Economically optimal configuration of onshore horizontal axis wind turbines

Renewable Energy 90 (2016) 469e480

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Economically optimal configuration of onshore horizontal axis wind turbines € ge Thomas Muche, Ralf Pohl*, Christin Ho €rlitz, 02763 Zittau, Germany Department of Economic Sciences and Business Management, University of Applied Sciences Zittau/Go

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 February 2015 Received in revised form 2 November 2015 Accepted 1 January 2016 Available online 17 January 2016

During the recent years, the use of new materials and enhanced production processes led to successively increasing sizing parameter e namely rotor diameter, rated power and hub height e of wind turbines as well as decreasing initial and running costs. The thereby risen market share shows that wind energy projects are an interesting field for financial investors. In this connection the question arises, how investors benefit from the mentioned technical developments. Regarding the utility maximization of investors in turbine projects, goal of this survey is to determine the economically optimal turbine configuration in contrast to the technically feasible by maximizing the net present value. For this purpose, a generalized approximation of the power curve of three-bladed, direct-driven, variable speed horizontal axis wind turbines is applied. For estimation of the economic parameters, a complex masscost-model is used for determination of the initial costs as a function of the sizing parameter. Thereby, a method is shown for adjustment of this model to several differences in price levels of the different turbine components. Furthermore, detailed relationships for running costs as well as the estimation of the cost of equity capital are shown. Due to the current importance of feed-in tariff, revenues for electricity sale depend on the German “Renewable Energy Act”. The calculation of economically optimal sizing parameter first is done for a reference site in conjunction with a sensitivity analysis to determine the most influential sizing parameter with respect to economic viability. Afterwards, a range of typically onshore wind speeds in Germany is considered. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Wind turbine Economic optimization Net Present Value Mass-cost-model Power curve

1. Introduction Over the past two decades, wind energy has rapidly evolved to the renewable energy with the highest market share worldwide [1] and an installed capacity of nearly 319 GW (2013) [2]. During this development, technological innovations in production processes and the use of new materials enabled turbine manufacturer to gradually enlarge the design properties of horizontal axis wind turbines (HAWT), namely rotor diameter, rated power and hub height, that are associated with the extraction of energy from the wind [3,4]. Actual, the most powerful available onshore turbine (2014) has a rotor diameter of 127 m with a hub height of 135 m and a rated power of 7.580 kW [5]. For offshore utilization, there exist

* Corresponding author. E-mail address: [email protected] (R. Pohl). http://dx.doi.org/10.1016/j.renene.2016.01.005 0960-1481/© 2016 Elsevier Ltd. All rights reserved.

prototypes with rated powers up to 8 MW and rotor diameters of 164 and 171 m, respectively [6,7]. However, the UpWind project even showed that a 20 MW turbine with a diameter of about 252 m is technically feasible in theory [8]. Besides upscaling of turbines, significant improvements in turbine efficiency could be observed during recent years, especially occurring in turbines designed for onshore utilization [9]. This development partly is a result of the long-term energy policy of several countries in Europe. Investors were offered incentives in form of subsidy programs, such as the German “Renewable Energy Act (EEG)”; the thereby generated market growth was accompanied by the technical progress mentioned above as well as the implementation of measures to reduce the costs for turbine manufacturing and operation. As a result, achievable annual energy production (AEP) of single turbines rose in context with decreasing production costs, enabling costs of energy production (COE) of large onshore HAWT that are now competitive with those of fossil fuels [1]. To summarize, by upscaling of HAWT up to the technical limitation, investors can expect higher remunerations from electricity sale. Since initial and

470

T. Muche et al. / Renewable Energy 90 (2016) 469e480 List of parameters and mathematical symbols Parameter

Symbol

Unit

Value/Range

Swept area Annual energy production Annual energy production (EEG reference) Pitch angle Equity beta Cash flow Additional investment costs Administration costs Insurance costs Investment costs Operation and maintenance costs Coefficient of performance Rental costs Running costs Rotor diameter Return on market portfolio Incoming payments Adjustment factor Cost factor Generator to rotor ratio Hub height EEG reference height Height to diameter ratio Annual rate of inflation Tip speed ratio Net present value Design angular speed of turbine Kinetic energy at hub height at vi Producer Price Index Rated power Air density Cost of equity Exchange rate Return of risk-free asset Return of market portfolio Tax rate Time of feed-in tarif Time Time of initial rate EEG reference wind speed in height h0 Average wind speed at hub height Cut-in wind speed Cut-out wind speed Rated wind speed Rated tip speed Roughness length

A AEP AEPref

m2 kWh kWh
deg

0

b bi

CFt cadInv cAdministration cInsurance cInv cOM cP,max cRental cRun D E(rm) Et fadjust fcost GRR h, h2 h0 HDR i

l NPV

ud Pi PPI Pr

rAir

rcoe rexchange rf rM s t T tini v0 v2 vci vco vr vts z0

running costs rise in the same time, however, there exists an economic optimum in contrast to the technically feasible turbine design. Since assessment of investment decision usually is done with net present value (NPV), the goal of this paper is to identify that combination of rotor diameter, hub height and rated power which is maximizing NPV with respect to the average wind speed. Evolved from the development of airplane propellers, previous research in optimization of HAWT can be divided into two main groups: the first generation of investigations considered the optimization of aerodynamic performance, usually by applying blade element momentum (BEM) theory [10]. Here, aerodynamically optimal geometric shape of the rotor is determined for a single design wind speed or, more important, for the maximization of the annual energy output of the turbine [11,12]. However, this method disregards the increasing loads and the costs thereby incurred. The second generation of optimization models therefore includes functional relations between loads (or mass) of single turbine components and costs [13]. By including fatigue and extreme loads, such a calculation can be used to optimize geometric shape and thickness of rotor blades, resulting in better aerodynamic performance simultaneously with reducing weight and costs [14]. However, these considerations are more helpful for turbine

