Optimisation of Hydrokinetic Turbine

8/18/2019 Optimisation of Hydrokinetic Turbine

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Optimisation of hydrokinetic turbine array layouts via surrogatemodelling

Eduardo Gonzalez-Gorbe~na*, Raad Y. Qassim, Paulo C.C. Rosman

Centro de Tecnologia, Cidade Universitaria, Ilha do Fundao, Bloco C, sala 203, CEP 21945-970, Rio de Janeiro, Brazil

a r t i c l e i n f o

Article history:

Received 20 March 2015

Received in revised form

5 November 2015

Accepted 15 February 2016

Available online 27 February 2016

Keywords:

Hydrokinetic energy

Turbine array layout

Surrogate based optimisation

a b s t r a c t

A procedure for the optimisation of hydrokinetic turbine array layout through surrogate modelling isintroduced. The method comprises design of experiments, computational uid dynamics simulations,

surrogate model construction, and constrained optimisation. Design of experiments are used to buildpolynomial and Radial Basis Function surrogates as functions of two design parameters: inter-turbine

longitudinal and lateral spacing, with a view to approximating the capacity factor of turbine arrayswith inline and staggered layouts, each of which having a xed number of turbines. For this purpose, two

scenarios have been used as case studies, considering uniform and non-uniform free-stream ows. Themajor advantage of this method in comparison to those reported in the literature is its capability to

analyse different design parameter combinations that satisfy optimality criteria in reasonable compu-tational time, while taking into account complex oweturbine interactions and different turbine types.

© 2016 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, in-stream hydrokinetic energy has drawn theattention of investors around the world. The large amounts of en-ergy found in river ows, tidal channels and ocean currents has

served as a strong motivation for research in optimising hydroki-netic turbine arrays with a view to making their commercialexploration viable. There are several issues concerning the designof turbine array layout, such as array size, complex oweturbine

interactions, different turbine types, and environmental impacts,which are necessary to be taken into account, in order to employsuitable optimisation methods. Vennell et al. [1] distinguish be-tween two scales of array optimisation: macro and micro array

design scales, where macro-design relates to the total number of

turbines in a farm, the number in each row and the number of rowsin the array. On the other hand, micro design is concerned with theindividual positions of the turbines within the array. This paper

focuses on the aspect of micro design of turbines arrayoptimisation.

In almost all reported research work on the turbine array layoutoptimisation problem, two approaches have been employed. In the

rst type of approach, highly simplied one dimensional tidal owmodels are employed [2e6]. In the second type of approach, morecomplex multidimensional tidal ow models are adopted [7e9].The simplied model approach possesses the appeal of simplicity;however, this approach cannot capture the complex nonlinear uid

ow interactions between turbines. In the second approach, more

realistic models are employed; however, they are so computa-tionally demanding that the whole design parameter space cannotbe explored, and as a consequence, only a limited number of

manually selected turbine array congurations are studied in thesearch for an optimal solution. Recently, a third approach has beenproposed [10], whereby a gradientebased optimisation method isdeveloped to solve the turbine array layout optimisation problem

for a given initial turbine array conguration. In the approach

presented by Funke et al. [10], a function (of the solution of theshallow water uid ow partial differential equations and of thedesign parameters, which comprise the location of turbines) is

optimised subject to constraints, which include the shallow water

uid ow partial differential equations. The approach of [10] pos-sesses several strong points: it combines macro and microarrangement optimisation, sound mathematical basis, relatively

fast computation times even for large turbine arrays, and simulta-neous determination of the uid velocity eld and the location of turbines. However, it is much more complex to implement, de-mands high computer storage capacity, it may encounter dif -

culties in obtaining a global optimum solution, and the parameter

* Corresponding author.

E-mail addresses: [email protected] (E. Gonzalez-Gorbe~na), qassim@

peno.coppe.ufrj.br (R.Y. Qassim), [email protected] (P.C.C. Rosman).

