T Diveux

P Sebastian

D Bernard

J R Puiggali

Horizontal Axis Wind Turbine Systems: Optimization Using Genetic Algorithms

Keywords: genetic algorithms, optimization, Mediterranean site, wind turbine

A method for the optimization of a grid-connected wind turbine system is presented. The behaviour of the system components is coupled in a non-linear way, and optimization must take into account technical and economical aspects of the complete system design. The annual electrical energy cost is estimated using a cost model for the wind turbine rotor, nacelle and tower and an energy output model based on the performance envelopes of the power coefficient of the rotor, C P , on the Weibull parameters k and c and on the power law coefficient ˛of the wind profile. In this study the site is defined with these three parameters and the extreme wind speed V max . The model parameters vary within a range of possible values. Other elements of the project (foundation, grid connection, financing cost, etc.) are taken into account through coefficients. The optimal values of the parameters are determined using genetic algorithms, which appear to be efficient for such a problem. These optimal values were found to be very different for a Mediterranean site and a northern European site using our numerical model. Optimal wind turbines at the Mediterranean sites considered in this article have an excellent profitability compared with reference northern European wind turbines. Most of the existing wind turbines appear to be well designed for northern European sites but not for Mediterranean sites.

Introduction

A wind turbine is a complex system working in a complex environment. It is composed of subsystems working in a tightly coupled way. The design choice of a component may affect the annual energy production of the system and may indirectly affect the other component costs and behaviour. Furthermore, wind turbine systems must be adapted to the specific meteorological and topographical characteristics of each site.

The aim of our study is to develop optimization tools for the design of wind turbines dedicated to a site. The objective function of the optimization is the cost of the generated electrical energy calculated by means of conceptual models, most of which come from Reference 1. These models use a set of parameters, called principal parameters, that define the configuration of a wind turbine system. The optimized values of these variables are determined by minimizing the objective function with an optimization tool well adapted to the nature of this function.

Most of the classical optimization methods are based on differential or variational calculus. Indeed, in most of the optimization problems the objective function appears to be continuous and derivable, and optimal values of the function may be calculated using iterative methods. These methods are limited to the treatment of optimization problems leading to continuous, derivable and well-defined objective functions. Since the mid-1950s, more difficult problems have been handled in the artificial intelligence field. These optimization problems are solved using knowledge-based systems, fuzzy logic, inductive learning, neural networks and genetic algorithms. [START_REF] Pham | Artificial intelligence in engineering[END_REF] Owing to their efficiency and robustness in the global optimization of design problems involving non-linear and non-explicit variables, genetic algorithms are used in this study as an optimization technique. [START_REF] Hugget | Global optimisation of a dryer by using neural networks and genetic algorithms[END_REF][START_REF] Diveux | Implantation d'un système éolien, optimisation par algorithmes génétiques[END_REF] The use of genetic algorithms is relevant since the design of wind turbine systems is a combinatorial problem and the number of possible configurations of a system is very substantial. The main objective of this article is to validate the use of genetic algorithms for wind turbine systems. Thus the class of grid-connected horizontal axis wind turbines which is considered in this article is the most widespread. The principal characteristics of the turbines considered in this article are presented in Table I (see Results and Discussion Section).

Wind turbine sites are characterized by the Weibull parameters k and c, by a constant power law coefficient ˛and by the extreme wind speed V max . The parameters k, c and ˛are used for the estimation of the annual energy output, and V max is involved in the design of several components through the estimation of extreme loads due to the wind. Other characteristics of the wind (direction, turbulence) are not taken into account in our model. However, despite this restrictive hypothesis, those four parameters are sufficient to identify interesting differences between optimal wind turbines for Mediterranean sites and northern European sites.

In this article the global process of our work is presented through the description of the models for the estimation of the objective function, and the principle of genetic algorithms. Then the results that validate the methodology and prove its usefulness are discussed. A sensitivity study is performed showing the relevance of the design parameters used and the benefits deriving from the optimization. Optimal wind turbines are determined for different sites in order to point out interesting classes of wind turbines that may be compared with existing wind turbines. Finally, interesting prospects of our work that are currently being developed are provided.

