Optimization of Wind Turbine Rotor Diameters and

Optimization of Wind Turbine Rotor Diameters and Hub Heights in a Windfarm Using Differential Evolution Algorithm Partha P. Biswas1 ✉ , P.N. Suganthan1, and Gehan A.J. Amaratunga2 (

2

)

1 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Singapore [email protected], [email protected] Department of Engineering, University of Cambridge, Cambridge, UK [email protected]

Abstract. In this paper some of the optimized wind turbine layouts in a wind‐ farm, as presented by many authors, have been chosen as basis for further eval‐ uation and study. The objective of most of earlier studies was to minimize cost per kW of power produced. This paper focuses from different perspective of optimization of turbine rotor diameters and hub heights to improve overall wind‐ farm efficiency at a reduced cost. Differential Evolution (DE) algorithm is employed to optimize rotor diameters and hub heights of turbines in a wind‐ farm. Keywords: Windfarm layout · Wind turbine rotor diameter · Hub height · Differential evolution (DE) algorithm · Efficiency · Cost function

1

Introduction

Windfarm layout is highly complex and several factors are involved in optimizing the layout in order to achieve maximum power at minimum cost. After Mosetti et al. [3] publication on wind turbine layout optimization in a windfarm, many papers have been published by various authors in pursuit of improving the results. Among various publi‐ cations, layouts proposed by Grady et al. [4] and Mosetti et al. [3] are considered here for further examination. The total power produced and published with the layouts proposed in these papers will be achieved, however with a better efficiency of the wind‐ farm with optimized turbine rotor diameters and hub heights. By optimizing wind turbine rotor diameters and hub heights in a staggered manner, we are practically considering effect of wakes in downstream of a row of wind turbines. Due to effect of several wakes by upstream turbines, capacity of rotors at downstream is not fully utilized. The concept presented in this paper is – get maximum out of the wind energy by the upstream turbine row and install turbine of smaller dimension at the under-utilized downstream row. This optimization in turn increases the overall efficiency

© Springer Nature Singapore Pte Ltd. 2017 K. Deep et al. (eds.), Proceedings of Sixth International Conference on Soft Computing for Problem Solving, Advances in Intelligent Systems and Computing 547, DOI 10.1007/978-981-10-3325-4_13

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of the wind farm and overall cost of installation is reduced. In a closely spaced layout of upstream row, the optimization is more effective. Rotor blades and Tower are major components for wind turbine contributing part of total cost. A comparative study on overall installation cost has been presented in the subsequent sections of this paper.

2

Windfarm Modelling

As in most of the other papers Jensen wake decay model [1, 2] is employed for consid‐ eration of the wind velocity inside the wake region. In the equations below for clarity k-th turbine is considered under the influence of single turbine at m-th location (Fig. 1). Assuming that the momentum is conserved in the wake, the wind speed in the wake region is calculated by–

⎡ ⎤ ⎢ ⎥ 2a uk = u0k ⎢1 − ( )⎥ ⎢ ⎥ x 1 + 𝛼m mk ⎥ ⎢ rm1 ⎦ ⎣

a=

rm1

1−

√ 1 − CT

2 √ 1−a = rm 1 − 2a

𝛼m =

0.5 ( ) hm ln z0

(1)

(2)

(3)

(4)

Where, u0k is the local wind speed at k-th turbine without considering the wake effect xmk is the distance between m-th & k-th turbine rm is the radius m-th turbine rotor rm1 is the downstream rotor radius of m-thturbine hm is the hub height of the m-thturbine 𝛼m is the entrainment constant pertaining to m-thturbine a is the axial induction factor CT is the thrust coefficient of the wind turbine, which represents the thrust exerted on the wind rotor by air. CT is considered same for all turbines. z0 is the surface roughness of windfarm.

The wake region is conical for the linear wake model and radius of the wake region is represented by wake influence radius defined as

Optimization of Wind Turbine Rotor Diameters and Hub Heights

Rwm = 𝛼m xmk + rm1

133

(5)

Local wind speed at k-th turbine is dependent upon the hub height of the turbine as velocity of wind changes with height from ground. Logarithmic law has been used here to represent the local wind speed.

