A semi-analytic method to optimize tidal farm layouts – Application to the Alderney Race (Raz Blanchard), France

Applied Energy 183 (2016) 1168–1180

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A semi-analytic method to optimize tidal farm layouts – Application to the Alderney Race (Raz Blanchard), France Ottavio A. Lo Brutto ⇑, Jérôme Thiébot, Sylvain S. Guillou, Hamid Gualous Normandy University, UNICAEN, LUSAC, EA 4253, 60 rue Max Pol Fouchet, CS 20082, 50130 Cherbourg-Octeville, France

h i g h l i g h t s
We present a new semi-analytic method designed to optimize the tidal farms’ layout.
The method is applied to the Alderney Race.
The tidal directional spreading strongly affects the power and the optimal layout.

a r t i c l e

i n f o

Article history: Received 8 June 2016 Received in revised form 2 September 2016 Accepted 24 September 2016

Keywords: Tidal farm Layout optimization Wake Ambient turbulence

a b s t r a c t The purpose of this paper is to present a semi-analytic model designed to optimize tidal farm layouts by maximizing the mechanical power production. A meta-heuristics method is used to find the turbine placement which minimizes the flow interaction between the turbines. The velocities in the wakes of turbines are simulated with an analytic model. The methodology is first applied to idealized cases: constant current magnitude and direction, and flow aligned with the turbines. Those preliminary tests permit to test the consistency of the results. In particular, they show that the optimal density of the devices grows with increasing turbulent intensities or increasing upstream velocity magnitude. The methodology is then applied to a site located in the Alderney Race (Raz Blanchard in French), situated between the Alderney Island and La Hague Cape (France). The results show that the optimal placement is influenced by the asymmetry of the tidal current and that the best layout is strongly dependent on the directional spreading of the current with respect to the predominant direction. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Marine Renewable Energies (MREs) are recognized as a resource to be harnessed for the production of electric power because it is an alternative to fuel resources. The oceans represent two-thirds of the Earth and they constitute a huge energy resource [1,2]. There are four main types of MREs: thermal energy, chemical energy, biological energy and kinetic energy (waves and currents). Among them, tidal current energy seems to be the most attractive because it is highly predictable. The European objective is to produce 20% of the energy from renewable sources by 2020. Tidal energy provides an opportunity to increase the energy mix in France. France has the second production potential of electrical energy from MREs, behind the UK. French production potential is between 5 and 14 TW h/year. The Alderney Race (Raz Blanchard in French), ⇑ Corresponding author. E-mail addresses: [email protected] (O.A. Lo Brutto), jerome.thiebot@ unicaen.fr (J. Thiébot), [email protected] (S.S. Guillou), hamid.gualous@ unicaen.fr (H. Gualous). http://dx.doi.org/10.1016/j.apenergy.2016.09.059 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.

situated between the Alderney Island and La Hague Cape (France), capitalizes about 50% of the national resource [3,4]. Maximizing the power extracted from a given tidal energy site requires finding the optimal density of devices and the most suitable type of layout (aligned, staggered or mixed configuration). In their synthesis and review paper, Vennell et al. [5] distinguish two main issues. The first issue, referred as macro-design of array, consists in determining the total number of turbines finding a compromise between the array power output and the power output of each machine. The second issue concerns the micro-design of arrays. It consists in determining the best array layout, the interrow spacing and the number of devices per row taking into account the flow interactions between the turbines (the wake/duct effects). As observations in the real environment with tidal stream array are not available today, micro-design studies rely on either flume experiments with scaled turbines or numerical simulations. Using scaled experiments with velocity conditions representing the hydrodynamics of the Alderney Race [6,7], Myers and Bahaj [4] suggested an inter-row (longitudinal) spacing of 15 diameters