Modelling

Wind turbines are complex systems that contain multiple components interacting through non-linearly coupled behaviours. A large number of design parameters and technological choices are required to define a wind turbine. The optimized design of a wind turbine well adapted to a specific site requires a complex economical and technical study. This study involves many different models. Our work is based on a single-criterion optimization method for which the objective function is the generated electrical energy cost. A synoptic diagram of the optimization method developed in this article is presented in Figure 1. The objective function is calculated using a wind turbine cost model and an energy annual production model, which have been developed using information from the literature. [START_REF] Harrison | Cost modelling of horizontal axis wind turbines (phase 2)[END_REF][START_REF] Collecutt | The economic optimisation of horizontal axis wind turbine design parameters[END_REF][START_REF] Wilson | Aerodynamic performance of wind turbines[END_REF] These models define the principal characteristics of a wind turbine system and include a set of equality constraints (geographical, material, aerodynamic and actualization variables, cut-in and cut-out wind speeds) and inequality constraints (hub height, rotor diameter and rotation speed). Figure 2 illustrates the different elements that constitute the two models.

Wind Turbine Cost Model

A global cost model of a wind turbine has been derived from cost models of all the components of the wind turbine and of some other costs of the project. These models are not presented here because they require a lot of parameters, but most of them can be found in References 1 and 4. For most of the components the models are derived from mechanical design laws and from the calibration of these models. Mechanical laws are used to calculate the weight of components, and calibration is performed by means of statistical data of component For the generator and electrical equipment the price is derived from the value of the nominal power. The total wind turbine cost is the sum of all the costs, and a calibration factor F WT allows us to use real wind turbine costs and take into account some unknown project parameters such as the manufacturer's margins:

C WT D F WT ð i C component i , F WT D 1Ð1 1 with C component i D C unit i ð f parameter 1 , parameter 2 , . . . , parameter ni 2
where C uniti is the unit cost per kg (¤).

The evaluation of the total cost of a project must take into account some additional costs due to:

ž land purchase and development of the site; ž transport of the wind turbine; ž installation of the wind turbine; ž foundations building; ž grid connection; ž remote monitoring; ž financing and insurance; ž miscellaneous costs (engineering, unexpected costs, etc.).

These additional costs have been estimated using data from the Nordex  society and have been interpolated as a function that depends on the diameter of the rotor and on the nominal power of the wind turbine:

C DV D i C additional cost i 3 with C component i D f P n , D R 4
The total investment cost C IT is the sum of the wind turbine cost and additional costs. Using the relation given in Reference 5, we can estimate the total annual cost C TA of the project. The actualization uses a factor a that depends on the economic life of the project, n, and on the required rate of return, r. The annual operation and maintenance costs are fixed to 2Ð5% of the investment cost.

C TA D a C 0Ð025 C IT 5 with a D r 1 1 C r n 6

Annual Energy Output Model

Optimal design of wind turbine systems goes through an estimation of their annual energy production. Indeed, this design always takes into account the cost of electrical energy, which appears to be the main design criterion of wind turbines. An energy production model has been built that involves both wind and turbine characteristics. Wind turbine technologies considered by the model are limited to horizontal axis wind turbines with two types of regulation:

ž pitch regulation with constant rotation speed or variable rotation speed; ž stall regulation with constant rotation speed. The estimation of the energy output is based on knowledge of the rotor performance or the rotor power profile. Figure 3 shows the power profiles of the different technologies used in our model. These performances are determined by a power coefficient C P , defined by the ratio of the power P of the rotor to the kinetic power of the wind crossing the rotation surface of the rotor ( 12 air S R V 3 , where S R is the swept area of the rotor). The calculation of C P by aerodynamic methods requires a lot of data, including the complete geometry of the blade, and should be carried out for each individual wind turbine. Here we use an analytical relation given in Reference 6 to calculate the performance envelopes of C P (the set of C P max ):