Fig. 1. Linear Wake Model - k-th turbine under the influence of single m-th turbine

( u0k = uref log

hk z0

)/

( ) href log z0

(6)

For comparison with earlier studies performed in [3, 4], reference height and wind speed at reference height are considered as, href = 60 m & uref = 12 m/s respectively; and z0 as 0.3. When one area is inside multiple wake flows, the velocity deficit will be enhanced. If we consider i-th turbine under the influence of multiple wakes, taking into account the wake flow superimposed effect, the wind speed at the position of i-th turbine can be written as [5, 6] ⎡ ui = u0i ⎢1 − ⎢ ⎣

Where,

√

( ) uij 2 ⎤ ⎥ 1− j=1 u0j ⎥ ⎦

∑NT

(7)

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u0i & u0j are the local wind speeds (free stream velocity) at i-th & j-th turbine respec‐ tively without considering the wake effect uij is the wind velocity at i-th turbine under the influence of j-th turbine NT is the number of turbines affecting the i-th turbine with wake effects. It is to be noted here that the variation in rotor diameters and hub heights are not of wide range and the turbine downstream are far enough to consider full wake effect instead of partial wake effect induced by the upstream turbine.

3

Previous Studies

Currently in this paper layouts of [3, 4] for Case-1 i.e. Constant Wind Speed and Fixed Wind Direction has been revisited. For power calculation both Mosetti et al. & Grady et al. made some approximation. Total Power equation for each turbine has been used as Pi = 0.3u3 for rotor radius of 20 m. As rotor radius in this paper has been used as variable the constant factor in power equation cannot be approximated. Without approx‐ imation the power equation becomes–

Pi = 0.5 ∗ 𝜌 ∗ 𝜋 ∗ r2 ∗ u3 ∗ CP ∕1000 kW

(8)

For air density ρ = 1.2254 kg/m3, rotor radius r = 20 m & rotor efficiency Cp = 0.4 Pi = 0.307976611*u3 kW with wind velocity u in m/s. 8 Turbine thrust coefficient, as in other papers, is considered as CT = i.e. approx. 9 0.8888. Table 1 summarizes the results obtained in previous studies with approximation and without approximation for Case-1: Constant Wind Speed and Fixed Wind Direction. In this paper the same calculated power or higher will be achieved with optimization of rotor diameter and hub height at a better overall efficiency of the wind farm.

4

Differential Evolution

Differential Evolution (DE), introduced by Storn and Price in 1996, is stochastic, popu‐ lation based optimization algorithm where the individuals in the population evolve and improve their fitness through probabilistic operators like recombination and mutation. These individuals are evaluated and those that perform better are selected to compose the population in the next generation. More details of recent updates can be found in paper [8]. Following are the steps involved in Differential Evolution algorithm. 4.1 Initialization The first step in the DE optimization process is to create an initial population of candidate solutions by assigning random values to each decision vector of the population. Such values must lie inside the feasible bounds (between maximum & minimum) of the deci‐ sion vector. We may initialize j-th component of the i-th decision vector as


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( ) (0) xi,j = xmin,j + randij [0, 1] xmax,j − xmin,j Where randij [0, 1] is a uniformly distributed random number lying between 0 and 1. 4.2 Mutation After initialization, DE creates a donor/mutant vector v(t) corresponding to each popu‐ i lation member or target vector xi(t) in the current iteration through mutation (the super‐ script ‘t’ denotes parameter at t-th iteration). There are quite a few strategies for muta‐ tion. The one used here is:

( ) (t) (t) (t) v(t) = x + F x − x i Ri Ri Ri 1

2

3

The indices Ri1 , Ri2 & Ri3 are mutually exclusive integers randomly chosen from the population range. The scaling factor F is a positive control parameter for scaling the difference vectors 4.3 Crossover Through crossover the donor vector mixes its components with the target vector xi(t) to = (u(t) , u(t) , … , u(t) ). Binomial crossover, which is form the trial/offspring vector u(t) i i,d i,1 i,2 adopted here, operates on each variable whenever a randomly generated number between 0 and 1 is less than or equal to a pre-fixed parameter Cr, the crossover rate. The scheme is expressed as:

{ u(t) i,1

=

v(t) if j = K or randi,j [0, 1] ≤ Cr, i,j (t) xi,j otherwise

Where K is any randomly chosen natural number in {1,2,….,d}, d being the dimen‐ sion of real-valued decision vectors. 4.4 Selection Selection determines whether the target (parent) or the trial (offspring) vector survives to the next iteration i.e. at t = t+1. The selection operation is described as: xi(t+1) =

{

( ) ( ) u(t) if f u(t) ≤ f xi(t) , xi(t) otherwise

Where f(.) is the objective function to be minimized.

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Application and Approach Using DE

For all practical purposes it is not possible to have as many variants of rotor diameters and hub heights as the number of turbines. The layouts considered for further evaluation in this paper have 30 & 26 turbines in a 2 km x 2 km windfarm area consisting of many 200 m x 200 m grids. 3 different values of both rotor diameters and hub heights are randomly selected from a defined range of these parameters. The selected values are then evenly distributed and assigned to all the wind turbines in the windfarm. Range of rotor diameter is input in DE as 36 to 44 m, while that of hub height is 54 to 62 m. It may be noted here that the selected values in DE are rounded off to calculate the objective function defined below. Power from each turbine inside wake is calculated using Eqs. (1) to (8) above. Efficiency of the windfarm is described as,

∑N 𝜂 = ∑N

i=1

i=1

Pi

(9)

Pi,max

Where, Pi, max is the maximum output from i-th turbine had there been no wake effect. The objective of the optimization: Maximize ‘η’ subject to ∑N i=1

Pi = Pdesired + 𝜀

(10)

Where, Pdesired is the total calculated power output as detailed in Table 1. ε is a positive error arbitrarily chosen in the range of 0 < 𝜀 ≤ 30 to facilitate the iteration process. Table 1. Recalculated power of previous layouts [3, 4] Parameter

No. of Turbines (N) Total Power Efficiency (%)

6

Grady et al. Reported 30

Mosetti et al. Calculated Reported (without approx.) 30 26

Calculated (without approx.) 26

14310 92.015

14667 91.867

12654 91.452

12352 91.645

Results and Discussion

The DE optimization for 30 turbine layout proposed by Grady et al. gives following output for Case 1: Constant Wind Speed and Fixed Wind Direction. Figures 2 and 3 below specify the selected rotor diameters and hub heights for the windfarm. Do take note of wind direction indicated by arrow in the figures.


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Fig. 2. Layout indicating Optimized Rotor Diameters

Fig. 3. Layout indicating Optimized Hub Heights

The DE optimization for 26 turbine layout proposed by Mosetti et al. gives following output for Case 1: Constant Wind Speed and Fixed Wind Direction. Figures 4 and 5 below specify the selected rotor diameters and hub heights for the wind farm proposed by Mosetti et al.

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Fig. 4. Layout indicating Optimized Rotor Diameters

Fig. 5. Layout indicating Optimized Hub Heights

It is obvious from the optimized layout is that the turbines with larger diameter and higher hub height tend to align in a row facing the oncoming wind. This helps to extract maximum power out of the wind. The turbines at the last row (farthest from oncoming wind) experience the least wind speed. Hence to recover power out of the wind, these turbines assume intermediate diameter and hub heights. The turbines in the middle have smallest rotor diameter and shortest hub height.