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and a lateral spacing of 3 diameters. Mycek et al. [8,9] also analyzed the wake effects between turbines placed in array using flume experiments. They highlighted the effect of the ambient turbulence on the wake characteristics and demonstrated that high turbulent intensities fasten the flow recovery and permit to reduce the longitudinal distance between two rows of machines. Flume experiments permits to understand the processes governing the flow interactions between turbines but only a limited numbers of idealized configurations can be tested and compared. Furthermore, those experiments have several limitations especially regarding the similitude of the Reynolds number. Computational Fluid Dynamics (CFD) is thus a complementary tool [10–14]. In CFD models of tidal farms, the turbines are represented with Blade Element Momentum (BEM) theory, Actuator Disk (AD) [15–20], Immersed Body Force Turbine [21] or frozen rotor method [22]. Because of their high CPU cost, only a limited number of turbine arrangements can be tested and compared. As only few array arrangements can be simulated, the turbine’s placement cannot be optimized using an iterative procedure evaluating the performance of the array configuration at each iteration. The only exceptions are the studies of Funke et al. [23] and González-Gorbena et al. [24]. Funke et al. [23] succeed in optimizing the placement of turbines using a CFD model combining a bi-dimensional flow model with a gradient optimization algorithm. They demonstrated that the power production obtained with an aligned layout can be increased by 33% when using their optimization algorithm. González-Gorbena et al. [24] optimized uniform turbine array layouts subjected to different flow conditions using the SurrogateBased Optimisation (SBO). Although the layout configuration is fixed (reducing the degree of freedom of the system), this method presents the main advantage of optimizing continuous and discrete design parameters simultaneously. The CFD studies dedicated to the design of tidal farm (e.g. [18,21–23]) are commonly carried out considering stationary hydrodynamics conditions or regular layouts. This simplification permits to gain insight in understanding the flow interactions between turbines. However, it is not suited to analyze the effect of realistic tidal currents (large range of tidal current magnitude (spring/neap tide conditions), and flow arriving obliquely with respect to the predominant direction. . .) on the micro-design. Noteworthy, Nguyen et al. [25] analyzed the influence of the current incidence with respect to the mean flow direction on the production of a tidal farm containing 10 turbines. They examined two scenarios based on a hypothetical tidal farm located in the Alderney Race. For the first scenario, realistic tidal current data are used (the velocity constantly vary in magnitude and direction); for the second scenario, the incidence of the current with respect to the turbine axis has been ‘‘switched off” (i.e. the flow remains parallel to the longitudinal axis of the turbines). The results showed that the temporal variation of the current direction has a very limited effect on the overall production of the tidal farm but that it strongly influences the power production of each device, especially when several devices are aligned. Nowadays, only few studies use realistic tidal hydrodynamics as input of model dedicated to optimize the array arrangement. Additional investigations are therefore required to understand the influence of realistic hydrodynamic on the flow interactions between turbines and on tidal farm layout. In the last few years, some authors investigated the possibility of using analytical models to assess the performances of tidal farm [26–28]. As far as we know, only Stansby and Stallard [29] used analytical methods to optimize the turbine arrangements of little tidal farms (two or three rows of devices). In their model, they used analytical equations validated with the experimental data of Stallard et al. [30]. This method successfully simulated the characteristics of the wakes measured experimentally. Studies of layout

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optimization are more advanced for wind turbines than for tidal turbines. They generally rely on the coupling between an optimization algorithm for finding the best turbine’s positions and a simple model for representing the wakes and their interactions [31]. The most popular optimization algorithms in wind farm literature are Genetic Algorithm (GA) and Particle Smarm Optimization (PSO) algorithms [32–37]. The PSO algorithm is generally preferred to GA because it converges faster and gives better results as demonstrated by Pookpunt and Ongsakul [38]. In wind turbines applications, the wake of wind turbines is generally simulated with an analytical model deriving from the earlier works of Jensen [39]. Analytical wake models are suited for optimization problem because they permit testing a large number of configurations which is required when searching the best layout with an iterative procedure. With the perspective to adapt the wind turbine methodology to tidal turbine applications, Lo Brutto et al. [28] developed a wake model based on the theory of Jensen [39]. This wake model has been validated using a dataset obtained with a coupled CFD-Actuator Disk method. In the present paper, we integrate this wake model in an optimization algorithm in order to optimize the placement of tidal turbines. The main objective of this paper is to present and validate a new optimization method for the tidal turbine positioning. The method is applied to several configurations with an increasing level of complexity permitting to give general guidelines as regards the placement of devices under different types of flows. In particular, we analyze the effects of the velocity magnitude, the turbulent intensity, the bi-directionality of the flow and the tidal current asymmetry. Finally, we apply the methodology using realistic tidal currents. The Section 1 is dedicated to the description of the tidal farm layout optimization problem. The Section 2 presents the characteristics of the hydrodynamics in the Alderney Race. The results of the optimization method are detailed in Section 3. 2. Solving the layout optimization problem We now present the layout optimization model starting with the parameters describing the tidal stream devices. When the incoming flow velocity U0 is lower than the cut-in speed vci, the torque exerted by the fluid on the turbine blades is insufficient to move the blades and the mechanical power P is nil. When the incoming fluid speed U0 exceeds the cut-in speed, the turbine starts to generate a mechanical power given by:

1 P ¼ qC P 2

pD2 4

!