C P max D 0Ð593 0 max ð p 0Ð67 1Ð48 C p 0Ð67 0Ð04 0 max C 0Ð0025 2 0 max 1Ð92 2 0 max ð p 1 C 2 0 max ð p C d C l 7 with 0 max D ω V des D R 2 8
The values of D R , N and V des completely determine C P max and thus the optimal power profile P opti D f V (see Figure 3) that simulates the power of a variable speed wind turbine. Optimal power curves may be determined using the following relations:

P opti V D KV 3 9 with K D 1 2 C P max D R /2 2 10
The power curve of horizontal axis pitch-regulated wind turbines with constant rotation speed (see Figure 3) between the cut-in wind speed V i and the nominal wind speed V n is well approximated by means of a straight line tangent to the optimal power curve (see Figure 3); thus

P V D KV 2 des 3V 2V des 11
Above V n and until the cut-out wind speed V f is attained, the wind turbine power remains fixed at the value P n .

For stall-controlled wind turbines we use the power curve of some Nordex  wind turbines. It bounds the value of some parameters, but is necessary since this type of wind turbine performance is difficult to generalize.

With the previous relations the annual electrical energy output is determined using the integration of the wind speed distribution and the corresponding energy output during 1 year:

E AP D 8760 1000 air 2 ð S R ð V f V i V 3 f V C P V Á GB V Á G V dV 12
Where E AP is the annual electrical energy output (kWh), S R is the swept area of the rotor (m 2 ), f V is the Weibull probability density function of wind speed, Á GB is the gearbox efficiency and Á G is the generator efficiency.

The Weibull probability density function of wind speed depends on the parameters k and c that determine the shape and intensity of the wind during 1 year on a site:

f V D k V V c k exp V c k 13
where k is the Weibull shape parameter (dimensionless) and c is the Weibull scale parameter (m s 1 ).

The parameters k and c are functions of the height of the wind turbine, since the site environment influences the wind distribution. The Weibull scale parameter is determined using a power law that involves wind speed increasing with hub height at vertical positions of the site:

c D c 0 Z/Z 0 ˛at hub height Z 14
where ˛is the coefficient of the power law and c 0 is the value of c at reference height Z 0 (generally 10 m). The Weibull shape parameter follows a law from Reference 7:

k Z D k 0 C k Z 15 where k Z D 0Ð008Z 0Ð08 for Z < 20 m 0Ð003Z C 0Ð02 for Z ½ 20 m
and k 0 is the value of k at reference height Z 0 . The gearbox and generator efficiencies are calculated using equations presented in Reference 1. They depend on the wind turbine power law, on the nominal power P n and on the type of regulation of the generator.

Optimization Variables and Constraints

The main factor that influences the design of wind turbines is the annual cost of the electrical energy produced by the wind turbine. These parameters are integrated in the mechanical laws and cost models presented above. The annual electrical energy cost is defined as the ratio of the total annual cost C TA and the annual electrical energy output E AP of the wind turbine:

C EP D C TA /E AP 16
Other factors, such as security parameters or environmental factors, also have great importance, but they are not used as optimization parameters in this article. These considerations introduce inequality constraints that bound the rotor diameter to the height of the wind turbine and limit the rotor speed. Indeed, in order to prevent security problems, the ground clearance between the blade tip and the ground is set to 15 m. Thus D R /2 C 15 Ä H hub Also, in order to limit noise levels and sound pollution, the maximum rotor tip speed is fixed at 80 m s 1 . Thus

V tip D 2 ND R /120 Ä 80 m s 1 18

Principal Parameters

The choice and number of principal parameters used for the optimization of wind turbine systems depend on the design constraints (physical constraints, technological constraints, design domain) and thus on the nature and complexity of the design model. Subsidiary parameters are determined by the model constraints or are fixed by the designer. The design model developed in this work uses eight principal parameters:

ž the number of blades p; ž the rotor diameter D R ; ž the hub height H hub ; ž the rotation speed of the rotor, N; ž the nominal power P n ; ž the design wind speed V des ; ž the type of regulation (stall or pitch); ž the type of generator (asynchronous with variable or constant speed).