Contents lists available at ScienceDirect

Renewable Energy

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / r e n e n e

http://dx.doi.org/10.1016/j.renene.2016.02.045

0960-1481/©

2016 Elsevier Ltd. All rights reserved.

Renewable Energy 93 (2016) 45e57

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set to be optimised must be continuous. This last one may representa serious weakness as it does not allow for the choice of different

turbine types at different locations as part of the output. The degreeof freedom in allowing for different turbine type choice, is crucialfor the assessment of the economic viability of sites with nonuniform depth. In such cases, the design parameter space includes

discrete variables depicting turbine size of different turbine types.This feature is typical of the classical equipment selection problemin operations research [11,12].

This paper is a continuation of the work presented by Gorbe~na

etal. [13], where a different approach of optimisation is introduced.In Ref. [13], the well established method known as Surrogate-BasedOptimisation (SBO) of computer experiments is applied to theoptimisation of uniform turbine array layouts under steady state

uniform ows. Where uniform array layouts are dened by twospacing parameters: longitudinal spacing and lateral spacing. Thismethod provides a less expensive surrogate model of a morecomplex model facilitating the exploration of the entire design

space. An additional advantage of this method is that it enables theoptimisation of continuous and discrete design parameters simul-taneously. As the number of design parameters increases, the

necessary number of computer experiments grows, resulting in the

main drawback of the SBO method. In the literature, there are anumber of papers where surrogates have been used to solvedifferent variants of the wind farm layout optimization problem.

Rodrigues et al. [55] used surrogates of wind roses to optimise thelayout of offshore wind farms. Zhang et al. [56] implemented aresponse surface-based cost model for wind farm design. InRef. [57] wind power assessment is approximated through surro-

gates, using wind parameters as design variables, to evaluate themaximum wind farm capacity factor for a specied farm size andinstalled capacity. Recently, Mehmani et al. [58] presented anapplication of the surrogate-based particle swarm optimisation for

large-scale wind farms.Except for the work of Funke et al. [10], most of the studies

related to hydrokinetic turbine array optimisation consider uniform

layouts positioned in uniform channels with at bed bottoms andwith uniform free-stream ows [6,8,13,14]. In the present work, thecapabilities of different surrogates are analysed to optimise uni-form turbine array layouts positioned in regular and irregularchannels to generate different ow conditions.

The paper is organised as follows. In Section 2, the SBO approachis presented and discussed. In Section 3, the problem formulation ispresented, and this is followed in Section 4 by the design of com-puter experiments. In Section 5 computer simulation results are

presented. In Section 6, surrogate model construction is carried out,which is then assessed and validated in Section 7. In Section 8, thesurrogate optimisation model is presented. Finally, conclusions andsuggestions for future work are presented in Section 9.

2. Surrogate based optimisation

Surrogate models are inexpensive approximations of more

complex computational models allowing fast exploration of thedesign space, therefore reducing the overall design process dura-tion and cost. Also known as response surfaces, metamodels, or

emulators, surrogate models are used for several tasks like: opti-misation, sensitivity analysis, design-space exploration and para-metric studies amongst others. There is a wide variety of metamodels available in the literature, each of which possessing

advantages and limitations. Popular methods for creating surrogatemodels include polynomial functions [15], Radial Basis Functions(RBF) [16] and kriging [17]; for a review of metamodel techniquessee Refs. [18e20]. Selecting the proper metamodel to be used de-

pends on the nature of the engineering problem under

consideration, as well as on the available data. For this reason,conducting a comparison analysis of the surrogate models that are

built with different methods is of great importance. Trying out all of them is beyond the scope of this work, and instead two specicmethods are employed and compared. The dif culty of surrogatemodelling consists in the construction of a reliable emulator using

the least possible number of samples. A typical ow diagramsummarising the Surrogate-Based Optimisation (SBO) procedure isshown in Fig. 1.

In the present paper, SBO procedure is applied to the optimi-

sation of turbine array layout, with two design parameters as in-dependent variables: longitudinal and lateral spacing, with a viewto approximating the capacity factor, CF , of turbine arrays withinline and staggered congurations, each of which is studied under

uniform and non-uniform ows. A second purpose is to analyse theadequacy of polynomial and RBF modelling methods for theirapplication to the turbine array layout problem.

