Optimal design of wind turbines on high-altitude sites based on improved Yin-Yang pair optimization

Energy 193 (2020) 116794

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Optimal design of wind turbines on high-altitude sites based on improved Yin-Yang pair optimization Dongran Song a, b, Junbo Liu a, b, Jian Yang a, b, Mei Su a, b, Yun Wang a, *, Xuebing Yang c, d, Lingxiang Huang c, d, Young Hoon Joo e a

School of Automation, Central South University, Changsha, China Hunan Provincial Key Laboratory of Power Electronics Equipment and Grid, Changsha, China XEMC Windpower Co., Ltd., Xiangtan, China d State Key Laboratory of Offshore Wind-power Technology and Testing, Xiangtan, China e School of IT Information and Control Engineering, Kunsan National University, Kunsan, South Korea b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 October 2019 Received in revised form 26 November 2019 Accepted 16 December 2019 Available online 18 December 2019

This study proposes a method to minimize the cost of energy (COE) of wind turbines on high-altitude sites, in which the parameters of rotor radius, hub height and rated power are optimally designed. Firstly, the COE model is built up, and the wind turbine optimization problem is formulated. Then, YinYang pair optimization is presented and improved to solve the optimization problem. Lastly, the proposed method is validated under typical wind resource distribution and different altitude scenarios. The results show that with the increase of altitude, the optimal COE increases, whereas the three optimized parameters present different trends of variation. It is indicated that by considering the influence of altitude, COE at a certain altitude can be reduced effectively. Meanwhile, the improved Yin-Yang pair optimization shows smaller iteration number of convergence and convergence time in comparison with particle swarm optimization and traditional Yin-Yang pair optimization. On this basis, sensitivity analysis of optimized parameters and optimal COE to wind resource parameters is carried out, and the influence of uncertainty of wind resource statistics on optimization results is explored. By doing that, it is shown that the rotor radius, hub height and COE are most sensitive to the change of mean wind speed, and rated power is sensitive to all three wind resource parameters. On the other hand, COE decreases with the increase of three wind resource parameters. These results can be used as a guide for the wind turbine design on the high-altitude site. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Onshore wind turbine Cost of energy Optimal design High altitude Sensitivity analysis

1. Introduction The rapid growth of energy demand and the rapid consumption of fossil fuels have led to the gradual shift of world energy development towards renewable energy [1]. Among all kinds of renewable energy, wind energy develops rapidly. According to statistics, in 2018, the total installed capacity of wind power in the world has reached 591 GW [2]. The global Wind Energy Council forecasts that the total installed capacity will reach 840 GW by 2022 [3]. As the cost continues to fall, wind energy has become one of the most efficient renewable sources [4]. However, compared with traditional energy technologies, wind energy still has a higher

* Corresponding author. E-mail address: [email protected] (Y. Wang). https://doi.org/10.1016/j.energy.2019.116794 0360-5442/© 2019 Elsevier Ltd. All rights reserved.

power generation cost. Therefore, it is necessary to further reduce the cost of wind energy and make it more competitive. Different evaluation indicators can be selected to estimate the energy cost. There are different techno-economic models presented to analyze the investment and construction feasibility of wind farm, such as the present value of cost (PVC) [5], the net present value (NPV) [6], the internal rate of return (IRR) [7], the payback period (PBP) [8], and the cost of energy (COE) [9]. Among them, COE is one of the most commonly used evaluation indicators. COE refers to the cost of producing 1 KWh electric energy, including annual energy production cost and annual energy production. At present, the COE model widely used in wind turbine optimization design mainly comes from the research result of “National Renewable Energy Laboratory (NREL) Wind Turbine Design Cost and Scaling Model” [10]. Optimal design of wind farm can effectively reduce the COE and

2

D. Song et al. / Energy 193 (2020) 116794

bring more economic benefits [11]. There are two common methods to optimize COE: wind farm layout optimization (WFLO) and wind turbine design optimization (WTDO) [12]. The WFLO is aimed at the overall optimization of wind turbines in the whole wind farm, via optimizing the layout of wind turbines. The WTDO is to optimize the single wind turbine, by optimal designing the turbine parameters. In recent years, there have been many attempts to reduce COE by WTDO, which has shown a significant impact on the economic benefit of wind farms [13,14]. In Ref. [15], the study of optimizing the rated wind speed and rotor radius of offshore wind turbines based on minimizing COE is conducted. Therein, an iterative method is proposed to search for the optimal rotor radius and rated wind speed corresponding to the minimal COE. In Ref. [16], the mathematical method of COE minimization for large-scale wind turbine is presented. Based on the techno-economic model given by NREL, the COE is expressed as a function of rated power and rated wind speed, and then the rated power and rated wind speed corresponding to the minimizing COE under the power curve model of wind turbine are analyzed. In these studies, COE has been effectively reduced by optimizing the design parameters of wind turbines to match the distribution characteristics of wind resource. Nevertheless, the above studies are aimed at low-altitude sites, without considering the impact of altitude on the optimal design of wind turbines, so they are not applicable to wind turbines on highaltitude sites. It is well known that the air density, air pressure and temperature will change with the increase of altitude. Existing research shows that the difference of air density between low-altitude and high-altitude sites is the main reason for the difference of energy production of wind turbines [17,18]. Compared with the altitude of 0 m, the air density is decreased by 17.67% at the altitude of 2000 m and 32.99% at the altitude of 4000 m [19]. On the other hand, the installation, maintenance and transmission of wind turbines on high-altitude sites are more difficult than the ones on low altitude sites, leading to a significant increase in wind turbine cost [20]. According to the technical specification released by China General Certification Center in 2013, special design is required for the wind turbines at the sites with altitudes higher than 2000 m [21]. In Ref. [22], the multi-objective energy-cost optimization design of variable speed wind turbines on high-altitude sites is presented. With the two objectives of minimizing annual production cost (APC) and maximizing annual energy production (AEP), the multiobjective particle swarm optimization (MOPSO) [23] is used to optimize the design parameter of wind turbines. It has been clear that the influence of altitude on turbine cost and energy production cannot be neglected, and that the energy cost can be further reduced by considering the site altitude into turbine designs. However, the study only focuses on the rotor radius and rated power, without considering the influence of hub height, which is an important design parameter. Meanwhile, there may be influence from uncertainty of wind resource distribution on the optimal results, which is not explored in the previous study. On the basis of the work in Ref. [22], this study further carries out the optimization design of wind turbines on high-altitude sites, and proposes an improved Yin-Yang pair optimization algorithm to minimize the COE. Compared with the previous researches, the main contribution of this paper is reflected into three parts: Firstly, a simplified COE model including the influence of the site altitude is established, which is a single-objective optimization problem. Secondly, an improved Yin-Yang pair optimization is proposed to solve the optimization problem expressed as a nonlinear function of three design parameters under constraints. Thirdly, the sensitivity of optimization parameters and COE to wind resource parameters is analyzed. In this study, the energy production of wind turbine on high-altitude sites is estimated by AEP, and the energy