Most of these parameters are classically used as principal parameters for the optimization of wind turbines. However, the power characteristic curve of the wind turbine may be determined using several different parameters. The parameter used in our work is the design wind speed V des . This parameter is the wind speed for which the wind turbine power is optimal. At constant rotation speed this parameter defines a unique point F (see Figure 3) that depends on the rotor diameter and rotation speed. Point F corresponds to the maximum tip speed ratio 0 max (equation ( 8)). The parameter V des greatly affects the power characteristic curve of the wind turbine by determining the slope of the power curve. Schematically, a variation in V des will result in a shift of the C p curve (Figure 4) and may be used to optimize the total energy retrieved from the kinetic energy of the wind.

V (m s -1 ) Cp Cp (V des =8 m s -1 ) Cp (V des =12 m s -1 )

Figure 4. Influence of the parameter V des on the power curve

A wind turbine configuration is defined with a set of discrete parameters (number of blades, type of regulation, type of generator) or positive real parameters (rotor diameter, hub height, rotor rotation speed, design wind speed, nominal power) evolving in a value space limited by physical or technological constraints (wind turbine cost model, annual energy output model) and design constraints (security constraints, environmental constraints). The aim of the wind turbine optimization is to find the best configuration of these parameters for a specific site by giving the minimum of an objective function. The optimization results presented in this article use the annual cost of the electrical energy produced by the wind turbine as the objective function. Owing to the nature of the design parameters and to the specificities of the design constraints, genetic algorithms have been used to solve this optimization problem.

Genetic Algorithms

Genetic algorithms, developed by Holland, [START_REF] Holland | Adaptation in Natural and Artificial Systems[END_REF] are powerful tools for the global optimization of problems defined through non-continuous variables and functions. They are well adapted to the solution of non-linear problems that lead to objective functions containing many local minima or maxima. [START_REF] Selig | Application of a genetic algorithm to wind turbine design[END_REF] Objective functions may be non-derivable, non-continuous or ill-defined (as the result of a complex algorithm for instance), and no gradient information is required to solve them. More to the point, genetic algorithms appear to be robust and easy to implement. Classical optimization methods, e.g. gradient, Newton or quasi-Newton methods, are not suitable for the optimization of such hard-to-solve problems.

Genetic algorithm implementation is based on the evolution of a population including several individuals (10 in our application) towards a best individual by using selection and natural reproduction processes. These processes are based on the Darwinian concept of natural selection 10 (see Figure 5). Individuals are defined through values of the principal parameter vector. Each principal parameter is called a gene, and individuals are completely defined by values of their own genes. Individuals evolve following an iterative

p i D 1 N ind r i 1 2 2 N ind 1 19
where N ind is the number of individuals in the population, r i is the rank of individual i and is the selection fitness equal to the average number of children, 2 [1; 2]. Individuals selected according to p i participate in the reproduction operations in order to create a new generation. These operations are the crossover operation (crossing of a few genes between two individuals) and the mutation operation (modification of genes). Crossover permits a good exploitation of the local value space, whereas mutation allows a renewal of the population and then an effective search of all the value space. The newly generated individuals are included in the generation only if the parameters check the inequality constraints. Otherwise, new crossovers or mutations are made. The number of tested individuals necessary to find the best turbine is determined statistically. The algorithm stops when this number is reached.

Figure 6 shows the evolution of a population of 10 individuals according to the objective function, which is the minimization of the electrical energy cost produced by a wind turbine. Convergence of the best individual and disparity of the individuals at each generation may be observed in the figure . A continuous line points out the best individual of each population.