EUR EUR EUR EUR EUR EUR e EUR EUR m % EUR e e kW/m m m e % e EUR s1 103 kW e kW kg/m3 % e % % % a h a m/s m/s m/s m/s m/s m/s m

0.48

[20,200]

[20,60] [20,200] 30

opt. 8

[100,10000] 1.225

1 e 20 8760 5.5 2 25 12 80 0.1

development instead of being an investment decision support. Other papers discuss the so called micro-siting and the optimized location of wind energy plants within a wind farm [15e17]. Differences in investment costs are estimated by single properties like rotor diameter [15], by relation to rated power [16] or by selection of given turbine types with hub height and location of single turbines [17]. Just two papers exactly discuss the question of economical turbine configuration, aimed in this paper [18,19], but the results do not allow a general appliance due to an outdated cost model [18] and the rotor diameter as the only viewed design parameter [19], respectively. Nearly all existing studies regarding this topic focus on minimization of the COE. In contrast, economical optimum is found by maximization of NPV in this paper. Furthermore, most authors neglect adjustment of the underlying mass-cost-model in [13], or use simple estimations for initial costs, thereby ignoring important relations of costs to turbine components. In this study, a large number of possible combinations of diameter, rated power and height of a three-bladed, pitch controlled, variable speed and direct driven HAWT is viewed, since this type of turbine is one of the most utilized onshore wind turbines worldwide [1]. However, to refrain from manufacturer data, a general description of the turbine

T. Muche et al. / Renewable Energy 90 (2016) 469e480

performance and economical aspects is needed. In particular, this is enabled by: - Estimation of AEP with a simplified model for power drain from the wind in conjunction with an approximation of a generalized characteristic curve - Determination of revenues by applying a national feed-in tariff as a prevalent alternative for electricity sale - Detailed adjusted mass-cost- model for the initial costs in combination with generalized relations for variable costs to estimate outgoing payments

471

years have to be found. Here, AEP plays an important role for the calculation of the achievable revenues and a fraction of the variable costs. To complete the estimation of cash flow, the initial costs are determined as relation to the considered design properties with the help of a mass-cost- model. Thirdly, the required costs of equity are estimated by Capital Asset Pricing Model (CAPM). Now, NPV can be calculated for different combinations of rotor diameter, rated power and hub height of the turbine, where cash flow varies in dependence of turbine properties. In doing so, economically optimal properties of the turbine are identified by maximizing NPV. 2.2. Estimation of the annual energy production

The outline of this paper is as follows. In Section 2 the methodology is described to determine the economically optimal turbine configuration. For this purpose, first the NPV as the objective function is depicted in conjunction with a description of the overall model structure. Afterward, the estimation of the achievable annual energy yield of the HAWT is shown with a focus on a generally valid approximation of the characteristic curve. For evaluation of profitability, the determination of the cash flow is described afterwards, followed by the estimation of the cost of equity capital. In section 3 the results of the calculation of the maximum NPV and the defined optimal configuration are discussed, completed by the conclusion of this work in section 4.

2.2.1. Power drain from wind energy For estimation of the NPV, information of the energy output of the HAWT as well as a description of dependence of the AEP on turbine configuration is needed first. Kinetic and therefore usable energy of the air mass can be described as:

E¼

1 m v2 2 Air Air

(2)

mAir and vAir are the mass and the velocity of the air, respectively. With density of the air rAir and the swept area of the turbine rotor A, the mass flow can be written as [23]:

m_ Air ¼ ArAir vAir

2. Determination of economically optimal turbine configuration

(3)

The power of the wind is obtained with:

1 r Av3 2 Air Air

2.1. Net present value and model structure

PWind ¼

NPV is used to identify optimal sizing parameter since this method usually is applied to decide between investment alternatives. In general, NPV being greater than zero indicates profitable investment decisions. In case of mutually exclusive investment decisions, the project with the larger NPV should be chosen.1 Furthermore, limits of profitable plant designs can be identified by NPV less than or equal to zero. NPV is calculated as follows [21]:

The swept area A of the rotor is calculated with the rotor diameter D:

NPV ¼

T X

CFt

t¼0

ð1 þ rcoe Þt

(1)

Applied to the addressed HAWT, cash flow CFt reflects the initial investment (t ¼ 0) necessary for plant building and the following cash flows, which result from annual revenue2 from electricity sale reduced by relevant running costs for the appropriate period t (t ¼ 1,..,20). The number of observation periods accords to typical lifetime of HAWT, which is assumed to be 20 years [22]. Time and risk preference are included in form of the so-called opportunity cost of capital, which is solely described by cost of equity rcoe and thus suppose a fully equity-financed investment for simplification. Several parameters have to be estimated to allow a valuation of profitability. Fig. 1 shows the structure of the overall model. First of all, AEP has to be estimated with respect to wind conditions as well as design properties of the HAWT, which are influencing the power drain from wind energy. The second step is the determination of the required economical parameters, where values for initial and running costs, revenues and their development in the following

1 This statement will be only valid if the compared projects have equal conditions or if the fictitious investment to be considered has a NPV of zero. See for further information e.g. Ref. [20]. 2 Terminology “revenue” and “costs” are terms of external accounting in the strict sense. Nevertheless, terms are used equivalently for internal accounting and cash flow components here.