production cost is estimated by APC. Considering the application scope of the proposed techno-economic model, the optimal design of wind turbines is conducted in plateau area with site altitude of 2000me4000 m. The improved Yin-Yang pair optimization algorithm is presented to solve the three-parameter single-objective optimization problem, and compared with traditional Yin-Yang pair optimization algorithm [24] and particle swarm optimization (PSO) algorithm [25]. Considering the uncertainty of wind resource distribution, sensitive analysis is carried out to explore the influence of wind resource parameters such as mean wind speed, wind shear coefficient on rotor radius, hub height, rated power, and COE. The optimization results reveal the relationship among the COE, wind resource parameters and optimized parameters, which can provide useful references for decision makers and designers. The remaining sections are structured as follows: in Section 2, a simplified COE techno-economic model is introduced for the wind turbines deployed on high-altitude sites. Section 3 introduces the improved Yin-Yang pair optimization algorithm and how to use the algorithm to solve the problem of WTDO. Section 4 carries out the result analysis and sensitivity analysis on optimization results. Section 5 concludes the study. 2. Proposed energy cost model of wind turbine on hignaltitude sites With the increase of altitude, the air density and pressure change. The impact on the cost and productivity of wind turbines cannot be ignored. The specific manifestation is that the cost increases and the productivity decreases, leading to the increase of energy cost. In order to maximize the economic benefit, it is necessary to introduce the COE model by adding the expressions related to altitude factors. This section mainly introduces the COE model of wind turbines installed on high-altitude sites, which consists of two parts: annual production cost (APC) and annual energy production (AEP). 2.1. Annual production cost Annual production cost (APC) refers to the cost of AEP of wind turbines. According to the techno-economic model given by NREL, APC includes wind turbine cost, infrastructure cost and operation and maintenance cost. For high-altitude sites, rising altitude leads to increased costs of wind turbines in all aspects, and thus the influence of altitude should be considered in estimating APC. APC is determined by the initial capital cost (ICC), the annual operating cost (AOE) and the fix charge rate (FCR). The specific expression is as follows [10]:

APC ¼ ICC  FCR þ AOE

(1)

where FCR refers to annual amount of debt, return on equity and some fixed expenses, which is set to 0.1158 per year [26]; AOE refers to the annual operation and maintenance cost, which is mainly related to annual energy production and rated power of wind turbine; and the cost data is based on the US dollar in 2002. Using the techno-economic model considering altitude factor, APC can be expressed as a function of altitude, AEP and three design parameters (R,HPr ):

APC ¼ fAPC ðhalt ; AEP; Pr ; R; HÞ

(2)

2.1.1. Initial capital cost Initial capital cost (ICC) consists of wind turbine cost and

D. Song et al. / Energy 193 (2020) 116794

infrastructure cost. The cost of wind turbine mainly consists of mechanical system (blade, gearbox, low speed shaft, main bearings, mechanical brake), electrical system (generator, power converter and electrical connection), control system (pitch system, yaw system, control safety system), auxiliary system (cooling system, hub, nose cone, mainframe, nacelle cover and tower). Infrastructure cost includes foundation, civil engineering, interface connection, project license, transmission, installation and so on. The cost model given by the NREL can only be applied to lowaltitude areas, which needs to be improved considering the impact of altitude factors. The improved initial capital cost model is shown in Table 1 [22]: According to Table 1, the function of wind turbine cost can be expressed as follow:

3

2.2. Annual energy production Annual energy production (AEP) refers to the annual energy output of wind turbines. The AEP model of onshore wind turbines and offshore wind turbines are almost the same, which are based on the power curve model [27] and the Weibull distribution model [28]. The difference lies in the difference of Weibull distribution scale parameter and shape parameter between land and sea, mainly due to the difference of roughness between land and sea. The expression of AEP is as follows:

AEP ¼ 8760ð1  hÞPm

(7)

where Pm is the hourly energy production, and h is the total loss of

Cturb ¼ 0:0537R3:5 þ 0:6698R3 þ 0:529R2:946 þ 0:0814R2:887 þ 3:0291R2:6578 þ11:55143R2:53 þ 4:5742R2:5025  0:06658R2:5 þ 0:59595pR2 H þ 46:1401R1:953

(3)

þ206:69R þ 18:1Pr1:249 þ 58:725Pr þ 156Pr ð1 þ halt =15000Þ þ 71378:59783

Similarly, the expression of infrastructure cost can be obtained as:

energy, including conversion loss, grid-connected loss and so on, which is set to 0.17 [29]. The hourly energy production is determined by the power curve model and the Weibull distribution model, and the probability

  CBoS0 ¼ 1:798  P 3r  0:052P 2r þ 124:24Pr þ 4:4325H1:1736 R1:1736 ð1 þ halt =15000Þ þ3:49  106 P 3r  0:02111P 2r þ 130:1Pr þ 481:3776H 0:4037 R0:8074

(4)

density of wind speed distribution is determined by the Weibull distribution model. The expression of Pm is as follows: Eqs. (3) and (4) show that the cost of wind turbine (Cturb ) and the cost of infrastructure (CBoS0 ) are related to the rotor radius, hub height, rated power and altitude. Therefore, when the altitude is determined, the cost of wind turbines and infrastructure can be expressed as functions of these three design variables. The relationship between ICC and Cturb and CBoS0 is as follows:

ICC ¼ Cturb þ CBoS0

PðvÞf ðvÞdv

(8)

0

where v is the wind speed, PðvÞ is the power curve equation, and fðvÞ is the probability density of wind speed distribution.

(5)

Therefore, Eqs. (3)e(5) show that the ICC is also a function of the rotor radius, hub height and rated power.

2.1.2. Annual operating cost The annual operating cost (AOE) consists of three parts: replacement cost, land leasing cost, and operation and maintenance cost. The improved annual operating cost is shown in Table 2 [22]. The specific expressions of AOE can be obtained from Table 2 as follows:

AOE ¼ ð17Pr þ 0:007AEPÞ  ð1 þ halt = 15000Þ þ 0:00108AEPð1  halt = 15000Þ

∞ ð

Pm ¼

(6)

It can be seen that AOE is a function of the rated power, AEP and altitude.