Results and Discussion

Other constraints limiting the optimization domain of our application are listed in Table I. Corresponding constraints define several equipment characteristics and technological choices. These choices have been made in order to remain close to the most commonly encountered technologies in wind turbine design. [START_REF] Timer | WECS blade airfoils-the NACA 63-4XX series[END_REF][START_REF] Harrison | Cost modeling of horizontal axis wind turbines-results and conclusions[END_REF][START_REF]Wind Energy, the Facts[END_REF] Indeed, the optimization process presented in this article is concerned with optimizing interactions between Optimization is performed by varying the principal parameters of a wind turbine configuration in a discretized domain. Table II presents the variation domain of these parameters and their corresponding variation step. Each principal parameter belongs to a finite set of possible values, so the possible configuration set of wind turbines is also a finite set. The design problem of wind turbines appears to be a combinatorial problem. This problem is non-polynomial owing to the non-linear structure of the model. [START_REF] Korte | Combinatorial Optimization, Theory and Algorithms[END_REF] The number of possible individuals of the design problems treated in this article is 647,360.

Principal Parameters of Wind Turbine Design

A sensitivity study was performed to determine the relevance of the principal parameters and to compare the performances of optimized wind turbines with those of reference wind turbines. The reference systems have the same characteristics as existing wind turbines that are summarized in Table III. It must be noted that the reference wind turbines are installed in the north of Europe and have been chosen because we know the characteristics of their components and the costs of the project. Each wind turbine uses a different control technology (stall with constant rotation speed, pitch with constant and variable rotation speed). Energy and costs of optimized configurations put forward here are calculated using our model and using wind and site characteristics typical of Mediterranean sites.

Tables IV-X present optimization results of simulations performed by fixing one parameter and varying the principal parameters. These results may be compared with optimized wind turbines obtained by varying all the parameters. Size characteristics of the optimized wind turbines (rotor diameter, hub height) are lower than those of the reference wind turbines, whereas parameters related to power (rotor rotation speed, design wind speed, nominal power) appear to be higher. As we shall see in the next subsection, this result is relevant to sites where the coefficient k is low and the coefficient c is high, which is typical of Mediterranean sites.

Optimized and reference configurations are presented in Tables IV-X in order to compare the characteristics of wind turbines installed in the north of Europe with those of wind turbines well-adapted to Mediterranean sites. Reductions in energy cost between reference and optimized configurations generally range between 10% and 30%, thus showing the necessity of specific wind turbines for Mediterranean sites. For high turbine rotation speed the optimization process reduces the size of the gearbox, which appears to be one of the most expensive devices of wind turbine systems. Table IV compares optimization results obtained for wind turbines with two and three blades with the performance of reference wind turbine 3. Cost reductions for the optimized wind turbines are close (35% and 37%), irrespective of the number of blades. The number of rotor blades has a low influence on wind turbine performance. The components optimized characteristics (dimensions, technological choices) of both wind turbines are also close. Three-blade wind turbines produce more energy but appear to be more expensive than two-blade ones. Thus two-blade wind turbines are more efficient on this site. Table V shows optimization results obtained for different values of the rotor diameter. The rotation speed and design speed of optimized wind turbines are reduced by increasing the rotor diameter. The air stream section, and thus energy production, increases with the squared rotor diameter and, as a consequence, energy may be produced with low characteristic speeds. However, the wind turbine nominal power increases with the rotor diameter. High-diameter turbines are expensive but produce a lot of energy, thus compensating for their high cost.

Table VI illustrates the influence of the hub height on wind turbine performance. Because of ground topography and roughness, wind speed and energy production increase with altitude. However, as the wind turbine tower becomes taller and more expensive, the increase in energy production is not sufficient to compensate the tower cost. Table VII presents optimized wind turbines determined for different blade rotation speeds. For low rotation speed of the blades, the rotor diameter and nominal power of the wind turbine must be high in order to maintain the annual energy production at a sufficient level.

Table VIII shows optimization results obtained by fixing design wind speeds. These speeds have a very strong influence on the wind turbine performances. Design wind speed is characteristic of the blade shape and is linked to the aerodynamic phenomena that occur at the blade profile level. The aerodynamic performance of the blades determines to a large extent the technological choices and the dimensions of the entire system. On this site a high design wind speed gives the best results, but this principal parameter is very much linked to the site properties.