(4)

p A ¼ D2 4

(5)

Hence convertible energy of the wind is proportional to the cubic velocity of the wind, to the square rotor diameter and the air density, depending on air temperature and atmospheric pressure [23,24]. For this study, air density is set constant to 1.225 kg/m3. Due to the large influence of wind speed on convertible energy, the hub height of wind turbines has to be considered. The average wind speed in hub height can be expressed as logarithmic ratio [25], where v0 is the wind speed at height h0, h2 is the hub height and v2 ¼ vAir:

  ln hz02 v2 ðh2 Þ ¼ v0 ðh0 Þ   ln hz00

(6)

Parameter z0 is the so-called roughness length, which describes the strength of turbulences in the atmospheric surface layer, depending on the profile and structure of surrounding landscape [26]. To allow a general statement for optimal rotor diameter, hub height and rated power of HAWT, this study focuses on the parameter for wind conditions of the EEG-reference location [27], which are depicted in Table 1. Thereby, Rayleigh-distribution is applied to approximate the wind speed distribution at the reference site [27], which can be written as follows:

p vi wðvi Þ ¼ exp 2 v22

p v2i  4 v22

! (7)

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T. Muche et al. / Renewable Energy 90 (2016) 469e480

Fig. 1. Calculation flow diagram.

Parameter

Value

Unit

therefore, is influencing evaluation of profitability due to an error in estimation of both the incoming and outgoing payments. Following equation is deployed to determine the AEP:

rAir

1.225 0.1 30 5.5

kg/m3 m m m/s

AEP ¼ T

Table 1 Site-specific parameters at reference location, EEG.

z0 h0 v0 (h0)

Zvco wðvi Þv3i Pðvi Þdv

(8)

vci

This commonly used model often provides a satisfying approximation (e.g. IEC 61400 [25], [28]), and is applied in this paper due to the lack of a frequency distribution of wind speed depending on measured data. Resulting from last mentioned point, AEP is considered to be constant over turbine lifetime and,

Here, T is time in hours of one year, vci and vco are the cut-in and cutout wind speed of the turbine, respectively. P(vi) is the power produced by the turbine in dependence of the wind speed vi. Since the extraction of kinetic energy causes a slowdown of wind speed in conjunction with an enlargement of the air volume [23], there exists a theoretical maximum of usable energy content with the

T. Muche et al. / Renewable Energy 90 (2016) 469e480

value of 16/27, the so-called Betz’ Law,3 which is described by the coefficient of power, cP,Betz ¼ 0.593. In practice, usually values for cP reach a maximum around 0.5 [24], in manufacturer data sheets often including mechanical and electrical losses besides the losses due to aerodynamic behavior of the blades. For variable speed and pitch-controlled HAWT, the power coefficient is a function of the tip speed ratio l and pitch angle b [29]. Hence the characteristic curve of converting mechanical energy of wind in electricity can be expressed with following equation:

Pðvi Þ ¼

1 r cp ðl; bÞAv3i 2 Air

(9)

P(vi) can be described as a piecewise defined function:

8 0 for vi < vci or vi > vco > > > > > > > > 1 rAir Acp;max v3 for vi ld  ud vci  vi  vr > i > R <2 Pðvi Þ ¼ 1 vi ld 3 > > r Acp ðlÞvi for > ud vci  vi  vr > > 2 Air R > > > > > 1 > : Pr for r Acp ðlÞv3i > Pr vr  vi < vco 2 Air

(10)

For wind speed below the cut-in wind speed, generated power is zero. Between two cases a distinction is drawn for the range of vi from vci to the rated speed of the turbine, vr: in the first case the turbine works in optimal conditions for tip speed ratio and with constant pitch angle, so the power coefficient equates the maximum obtainable value cPmax. In the second region rotational speed limitation result from increasing electrical torque; the tip speed ratio is lower than the optimum value ld, and the performance coefficient decreases; b is also set constant. If the wind speed exceeds the rated velocity of the turbine, speed limitation is carried out with the pitch system; in this case the generated power is equal the rated power. Finally, P(vi) equals zero for wind speeds above cut-out- wind speed. 2.2.2. Approximation of the characteristic curve There exist various expressions to describe the power curve of a wind energy converter, among them numerical approximations for the power coefficient as a function of tip speed ratio l (and ld for the value the turbine is designed for) and pitch angle b [29e32]. The differences between these expressions and between different wind turbine types are important in case of calculating the annual energy yield. For this survey the following equations are used to estimate the power coefficient [33]:

    c c5 þ c6 l cp ðl; bÞ ¼ c1 2  c3 b  c4 exp li li li ¼

l¼

1 lc7 b

1 

c8 b3 þ1

vts vi

(12)

(13)

here, vts is the tip speed of the turbine, which is set to 80 m/s [24]. The design angular speed is determined by:

ud ¼

vts R

(14)

The design tip speed ratio ld is set to 8 and the pitch angle b to

3

Obtained by Betz and Lanchester [21].