2.2.1. Power curve model The power curve model determines the energy production of wind turbines on specific sites [30]. Different wind turbines have different power curves. Considering the hardware limitations, there are three types of wind turbine design, including the one with power limit, the one with the speed and power limit, and the one with power and torque limit [31]. The former type has better performance than the latter two, and its power curve is shown in Fig. 1. In this research, the wind turbine only with the power limitation is considered. In addition to rated power and geometric parameters, the setting of cut-in and cut-off wind speed values will also lead to different productivity. The conventional power curve model is designed for the wind turbine at low-altitude site, which only describes the relationship between wind speed and output power of wind turbines. The power curve model is shown in Fig. 1, and the power curve is formulated as follows [32]:

4

D. Song et al. / Energy 193 (2020) 116794 Table 1 Initial capital cost considering altitude. Type

Cost model (Unit:$)

Mechanical system Blade

ð0:4019R3 þ 2:7445R2:5025  955:24Þ=0:6

Gearbox

18:1Pr 1:249

Low speed shaft

0:011ð2RÞ2:887

Main bearings

ð0:712R=75  0:01177Þ*ð2RÞ2:5 2:188Pr  0:1255

Mechanical break Electrical system

65Pr  ð1 þ halt =15000Þ 79Pr  ð1 þ halt =15000Þ 45Pr

Generator Power converter Electrical connection Control system Pitch system

0:48ð2RÞ2:6578

Yaw system

0:0678ð2RÞ2:964 50,000

Control safety system Auxiliary system

12Pr  ð1 þ halt =15000Þ

Cooling system Hub

2:0ð2RÞ2:53 þ 24141:275 206:69R  2899:185

Nose cone Mainframe

11:917ð2RÞ1:953 11:537Pr þ 3849:7

Nacelle cover Tower

0:59595pR2 H  2121

Infrastructures Foundation

303:24ðpHR2 Þ0:4037

Roads civil work

Pr ð2:17 *106 Pr 2  0:0145Pr þ 69:54Þð1 þ halt =15000Þ

Interface connections

Pr ð3:49 *106 Pr 2  0:0221Pr þ 109:7Þ

Engineering permits

Pr ð9:9 *104 Pr þ 20:31Þ

Transportation

Pr ð1:581 *105 Pr 2  0:0375Pr þ 54:7Þð1 þ halt =15000Þ

Installation

1:965ð2HRÞ1:1736 ð1 þ halt =15000Þ

. Pf ðvÞ ¼ rpR2 Cp;max v3 2

Table 2 Annual operating cost considering altitude. Type

Cost model (Unit:$)

Levelized replacement cost bottom lease cost O&M

17Pr ð1 þ halt =15000Þ 0:00108AEPð1  halt =15000Þ 0:007AEPð1 þ halt =15000Þ

(10)

where r is the air density, R is the rotor radius, Cp;max is the maximum power coefficient which is set to 0.48 in this study [33]. The wind turbine is in rated power state at rated wind speed. The expression of rated wind speed of wind turbine is as follows:

. 1=3  vr ¼ 2Pr rpR2 Cp;max

(11)

According to Eqs. (9)e(11), it can be seen that the power curve can be expressed as a function of wind speed and air density when the maximum power coefficient Cp;max is known.

ðv < vc Þ 0 ðvc  v  vr Þ  P ðvÞ PðvÞ ¼ f vr  v  vf Pr   0 vf < v

(9)

where vc is the cut-in wind speed, vr is the rated wind speed, vf is the cut-off wind speed, Pr is the rated power, and Pf ðvÞ is the partial power at below the rated wind speed.where the cut-in wind speed, cut-off wind speed and rated wind speed are known, the power curve is mainly determined by Pr and Pf ðvÞ. The expression of Pf ðvÞ is as follows:

Pðv; rÞ ¼

0 8 > > > < Pf ðv=vr Þ > > > :

Pr 0

ðv < vc Þ . 1=3  vc  v  2Pr rpR2 Cp;max  . 1=3  2Pr rpR2 Cp;max  v  vf   vf < v 



(12) It can be seen from Eqs. (11) and (12) that with the increase of altitude, the air density decreases. When the rated power and rotor radius remain unchanged, the rated wind speed increases, resulting in a decline in energy production.

D. Song et al. / Energy 193 (2020) 116794

5

Power (kw) Rated power

Region 3

Region 4

Region 2

Region 1

Cut-in speed(vc)

Cut-out speed(vf)

Rated speed(vr)

Wind speed (m/s)

Fig. 1. Power curve of wind turbine.

According to Eqs. (7)e(11), the specific expression of AEP is as follows:

AEP ¼ 1744:992rpR2

vðf

 vðr v3 f ðvÞdv þ vc

 v3r f ðvÞdv

(13)

vr

(14)

where k is the shape factor, and c is the scale factor. With the increase of hub height, the wind speed will increase and the corresponding scale factor and shape factor will increase. The scale factor has the following relationship with hub height [36]:

c ¼ c0 ðH=H0 Þ

a

(15)

where H is the hub height, H0 is the reference height, c0 is the scale parameter in reference height, a is the Hellmann exponent, which indicate the level of wind shear. Generally, a is different under different geographical conditions [37]. When the shape factor k and the mean wind speed vm are known, the c0 can be obtained by the following formula:

(16)

where Gð Þ is the gamma function. The shape factor is generally expressed as follows:

k ¼ k0 ½1  0:088 lnðH0 = 10Þ=½1  0:088 lnðH = 10Þ

2.2.2. Weibull distribution model The most commonly used method for estimating the probability density of wind speed distribution is Weibull distribution. Rayleigh distribution is a special case when the shape factor of Weibull distribution is 2 [34]. The Weibull probability density distribution equation is used to represent the distribution characteristics of wind resource parameters. The expression is as follows [35]:

  f ðvÞ ¼ ðk = cÞðv=cÞk1 exp  ðv=cÞk

c0 ¼ vm =Gð1 þ 1 = kÞ

(17)

where k0 is k at reference height. According to Eqs. (7)e(17), the annual energy production (AEP) of wind turbines can be expressed as a function of altitude, wind resource parameters (c0 , k0 , a) and three design parameters (R, H, Pr ).

