurocode 1 Actions on structures - Part 1-4 General actions - Wind actions ENV 1991-1-4 2005CEN European Committee for standardization Brussels 2010

41 Morfgan EC Lackner M Vogel RM Baise LG Probability distributions for offshore wind speeds Energy Conversion and Management 52(1) 15-26 2011

42 Holy J Hladky M Mlady O Kolar L Jaros M Estimation of frequency of occurrence ofextreme natural external events of very high intensity on the base of (non)available data ProbabilisticSafety Assessment (PSA) of Natural External Hazards Including Earthquakes WorkshopProceedings Prague Czech Republic 17-20 June 2013

43 Hov Oslash Cubasch U Fischer E Houmlppe P Iversen T Kvamstoslash NG Kundzewicz ZW

Rezacova D Rios D Santos FD Schaumldler B Veisz OZerefos C Benestad R Murlis JDonat M Leckebusch GC Ulbrich U Extreme weather events in Europe preparing for climatechange adaptation Oslo Norwegian Meteorological Institute 2013

44 Kost C Mayer JN Thomsen J Hartmann N Senkpiel C Philipps S Levelized cost ofelectricity- renewable energy technologies Fraunhofer Institute for Solar Energy Systems (ISE)Freiburg Germany 2013



18

FIGURES

Figure 1 Flowchart of the life-cycle cost analysis

Step 1 Definitions

Numerical modelsUncertainties (epistemic amp aleatory)IMs intensity measuresEDPs Engineering demand parametersLS limit states (Table 1)

Basic limit state costs (Table 2)Nsim number of stochastic fields

Step 2 Incrementral analysis

Step 3 Calculate LCC

For n=1 to Nsim

Calculate annual exceedance

probability of the ith damage level

0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median

R e l a t i o n b e t w e e n

E D P s - I M s

For i=1 to LS

Calculate exceedance probabilitygiven occurrence (Eq 5)

Calculate l imit state dependent cost(Eq 3)

Calculate probabilistic

characteristics of life-cycle cost



19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)

Figure 2 Stochastic field for the frequent wind hazard level and different wind directions (a) 0o (b) 30o(c) 60o (d) 90o (e) 120o (f) 150o (g) 180o (h) 210o (i) 240o (j) 270o (k) 300o and (l) 330o



21

(a) (b)

Figure 3 Stochastic field for (a) the rare and (b) the extreme wind hazard level



Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)

e l e v a t i o n ( m )

22

he terrain of the examined wind park produced

(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)

ity curve for tower height (a) H=80m and (b) 100m



Figure 6 Test

Figure 7 Test case A - Contributio

24

ase A - Positions of wind towers with height 80 m

of the limit state cost components of the 41 wind t wers of the park



25

Figure 8 Test case B - Positions of wind towers with height 80 m (green) and height 100 m (red)

Figure 9 Test case B - Contribution of the limit state cost components of the 41 wind towers of the park



Tables

Table 1 Drift ratio and floor acceleration limits for bare moment resisting frames

Limit State Drift () [20]

(I) - None θ le030

(II) - Slight 030ltθ le

036(III) - Light 036lt θ le056

(IV) - Moderate 056lt θ le120

(V) - Heavy 120lt θ le300

(VI) - Major 300lt θ le800

(VII) - Collapsed θ gt800

Table 2 Limit state cost - calculation formulas in monetary units (MU corresponding to Dollars or Euro)

Cost Category Calculation Formula Basic Cost [44]

Damagerepair (Cdam) Replacement cost times number of towers times mean damage index 06times10 MUitem (H=80m)08times106 MUitem (H=100m)

Loss of contents (Ccon)Unit contents cost times number of turbines times mean damageindex

12times106 MUitem

Rental (Cren) Rental rate times gross leasable area times loss of function 50times10 MUmonthKm

Income (Cinc) Income rate times number of turbines times down time50times10 MUyearitem (H=80m)60times104 MUyearitem (H=100m)

Table 3 Limit state parameters for cost evaluation

Limit

State

FEMA-227 [25] ATC-13 [24] Mean damage

index () Loss of

function () Down time

() (I) - None 0 0 0 (II) - Slight 05 09 09 (III) - Light 5 333 333 (IV) - Moderate 20 124 124 (V) - Heavy 45 348 348 (VI) - Major 80 654 654 (VII) - Collapsed 100 100 100



1

Stochastic Life-Cycle Cost Analysis of Wind Parks

Nikos D Lagaros1+

Matthew G Karlaftis Maria K Paida

+

+ Institute of Structural Analysis amp Antiseismic ResearchDepartment of Structural Engineering



e-mail nlagaroscentralntuagrmarypaidagmailcom

Department of Transportation Planning andEngineering



e-mail mgkcentralntuagr

Abstract We develop a life-cycle cost model for assessing wind parks implementing the model

requires calculation of cost components that are related to wind tower structural performance formultiple wind hazard levels We compute the structural capacity of the wind towers by means ofnonlinear static structural analysis for three wind hazard levels then the limit state dependentand life-cycle costs for the wind park are calculated based on the proposed model The wind loadfor each wind hazard level is based on actual collected data and is generated probabilisticallyApplication of the proposed life-cycle cost analysis model is tested for a wind park with knowncharacteristics (number and location of wind towers wind potential and so on)

Keywords Life-cycle cost assessment stochastic wind load nonlinear static structural analysiswind parks