Table 2 Coefficients for cP(l,b). c1

c2

c3

c4

c5

c6

c7

c8

0.5175

116

0.4

5

21

0.0068

0.02

0.0035

0
(deg). The coefficients are shown in Table 2. Due to obtaining the characteristic curves from mean values of sets of measurements, a smooth shape can be found in data sheets of several manufacturers (e.g. Ref. [5]). Deviation in AEP between approximation of the power curve with the equation mentioned above and manufacturer data amount under 1% for an Enercon E822300. This approximation therefore seems to be practicable for the determination of the AEP to minimize the resulting error in estimation of revenues and running costs, which largely depends on the energy yield of the turbine.

2.2.3. Model constraints For calculation of AEP first the quantities for hub height, rotor diameter and rated power of the HAWT are needed to define the number of turbine configurations investigated in this study: H2 [20,200] and D2[20,200], both equidistantly distributed with distance 1 m, and Pr2[100,10000], equidistantly distributed with 100 kW. The upper boundaries for the sizing parameter are each chosen larger as the size of commercial available turbines and prototypes, respectively. Hereby, the boundary for the hub height is set significantly larger to reflect the actual trend to increasing hub heights for onshore turbines due to the thereby obtainable better wind conditions. From complete enumeration is abstained due to the large number of variables and associated processing power and memory capacity. On that reason, two additional technical constraints should be introduced, which furthermore set technically reasonable limits: the diameter to height ratio (HDR) and the generator to rotor ratio (GRR). As empirical investigation showed, for onshore wind turbines HDR reaches values from 1.2 to nearly 1.8; for offshore turbines the range of HDR is 1.0e1.4 due to the smaller surface boundary layer4 [23]. To avoid cutting a potentially optimal solution, HDR is set from 1.0 to 2.0. Second heuristic boundary is done with the GRR by applying following equation [33]:

GRR ¼ (11)

473

Pr þ b D

(15)

Since optimal GRR largely depends on site-specific average wind speed in hub height, optimal values of this linear approximation ranges from about 30 for an average wind speed in hub height of 6 m/s to about 43 for average wind speeds of 10 m/s with parameter b equals 1300.5 Once again, this range is extended to GRR2[20,60] to examine a wide range of possible plant configurations. Table 3 exemplarily shows the resulting ranges for rotor diameter (with an increment of 1 m) and rated power (with increment 100 kW) subject to hub height of the turbine by application of the constraints mentioned above:

4 Lower values for roughness length in conjunction with lesser intensity of turbulences. 5 In Ref. [33] a method for optimization of the rotor to generator ratio, subject to average wind speed, is described. Parameter b can be chosen independently. Since this approximation is only a constraint in this study, for further information please refer to [33].

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T. Muche et al. / Renewable Energy 90 (2016) 469e480

Table 3 Ranges for diameter and rated power subject to hub height. Hub height

Rotor diameter

Rated power

[m]

[m]

[kW]

50 100 150 200

[25,50] [50,100] [75,150] [100,200]

[100,2200] [100,5700] [200,9200] [700,10000]

2.3. Determination of economical parameters 2.3.1. Revenues Based on the estimated AEP, annual revenues from electricity sale are determined. Depending on country-specific possibilities of sale and investors decision, revenues can be generated in different ways6: - feed-in to national grid - selling on spot and futures market - long-term power purchase agreements



0:0913,AEP; 0:0472,AEP;

0 < t  16 16 < t  20

 Adjustment to national currency with the help of the exchange rate rexchange for 2002  Adaption of differences in price level on national market in 2002 (fadjust)  Fitting to current prices because of changed price level since 2002 (PPI) Taking all these elements into account, necessary adjustments for the single components j can be summarized to a so-called cost factor fcost:

fcost;j ¼

Despite this wide range of sale opportunities, feed-in is currently of particular importance, since various European countries, for instance Germany, promote wind energy in form of a feedin tariff.7 In Germany, the EEG allows investors of wind energy plants to feed-in generated electricity to a fixed tariff guaranteed for 20 years. Besides that long-term price certainty, the tariff offers further investment incentive by dividing remuneration into a high initial rate during the first five years and a subsequent basic rate. In this context, time of initial rate will even extend, if wind yield of investment location is less than 150% of average reference yield [27]. Assuming an entry into service at reference location in January 2014,8 revenues can be written in following relations to AEP:

Et ¼

model, adjustment will be necessary. While specific component masses and production methods are accepted to be unchanged, current national prices have to be taken into account at this. For this purpose, the existing model offers a detailed list of key materials for single components allocated to suitable Producer Price Indexes (PPI) [13]. Based on this information an adjustment in up to three respects has to be incorporated (see for detailed analysis of differences [35]):

(16)

2.3.2. Mass-cost-model for investment costs In addition to revenues, investment costs for constructing and running costs while time of operation have to be determined. In view of the fact, that these costs depend on the sought plant properties [24], an appropriate functional correlation between costs and properties has to be found in order to allow a generalized estimate within optimization. The development of such a relation is, however, fairly complex [33]. Fingersh et al. show at least one approach for initial costs in form of an empirical inferred masscost-model [13]. Referring to this, initial costs are estimated by determining the mass of required key materials for constructing the single plant components under certain properties and, subsequently, valuing these masses by means of prices. However, due to the age and country-specific orientation of the origin mass-cost-