3. Problem formulation

In order to solve an optimisation problem, it rst needs to be

formulated in such a manner that the objective function, design

parameters and constraints are dened. This is a crucial step, whichneeds special attention. The consideration of optimising uniformarray layouts may be reduced to a two-dimensional horizontal

optimisation problem. Therefore, the optimisation problem can beformulated as a function of two design parameters: longitudinaland lateral distance between rows ( x1 ¼ DX) and columns ( x 2 ¼ DY)of turbines, respectively. This consideration implies that all tur-

bines within the array are positioned at the same depth.It is important to notice that even though, in this particularcase,

we focus in the optimisation of uniform array layouts, this methodis applicable for more sophisticated problems. The problem may be

sophisticated by adding new design parameters, for example, butnot only, the inclusion of the vertical dimension ( x 3 ¼ DZ), differentturbine rotor diameters ( x5 ¼ D1; x6 ¼ D 2; x7 ¼ D 3), setting a range

for the number of turbines to be installed, etc.Once the design parameters are dened, it is necessary to

specify which is the dependant variable. In this work, the turbinearray layout has been optimised to maximise prot from electricenergy production. Energy production of the array may be quan-

tied through the Capacity Factor, CF , ratio, see Eq. (4), that is thepower produced over a period of timedivided by the rated power of the turbines. In this way, considering the simulations are time in-dependent, the surrogate model, ^ y ( x ), will represent the instan-

taneous Capacity Factor of the array as a function of longitudinaland lateral spacing between turbines.

4. Design of computer experiments

The design of computer experiments consists in statisticalmethods to establish combinational relationships of input variablesinvolved in the optimisation problem. The objective of a design plan

consists in minimising the error between the surrogate and com-puter models using the smallest possible number of simulations.Therefore, the main challenge resides in an adequate description of

the design space.In the literature, there exists extensive work on this subject:

introducing, analysing, and comparing different methods, c.f.,[18,21e23]. Among the established methods of design of experi-

ments, the Latin Hypercube Design [24] is one of the most widelyused sampling methods. Latin Hypercube Designs (LHD) mayfollow strategies for improving its performance over the designspace. Some of these strategies consist in adopting some optimality

criteria; a review of several of these methods may be found in

E. Gonz alez-Gorbe~na et al. / Renewable Energy 93 (2016) 45e57 46


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Refs. [18,25]. For the purpose of this paper, a maximin distancecriterion is adopted to dene one set of experimental data, as it

provides better designs to estimate regression parameters [59].This criterion maximise the minimum Euclidean (or [ 2-norm)distance between sample points [26].

As dened in Section 3, the design variables are two: longitu-

dinal ( x1 ¼ DX) and lateral ( x 2 ¼ DY) distance between rows andcolumns of turbines, respectively. Now, the number of computerexperiments, n, to train the surrogate needs to be specied. As anempirical rule, the number of data collection points in a sampling

plan should be around ten times the number of design variables, s;c.f., [27]. Other authors [28] state that the number of sample pointsmust be at least equal to or greater than the number of modelparameters to be estimated. Following the above recommendation,

in this work a sample plan of n ¼ 20 has been selected. Consideringphysical and operational restrictions, the limits of each variable areset to:

10D x1 30D and 2D x2 4D:

Where D is the turbine rotor diameter.Fig. 2 shows the design of experiments used in this work. The

selected sample plan design includes two sets of experimental data.The rst data set consists of an approximate maximin LHD (n, s),with n ¼ 11 and s ¼ 2 (blue points) (in web version), which areobtained using the built-in function lhsdesign available in Matlab

[60]. The second data set consists of 9 experimental points (red

points), which intend to obtain information of the variables spaceend and middle values and help improve the surrogate tness, as

noted by Ref. [13]. The sample values (u ¼ 1, 2, …, n) for each inputvariable (i ¼ 1, 2, …, s) are normalised on the interval [1, 1].

5. Computer simulation

Computer simulation runs have been carried out using thestudent version of the CFD software ANSYS 15.0 FLUENT [29] tosolve three-dimensional ows. Turbines are modelled as actuatordisks, assuming a horizontal axis turbine type, this being one of the

simplest representations of a rotor, which provides reliable results

with affordable computational expense [30e32]. In FLUENT, this isachieved using a porous-jump boundary condition [29].