Table IX illustrates the influence of the nominal power of wind turbines on the produced energy cost. For low nominal powers, stall regulation appears to be more efficient than pitch regulation. Indeed, even for low-power turbines, pitch regulation remains expensive. It must be noted that, despite the low nominal power of the 300 kW wind turbine and owing to the high wind speeds of the site investigated, optimized wind turbines appear to be better adapted to this site than reference turbines. Optimized turbines are indeed well adapted to high wind speeds.

Table X presents optimization results for stall-or pitch-regulated wind turbines with variable or constant rotation speeds. These results show that, on the site considered, the regulation has a low influence on the characteristics of the optimized systems. Produced energy costs are close and, for reasons discussed below, stall wind turbines have a slightly lower nominal power.

The sensitivity of the produced energy cost to variations in the principal parameters is summarized in Figures 7 and8. The cost reductions range between 3% and 38%, while the principal parameters vary between 60% and 150% of their values. These figures point out the influence of the principal parameters, which appears to be complex and non-linear, arising from interactions between components which have complex behaviours. Owing to this complexity and to the sensitivity of wind turbine performance to the principal parameters used in our model, global optimization of wind turbines appears to be highly profitable. This optimization process is linked to the installation site. The wind and site topographic characteristics also have a major influence on wind turbine performance.

Influence of Installation Site on Wind Turbine Design

About 80% of the installed power of wind turbines in Europe was located in northern Europe in 1999. However, Mediterranean sites have a great potential that remains underused owing to local politics of the Mediterranean countries during the 1980s and the 1990s. Environmental politics have been changing and these countries are interested in increasing their wind turbine parks. Our applications are mainly focused on the analysis of the design of wind turbines for Mediterranean sites.

In our model, installation site characteristics are taken into account through the annual distribution function of the wind and through the ground roughness coefficient. Mediterranean sites are characterized by Weibull parameters k which are smaller and parameters c which are larger than the corresponding parameters for northern Europe. Parameter k usually ranges between 1 and 2 and velocity c is around 9 m s 1 at 10 m height in Mediterranean sites, while k is usually around 2 and velocity c ranges between 5 and 7 m s 1 at 10 m height in northern European sites. Wind speed distributions may be far from a Weibull function and, owing to uneven reliefs in these regions, most Mediterranean sites have singular wind profiles. More to the point, in the presence of obstacles the wind shear coefficient ˛is stronger for Mediterranean sites, where it ranges between 0Ð16 and 0Ð18, than for northern European sites, where it is about 0Ð10. This coefficient also varies with height and wind sector. Simulation results presented as an application in this article are based on Weibull distributions.

Tables XI-XIII illustrate the influence of the Weibull parameters for wind turbines optimized for k values between 1Ð2 and 2 and for c values between 6 and 12 m s 1 . The main trends that can be observed are linked to the size of wind turbines and to the energetic potential of sites. Indeed, wind turbines well adapted to the Mediterranean regions appear to be smaller (D R and H hub ) than northern European wind turbines. The hub height and rotor diameter of Mediterranean turbines are small, but their power speed and parameters (N, V des , P n ) appear to be greater. The kWh cost is much smaller for Mediterranean sites, varying between 2Ð9¤cent for k D 1Ð2 and c D 12 m s 1 and 4Ð4¤cent for k D 2 and c D 8 m s 1 . Energy potential is greater for Mediterranean sites than for north European sites. Optimal design of wind turbines leads to small devices adapted to optimal energy capture and mechanical resistance in strong winds. This result is illustrated by Figure 9, showing that the smaller k is and the greater c is, the more important is the ratio P n /S R .

The influence of the wind shear coefficient ˛is presented in Figure 10. This coefficient has a great influence on the produced energy cost through the hub height and wind potential. Figure 10 shows the evolution of the As can be seen in Reference 12, optimal hub heights corresponding to the minima of the curves presented in Figure 10 increase with the wind shear coefficient. However, the produced energy cost goes through a maximum value, since the tower cost increases with the hub height and wind speed.