6 Several literature of the recent past also consider alternative usage in form of owners-consumption. Analysis of such consumption is still in a very early stage and is currently restricted to micro generation and domestic application and, thus, is neglected in the remainder of this paper. 7 For an overview of past and current promotion strategies see Ref. [34]. 8 While time of survey, Renewable Energy Act was amended with validity from August 1st, 2014. However, there are provisions to safeguard existing standards and, thus, validity of above mentioned promotion for plants entered into service before July 31st, 2014.

fadjust $PPIj rexchange

(17)

Extending the origin model on this cost factor results in the following generalized mass-cost formulas which distinguish between investment costs cInv of the mentioned components, additional investment costs cadInv and corresponding cost factors (Table 4). Because the survey will concentrate on Germany as investment location, German price level and development as well as relations have to be considered. While the exchange rate rexchange is quiet simply available as financial market data (e.g. Ref. [36]), the adjustment factor fadjust has to be estimated by comparing and setting the average cost level for wind energy plants on US and German market in 2002 into relation [37,38]. PPIs are estimated with the help of the average price level of the key materials in year 2002 and 2013, which are taken from national PPIs [39e41]. Assembling the single elements of the differentiated cost factors and plugging into the functional relation of Table 5 leads to an aggregated estimate for cInv and cadInv as follows:

cInv ¼ 2:285R3 þ 2:6689R2:9158 þ 16:7472R2:5025 þ 0:0055D3:5 þ 0:0866D2:964 þ 0:0151D2:887 þ 0:6236D2:6578  0:0137D2:5 þ 2:545D1:953 þ 947:9142D0:85 þ 139:6991D þ 0:00002046Pr3  0:0485Pr2 þ 547:1451Pr þ 2:9756ðDhÞ1:1736 þ 1:1357Ah þ 72997:4398 (18) cadInv ¼ 0:000007961Pr3  0:05145443Pr2 þ 281:99Pr þ 439:6101ðAhÞ0:4037

(19)

2.3.3. Running costs Having a closer look on running costs, the majority thereof occur due to [22]:    

Operation and maintenance Land rental Insurance and taxes Management and administration

T. Muche et al. / Renewable Energy 90 (2016) 469e480

475

Table 4 Adjusted mass-cost- models for turbine components and corresponding cost factors. Component Investment costs Blade Hub Pitch & bearings Spinner/Nose cone Low-speed shaft Bearings Brake Generator Electronic Yaw drive Mainframe Platform & railing Electrical connection Hydraulic & cooling Nacelle cover Control, safety system Tower Transportation Assembly & installations Additional investment costs Foundation Roads, civil works Electrical interface Engineering & permits

Costs

fcost,j

(1.6746 R33980.1667) fcost,j,1 þ 11.4354 R2.5025 fcost,j,2 (1.7661R2.9158 þ 24141.275) fcost,j 0.4802 D2.6578 fcost,j (103.045D2899.185) fcost,j 0.01D2.887 fcost,j (0.0043 D3.5e0.01069 D2.5) fcost,j (1.9894 Pr0.1141) fcost,j 219.33 Pr fcost,j 79 Pr fcost,j 0.0678 D2.964 fcost,j 627.28 D0.85 fcost,j 1.3355 D1.953 fcost,j 40 Pr fcost,j 12 Pr fcost,j (11.537 Pr þ3849.7) fcost,j 35000 fcost,j (0.5960 A h2121) fcost,j (0.00001581 Pr 3e0.0375 P2r þ54.7 Pr) fcost,j 1.965 (D h)1.1736 fcost,j

1.36451/1.4645 1.51115 1.29874 1.35571 1.51115 1.27965 1.66512 1.27461 1.32963 1.27713 1.51115 1.90573 1.37166 1.49379 1.35571 1.27717 1.90573 1.44971 1.29431

303.24 (A h)0.4037 fcost,j (0.00000217 Pr 3e0.0145 P2r þ69.54 Pr) fcost,j (0.00000349 Pr 3e0.0221 P2r þ109.7 Pr) fcost,j (0.0000994 P2r þ20.31 Pr) fcost,j

1.51429 1.51429 1.33949 1.46451

Table 5 Sizing parameter with the highest NPV in descending order. Nr.

1 2 3 4 5 6 7 8 9 10

H

D

Pr

NPV

AEP

GRR

HDR

[m]

[m]

[kW]

[EUR]

[kWh]

[kW/m]

[e]

[EUR]

136.00 135.00 137.00 134.00 138.00 133.00 139.00 135.00 136.00 137.00

133.00 133.00 133.00 133.00 133.00 133.00 133.00 135.00 135.00 135.00

4100.00 4100.00 4100.00 4100.00 4100.00 4100.00 4100.00 4200.00 4200.00 4200.00

3,525,896.34 3,525,888.70 3,525,712.68 3,525,686.81 3,525,340.58 3,525,287.67 3,524,782.86 3,524,589.75 3,524,516.01 3,524,246.08

14,914,838.63 14,890,839.29 14,938,638.57 14,866,637.46 14,962,242.13 14,842,229.96 14,985,652.28 15,302,181.83 15,326,772.59 15,351,158.85

40.60 40.60 40.60 40.60 40.60 40.60 40.60 40.74 40.74 40.74

1.02 1.02 1.03 1.01 1.04 1.00 1.05 1.00 1.01 1.01

8,903,590.51 8,884,819.97 8,922,362.34 8,866,050.72 8,941,135.47 8,847,282.76 8,959,909.91 9,211,443.52 9,230,742.56 9,250,042.93

Total investment

The optimal turbine configuration and corresponding values for NPV and AEP are highlighted in bold.