AEP ¼ fAEP ðhalt ; c0 ; k0 ; a; Pr ; R; HÞ

(18)

2.3. Simplified COE expression The COE model considering altitude is divided into two parts: annual production cost (APC) and annual energy production (AEP). The relationship among them is as follows [10]:

COE ¼

APC AEP

(19)

According to Eqs. (2) and (18), APC and AEP are non-linear functions related to rotor radius, hub height and rated power. Simplified expressions of initial capital cost, annual operating cost and annual energy production have been obtained. The expression of COE can be further simplified and expressed as follows:

6

D. Song et al. / Energy 193 (2020) 116794

COE ¼

0:00622R3:5 þ 0:07756R3 þ 0:0612R2:946 þ 0:009426R2:887 þ 0:3508R2:6578  ð vr  ð vf 1744:992rpR2 v3 f ðvÞdv þ v3r f ðvÞdv vc

2:53

1:3377R þ

vr

 0:00771R þ 0:06901pR2 H þ 5:343R1:953  ð vr  ð vf 1744:992rpR2 v3 f ðvÞdv þ v3r f ðvÞdv

þ 0:5297R

2:5025

2:5

vc

vr

23:9347R þ 55:7435H R þ 4:041  107 Pr3  0:002445Pr2 þ 2:096Pr1:249  ð vr  þ ð vf 1744:992rpR2 v3 f ðvÞdv þ v3r f ðvÞdv 0:4037 0:8074

vc

vr

(20)

15:0656Pr þ 6:8004Pr þ 0:00108AEPð1  halt =15000Þ þ 8265:6416  ð vr  þ ð vf v3 f ðvÞdv þ v3r f ðvÞdv 1744:992rpR2 

vc

vr

 0:2082Pr3  0:006Pr2 þ 49:4518Pr þ 0:5133H1:1736 R1:1736 ð1 þ halt =15000Þ  ð vr  þ ð vf v3 f ðvÞdv þ v3r f ðvÞdv 1744:992rpR2 vc

vr

þ0:007ð1 þ halt =15000Þ

In Eq. (20), both cut-in and cut-off wind speeds are constant, and COE is only related to altitude, rotor radius, hub height and rated power when wind resource parameters are determined. Therefore, at a certain altitude, COE can be minimized by choosing the appropriate rotor radius, hub height and rated power.

COEmin ðR; H; Pr Þ ¼ minðCOEðR; H; Pr ÞÞ o n s:t: Rmin  R  Rmax ; Hmin  H  H max ; P min  Pr  P max r r 1:5R < H < 3R 20R < Pr < 40R (21)

3. Energy cost optimization of wind turbine on high-altitude sites The main work of this section is to formulate the optimization problem, select the appropriate fitness function, and introduce the improved Yin-Yang pair optimization. The optimization parameters in this study are rotor radius, hub height and rated power. The optimization objective is to minimize COE, which is a typical singleobjective optimization problem. In order to formulate the optimization problem, the COE model considering altitude factor given in Section 2 is used. Three optimization parameters are selected as design variables and COE minimum as fitness function. In this study, the Yin-Yang pair optimization is used to search the optimal parameters corresponding to the minimization of COE, and the YinYang pair optimization is improved from four aspects to speed up its convergence.

3.1. Optimizing problem expression Eq. (20) shows that COE can be expressed as non-linear function of the three design parameters (R, H, Pr ). Considering the manufacturing process, there are structural constraints among the hub height, rated power and rotor radius of wind turbines. In order to satisfy the structure constraint, hub height is set to 1.5e3 times of rotor radius, and rated power is set to 20e40 times of rotor radius [38]. In order to formulate this optimization problem, the fitness function and structure constraint are presented as follows:

where Rmin and Rmax are the smallest and largest rotor radius, H min and H max are the smallest and largest hub height, P min and P max are r r the minimum and maximum rated power. 3.2. Optimization method The optimization method is designed to solve the optimization problem. The common classical optimization methods are gradient descent method, Newton’s method and conjugate gradient method. The classical optimization method relies on the mathematical performance of the problem in solving optimization problems. The convergence speed and optimization results of some special problems, such as Non-deterministic Polynomial (NP) problem, are not ideal. The heuristic optimization algorithm effectively solves this problem. Traditional heuristic optimization algorithms include PSO algorithm [25,45], genetic algorithm [39], ant colony algorithm [40] and Grey Wolf algorithm [41]. Yin-Yang pair optimization algorithm is a new heuristic optimization algorithm, which has higher efficiency and better convergence effect than traditional heuristic optimization algorithm in solving single-objective optimization problems [42], and many literatures have proved that YinYang pair optimization algorithm can effectively solve the optimization problem of renewable energy system, such as solar power plant optimization problem [43] and Stirling engine systems optimization problem [44]. 3.2.1. Yin-Yang pair optimization Yin-Yang pair optimization algorithm is not based on specific laws or phenomena, but on the Yin-Yang philosophy of power balance between the two sides in real conflict. By making the two

D. Song et al. / Energy 193 (2020) 116794

sides of the contradiction compete and complement each other, the optimal solution of the problem is finally determined effectively. Yin-Yang pair optimization algorithm was first proposed in 2015, which is based on maintaining a balance between exploration and exploitation of the search space. It is a low-complexity optimization algorithm which works with two points and generates additional points depending on the number of decision variables in the optimization problem. In solving the optimization problem, two points are generated by initialization. According to the number of design variables, two points are split to search the global optimal solution. There are three user-defined parameters: minimum value of archive range Imin , maximum value of archive range Imax and expansion coefficient a. The Yin-Yang pair optimization algorithm is structurally divided into the following parts:

7

Start

Set , , ,termination criteria and variable bounds, and set = 0

Initialize two random points { 1

1

={

1 2 3 1, 1, 1}

1 2 3 2 , 2 , 2 },

and 0

= 0.5,

= 0.5. Generate I between

2

and

2

=

1, = 1,2,3. Initialize

1, 2

and

No 2

is fitter than

(1) Initialization:

1

?

Yes

Step 1: Determine the user defined parameters (Imin 、Imax and

Interchange

Step 2: Generate two random points P1 and P2 , which are Ddimensional; Initialize search radius coefficients d1 and d2 , which are set to 0.5; Generate a random integer I between Imin and Imax and set i ¼ 0. Step 3: calculate the fitness of P1 and P2 .

and

1

a), number of design variables (D) and their boundary conditions.

Store

1

and

2,

2

1

and

2,

t = t+1

in an archive, j=1

Execute splitting function s (23), (24) and (25) to

(2) External circulation: Step 1: If the fitness value of P2 is better than P1 , exchange P2 with P1 and d2 with d1 at the same time. Step 2: Archive P1 and P2 , and i ¼ i þ 1. Step 3: Split P1 and P2 to produce 2D points, and the point with the best fitness is selected to replace the point used for splitting. Step 4: Whether the termination condition is satisfied or not.