Comparison of Optimized Wind Turbines and Commercial Wind Turbines

The optimized wind turbines presented in Tables XI-XIII are compared with commercial wind turbines in Figure 11. This figure points out the cost of optimized wind turbines and compares these costs with those of installed systems. Calculated costs of the reference wind turbines used in the sensitivity study are realistic compared with the costs of existing wind turbines, and our wind turbine cost model leads to realistic results. The application of this model to the optimization of wind turbine design lowers the cost of existing wind turbines. The objective function of the optimization, which is the minimization of the produced energy cost, may be attained by decreasing wind turbine costs or by increasing the amount of energy produced on the installation site. The optimization process leads to enhanced wind turbine performance by improving both factors.

The commercial wind turbines used in this article as references are well designed for northern European sites (for k values around 2), but Tables XI-XIII show that performance enhancement is more important for small values of k (k values around 1Ð2). Optimization of wind turbine design remains profitable for Mediterranean devices. Table XIV illustrates the performance of a wind turbine optimized for a Mediterranean configuration on a site that has typical northern European characteristics. In this configuration the energy produced by the wind turbine is much more expensive than the energy produced by reference wind turbine 3 . The reference wind turbine performance can also be improved, as the kWh cost may be lowered to 7Ð8¤cent.

Conclusion and Further Work

A method for optimizing a wind turbine system for a specific site has been presented in this article. This optimization involves a model for an estimation of the cost of electrical energy output. Using this model, a sensitivity study has shown the principal parameter influence on the produced energy cost. The optimization method is based on the use of genetic algorithms, as they appear to be suitable for the optimization of problems involving non-continuous, non-derivable and ill-defined variables and functions. A study with different Weibull parameters has been performed. It shows that wind turbine characteristics for Mediterranean sites are very different from those for northern European sites: according to our model, optimal wind turbines are smaller for Mediterranean sites and have larger power parameters. Produced energy with wind turbines optimized for Mediterranean sites is cheap. Improvements in the model are being developed that allow for better consideration of existing technologies that are not taken into account in our model. They are focused on: ž introduction of other types of components, e.g. generators with other rotation speeds (750, 1500 rpm), dual-speed generator, direct drive generator (variable speed) or active stall regulation; ž elaboration of a control strategy model for wind turbines (starting, cut-in and cut-out wind speed, stopping) and optimization of this strategy according to the wind characteristics; ž improvement in site characterization by introducing factors related to accessibility, development (difficult for Mediterranean sites), foundation and grid connection. tip speed ratio at maximum power coefficient air air density (kg m 3 )

Appendix: Notation

x density material of x (kg m 3 )

x admissible stress of material x (Pa) ω angular rotation speed of rotor (rad s 1 )

Figure 2 .

 2 Figure 2. Global schema of the model

FFigure 3 .

 3 Figure 3. Power curve models

Figure 5 .Figure 6 .

 56 Figure 5. Global schema of the optimization process that uses genetic algorithms

Figure 7 .Figure 8 .

 78 Figure 7. Influence of the principal parameters on the cost of a kWh

Figure 9 .Figure 10 .

 910 Figure 9. Ratio P n /S R versus Weibull law parameters

Figure 11 .

 11 Figure 11. Wind turbine costs-comparison between optimized and commercial wind turbines Table XIV. Comparison of the cost of a kWh for k D 2, c D 6 m s 1 and ˛D 0Ð12 (k, c and ˛are defined at 30 m height) p D R (m) H hub (m) N (rpm) V des (m s 1 ) P n (kW) Cost of kWh (¤cent) Optimized WT ofTable 6 2 35 35 40 14 900 11Ð13 Reference WT 3 3 48 70 28 9 600 8Ð69 Optimized WT 2 55 45 15 9 600 7Ð77

Table I .

 I Main technological choices We do not perform optimization of existing blade geometry, mechanical or electrical equipment. Our investigations are limited to the optimization of the global performance of systems made with elements having known and predictable behaviours.