In order to study the capabilities of the proposed surrogate

models, two channel geometries, both with the same length (x-axis), are studied for inline and staggered turbines array layouts. A

at bottom channel with constant width (y-axis) along its length,

illustrated in Fig. 3 and Fig. 4, has been used to optimise layoutswith steady state free stream uniform ow conditions. To avoid theeffect of lateral boundaries on turbine performance, for each of thesimulations on the at bottom channel, the distance between the

turbines located at the sides of the array and these boundaries isxed in 6D. On the other hand, an irregular channel, presented inFig. 5 and Fig. 6, has been used for the same purpose but this time,under steady state free stream non-uniform ow. The irregular

channel has a variable width along its length but its geometry is

Fig. 1. Simulated Based Optimisation ow chart.



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maintained constant for all simulations. The geometries of thechannels are kept as simple as possible to guarantee that the

number of elements does not exceed the limits by the ANSYSFLUENT student version.

The number of turbines (N t ) are given and xed. For the inlineand staggered arrays 6 and 8 devices are considered, respectively.

Its vertical position (z-axis) is also xed, as well as the position of the centreline of the turbines. The arrangement of the inline arrayconsists of two rows with three turbines each. In the case of thestaggered array, there are three rows, with two turbines in the

second row, and three turbines in each of the rst and third rows.Turbines (T 1, T 2, …, T Nt) are numbered from top to bottom and fromleft to right, starting at the top left turbine. For an illustration of theturbine arrangement in each case, see Figs. 3e6.

A constant owvelocity of 2.5 m s1 is prescribed at the inlet forthe regular channel. In the case of the irregular channel, a sinu-soidal ow velocity in [ms1] is given as an input signal at the inletboundary, see Fig. 7.

The outputs of the computer simulation are averaged surfaceintegrals of velocity magnitudes for each porous disk; i.e.,

Z S

U dd AT ¼ U d AT ¼Xni¼1

ud;iaT ;i (1)

where, AT is the cross sectionarea of the turbine rotor, U d represents

the averagedow velocity at the rotor, and ud,i is the ow velocity ati-th surface facet, with area aT,i, that forms the disk surface. By

knowing AT , U d is obtained for each disk. Then, the instantaneousCapacity Factor, CF , of the j-th array layout is calculated using Eqs.(3) and (4), and ow velocities at each turbine rotor are obtainedfrom the CFD simulations; i.e.,

P i ¼ 1

2rC PL AT U

3d;i (2)

Fig. 2. Sampling plan using LHD maximin criterion for n ¼ 11 and s ¼ 2 (blue points)

and 9 extra points (red points). (For interpretation of the references to colour in this

gure legend, the reader is referred to the web version of this article).

Fig. 3. Top view of the regular channel geometry used for uniform ow CFD simulations. In the gure, the turbine congurations are shown: a) inline layout (top) and b) staggered

layout (bottom) with x1 ¼ 20D and x2 ¼ 2D.

Fig. 4. Front view of the regular channel geometry used for uniform ow CFD simulations. In the gure, the turbine congurations are shown for: a) inline layout (top) and b)

staggered layout (bottom) with x1 ¼

20D and x2 ¼

2D.



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CF j ¼

PNt i

P i

PNt

i

P r ;i

(3)

where, P i and P r,i represent the power output and rated power of the i-th turbine of a set of Nt turbines, respectively; r depicts waterdensity, taken to be 998.2 kg m3; nally, C PL depict the power

coef cient expressed in terms of the local velocity U d. Through the1-dimensional linear momentum actuator disc theory in an innitemedium an upper bound for the theoretical maximum powerextractable, the known Betz's limit [33], is obtained, with C P ¼ 16/

27 (C PL ¼ 2). However, it is worth noting that for a turbine in openchannel ows [34], the Betz upper bound may well be exceededdue to constrained ow conditions, but a series of requirements

have to be achieved [35]. In this study, horizontal axis turbine typeswith 20 m rotor diameters are considered to have a rated power of 2.58 MW at a ow velocity of 3 m s1. Detailed information about

the CFD simulations and validation of the porous-jump boundary

condition method may be obtained in Ref. [36].