No

Is the fittest new point fitter than

?

Yes

Interchange the fittest new point and

(3) Internal circulation:

No

j=2 ?

j=j+1

Step 1: When i ¼ I is satisfied, the archive is processed, the best point in archive is selected to replace the P1 , and the second best point in archive is selected to replace the P2 . Step 2: The search radius is updated, the search radius of P1 point decreases, and the search radius of P2 point increases, as shown in Eq. (22). Step 3: Clean up the archives, and produce a random integer I between Imin and Imax , set i ¼ 0.

No

Yes

t=I ? Yes

Replace 1 with the best point in archive, replace 2 with the second best point in archive and set t = 0. Generate I between and , update 1 and 2 based on function (22).

(4) Splitting:

No

Step 1: Generate a random number t between 0 and 1. Step 2: If t < 0:5, the first splitting method is used to generate 2D points, as shown in Eq. (23). If t  0:5, the second splitting method is used to generate 2D points, as shown in Eq. (24). Step 3: Calculate the fitness values of these 2D points, and the point with the best fitness is selected to replace the points used for splitting.

Is the termination criteria satisfied ?

Yes

End Fig. 2. Flow chart of improved Yin-Yang pair optimization.

8 d1 < d1 ¼ d1  :

a d2 d2 ¼ d2 þ a

(22)

where d1 and d2 is the search radius coefficients of P1 and P2 , the maximum of d2 is 0.75, and a is expansion coefficient.

j

Sj ¼ Sj þ r d SjDþj ¼ Sj  r d where j ¼ 1; 2; 3…D.

(23)

8

D. Song et al. / Energy 193 (2020) 116794

 . pffiffiffi 2 Sik ¼ Sj ±r d

(24)

where j ¼ 1; 2; 3…D, k ¼ 1; 2; 3…2D. A random binary matrix B of 2D  D, when Bjk is positive, Eq. (24) takes þ, and when Bjk is negative, Eq. (24) takes -. 3.2.2. Improved Yin-Yang pair optimization Eq. (21) shows that the single-objective optimization problem is with three-dimensional variables. Aiming at solving this problem, the Yin-Yang pair optimization algorithm is improved. To facilitate identification, the improved algorithm is named 3D-YYPO. Compared with the traditional YYPO, 3D-YYPO has been improved from four aspects, as follows: (1) The initial value of Imax is set to 10, Imax is decreased by 4 when the archive is updated, and the minimum value of Imax is 3. This can reduce the number of archive update and accelerate the search speed in the early stage. (2) The initial value of a is set to 10, and the minimum value is 4 as the iteration number decreases. In this way, the searching range of best point can be reduced quickly near the global optimum, and the searching range of sub-advantages can be enlarged rapidly, so as to accelerate the convergence speed. (3) Comparing the best fitness points after each splitting with the original point for splitting, if the best fitness point after splitting is better than the original point, it will replace the original point with the best fitness point after splitting. It ensures that the P1 point of each generation is the current best point. This method can accelerate the convergence speed. (4) Adding a new splitting method for the three-dimensional problem. The search range of the third method is between the first and the second splitting methods. As shown in Eq. (25), the probability of using the three splitting methods is the same, which can enhance the search ability and accelerate the convergence speed in dealing with the threedimensional problem.

S ¼ S þ A:*d

1 B 1 B B r3 A¼B B r4 B @ 0 0

r1 r2 0 0 1 1

Fig. 4. Rotor radius convergence trend at 3000 m altitude.

(25)

where matrix A refers to the search range of three planes XY, XZ and YZ, expressed as follows:

0

Fig. 3. COE convergence trend at 3000 m altitude.

1 0 0 C C 1 C C 1 C C r5 A r6

Fig. 5. Hub height convergence trend at 3000 m altitude.

where r1  r6 takes any random number in [-1,1]. 3.2.3. Energy cost optimization of wind turbine based on improved Yin-Yang pair optimization algorithm In this study, the improved Yin-Yang pair optimization

Table 3 Parameter settings of three algorithms. YYPO Imin Imax a ¼ T ¼

¼ 1 ¼ 3 10 200

Fig. 6. Rated power convergence trend at 3000 m altitude.

3D-YYPO

PSO

Imin ¼ 1 Imax ¼ 10 a ¼ 10 T ¼ 200

c1 ¼ 0:8; c2 ¼ 2:0 wmax ¼ 0:9; wmin ¼ 0:4 xsize ¼ 50 T ¼ 200

algorithm is used to search the optimal parameters corresponding to minimum COE. The improved algorithm has stronger search ability and better convergence effect. The flow chart of the improved Yin-Yang pair optimization algorithm is shown in Fig. 2.

D. Song et al. / Energy 193 (2020) 116794

To deal with the single-objective optimization problem, three user-defined parameters, variable boundaries and the maximum number of iterations are given first, and then initialization is started. The initialization begins with the generation of two random points P1 and P2 satisfying the structural constraints and evaluating their fitness. Generate search radius coefficients d1 and d2 related to two points respectively. The number of archive updates is randomly generated between Imin and Imax , and then iteration starts. In the iteration process, the fitness values of P1 and P2 are compared first. If P2 is fitter than P1 , the points as well as their corresponding d values are interchanged. Both the points are stored in the archive. Set t ¼ 0, and t ¼ t þ 1. Subsequently, start the splitting and archiving, which are the two key steps in this algorithm. Three splitting modes are applied in the splitting process, as shown in Eqs. (23)e(25), to search the three-dimensional search range in line, plane and space. Pi ði ¼ 1; 2Þ produces six splitting points. Comparing the fitness value of the best splitting point with that of Pi , if best splitting point is fitter than Pi , Pi will be replaced by this point, and if the best splitting point is not fitter than Pi , Pi does not change. The archive stage is initiated by t ¼ I. Replace P1 with the best point in archiving, and replace P2 with the second-best point in archiving. The search radius coefficients are updated as Eq. (22), and then the archive matrix A is cleared. Generate I between Imin and Imax , and set t ¼ 0.

9

Fig. 7. The trend of AEP with Altitude under Optimal Parameters.