U 30

ð1Þ

where q is the density of the fluid, Cp is the power coefficient and D is the turbine diameter. When the tidal speed exceeds the nominal speed vr, the power is rated to the nominal power Pr [40,41]. Following the reference work of Jensen [39], the spatial distribution of the velocity in the wake of a turbine (2) can be obtained from a balance of momentum [42,43]:

pr20 U w0 þ pðr2  r20 ÞU 0 ¼ pr2 U w

ð2Þ

where Uw the velocity in the wake of the turbine at a downstream distance x (along the longitudinal axis), r is the wake expansion which is assumed to be linear in x and Uw0 is the minimum speed according to Betz limit:

U w0 ¼ U 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  CT

ð3Þ

In Lo Brutto et al. [28], the wind turbine model of Jensen was adapted to represent the flow in the wake of a marine turbine. Hydrodynamic data obtained from CFD simulations were used to develop the wake model [25,44]. The analysis of the CFD data indi-

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cate that the wake expansion of marine turbine is exponential in x and that it is strongly influenced by the ambient turbulence I0 (as already demonstrated by laboratory experiments or CFD simulations [8,9,45]). A law for the wake expansion has thus been proposed (4). This formulation is retained here.

In the marine environment, the current varies constantly in magnitude and direction. Decomposing the total output of a tidal farm into several current directions and magnitudes, the power produced by Nt turbines is:

   r x r ¼ 0 cðI0 Þ 5:58 1  e0:051D þ 1:2 2:59

Pfarm ¼

ð4Þ

where x is the distance downstream the turbine and c(I0) = 15.542 I20 + 21.361 I0 + 0.2184. Following [34,38,46], the velocity deficit at the tidal turbine j is given by (5) when it is dived in the wakes of turbines located upstream.

 2 X  2 Nt uj Aov er;ij uij 1 ¼ 1 U0 AT U0 j¼1

for i 2 WðjÞ

ð5Þ

where W(j) indicates the set of turbines whose wakes affect the turbine j, uij is the resulting velocity due to wake effect caused by turbine i on turbine j, Aover,ij is the overlapping area of the wake of turbine i on turbine j (Fig. 1), AT is the rotor surface and Nt is the number of tidal turbines in the tidal farm. According to Eq. (2), we obtain:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi    1  1  CT uij ¼ 1   U0 rij =r0 2

ð6Þ

where rij is the wake expansion depending on the distance xij between turbines i and j. Turbine j is in the wake of turbine i (i.e. i 2 W(j)) if and only if:

xij < 0 & jyij j  r 0 < rij

ð7Þ

where yij is the lateral distance between turbines i and j. Finally, the partial wake area Aover,ij is calculated with: 8



sinð2hw;ij Þ sinð2h Þ 2 > þ r 20 h0;ij  2 0;ij ; 8jyij j 2 ½rij  r 0 ; rij þ r0  > < Aov er;ij ¼ rij hw;ij  2 Aov er;ij ¼ 0 8jyij j P rij þ r 0 > > : Aov er;ij ¼ A0 8jyij j P rij  r 0