In contrast to the calculation of investment costs, running costs are estimated using simple relations to the first mentioned or annual yield. For example, the insurance rate often follows a fixed percentage of the insured value or investment costs just as land tenancy is often orientated on AEP and its remuneration (see for example [24]). However, significant variations between countries, regions and sites are found [22], so that a functional relation for running costs cannot be formulated generalized but has to be derived individually depending on national conditions. In order to capture scheduled and especially unscheduled operation and maintenance costs cOM, a full-service contract is assumed. The contract ensures technical availability of wind energy plants up to 15 years by assigning any operation and maintenance to contracting partner, e.g. turbine manufacturer [24]. With the risk involved in the investment, the majority of investors sign a full agreement with the manufacturer [42]. The contract fee depends on the annual energy yield of the respective wind energy plant and is calculated individually for every plant [43]. Due to a lack of manufacturers' information, available fee data of a realized contract are used. There, costs for a full-service contract are determined by [44]:

cOM ¼ 0:0105AEP

(20)

Other running costs, in detail land rental cRental, insurance cInand administration costs cAdministration, are formulated similar by falling back on existing bibliographical reference, which reflects German prices [3,24] 9,10,11:

surance

cRental ¼ 0:06Et

(21)

cInsurance ¼ 0:005cInv

(22)

cAdministration ¼ 0:0075cInv

(23)

To summarize, running costs crun can be estimated as follows:

9 A spread of 6 to 8% for good sites close to the German coast is mentioned in [24]. Because of the inferior characteristics of the assumed reference site the minimum of 6% is used. 10 Hau propose a rate of about 0.55%, which includes insurance against machine breakage, loss-of-profit insurance and liability insurance. Moreover he gives information about fewer costs in case of full-service contract, so that the rate was reduced. 11 The arithmetic average of the spread mentioned is used.

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T. Muche et al. / Renewable Energy 90 (2016) 469e480

crun ¼ 0:0105AEP þ 0:06Et þ 0:0125cInv

(24)

In doing so, however, the problem of current price level and probability of changes during plant lifetime has to bear in mind. While revenues are fixed and initial costs just occur at the beginning, it seems to be unreliable to set running costs constant during the period of 20 years. The more likely is an increase greater than or equal to rate of inflation. Due to no further information, average annual increase equal to the forecasted long-term annual rate of inflation i of 2.0% is assumed [45]. Hence, running costs develop as follows:

crun;t ¼ ð0:0105AEP þ 0:06Et þ 0:0125cInv Þð1 þ iÞt

(25)

2.3.4. Cost of equity capital As usual, the required costs of equity rcoe are estimated by CAPM with following formula [46]:

h i rcoe ¼ rf þ bi Eðrm Þ  rf

(26)

In this context, the return on risk-free asset rf and expected return on market portfolio E(rm) are valued by using the current running return of listed Federal government securities issued by the German Bundesbank [47] and the average of monthly continuous stock yields of CDAX from January 1970 to July 2013 ultimo, respectively. In view of equity beta bi, monthly capital market data, in detail, continuous stock yield of the German sub-index DAX Utilities is used. Recourse on single comparable, listed companies as common practice was not possible, nevertheless, wind energy plants, which generate and retail electricity, resemble most likely a supplier. Thus, their inherent risk fairly equals the considered subindex’ risk. However, estimated equity beta by a sub-index of companies probably partly debt-financed also represents implied financial risk besides the sought business [48] and, hence, requires unlevering (see[16] for further detail). Thereby, a market cap-weighted share of the single companies,12 simplified depicted by book values, and a tax rate s of 30% is assumed. In doing so, nominal cost of equity of 3.34% results13. Adjustment for inflation rate has to be neglected since cash flows are considered from a nominal point of view. 3. Results For the calculation of the optimal plant design according to the EEG-reference location, the equations presented in section 2 were implemented in MATLAB. The 10 combinations of the investigated plant properties with the highest NPV are shown in Table 5, where the maximum NPV and the corresponding design properties are accentuated. The maximum NPV with the technical and economic assumptions mentioned above is reached with sizing parameters which actually are state of the art in turbine manufacturing. While diameter of 154 m and rated power of 7580 kW, respectively, are even commercially available, the optimal hub height of 136 m is barely exceeding the technical boundary or, to be more precise, the largest realized hub height of 135 m. However, by comparison of

12 Since companies of the DAX are weighted by their market value [49], it is also assumed for its sub-indices. 13 According to the described procedure, cost of equity is based on a risk-free return of 1.56%, a return on market portfolio of 6.58% and unlevered equity beta of 0.3543.

the 10 results with the highest NPV it is recognizable that each of the three considered design parameters has a differently strong influence on the trend of the NPV. The first 7 combinations only vary in hub height with a constant rotor diameter and rated power and, thus, with a constant GRR, indicating that little variation of the hub height has not the largest influence of the economic viability. To quantify the effect of changes in design parameter on NPV, sensitivity analysis is carried out, where every sizing parameter is varied between 50% and þ50% separately while the two other parameters remain constant (Table 6). Fig. 2 shows the trend of NPV due to relative changes of design parameter. The dotted line shows the development of NPV according to changes in tower height, having the lowest impact on profitability of investment in turbine projects. Although the power of the turbine is proportional to the cubic velocity of the wind, changes in tower height have only a minor impact due to the logarithmic wind profile. The most influential sizing parameter, depicted with the solid line, is the rotor diameter by reason of the dependence of convertible energy on the square diameter on the one hand and the large influence on the initial costs on the other hand. Previous consideration is made for wind speed at EEG-reference site, the resulting AEP and the thereby generated revenues from feed-in tariff. However, the revenues from the EEG feed-in tariff are varying due to changes in average wind speed of the project site. For that reason, survey is extended to average wind speeds v2 from 3.5 m/s in 30 m height for parts of Central Germany to 6.75 m/s for coastal regions. For this purpose, shortening or extension of initial rate of feed-in tariff tini can be written as a ratio of the AEP of the project site and the AEP of the reference location AEPref [27]:

tini

1 AEP ¼ 5a þ a 1:5  6 AEPref

!