6. Surrogate construction

After the sampling plans have been computationally imple-

mented, the next step involves selecting an approximating func-tional form, ^ yi ( x ), to use as a surrogate of complex computersimulations, yi ( x ), in this case representing the CF j of each turbine

array layout. The majority of metamodels can be dened with alinear combination of a set of basis functions { B1 ( x ), B 2 ( x ), …, Bn( x )}. Thus, the general form of a metamodel can be expressed as:

b yið x Þ ¼ b1B1ð x Þ þ b2B2ð x Þ þ … þ blBlð x Þ (4)where b ¼ fb1; b2;…; bng are coef cients that need to beestimated.

Therefore each computer experiment is approximated by a sumof basis functions plus an error, ε,

yið x Þ ¼ b yið x Þ þ εi (5)As mentioned before, there are multiple candidates, each withits advantages and limitations. In order to choose the right surro-gate method, Santos [37] describes a series of criteria based on theessence of the problem. In this work, we focus on two specicmethods: polynomials and RBF, which are discussed below.

6.1. Polynomial functions

Polynomial functions are the most popular type of metamodels.

They have been widely used in engineering problems because theyare relatively easy to construct [38]. The main disadvantage of loworder polynomial models is the inadequacy to represent highlynon-linear responses, while high order polynomials are more sus-

ceptible to be unstable [19]. As the number of basis functions

Fig. 5. Top view of the irregular channel geometry used for non-uniform ow CFD simulations. In the gure, the turbine congurations are shown for a) inline layout (top) and b)

staggered layout (bottom) with x1 ¼ 20D and x2 ¼ 2D.

Fig. 6. Front view of the irregular channel geometry used for non-uniformow CFD simulations. In the gure, the turbine congurations are shown for: a) inline layout (top) and b)

staggered layout (bottom) with x1 ¼ 20D and x2 ¼ 2D.

Fig. 7. Flow velocity pro

le at inlet of the irregular channel.



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increases with the number of input variables, s, and the degree of polynomial, low order polynomial models are preferable to repre-

sent linear or slightly non-linear responses because they can pro-vide accurate predictions from small data sets.

In this work, the polynomials used are second order, as in Eq. (7),and third order, as in Eq. (8)

b yð x Þ ¼ bb0 þ Xsi¼1 bbi xi þ Xs

i¼1 bbii x2i þ Xs1

i¼1Xs

j¼iþ1 bbij xi x j; i ¼ 1; 2;

(6)

b yð x Þ ¼ bb0 þ Xsi¼1

bbi xi þ Xsi¼1

bbii x2i þ Xs1i¼1

Xs j¼iþ1

bbij xi x j þ … þXs1i¼1

Xs

j¼iþ1

bb2ij x2i x j þ Xs1i¼1

Xs j¼iþ1

bb2 ji xi x2 j þ Xsi¼1

bb3ii x3i ; i¼ 1; 2;

(7)

Polynomial parameters, bbi, are calculated through the leastsquares procedure [15], which minimises the sum of the squareerrors,

LS ¼ εTε ¼ ðy BbÞTðy BbÞ ¼Xni¼1

8


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For each of the simulations scenarios and layouts, results formetamodel tting are summarised in Fig. 9. Note that values of RMSE are normalised by the range of observed data [ ymin, ymax] andthe values of MAX represent units of Capacity Factor.

Results for each scenario and surrogate are represented graph-ically (see Figs. 10e17) as contour plots to help visualize the shapeof the response surfaces.

8. Surrogate optimisation

Once the surrogate is builtand validated, it can be used to obtainan optimal solution subject to a series of constraints. A general form

to express a constrained maximisation optimisation problem is

Maximise f 0ð x ÞSubject to : g ið x Þ g i; i ¼ 1;…; m

h jð x Þ ji; j ¼ 1;…; k

where the vector x { x1, …, xs} is the set of design variables of the

problem, the function f 0: R n/ R is the objective function, the

functions g i and hi: R n/ R, i ¼ 1, …,m and j ¼ 1, …,k are the

inequality and equality constraint functions, and the constants b1,

…,bm and d1, …,dk are the bounds and values of the constraints. Insurrogate optimisation, the objective function is given by the vali-dated metamodel, i.e., f 0 ¼ ̂y ( x ).