d¼1 v ¼1

where

ð9Þ

ij

ð10Þ

j¼1

where D and V are the numbers of current directions and magnitudes used to discretize the hydrodynamic forcing, fd,v is the probability of the current occurring conjointly in the vth bin of current magnitude and the dth bin of current direction. Pd,v,j and ud,v,j are the power and the current magnitude at the j-th turbine for current conditions being in the vth bin of current magnitude and the dth bin of current direction. This model does not simulate the global reduction of the flow field induced by the array. This global flow perturbation is referred as ‘‘array-effect”. Vennell et al. [5] demonstrated that this effect is significant when the array is large that is to say when the area swept by the blades of the turbines exceeds 2–5% of the channel’s cross-sectional area. Developers intend to optimize the layout of farms in order to maximize the profit. Several wind farm applications such as Aytun Ozturk and Norman [47] thus use the total profit as a cost function. The estimate of the profit is however highly dependent on the market. Furthermore, as tidal industry is at a pre-commercial stage, there are many uncertainties as regards the cost of cabling, maintenance operations, construction, permitting and license. . . Some authors overcome those difficulties using assumptions on the cost of maintenance operations or logistic, the discount rates, the fees applied by the governments [24,48–50]. Instead of using such assumptions, we opted for the cost function (11). This function has been proposed by Mosetti et al. [51] for wind farms applications but it is independent of the technology (wind or tidal energy) as it only represents the economy of scale in large arrays.

cost ¼ Nt ð8Þ

8  2 2 2   rij þyij r0 > 1 > > ; 8jyij j 2 rij  r 0 ; rij þ r 0 < hw;ij ¼ cos 2jyij jrij  2 2 2   > rij yij r0 > > : h0;ij ¼ cos1 2jy jr0 ; 8jyij j 2 rij  r 0 ; rij þ r0

Nt D X V X X f d;v P d;v ;j ðud;v ;j Þ

  2 1 0:00174N2t þ e 3 3

ð11Þ

Eq. (11) gives a non-dimensional cost of the farm and only depends on the number of devices, as underlined in [52–55]. It represents the ratio between the total cost and the cost of a single machine (without economy of scale). The function to optimize (or fitness) is therefore expressed as the minimization of the cost per unit power produced:

  cost Obj ¼ min Pfarm

ð12Þ

A tidal array optimization is thus governed by three interconnected parameters: the number of machines, the installation cost and the production. The key issue is to find a good compromise between reducing the wakes’ interaction (which is easier when the number of machines is low), having enough devices to obtain a sufficient overall production and lowering the installation costs. The main constraint of the problem concerns the position of the turbines. Considering that the turbines should be placed in a rectangular domain measuring Xfarm and Yfarm in the x and y directions, the first constraint is:

0 6 xi 6 X farm 0 6 yi 6 Y farm

ð13Þ

where (xi; yi) represent the positions of the turbine. Assuming that turbines should respect a minimum clearance dmin, the second constraint is:

dij ¼ Fig. 1. Overlapping area between the wake from the upstream turbine and a downstream turbine (after [38]).

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2ij þ y2ij 6 dmin

where dij is the distance between two turbines i and j.

ð14Þ

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Fig. 2. Spatial distribution of the averaged power density in the Alderney race. The iso-lines represent the depth with respect to the mean sea level. The center of the tidal farm is represented by the circle.

Fig. 3. Frequency of occurrence of the current direction and magnitude. The directions are given with respect to the tidal farm frame (a rotation of 20° was applied to the regional model data).

Power curve 600

Pmec (kW)

500 400 300 200 100 0

0

1

2

3

4

Tidal speed (m/s) Fig. 4. Power curve of the tidal turbines used in the algorithm.

In this study, the clearance is set to 5D, which allows a minimum recovery behind an upstream turbine [5]. The representation of the turbine coordinates is one of the main issue in the farm layout optimization. There are two possibilities: using real coordinates or using a grid representation with discrete coordinates [32]. The first approach (real coordinates) has the advantage of providing numerous degrees of freedom to the turbines’ position. However, as stated by Rodrigues et al. [56], it is difficult to optimize simultaneously the number of turbines and their positions when using real coordinates. The reason is that the convergence and the optimization performance are affected when using large variable-domain size of continuous variables. To overcome this limitation, the number of turbines is generally imposed and only the turbine positions are optimized.

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The discrete representation approach consists in discretizing the farm using a grid with fixed cell size. This technique was first proposed by Mosetti et al. [51], who discretized the domain occupied by the tidal farm with cells having lateral and longitudinal dimensions corresponding to the minimum clearance dmin. With this representation, each cell represents a possible location for a turbine (when the cell is occupied by a turbine, the turbine is located in the middle of the cell). The advantage of this method is that the

constraints on the turbine’s locations (minimum clearance, turbine located in the domain) are automatically respected thanks to the encoding in discrete variables. This fastens the convergence and allows to optimize the number of turbines which is of great importance during the Front End Engineering Design (FEED) phase of a project [57]. Furthermore, as the computational cost is reduced (in comparison to the real coordinates technique), it is possible to use complex (more realistic) hydrodynamic data as input.