100 0:75

(27)

tini is rounded to the nearest whole number to facilitate matrix calculation. While the constraints for GRR remains constant, HDR is set to [0.7,1.6] in view of the results of reference location. The economic optimal sizing parameters for the range of the considered wind speeds are depicted in Table 7, where the results for the considered EEG-reference location are highlighted. Generally, increasing NPV of investments in wind energy projects is reached with increasing wind speeds and simultaneously increasing sizing parameter. Furthermore, GRR is of vital importance, which mainly depends on average wind speed, which in turn is a function of the tower height [18,33]. Since approximation of the GRR is not linear for rated power below 1 MW, the resulting values of GRR for wind speed from 3.5 to 4 m/s are invalid [33]. For sites with such disadvantage wind conditions, economic viability is also not ensured. As stated above, maximization of NPV is used to determine the optimal turbine configuration instead of minimization of COE. To compare the results of these two different approaches, levelized costs of electricity (LCOE) are calculated as follows [50]:

P c cInv þ cadInv þ Tt¼1 run;t t ð1þrcoe Þ LCOE ¼ PT AEPt

(28)

t¼1 ð1þrcoe Þt

Fig. 3 shows the trends of NPV and LCOE, respectively, as a function of the average wind speed for both optimization approaches (Table 8). The two solid lines are the results for maximizing NPV, while the dotted lines are illustrating the development by minimizing LCOE (NPVcom and LCOEcom, respectively). It is clear to see, that maximizing NPV leads to significantly higher economic viability, while LCOE are slightly higher. To illustrate this effect, Fig. 4 shows the development of rotor diameter and

T. Muche et al. / Renewable Energy 90 (2016) 469e480

477

Table 6 Results of sensitivity analysis. Diameter Rel. Change

Hub height

Abs. Change

NPV

Rated power

Abs. Change

NPV

Abs. Change

NPV

[%]

[m]

[EUR]

[%]

[m]

[EUR]

[%]

[kW]

[EUR]

[%]

¡50 ¡40 ¡30 ¡20 ¡10 0 10 20 30 40 50

66.50 79.80 93.10 106.40 119.70 133.00 146.30 159.60 172.90 186.20 199.50

78,888.07 836,150.06 1,692,115.01 2,632,244.64 3,253,991.92 3,526,102.14 3,306,918.03 2,856,496.14 1,927,983.86 831,431.18 910,254.47

2.24 23.71 47.99 74.65 92.28 100.00 93.78 81.01 54.68 23.58 25.81

68.00 81.60 95.20 108.80 122.40 136.00 149.60 163.20 176.80 190.40 204.00

2,839,172.17 3,134,425.70 3,327,473.59 3,446,098.43 3,508,233.66 3,526,102.14 3,508,393.72 3,461,498.47 3,390,246.93 3,301,343.05 3,188,832.90

80.52 88.89 94.37 97.73 99.49 100.00 99.50 98.17 96.15 93.63 90.44

2050.00 2460.00 2870.00 3280.00 3690.00 4100.00 4510.00 4920.00 5330.00 5740.00 6150.00

1,500,374.91 2,195,002.40 2,813,638.17 3,176,879.94 3,398,242.02 3,526,102.14 3,344,922.74 3,051,286.62 2,599,265.48 1,869,010.20 985,491.15

42.55 62.25 79.79 90.10 96.37 100.00 94.86 86.53 73.71 53.00 27.95

Fig. 2. NPV according to relative changes of single sizing parameter.

Table 7 Optimal sizing parameters as a function of annual average wind speed. v2

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5 6.75

H

D

Pr

NPV

LCOE

AEP

GRR

HDR

tini

[m]

[m]

[kW]

[EUR]

[EUR/kWh]

[kWh]

[kW/m]

[-]

[a]

43 41 63 98 106 138 117 140 136 133 125 121 149 164

29 35 51 91 117 118 125 130 133 138 138 138 138 140

100 100 300 1400 2300 3200 3600 3900 4100 4400 4400 4400 4400 4500

171,497.18 137,628.73 101,303.75 143,797.24 743,744.48 1,563,830.97 2,527,825.99 3,134,864.01 3,525,896.34 4,148,281.21 4,225,420.53 4,610,234.87 4,764,771.09 4,733,661.82

0.167 0.130 0.101 0.088 0.084 0.079 0.075 0.072 0.068 0.066 0.063 0.060 0.059 0.058