3.3. Method validation and result discussions Fig. 8. The trend of APC with Altitude under Optimal Parameters.

In this section, the rotor radius, hub height and rated power of wind turbines at different altitudes are optimized by employing the improved Yin-Yang pair optimization algorithm. In order to verify the effectiveness and superiority of the improved algorithm, the traditional Yin-Yang pair optimization and PSO algorithms are also used to solve this problem. Based on the optimization results, sensitivity analysis of the optimized parameters and COE to the wind resource parameters is carried out.

parameters of PSO algorithm, wmax and wmin are inertia factors, and the xsize is population size. The above three algorithms are compared in terms of convergence iteration times, convergence time and execution time, so as to judge the advantages and disadvantages of these three algorithms in solving the optimization problem.

3.4. Parameter settings

3.5. Result discussion

The design parameters in this study are rotor radius (R), hub height (H) and rated power (Pr ). Considering that the cost model used is aimed for large-scale wind turbines on high-altitude areas, the range of R is 20me55 m, the range of H is 10me150 m and the range of Pr is 100kw-3000kw, the cut-in and cut-off wind speeds are 3:0m=s and 25:0m=s, respectively. The reference height is set at 20 m, the shape factor k0 is a typical value of 2 [46], and the scale factor c0 can be obtained by Eq. (16). The mean wind speed vm at the reference height is 7:23m=s. Because the wind shear coefficient is small on high-altitude flat area, the wind shear coefficient is 0.1 [37]. The maximum power coefficient is set at 0.48 [47]. The initial altitude value is set at 2000 m, increasing 500 m each time, the maximum altitude is 4000 m. The parameter settings of 3D-YYPO, YYPO and PSO are shown in Table 3.where Imin and Imax are the minimum and maximum values of archive range, a is expansion coefficient, T is maximum iteration number, c1 and c2 are

Three different algorithms are used to optimize the design parameters of wind turbines, and the optimal parameters corresponding to the minimum COE at different altitudes are obtained. Firstly, the variation trends of the optimal parameters and COE with altitude are analyzed, and then the efficiency of the three algorithms in solving this optimization problem is compared. 3.5.1. Optimization results under three optimization algorithms By using the above three optimization algorithms to optimize the rotor radius, hub height and rated power of wind turbines, the minimum COE and its corresponding optimization parameters can

Table 4 AEP and APC under optimal parameters at different altitudes. Site altitude [m]

AEP [kWh]

APC [$]

COE [$/kWh]

2000 2500 3000 3500 4000

4,287,330.5 4,155,058.6 4,021,213.2 3,889,126.4 3,749,960.7

210779.23 210887.56 210745.12 210515.77 209705.02

0.049163 0.050754 0.052408 0.054129 0.055921

Fig. 9. Minimum COE changes with Altitude under Optimal Parameters.

10

D. Song et al. / Energy 193 (2020) 116794

be obtained. By setting the maximum number of iterations to 200 times, the plot of COE versus the iteration number is shown in Fig. 3. When using 3D-YYPO, COE converges fastest, and it has basically converged after 72 iterations; PSO converges slowest, and after 170 iterations COE converges basically. YYPO converges faster than PSO and slower than 3D-YYPO, and it has basically converged after 157 iterations. The convergence trend of COE at other altitudes is similar to that at 3000 m. According to Fig. 3, it is obvious that COE converges to a certain value after several iterations. As to 3D-YYPO, the initial COE value is 0.0529$/kWh, the COE value is 0.052409$/kWh after 37 iterations, and the COE value converges to 0.052408$/kWh after 72 iterations. Similarly, as to YYPO and PSO, the initial COE value is 0.05670$/ kWh and 0.05267$/kWh, respectively. After 59 iterations and 118 iterations, the COE value optimized by YYPO and PSO reaches 0.052409$/kWh. COE optimized by YYPO and PSO is converged to 0.052408$/kWh after 157 and 170 iterations, respectively. It is shown that these three algorithms can effectively solve this optimization problem. The convergence trends of R, H and Pr at the altitude of 3000 m based on three optimization algorithms are show in Figs. 4e6. Compared with the other two algorithms, COE can converge to the minimum quickly when using 3D-YYPO. At the same time, the optimization parameters can converge to the optimal solution quickly at different altitudes. Combining these three graphs, it can be seen that the optimization parameters have converged to the optimal solution after 72 iterations using 3D-YYPO, while the optimization parameters have converged to the optimal solution after 170 and 157 iterations using PSO and YYPO respectively. It can be seen that when COE basically converges to the minimum, the corresponding three optimization parameters also converge to the optimal values. The convergence trend of the three optimization parameters at other altitudes is similar to that at 3000 m, but the convergence value is different. As can be seen from the above Figs. 3e6, the optimization parameters and COE of three different algorithms can converge to similar results, but the convergence speed is different. It is obvious that 3D-YYPO has better performance than the other two algorithms. 3.5.2. Optimal parameters and COE condition under different altitude According to the optimal rotor radius, hub height and rated power of the wind turbine, APC, AEP and COE values at different altitudes are obtained and shown in Table 4. To further present the relation between the optimization results and the site altitude, the

comparison results taking the ones at altitude of 2000 m as the benchmark are illustrated in Figs. 7 and 8. The AEP and APC corresponding to the optimal parameters is shown in Figs. 7 and 8, respectively. The AEP decreases with the increase of altitude, whereas the APC is first increase and then decrease. From Fig. 7, it can be seen that the AEP varies almost linearly with altitude, and the AEP decreases with the increase of altitude. The AEP is decreased by 12.5% when the altitude is 4000 m compared with that when the altitude is 2000 m. By comparison, Fig. 8 shows that the APC does not change linearly with altitude, and that the influence of altitude on APC is less than that of AEP. Specifically, APC is increased by 0.04% at 2500 m, and decreased by 0.51% at 4000 m. According to Figs. 7 and 8, AEP and APC have different trends with the increase of attitude: 1) For AEP, the increase of altitude leads to the decrease of air density and the decrease of the rated power corresponding to minimum COE. Although the rotor radius increases with the increase of altitude, its variation cannot meet the requirement of increasing rated power, which leads to the increase of rated wind speed. The increase of rated wind speed and the decrease of rated power are the main reasons for the decrease of AEP with the increase of altitude. Rated wind speed and rated power change almost linear with altitude, which result in almost linear change of AEP with altitude. 2) For APC, the cost of wind turbine increases with altitude rising, while the cost of operation and maintenance decreases, mainly due to the reduction of land leasing cost. The influence of altitude on the initial capital cost decreases with the increase of altitude, while the influence of altitude on the operating cost increases with the increase of altitude. Therefore, when altitude rises from 2000 m to 2500 m, the initial capital cost increases more than the operating cost decreases, which leads to the increase of APC. With the further increase of altitude, the increasing trend of ICC becomes smaller, and the decreasing trend of AOE becomes larger, which result in the decrease of APC. From the optimization results, it can be seen that the minimum COE value increases with the altitude increase. Fig. 9 shows the variation trend of COE with altitude, and COE varies linearly with altitude. At the altitude of 4000 m, the COE is 0.055921$/kWh, which is increased by 13.75% when compared with that at the altitude of 2000 m. For the same wind turbine, the low air density at high altitude site leads to the decline of energy production. To