Fig. 5. Best layout configurations (left) and velocity contour of the best layout (right) for case (a) at U0 = 2.5 m/s and different values of ambient turbulence I0.

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Weighting the advantages and the drawbacks of the two methods we opt for the discrete representation in this work. 3. The hydrodynamics of the Alderney Race We now present the current data we use as input of the optimization model as well as the site of application. The input data of the analytic model are provided by a regional hydrodynamic model covering the English Channel. The latter was

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primarily designed to assess the tidal resource and to analyze the effect of tidal turbines on the physical conditions of the Alderney Race [58]. It uses the finite element method to solve the shallow water equations [59]. The regional model results have been validated using 4 time-series (90 h of measurements) of depth-averaged current velocities measured with Acoustic Doppler Current Profiler as well as low tide and high tides measured with 3 tidal gauges [60]. The model performance is such that the mean phase lag between measurements and model results is

Fig. 6. Best layout configurations (left) and velocity contour of the best layout (right) for case (a) at U0 = 3 m/s and different values of ambient turbulence I0.

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smaller than 16 min. The mean errors as regards the tidal amplitude and the current magnitude are smaller than 0.47 m (representing 9% for mean tidal amplitude of 5.19 m) and 0.02 m/s respectively [58]. The Alderney Race is characterized by a high tidal resource combined with suitable depth for tidal turbine deployment. Fig. 2 represents the Averaged Power Density APD (0.5qU3) which is a good estimate of the tidal resource. For the present purpose, we choose a hypothetical site for applying the array layout optimization methodology. The center of the tidal farm was chosen such that the depth (as regards the mean sea level) is 50 m and the tidal

resource is high. The site is located at (2.019°W; 49.72°N) (Fig. 2). The averaged power density APD in the middle of the tidal farm is 4.03 kW/m2. It is noticeable that there are no significant changes in hydrodynamics conditions within the tidal farm because the size of the latter is small (50D  50D) with regards to the spatial variations of the current field. We thus assume that the velocities in the free stream of the tidal farm (at the boundaries of the analytical model) are equal to that calculated by the regional model in the middle of the tidal farm. Obviously, this simplification should be revised when considering a very large tidal farm.

Fig. 7. Best layout configurations (left) and velocity contour of the best layout (right) for case (a) at U0 = 3.5 m/s and different values of ambient turbulence I0.

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O.A. Lo Brutto et al. / Applied Energy 183 (2016) 1168–1180 Table 1 Results for case (a). U0 (m/s)

I0 (%) 5

10

15

2.5

Nt Pfarm Dxaver /D Dyaver /D

20 6.533 45.00 5.00

40 12.712 15.00 5.00

42 14.159 14.06 7.97

3.0

Nt Pfarm Dxaver /D Dyaver /D

20 11.289 45.00 5.00

40 21.977 15.00 5.00

42 22.925 14.06 7.16

3.5

Nt Pfarm Dxaver /D Dyaver /D

30 18.85 22.5 5

55 34.46 9.89 7.56

100 62.83 5 5

Prior to integrate the current data of the regional model in the optimization model, the regional data were processed. Firstly, we defined the orientation of the tidal farm frame with respect to the North and synthetized the data into a D  V matrix containing the probability of occurrence fd,v of a given range of velocity and direction. As regards the orientation of the tidal farm frame, we search the current direction which gives the maximal power density. It is obtained by weighting the current direction by the cubed velocity. At the chosen site, the optimal orientation with respect to the North is 20°. The currents are maximal and oriented towards the North-Northeast around high tide and they flow towards the South-Southwest around low tide. Fig. 3 illustrates the frequency of occurrence of the current as a function of the current direction and magnitude. These histogram shows that the greatest and the most frequent current velocities correspond to a narrow range of directions (around 0° and 180° in the local frame).