157,382.19 244,125.36 760,050.44 3,545,641.79 6,702,986.47 9,083,790.73 10,764,947.82 13,276,240.05 14,914,838.63 17,077,293.49 17,926,162.96 18,825,456.18 20,475,625.16 22,191,327.75

e e e 29.67 30.77 38.14 39.20 40.00 40.60 41.30 41.30 41.30 41.30 41.43

1.48 1.17 1.24 1.08 0.91 1.17 0.94 1.08 1.02 0.96 0.91 0.88 1.08 1.17

20.00 20.00 20.00 20.00 20.00 20.00 20.00 18.00 16.00 15.00 13.00 12.00 11.00 10.00

The optimal turbine configuration (by maximizing NPV) for the EEG- reference location are highlighted in bold.

rated power due to variation of the average wind speed. Maximizing NPV leads to increasing diameter and rated power, whereby increasing initial and running costs are compensated by larger

revenues due to the larger power drain from the wind. By minimizing LCOE, revenues are neglected and rotor diameter and rated power (Dcom and Pcom), respectively, are nearly constant and show

478

T. Muche et al. / Renewable Energy 90 (2016) 469e480

Fig. 3. Development of NPV and LCOE as a function of average wind speed.

Table 8 Sizing parameters, resulting from minimization of LCOE. v2

3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5 6.75

H

D

Pr

NPV

LCOE

AEP

GRR

HDR

tini

[m]

[m]

[kW]

[EUR]

[EUR/kWh]

[kWh]

[e]

[e]

[a]

89 82 76 91 85 79 74 89 84 79 68 70 68 65

77 77 77 77 77 77 77 78 75 75 74 75 74 74

700 700 700 1000 1000 1000 1000 1400 1300 1300 1300 1300 1300 1300

674,272.86 410,551.63 157,976.98 115,714.94 425,674.20 721,275.61 937,183.17 1,426,400.04 1,445,057.13 1,513,968.51 1,464,538.50 1,625,347.12 1,528,570.29 1,391,508.42

0.123 0.108 0.097 0.088 0.080 0.074 0.069 0.065 0.061 0.058 0.055 0.052 0.050 0.048

1,461,802.72 1,659,849.96 1,855,195.51 2,480,456.49 2,731,111.22 2,967,936.70 3,200,207.93 4,261,771.42 4,225,258.21 4,486,073.42 4,558,322.39 4,969,827.53 5,151,901.89 5,378,659.57

e e e 29.87 29.87 29.87 29.87 34.62 34.67 34.67 35.14 34.67 35.14 35.14

1.16 1.06 0.99 1.18 1.10 1.03 0.96 1.14 1.12 1.05 0.92 0.93 0.92 0.88

20.00 20.00 20.00 20.00 20.00 20.00 19.00 18.00 16.00 14.00 12.00 11.00 9.00 7.00

The optimal turbine configuration (by minimizing LCOE) for the EEG- reference location are highlighted in bold.

Fig. 4. Development of diameter and rated power as a function of average wind speed.

T. Muche et al. / Renewable Energy 90 (2016) 469e480

only a minor increase, respectively. Although the results are suitable to give incentives for the economically optimal turbine configuration according to a specific site, however, some aspects had to be neglected and, therefore, the results do not allow a generally valid statement for investors. First of all, numerous turbines are available on the market, differing in design parameters like cut-in velocity, design tip speed and, in general, the performance, described by the power curve, optimized for different wind speeds. Wind resource on different sites, characterized by roughness length and wind speed distribution, is also varying over a wide range, resulting in deviation of AEP and, thus, in different revenues. In the context of the strong dependence of convertible energy on wind conditions, wind resource assessment is of vital importance [51]. In doing so, long-term wind speed in hub height and the resultant AEP has to be accurately estimated by collecting data about annual wind speed distribution. Because of the variation of wind speed from year to year over turbine lifetime, uncertainty in determination the AEP for different years is inevitable [28]. This uncertainty influences the estimation of the cash flow. Due to the complicated process of wind resource assessment and the lack of measured data, estimation of error in determining AEP and cash flow, respectively, is neglected in this study. To avoid the difficulties in wind resource assessment, randomness and forecast uncertainty could be considered by application of stochastic processes. In this context, various possible developments over time have to be taken into account. Furthermore, cost of debt is neglected in this survey, however, it has to be considered in case of an individual investment decision. Although considering debt and equity financing simultaneously improves representation of reality, generalized estimation of cost of capital will become more difficult due to individual range of available types of debt financing and to some extent additional finding of appropriate risk-adjusted cost of debt.

4. Summary In view of utility maximization for an investor in wind turbine projects and the successive enlargement of wind turbines, economically optimal turbine sizing parameters have to be determined by maximizing the NPV. By application of a generalized description of the power drain from kinetic energy of the wind, detailed models for initial and running costs and the German feedin tariff it could be shown that the economic optimum for the considered average wind speeds are reached with rotor diameter and rated power that are state of the art currently. Only the values for hub heights e especially for high average wind speeds e are exceeding the hub heights of commercially available turbines. However, it is shown that deviation of the hub height from the determined optimum has the lowest impact on the economic viability. Furthermore it could be shown that minimizing LCOE leads to significantly smaller design parameter and lower economic viability. Although the results are suitable to provide an indication for an optimal investment decision, universal statement for optimal sizing parameter cannot be given. Accurate, individual assessment of available turbines in dependence of site-specific wind conditions and of given economically condition has to be done in order to find the most profitable turbine configuration.

Acknowledgment The authors want to acknowledge the financial backing by the European Social Fund application number: 100097891, a financial instrument of the European Union.

479

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