In the turbine array layout optimisation problem for a com-mercial scale project, a series of constraints types arise (e.g.: eco-nomic, environmental, zoning, structural loads, etc.) that need to besatised. As these constraint types depend on the specic nature of

each project, it is not the objective of this work to present a uniqueoptimisation problem, but to formulate and include a series of constraints to serve as a prototype for more complex optimisationproblems. In Li et al. [46], an integrated model for estimating en-

ergy cost is presented, which considers many aspects of a tidal

current turbine farm, where each of them are extensivelydiscussed.

The following prot optimisation model constitutes an examplefor both, inline and staggered, array layouts. The model seeks tomaximise annual prot. Constraints take into account the price of electricity, operational and maintenance costs (O&M), an annual

rental fee for submerged land occupied by the project, turbineclearance regions and maximum area occupied by the turbine farm.Considering as a reference data from different countries around theworld, a xed price of electricity is set at 0.25$/kWh [47]. On the

other hand, O&M costs are proportional to the sum of the linelengths (L) joining all turbines in the array. This procedure assumesO&M cost is proportional to the maintenance vessel travel distance;i.e. the larger the farm is, the higher the O&M cost is. A value of

0.025$/kWh is dened as a lower bound for O&M costs. This valueis approximated, and it is based on typical O&M costs for offshorewind energy farms in Europe [48]. It is assumed that an annualrental fee of $0.025 per square metre of submerged land occupied

by the project, AF [49], is incurred. A detailed nancial examplecomprising CAPital EXpenditure (CAPEX), OPerational EXpenditure(OPEX) and Levelised Cost of Energy (LCOE) costs fora test stage of a

1 MW tidal turbine is available in Ref. [50]. Other economical

studies involving marine renewable energies are available inRefs. [51e53]. The resulting optimisation model is mathematicallyformulated as follows:

MAX ðPrice of electricity O&M costsÞ XNt

i

P i$t

ðcost of leasingÞ (18)

Subject to

XNt

i

P i$t ¼

PNt i

P i

PNt

i

P r ;i

$

XNt

i

P r ;i$t ¼ CF ð x1; x2Þ$

XNt

i

P r ;i$t

¼ b yð x Þ$XNt i

P r ;i$t (19)

t ¼ 8760h (20)

Price of electricity ¼ 0:25$=kWh (21)

O&M cost½$=kWh ¼ ð0:025=560ÞL½m þ 0:0089 (22)

Cost of leasing ½$=year ¼ 0:025$=

m2$year$ AF

hm2

i (23)

Di ¼ 20m ci ¼ 1;…; N T (24)

T 1ð200; y þ x2Þ; T 2ð200; 200Þ; T 3ð200; y x2Þ2i ¼ 1;…; N T (25)

10D x1 15D (26)

20D x1 30D (27)

Inline array:

N t ¼ 6 (28)

Fig. 8. Sampling plan points (blue) and testing points (red). (For interpretation of the

references to colour in this gure legend, the reader is referred to the web version of

this article).



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L½m ¼ x1 þ 4 x2 (29)

AF ¼ ð2$5D þ x1Þ$ð2$4D þ 4 x2Þ 0:256km2 (30)

T 4ð200 þ x1; y þ x2Þ; T 5ð200 þ x1; 200Þ; T 4ð200 þ x1; y x2Þ2i

¼ 1;…; N T

(31)

Staggered array:

N t ¼ 8 (32)

L½m ¼ 2

x21 þ x22

0:5þ 5 x2 (33)

AF ¼ ð2$5D þ 2 x1Þ$ð2$4D þ 5 x2Þ 0:448km2 (34)

Eq. (18) denes the objective function to be maximised, whichrepresents the annual revenue. Eq. (19) expresses how the annualenergy output is calculated by combining the desired surrogate ^ y

Fig. 9. Normalised root mean square error (top) and absolute maximum error (bottom) obtained with the surrogates for each array layout out and scenario.