4. Description of the tested configurations All computations have been carried out using the same turbine parameters. We use 10 m-diameter turbines placed 25 m above the bottom. We assume that the rotors are always oriented perpendicularly to the tidal direction [34,61]. This assumption corresponds to the cases of turbines with a yaw mechanism or a tethering system allowing the device to face the flow constantly [62]. According to the local range of velocity values, the cut-in velocity is set to 1 m/s and the rated velocity is 3 m/s (Fig. 3). There is no cut-out speed and the power coefficient CP is set equal to its maximum value, i.e. 0.59 according to the Betz theory. The power curve is shown in Fig. 4. The area occupied by the tidal farm is

500 m  500 m. The farm is placed perpendicularly to the prevailing tidal current direction [63]. The tidal farm has been discretized using 100 cells. As the clearance dmin is 5D (=50 m), each cell measures 5D  5D and the tidal farm dimension is 50D  50D. The maximum number of turbines is 100 corresponding to a maximum area swept by the blades of the turbines of 7850 m2. The Alderney Race’s cross sectional is 5.2  105 m2 at the study site. The ratio between the two areas is 1.51%. Thus, according to Vennell et al. [5], the array can be considered as small which implies that the global flow field reduction can be neglected. The PSO algorithm of Pookpunt and Ongsakul [38] is used to optimize the tidal farm layout. The algorithm and its setup are described in Appendix A. In wind farm applications, three reference cases are used to check the correct functioning of the optimization algorithms: (i) constant wind (constant in magnitude and direction), (ii) constant magnitude and variable direction and (iii) variable magnitude and direction (obtained from wind rose using a Weibull distribution [51]. Similar test cases are also retained here. The first case (case (a)) corresponds to an upstream velocity constant in magnitude and direction (current parallel to the x-axis). The aim of this test is to assess the effect of the velocity magnitude (three velocities have been tested: 2.5 m/s, 3 m/s and 3.5 m/s) and of the ambient turbulent intensity on the density of machines (three ambient turbulent intensities have been tested: 5, 10 and 15%). The second test (case (b)) consists in imposing a sinusoidal velocity magnitude representing the temporal variation of a tidal flow. The effect of the flow bi-directionality is analyzed comparing a case where the flow is unidirectional (case (b1)) to a case where the current direction reverses twice per tide (case (b2)). For the third case (case (c)), we use the temporal evolution of the velocity magnitude calculated by the regional model and we set the current direction parallel to the predominant direction (no directional spreading, the flow reverses twice per tide but remains parallel to the x-axis). The aim of case (c) is to assess the influence of the tidal current asymmetry. Finally, the case (d) consists in imposing the hydrodynamics conditions calculated by the regional model (flow varying in magnitude and direction). The results are analyzed using the averaged axial distance Dxaver between two consecutive turbines having the same y-coordinate, the averaged radial distance Dyaver between two consecutive turbines having the same x-coordinate, and the performance rate R which is the ratio between the production and the resource:

R¼ Fig. 8. Tidal velocity profiles used in case (b).

Pfarm APD  Afarm

where Afarm is the surface of the tidal farm.

ð15Þ

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5. Results 5.1. Case (a): Effect of the velocity magnitude and of the turbulent intensity The objective of the first case is to analyze the effect of both the current magnitude and the ambient turbulence on the density of devices. The layouts and the spatial distribution of the speed magnitude are shown in Figs. 5–7. The indicators are given in Table 1. Whatever the current magnitude, most turbines face with the free stream and are placed along lines perpendicular to the flow direction. The averaged lateral distance Dyaver between turbines in the same row is close to the clearance distance (the averaged value of Dyaver/D is 5.85, see Table 1) which indicates that the algorithm tends to place an elevated number of turbines in the same rows. Figs. 5 and 6 show that the ambient turbulence strongly affect the machines density, especially when the tidal speed is lower than the rated speed of the turbine. When the ambient turbulence is low (5%), the longitudinal distance Dxaver/D is 45 whereas it is only 15 and 14.06 for turbulent intensity values of 10 and 15%, respectively. This difference is due to the fact that great turbulent intensities fasten the flow recovery permitting to reduce the longitudinal distance between two turbines. Noteworthy, when the incoming flow magnitude is greater than the nominal speed (Fig. 7), a great density of machines can be deployed as the production remains high even for turbines dived in the wakes of the machines located upstream. The velocity contours obtained with ambient turbulence rate of 5% (top of Figs 5–7) suggest that using a staggered configuration would permit slightly increasing the power. This underlines one limitation of the method: the accuracy of the turbines’ position is bounded by the cell size. Here, we use a cell size of 5D. It is thus impossible to move the turbines a 2.5D away from the center of the cell (which would be required to obtain a staggered layout). 5.2. Case (b): Effect of the flow bi-directionality The objective of case (b) is to show the effect of the flow bidirectionality on the tidal farm layout. To this end, we compare a simulation with a unidirectional flow (the flow comes from the eastern side of the domain) to a simulation with a bi-directional flow (the flow comes alternatively from the eastern and the western sides of the domain). We use a sinusoidal current magnitude