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Adopting different rotor diameters and hub heights essentially increases the overall efficiency of the windfarm as can be seen from Tables 2 and 3. Though some of the rotor radii and hub heights have increased from earlier studies considering fixed rotor radius of 20 m and hub height of 60 m, many have decreased as well to bring down the overall investment cost which has been discussed in subsequent clause here. It is to be noted here that the above solutions are not unique and different combinations of variable radii and hub heights can be obtained by running the optimization program several times. Table 2. Present layout vs Grady et al. layout [4] Parameter

No. of turbines (N) Total power Efficiency (%)

Grady et al. Reported

Calculated (without approx.) 30 14667 91.867

30 14310 92.015

Optimized layout (using differential evolution) 30 14692 93.064

Table 3. Present layout vs Mosetti et al. layout [3] Parameter

No. of turbines (N) Total power Efficiency (%)

7

Mosetti et al. Reported

Calculated (without approx.) 26 12654 91.452

26 12352 91.645

Optimized layout (using differential evolution) 26 12680 92.612

Cost Model

The cost model introduced by Mosetti et al. accounts for only one variable, number of wind turbines. The cost for a windfarm having N turbines can be expressed as – Costbase = N

(

2 1 −0.00174N 2 + e 3 3

)

(11)

In a wind turbine, Turbine Blades and Tower account for 17.7% & 21.9% cost respectively [11]. Wind turbine costs about 75% of total installation cost of a windfarm setup [10]. For rotor radius change in the range of ± 10% from base radius of 20 m, 1% change in rotor radius affects the blade cost by about 3% [9]. Tower cost varies propor‐ tional to its height [7]. Considering variations in rotor diameters and hub heights, and to compare the cost with previous studies the modified cost model, taking into account the weight age of rotor cost and tower cost w.r.t. total installation cost, can be represented as: ] [ 1 ∑v2 1 ∑v 1 0.0039825xi Ni + 0.0016425xj Nj Cost = Costbase 1 + i=1 j=1 N N

(12)

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Where,

Costbase = Calculated cost using Eq. (11) above xi = % change in rotor radius for i-th variant of turbines from Rbase = 20 m, ‘−ve’ for reduction, ‘+ve’ for increase in radius xj = % change in hub height for j-th variant of turbines from Hbase = 60 m, ‘−ve’ for reduction, ‘+ve’ for increase in height Rbase = Rotor radius of turbine used in previous study = 20 m Hbase = Hub height of turbine used in previous study = 60 m v1 = Number of variants of turbines having rotor radii different from Rbase v2 = Number of variants of turbines having hub heights different from Hbase Ni = Number of turbines of i-th variant, turbine rotor radius different from Rbase Nj = Number of turbines of j-th variant, turbine hub height different from Hbase. Using Eq. (12) above the Cost/kW of power generation is calculated and tabulated below (Tables 4 and 5). Table 4. Cost per kW comparison (for layout proposed by Grady et al.) Parameter

Grady et al. Reported

Total power Cost/kW

14310

14667

Optimized layout (with different turbine rotor diameters & hub heights) 14692

1.5436 × 10−3

1.506 × 10−3

1.4939 × 10−3

Calculated (without approx.)

Table 5. Cost per kW comparison (for layout proposed by Mosetti et al.) Parameter

Total power Cost/kW

8

Mosetti et al. Reported

12352

12654

Optimized layout (with different turbine rotor diameters & hub heights) 12680

1.6197 × 10−3

1.5810 × 10−3

1.5678 × 10−3

Calculated (without approx.)

Conclusion

The selection of hub heights and rotor diameters in a staggered manner helps improve overall efficiency of the windfarm; also offers an economical advantage when wind speed and direction are mostly fixed. A supposition made in the study is of constant thrust co-efficient for different rotor diameters and this can be achieved with proper design of rotor. Similar to earlier studies wind speed at respective hub height is consid‐ ered as local wind speed contributing to the power output from that wind turbine. An optimized layout found using Genetic or Greedy or any other algorithms can further be


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reviewed in terms of selection of rotor diameters and hub heights. The effectiveness of different hub heights and rotor diameters for more practical cases when wind speed and direction are variable, remain topic for further studies. Acknowledgements. This project is funded by the National Research Foundation Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) program.

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