Fig. 10. Quadratic polynomial capacity factor surfaces responses for arrays in Channel-1, inline (left) and staggered (right) layouts.



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( x ) with Eq. (3). Constraints (20e27) are common for both types of array layouts, while constraints (28e31) are specic for the inline

array and constraints (32e37) are particular for the staggeredlayout. Constraint (20) denes the time in hours for which theprot is calculated. Constraints (21e23) set the price of electricity,

O&M costs and project area leasing, respectively. Constraint (24)species the rotor diameter, D, of the i-th turbine in metres [m].

Local coordinates (x, y) dene turbine position, (T i (x, y) ci ¼ 1, …,

N T ). Constraint (25) denes the position of the rst row of turbines(T 1, T 2 and T 3), while constraints (31) and (35) denes the positionfor the rest turbines in each array. Constraints (26) and (27) deneestablished regions of turbine clearance; for example, these regions

may be associated with waterways. Constraints (28) and (32)

establish the number of turbines for each array type. Constraints(29) and (33) express the sum of the line lengths (L) joining all

turbines in the array. Finally, constraints (30) and (34) represent themaximum area AF that an array may have.

The optimisation model is solved through the enumeration

method [54]. This method consists in evaluating all possible com-binations of the design parameters space, then the solution that

satises all conditions is selected. For this purpose, the design pa-

rameters are discretised in increments of 0.25D for x1 and of 0.025Dfor x 2. The results obtained for the above optimisation model foreach modelling scenario and surrogate are summarised in Table 1.

In general, the results of x1 and x 2 obtained for each of the

surrogates implemented in common scenarios are very similar. As

Fig. 11. Cubic polynomial capacity factor surfaces responses for arrays in Channel-1, inline (left) and staggered (right) layouts.

Fig. 12. Linear RBF capacity factor surfaces responses for arrays in Channel-1, inline (left) and staggered (right) layouts.

T 4ð200 þ x1; 200 þ 0:5 x2Þ; T 5ð200 þ x1; 200 0:5 x2Þ;T 6ð200 þ 2 x1; 200 þ x2Þ; T 7ð200 þ 2 x1; 200Þ; T 8ð200 þ 2 x1; 200 x2Þ2i ¼ 1;…; N T

(35)



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expected, for the uniform ow scenario, optimum conguration forinline array layouts demand larger longitudinal distances betweenrows, while for staggered arrays this distance is shorter. Differencesin longitudinal distance between inline and staggered arrays are of

the order of 20D, while differences of lateral distance between bothcongurations are up to 0.6D, this being shorter for the inline lay-outs when compared with the staggered congurations. On theother hand, arrays under non-uniform ows present the same

pattern for longitudinal distance with respect to the previouslymentioned ow condition but with orders of magnitude lower(~10D). Lateral distance between arrays under non-uniform oware practically the same with values of 4D.

Regarding ef ciency, staggered arrays have a higher powerdensity than inline arrays when they are under uniform ows, asthe former produce more power in smaller regions than the latter.The main reason for this is that downstream turbines benet from

the stream accelerated through the gaps between upstreamturbines.

While the results obtained for arrays under uniform ow can begeneralized for at channels non-constrained by side walls, the

optimum array congurations obtained for the channel under non-uniform ow are specic for this particular case.

9. Discussion and recommendations

Through this paper, we have presented a different approach for

the turbine array layout optimisation problem. Two variations of Polynomial and RBF surrogate methods are employed to t anobjective function dependent on two variables: longitudinal andlateral spacing. The methodology is applied to two different sce-

narios, considering uniform and non-uniform ow, with theobjective to assess the capacities of each surrogate.

As a consequence of having a greater number of turbine rows,

results from the surrogate

tting validation evidence a greatersensitivity of the response results for staggered layouts than forinline ones. Similarpattern occurs with the cases with non-uniform

ow, as the irregular channel scenario increases non-linearity inthe objective function response.

For both ow conditions and layouts, Radial Basis Functionsdenote an overall better performance than polynomial models in

Fig. 13. Thin Plate Spline RBF capacity factor surfaces responses for arrays in Channel-1, inline (left) and staggered (right) layouts.