1

INTRODUCTION

Renewable energy sources (RES) or mild forms of energy offer exploitable energy derived from

natural processes such as solar radiation movement of water wind and geothermal energy 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 Wind turbines are used to produce electricity windmills are used to produce

mechanical power and wind pumps are used for water pumping Since the early 1980rsquos many

wind parks have been constructed and significant work has been published related to theassessment of wind park efficiency

The principles of life-cycle cost analysis (LCCA) are based on economic theories and are

related to the monetary estimation of possible losses due to events with random occurrence and

intensity during a projectrsquos service life In the past several studies concerning such analyses of

1 Corresponding author



2


































3






























4





( ) ( ) ( ) LC IN LS C t C C t = + s s s (1)





























5


studied


i EDP i i i i


where i


conC is related to






damC and i





renC and i

incC refer to the















limit states

( )

1

( ) 1 N

t i EDP EDP

LS LS i

i

C t e C Pλ ν

λ

minus

=

= minus sdotsum s

(3)

where

1( ) ( ) EDP



and

1( ) ln[1 ( )]





6

wherei EDP


iP denotes





damage level( )
























( ) [ ]0 1 ( )P x y P f x y= + (6)





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





2


































3






























4





( ) ( ) ( ) LC IN LS C t C C t = + s s s (1)





























5


studied


i EDP i i i i


where i


conC is related to






damC and i





renC and i

incC refer to the















limit states

( )

1

( ) 1 N

t i EDP EDP

LS LS i

i

C t e C Pλ ν

λ

minus

=

= minus sdotsum s

(3)

where

1( ) ( ) EDP



and

1( ) ln[1 ( )]





6

wherei EDP


iP denotes





damage level( )
























( ) [ ]0 1 ( )P x y P f x y= + (6)





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





3






























4





( ) ( ) ( ) LC IN LS C t C C t = + s s s (1)





























5


studied


i EDP i i i i


where i


conC is related to






damC and i





renC and i

incC refer to the















limit states

( )

1

( ) 1 N

t i EDP EDP

LS LS i

i

C t e C Pλ ν

λ

minus

=

= minus sdotsum s

(3)

where

1( ) ( ) EDP



and

1( ) ln[1 ( )]





6

wherei EDP


iP denotes





damage level( )
























( ) [ ]0 1 ( )P x y P f x y= + (6)





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





4





( ) ( ) ( ) LC IN LS C t C C t = + s s s (1)





























5


studied


i EDP i i i i


where i


conC is related to






damC and i





renC and i

incC refer to the















limit states

( )

1

( ) 1 N

t i EDP EDP

LS LS i

i

C t e C Pλ ν

λ

minus

=

= minus sdotsum s

(3)

where

1( ) ( ) EDP



and

1( ) ln[1 ( )]





6

wherei EDP


iP denotes





damage level( )
























( ) [ ]0 1 ( )P x y P f x y= + (6)





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





5


studied


i EDP i i i i


where i


conC is related to






damC and i





renC and i

incC refer to the















limit states

( )

1

( ) 1 N

t i EDP EDP

LS LS i

i

C t e C Pλ ν

λ

minus

=

= minus sdotsum s

(3)

where

1( ) ( ) EDP



and

1( ) ln[1 ( )]





6

wherei EDP


iP denotes





damage level( )
























( ) [ ]0 1 ( )P x y P f x y= + (6)





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





6

wherei EDP


iP denotes





damage level( )
























( ) [ ]0 1 ( )P x y P f x y= + (6)





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





7







as N rarrinfin [33]

( ) summinus

=

+=

1

0

)()( )]cos([2 N

n

innn



nφ which is


κ κ ∆=)(2 n ff n S A (8)

with

κ κ ∆= nn (9)

N

uκ κ =∆ (10)





as

( ) κ κ σ κ

b

ff bS minus

= e4

1 232 (11)






field







8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





8
























41 Description







9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





9
































10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





10

( )

ln

ln

o

g

ref

o

h

zu h u

h

z

=

(12)









using Eq (12)





0

0

1

g M

R L L h p p

T

sdot


(13)






p M d

R T

sdot=

sdot (14)




21( )

2w d






11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





11

5 NUMERICAL RESULTS
























( ) ( ) k


gt = (16)




Table 1








12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





12




















( )

k k 1 x

λ k x

e 0f x λ k λ λ

0 0

x

x

minus minus

gt =

lt

(17)











13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





13

52 Test case A













( )i i i i i


θ = + + + (18)







( )1

( )T N

wind park j

LS LS

j

C t s C t s=

= sum (19)















14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





14






53 Test case B




















C

k










15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





15









income loss cost








ACKNOWLEDGMENTS






REFERENCES




of Green Energy 4(3) 233-247 2007




Social Change 76(1) 178-191 2009



16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





16







rd












18












17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





17




29




















18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





18

FIGURES


Step 1 Definitions





For n=1 to Nsim



0 05 1 15 2 25 302

04

06

08

1

12

14

16

18

2

EDP

I M

single records

MSDA 1684 fractiles

MSDA median


E D P s - I M s

For i=1 to LS







19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





19

(a) (b)

(c) (d)

(e) (f)



20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





20

(g) (h)

(i) (j)

(k) (l)




21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





21

(a) (b)




Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





Figure 4 T

2000

4000

6000

8000

0

200

400

600

northing (m)


22


(a)

01000

2000

30004000

50

0 easting (m)

006000



Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





Figure 5 Capac

23

(b)




Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





Figure 6 Test


24





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





25





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of





Tables



(I) - None θ le030











12times106 MUitem




Limit

State


index () Loss of



Documents

Stochastic Life-Cycle Cost Analysis of Wind Parks