Table 5 Optimal results obtained by different optimization at different altitudes. Altitude [m]

Algorithm

Rotor radius [R/m]

Hub height [H/m]

Rated power [Pr /WM]

COE [$/kWh]

2000

3D-YYPO YYPO PSO 3D-YYPO YYPO PSO 3D-YYPO YYPO PSO 3D-YYPO YYPO PSO 3D-YYPO YYPO PSO

36.8326 36.8330 36.8325 37.0583 37.0577 37.0707 37.2712 37.2719 37.2724 37.4725 37.4692 37.4720 37.6580 37.6593 37.6573

65.7190 65.7154 65.7184 65.1300 65.1238 65.1235 64.5157 64.5150 64.5173 63.8936 63.9134 63.8895 63.2495 63.2525 63.2540

1.1947 1.1947 1.1946 1.1686 1.1686 1.1695 1.1418 1.1418 1.1419 1.1143 1.1141 1.1143 1.0860 1.0861 1.0860

0.049163 0.049163 0.049163 0.050754 0.050754 0.050754 0.052408 0.052408 0.052408 0.054129 0.054129 0.054129 0.055921 0.055921 0.055921

2500

3000

3500

4000

D. Song et al. / Energy 193 (2020) 116794

11

Table 6 The statistic results of execution time and convergence time by three different algorithms. Algorithm

3D-YYPO YYPO PSO

Execution time [m]

Convergence time [m]

Iteration number of convergence

Max

Min

Mean

Max

Min

Mean

Max

Min

Mean

7724 7804 17811

7660 7696 17910

7692 7750 17861

3157 7133 15589

2107 4989 13359

2632 6160 14474

82 183 175

55 130 150

69 157 163

YYPO and YYPO is close, 7692 s and 7750 s, respectively, but the execution time of PSO is more than twice as long as other two algorithms. In terms of the overall performance, PSO has the lowest efficiency in dealing with this problem compared with other two algorithms, and 3D-YYPO has the best performance. The above results prove that the improved Yin-Yang pair optimization algorithm is better than the traditional PSO and YYPO algorithms in dealing with the optimization problem.

3.6. Sensitive analysis

Fig. 10. Sensitivity of rotor radius to wind statistics.

maintain the same energy production, the rotor radius needs to be increased, which will lead to an increase in the cost of wind turbines. Because of the worse geographical conditions in high altitude areas, the cost of wind turbines and infrastructure increases. As a consequence, the minimum COE increases with the increase of altitude.

3.5.3. Comparison of optimization results under three optimization algorithms The optimal parameters obtained by different optimization algorithms at different altitudes are shown in Table 5. The minimum COE value and corresponding optimization parameters are obtained by the three optimization algorithms are very close. When the altitude increases from 2000 m to 4000 m, rotor radius varies from 36.83 m to 37.65 m, hub height varies from 65.71 m to 63.25 m, rated power varies from 1.194 MW to 1.086 MW, and COE increases from 0.049163 kWh to 0.055921 kWh. The optimized design parameters by the three algorithms show the same trend: with the altitude increase, the rotor radius increases, whereas the hub height and rated power decrease. Because the wind shear coefficient is small in this study, the increase of hub height will not bring about big benefit in increasing AEP, but cause the raise of the APC. Therefore, the optimal design is achieved by reducing hub height appropriately. In Table 5, the contradiction between the increased rotor radius and the declined rated power can be explained by the fact that the slight increase of the rotor radius cannot remove the influence of the reduction of air density, leading to the decline of the rated power. According to the optimization results obtained, it has been proved that the 3D-YYPO, YYPO and PSO can effectively solve the optimization design problem of wind turbines. However, by comparing the iteration number of convergence, execution time and convergence time, the three algorithms present obvious differences. Table 6 shows the iteration number of convergence, execution time and convergence time of the three algorithms in 10 runs. The average convergence iteration times of 3D-YYPO, YYPO, and PSO is 69, 157, and 163, respectively, and the average convergence time is 2632 s, 6160 s, and 14474s, respectively. The execution time of 3D-

The distribution of wind statistics is uncertain, such as shape factor k, scale factor c, mean wind speed vm and wind shear coefficient a. Their values are related to atmospheric conditions, seasonal time and topographic conditions. The variation of wind resource statistics has an uncertain influence on the optimized parameters and COE. Therefore, it is necessary to analyze the sensitivity of the optimal design parameters and COE to the wind statistics, and to explore the influence of the variation of each wind statistics on the rotor radius, hub height, rated power and COE. In the sensitivity analysis, a single-variable analysis method is used, and the sensitivity of three optimal design parameters and COE to the variable is tested. All of the three optimal design parameters and the optimal COE are obtained by using 3D-YYPO method. When a single wind statistic variable is tested, other variables are set to baseline values. By using the method of single variable analysis, the influence degree of each variable on optimized parameters and COE is determined. The optimized parameters and COE are taken as the reference values when altitude is 3000 m. The reference values of wind resource parameters are set as follows: mean wind speed vm ¼ 7:23m=s; shape factor k ¼ 2; wind shear coefficient a ¼ 0:1. The sensitivity of optimized parameters and COE to various wind resource statistics (shape factor, mean wind speed and wind shear coefficient) is analyzed. The sensitivity of rotor radius to wind statistics is shown in Fig. 10. When the shape factor, wind shear coefficient, and mean wind speed vary from 0.2 times to 1.8 times of reference value, the rotor radius varies from 30.78 m to 37.94 m, 28.21 me37.59 m, and 54.96 me28.93 m, respectively. Therefore, the rotor radius is most

Fig. 11. Sensitivity of hub height to wind statistics.

12

D. Song et al. / Energy 193 (2020) 116794

Fig. 12. Sensitivity of rated power to wind statistics.