Table 2 Results for case (b). Scenario (b)

Nt

Pfarm (MW)

Fitness

Dxaver (D)

Dyaver (D)

R (%)

Unidirectional tide Bidirectional tide (Xfarm = 50D) Bidirectional tide (Xfarm = 55D)

50 50

18.99 18.98

1.7671  106 1.7671  106

11.25 11.25

5.00 5.41

0.59 0.59

52

19.76

1.7621  106

11.90

6.00

0.61

with an amplitude of 3.5 m/s (Fig. 8). The turbulence intensity I0 is set to 15% in agreement with available measurements at comparable tidal sites [64,65]. The results obtained with the unidirectional flow are shown in Fig. 9 and Table 2. Similarly to the case (a) (constant current magnitude), the machines are placed in lines perpendicular to the flow. The results obtained with the bi-directional flow are shown in Fig. 10a and Table 2. As expected, the output and the rate R are similar to the unidirectional case (18.98 MW and 0.59 respectively). The only difference between the unidirectional and the bi-directional flow concerns the position of the machine of the central row. Whereas all the machines are placed at x/D = 27.5 when the flow is unidirectional, the machines are placed on both sides of the central line when the flow is bidirectional (5 machines at x/D = 22.5 and 5 machines at x/D = 27.5). An additional simulation with a different longitudinal discretization show that the results are perfectly symmetric when the number cells is impair (Fig. 10b). The symmetry of the results suggests that the algorithm correctly operates when the flow comes from several directions. 5.3. Case (c): Effect of the tidal current asymmetry The third test aims at analyzing the effect of the tidal flow asymmetry on the layout of the tidal farm. The input data were extracted from the regional model (Fig. 2). The current directions at flood and ebb were assigned to the value 0° and 180° in the local frame (i.e. the current remains parallel to the x-axis). The Fig. 11 represents the histogram of current magnitude at flood and ebb. The current asymmetry is clearly visible. The flood and ebb APD are 4.09 kW/m2 and 3.96 kW/m2 respectively. The results are shown in Fig. 12 and in Table 3. Once again, the layout is such that the tidal turbines are positioned along lines perpendicular to the main flow direction. The averaged radial distance Dyaver/D between two consecutive turbines is 7.16 which is close to the clearance distance dmin/D (5). The averaged axial distance Dxaver/D between two consecutive turbines is 14.06 which is in line with the longitudinal distance suggested by Myers and Bahaj [4] who determine the optimal longitudinal distance (15D) in the Alderney Race from the results of scaled flume experiments. The longitudinal and lateral spacing between the turbines are similar to those obtained with a constant flow magnitude of 2.5 m/s and 3 m/s (case (a)). It suggests that, although the highest velocities have a low frequency of occurrence (velocity magnitude exceeds 2.5 m/s only 20% of the time here), they are the most influential on the characteristics of layout. 5.4. Case (d): Layout optimization with realistic tidal flows

Fig. 9. Best layout configuration for case (b) with a unidirectional tide (U0 = 3 m/s).

We now apply the optimization algorithm using realistic tidal flows. The layout obtained with hydrodynamics extracted from the regional model (Fig. 3) is represented in Fig. 13. In comparison to the results of the previous case (case (c)) where the directional spreading has been ignored, Fig. 13 shows that the variation of the current around the predominant direction strongly affect the layout. The turbines layout of Fig. 13a is more scattered than the

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Fig. 10. Best layout configuration for case (b) with a bidirectional tide: (a) Xfarm = 50D; (b) Xfarm = 55D.

Fig. 11. Histograms of current magnitude at flood and ebb. The data are extracted from the regional model (Fig. 2).

Fig. 12. Best layout configuration for scenario (c).

1177

1178

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Table 3 Results for scenario (c). Nt Scenario (c)

42

Pfarm (MW) 7.00

Fitness 4.09510

6

Dxaver (D)

Dyaver (D)

R (%)

14.06

7.16

0.69

Fig. 13. Best layout configurations for scenario (d).