Fig. 14. Quadratic polynomial capacity factor surfaces responses for arrays in Channel-2, inline (left) and staggered (right) layouts.



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terms of NRMSE. In the regular channel case scenario, the quadraticpolynomial provides better tting and stability than the cubicpolynomial. When nonlinearities increase, RBF achieve superior

results of NRMSE, even though MAX error are high, as for thescenario of staggered layout in an irregular channel. In order tominimise MAX error results, adaptive sampling designs aremethods to consider. In conclusion, if the problem to be modelled is

highly nonlinear and involves an important number of variables,the use of RBF is preferable due to their exibility despite beingmore computationally demanding. On the other hand, polynomialmodels may be a choice for problems with fewer variables and

smoother behaviour.Through optimisation application, an example of how to explore

the surrogate taking into account a series of constraints consideringgeometric and costs limitations has been shown. Results for the

various surrogates present a good agreement among them.Therefore, small differences in CF can lead to signicant protvariations at the end of a year.

Advanced computational modelling techniques of hydrokinetic

energy turbines array may be adopted to increase the approxima-tion with real conditions. The geometry and rotation of the turbinesunder unsteady open channel ow conditions can be modelled. As

the complexity of computational modelling increases, simulationsbecome more time demanding. It is for these cases that themethodology presented in this paper becomes even moreappealing, as it provides a surrogate of the underlying model,

which is easier to work with for design space exploration andsensitivity analysis. The main dif culties of the method are toformulate the problem in terms of a few number of design variablesand to obtain a reliable surrogate using as less as possible computer

simulations.Future research is directed to implement the SBO approach for

the hydrokinetic array layout problem considering continuous anddiscrete design parameters, like space coordinates for turbine

positioning and rotor turbine diameter size, as well as to develop amore sophisticated optimisation model, including the three-dimensional free positioning of turbines and considering environ-mental constraints as the inuences in sediment dynamics.

Fig. 15. Cubic polynomial capacity factor surfaces responses for arrays in Channel-2, inline (left) and staggered (right) layouts.

Fig. 16. Linear RBF capacity factor surfaces responses for arrays in Channel-2, inline (left) and staggered (right) layouts.



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Acknowledgements

The corresponding author wishes to acknowledge the post-

doctoral fellowship provided by the following funding agency fromBrazil: Conselho Nacional de Desenvolvimento Cientíco e Tec-nologico (CNPq) (Grant number: 500177/2014-7).

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Fig. 17. Thin Plate Spline RBF capacity factor surfaces responses for arrays in Channel-2, inline (left) and staggered (right) layouts.

Table 1

Results from the optimisation model.

Flow scenario Array layout Surrogate x1 [D] x2 [D] CF [%] Prot [M$/year]

Uniform Inline Quadratic poly. 30.00 2.000 46.50 13,043

Uniform Inline Cubic poly. 30.00 2.275 47.38 13,225Uniform Inline RBF-Linear 30.00 2.075 46.98 13,161

Uniform Inline RBF-TPS 29.25 2.125 47.82 13,425

Uniform Staggered Quadratic poly. 10.00 2.500 56.22 21,466

Uniform Staggered Cubic poly. 10.00 2.500 56.41 21,538

Uniform Staggered RBF-Linear 10.00 2.600 56.51 21,527

Uniform Staggered RBF-TPS 11.25 2.225 58.57 22,276

Non-uniform Inline Quadratic poly. 21.50 3.600 64.52 18,258

Non-uniform Inline Cubic poly. 22.75 4.000 69.83 19,521

Non-uniform Inline RBF-Linear 21.75 3.725 69.83 19,700

Non-uniform Inline RBF-TPS 22.50 3.950 71.15 19,930

Non-uniform Staggered Quadratic poly. 10.00 4.000 59.88 22,052Non-uniform Staggered Cubic poly. 10.00 4.000 62.49 23,011

Non-uniform Staggered RBF-Linear 12.50 3.925 61.75 22,311Non-uniform Staggered RBF-TPS 11.00 3.800 62.51 22,944


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