Fig. 13. Sensitivity of COE to wind statistics.

sensitive to the mean wind speed. With the change of mean wind speed, rotor radius varies in the range of 0.78e1.48 times reference value. When the mean wind speed is 0.2 times of the reference value, the rotor radius is increased by 48%. It can be seen that the rotor radius increases with the increase of shape factor, and decreases with the increase of mean wind speed. When wind shear coefficient varies from 0.2 to 1.2 times of reference value, the rotor radius increases from 28.2139 m to 37.5980 m. When wind shear coefficient varies from 1.2 to 1.8 times of reference value, the rotor radius decreases from 37.5980 m to 37.0329 m. Reason of the two different variation trends is that compared with the results with lower wind shear coefficient, the effect of properly reducing rotor radius on energy production is less than that on wind turbine cost. The sensitivity of hub height to wind statistics is shown in Fig. 11. When the shape factor, wind shear coefficient, and mean wind speed vary from 0.2 times to 1.8 times of reference value, the hub height varies from 46.17 m to 66.63 m, 42.32 me110.69 m, and 135.41 me43.40 m, respectively. It is obvious that the mean wind speed and wind shear coefficient influence the hub height most. Especially when the mean wind speed is small, and the hub height is increased by 110%. With the change of wind shear coefficient, hub height varies in the range of 0.66e1.75 times of the reference value. The change of shape factor has the smallest impact on hub height, which varies in the range of 0.72e1.03 times of the reference value. It can be seen that the hub height increases with the increase of shape factor. When wind shear coefficient varies from 0.2 to 1.5 times of reference value, the hub height increases from 42.3209 m to 112.7018 m. When wind shear coefficient varies from 1.5 to 1.8 times of reference value, the hub height decreases from 112.7018 m to 110.6934 m. Reason of the two different variation trends is that in comparison with the results with lower wind shear coefficient, the effect of properly reducing hub height on energy production is less than that on wind turbine cost.

The sensitivity of rated power to wind statistics is shown in Fig. 12, it is obvious that rated power is sensitive to the three wind resource statistics. While the shape factor, wind shear coefficient, and mean wind speed vary from 0.2 times to 1.8 times of reference value, the rated power varies from 0.893 MW to 1.311 MW, 0.564 MWe1.502 MW, and 0.781 MWe1.376 MW, respectively. Among the three wind resource parameters, the change of wind shear coefficient has the greatest impact on rated power, which varies in the range of 0.49e1.3 times of the reference value. The sensitivity of COE to wind statistics is shown in Fig. 13. While the shape factor, wind shear coefficient, and mean wind speed vary from 0.2 times to 1.8 times of reference value, the COE varies from 0.16149$/kWh to 0.04881$/kWh, 0.05762$/kWh to 0.04556$/kWh, and 9.08611$/kWh to 0.03461$/kWh, respectively. The COE decreases with the increase of three wind statistics. When the three wind resource parameters are larger than the reference value, the change of COE is small. It is obvious that the mean wind speed influences the COE most especially when it is small. With the change of mean wind speed, COE varies within 0.66e173.4 times of reference value. At 0.2 times of the reference mean wind speed, the COE value becomes 173.4 times of the reference value. It shows that the energy cost is too high in sites with small mean wind speed. With the change of shape factor, COE varies within 0.93e3.1 times of reference value. Sensitivity of COE to wind shear coefficient is smaller than other two wind statistics, and COE only changes within 0.87e1.1 times of the reference value. Based on the above sensitivity analysis, it is found that wind statistics have noticeable influences on the selection of rotor radius, hub height and rated power of wind turbine, especially when the wind statistics are small. When the mean wind speed and shape factor are small, the minimum COE rises rapidly, which indicates that the wind farm should be located in the area with larger mean wind speed and shape factor in order to achieve more economic benefits. The rotor radius is mainly affected by the mean wind speed, the hub height is mainly affected by the mean wind speed and the wind shear coefficient, and the rated power is sensitive to the three wind statistics. 4. Conclusion In this study, a new single-objective optimization algorithm is proposed for wind turbine design optimization on high-altitude sites, which searches for the optimal rotor radius, hub height and rated power corresponding to minimum COE at different altitudes. Result comparisons among three different optimization algorithms have proven that the improved Yin-Yang pair optimization algorithm has better performance in solving the single-objective optimization problem of wind turbine. Sensitivity of optimized parameters and COE to wind resource parameters is analyzed. The important conclusions are summarized as follows: C With the increase of altitude, the optimal rotor radius increases, the optimal hub height and optimal rated power decreases, and the minimum COE increases. When altitude increases from 2000 m to 4000 m, the minimum COE increases from 0.049163 $/MWh to 0.055921 $/MWh, the rotor radius increases from 36.83 m to 37.65 m, the hub height decreases from 65.71 m to 63.24 m, and the rated power decreases from 1.194 MW to 1.086 MW. C With the increase of altitude, the APC and AEP corresponding to the optimal parameters show a decreasing trend, but the influence of altitude on APC is less than that of AEP. The AEP at 4000 m is decreased by 12.5% compared with that at 2000 m, while the APC at 4000 m is decreased by 0.5% compared with that at 2000 m. The AEP varies almost

D. Song et al. / Energy 193 (2020) 116794

linearly with altitude, but the APC does not change linearly with altitude. It is indicated that the main reason for the increase of COE with altitude is the change trend between AEP and APC. C It is the first time that Yin-Yang pair optimization algorithm is applied to solve the wind turbines design and optimization problem. By comparing 3D-YYPO with YYPO and PSO, 3DYYPO has the fast convergence speed. The average convergence iteration times of 3D-YYPO, YYPO and PSO is 69, 157 and 163, respectively, whereas the average convergence time is 2632 s, 6160 s and 14474 s, respectively. Therefore, the convergence iteration number and convergence time of 3DYYPO are much less than that of other two algorithms. C Sensitivity analysis shows that wind resource parameters have a great influence on the optimal parameters and energy cost of wind turbines. From 0.2 times to 1.8 times of the reference wind resource parameters, the COE varies from 0.87 times to 173.4 times of the reference value, the rotor radius varies from 0.76 times to 1.48 times of the reference value, the hub height varies from 0.72 times to 2.1 times of the reference value, and the rated power varies from 0.49 times to 1.3 times of the reference value. In order to reduce COE, wind farms at high-altitude sites should be located in the areas with high mean wind speed and shape factor. In our future work, other uncertainties, such as wind turbine cost and civil work, will be considered. Moreover, the stochastic optimization method can be used to optimize the design of wind turbines for specific wind farms at high altitudes using different techno-economic model. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant 61803393 and 51777217, the Project funded by China Postdoctoral Science Foundation under Grant 2017M622605, and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, NRF-2016R1A6A1A03013567.

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