Table 4 Results for scenario (d).

Scenario (d)

Nt

Pfarm (MW)

Fitness

Dxaver (D)

Dyaver (D)

R (%)

49

8.744

3.765106

10.77

8.97

0.868

one represented in Fig. 12. There is no preferable direction as found in the previous cases. The averaged axial distances Dxaver/D decreases (14.06 without directional spreading and 10.77 with directional spreading) and the averaged radial distances Dyaver/D increases (7.16 without directional spreading and 8.97 with directional spreading). Interestingly, the averaged power increase by 11.25% meaning that the variation of the current with respect to the predominant direction permits to capture more power (see Table 4). This gain is imputed to a greater mixing of the wake when the current change in direction, which tends to fasten the flow recovery.

6. Conclusions A semi-analytical method aiming at optimizing the arrangement of tidal turbines has been presented. It uses a Particle Swarm Optimization (PSO) algorithm coupled to an analytical model to represent the interaction between the wakes of the turbines. The methodology was applied to different flow conditions. The idealized flow conditions permits to gain insight into the results of the algorithm. When the flow remains parallel to the longitudinal axis, most turbines face with the free stream and are placed along lines perpendicular to the flow. In this configuration, the effect of the ambient turbulence on the longitudinal spacing is very important. The greater is the turbulent intensity, the faster is the flow recovery and the smaller is the longitudinal spacing between two consecutive rows. The application of the algorithm to scenarios with realistic hydrodynamics conditions show that the variation of the current with respect to the main flow direction strongly affects the layout as the turbines arrangement becomes more scattered. Noteworthy, the directional spreading of the tidal current increases the production of the tidal farm by 11% (in comparison to a similar case without tidal spreading). Future investigations

will concern the use of other optimization algorithms, and the development of a more realistic cost function. Acknowledgments The authors warmly thank the Normandy Region for funding the PhD thesis of the first author. Appendix A. Particle Swarm Optimization algorithm Particle Swarm Optimization (PSO) is a population-based metaheuristic optimization method (developed by Eberhart and Kennedy [66]) inspired by the social behavior of fish and bird. A particle is characterized by a position, a velocity and the objective function value of its position. Each particle moves towards its best previous position according to the global neighborhood and towards the best particle in the whole population. Furthermore, according to the local variant, each particle moves towards its best previous position and towards the best particle in its restricted neighborhood [67]. The first step of the PSO algorithm consists in creating randomly the population. In this paper, the population is composed of 300 binary encoded solutions. Subsequently, the members of the population are evaluated through the fitness function in order to find the fitness value for each particle and the best particle in the whole swarm. Once the evaluation is achieved, the speed and position of each particle is updated according to the following equation:



v tþ1 ¼ xv t þ r1 c1 ðpi;t  xt Þ þ r2 c2 ðpg;t  xt Þ xtþ1 ¼ xt þ v tþ1

ðA1Þ

where vt is the velocity at the time t, xt is the position of the particle, pi,t (the ‘‘personal guidance”) is the best previous position, pg,t (the ‘‘Global guidance”) is the best candidate solution for the entire pop-

O.A. Lo Brutto et al. / Applied Energy 183 (2016) 1168–1180

ulation, r1 and r2 are the real numbers randomly generated between 0 and 1 and x is an inertia weight coefficient, c1 and c2 are acceleration coefficients. In this paper, the method of Pookpunt and Ongsakul [38] was used. The inertia weight coefficient x are equal to 1 while the acceleration coefficient c1 and c2 linearly change with time (c1 changes from 2.5 to 0.5 and c2 changes from 0.5 to 2.5) in a manner that, during the first iterations, particles are allowed to diversify the search thanks to the large cognitive component and small social component, and that, during the last iterations, particles intensity the search around the global optimum thanks to a smaller cognitive component and a larger social component. The last step of the iteration is the updating of the population. The sigmoid function is used to scale the velocities between 0 and 1:

sigmoidðv tþ1 Þ ¼

1 1 þ ev tþ1

ðA2Þ

Particles’ positions are updated comparing the sigmoid function (depending on velocity value) of each bit with a random number nr which value is in the range [0, 1]:

 xtþ1 ¼

1; ifnr < sigmoidðv tþ1 Þ 0; otherwise

ðA3Þ

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