	Number of wind turbines on site	1
	Availability	100%
	Rotor	Horizontal axis
	Blades	2 or 3
		Rigid blades
		Aerofoil: NACA 63Ð4
		Thickness/chord ratio D 0Ð18
		C l /C d = 120
		Stall or pitch regulation (hydraulic actuator)
		Tip brake mechanism for stall regulation
	Blade spar and aerofoil	Glass-reinforced polyester
	Hub	Rigid for 3 blades
		Teeter for 2 blades
	Drivetrain	Integral
	Low-speed shaft	Constant diameter
		Internal diameter/external diameter D 0Ð5
		Length D 0Ð05D R
	Mechanical brake	Located on high-speed shaft for pitch regulation,
		on low-speed shaft for stall regulation
	Gearbox	2 stages epicyclic: ratio Ä 40
		3 stages (2 epicyclic, 1 parallel): 40 < ratio Ä 80
		3 stages epicyclic: 80 < ratio
	Generator	Asynchronous generator
		Constant speed: N G D 1500 rpm
		Variable speed: N Gmax D 1500 rpm
	Electrical cables	4 copper sections, 3 cores
	Tower	Tapered steel sections
	Cut-in and cut-out speeds	3 and 25 m s 1
	Table II. Variation range of principal parameters
	Parameter	Domain of variation
	p	2 or 3
	D R	20-80 m (step of 5 m)
	H hub	25-95 m (step of 10 m)
	N	15-65 rpm (step of 5 rpm)
	V des	6-15 m s 1 (step of 1 m s 1 )
	P n	300-2000 kW (step of 100 kW)
	Type of regulation	Stall or pitch
	Type of generator	Asynchronous constant or variable speed
	wind turbine components.

Table III .

 III Principal parameters and optimization results for reference wind turbines

		Table V. Optimization results: influence of the rotor diameter
	Parameter	Reference	Optimized	Optimized	Optimized
				WT 3	D R D 20 m	D R D 40 m	D R D 60 m
	p					3		2	2	2
	D R (m)		48		20	35	60
	H hub (m)		70		25	35	45
	N (rpm)		28		65	40	15
	V des (m s 1 )				9		15	14	12
	P n (kW)		600		400	900	2000
	Type of regulation	Pitch		Pitch	Pitch	Pitch
	Type of generator	Var. speed		Var. speed	Var. speed	Var. speed
	Energy (kWh year 1 )	1 .9 ð 10 6		0.5 ð 10 6	1.5 ð 10 6	4.2 ð 10 6
	Cost of WT (M¤)			0 Ð72		0Ð11	0Ð34	1Ð05
	Cost of project (M¤)			0 Ð84		0Ð17	0Ð43	1Ð25
	Cost of kWh (¤cent)			6Ð3		4 Ð8	4 Ð0	4 Ð3
	Cost reduction (%)					23	37	32
		Parameter		Reference		Reference	Reference
	WT 1 Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m. WT 2 WT 3
		p				3			3	3
		D R (m) H hub (m)	43 50 Table VI. Optimization results: influence of the hub height 47 48 60 70
		N (rpm)				27			26	28
	V des (m s 1 ) Parameter P n (kW)	8 600 Reference WT 3	Optimized WT	8 660	Optimized H hub D 50 m 600 9	Optimized H hub D 80 m
		Type of regulation			Stall		Pitch	Pitch
	p	Type of generator	Const. speed 3	2	Const. speed	2	Var. speed	2
	Energy (kWh year 1 ) D R (m) Cost of WT (M¤) H hub (m) Cost of project (M¤) N (rpm) Cost of kWh (¤cent) V des (m s 1 ) P n (kW)	48 70 28 9 600	1 .6 ð 10 6 0 Ð50 0 Ð61 5Ð5	35 35 40 14 1000	1.8 ð 10 6 0Ð61 0Ð81 6 Ð4	35 50 40 14 1000	1.9 ð 10 6 0Ð72 0Ð84 6 Ð3	35 80 40 15 1100
	Type of regulation		Pitch		Pitch	Pitch	Pitch
	Type of generator		Var. speed	Var. speed	Var. speed	Var. speed
	Energy (kWh year 1 )	1 .9 ð 10 6	1.5 ð 10 6	1.7 ð 10 6	1.9 ð 10 6
	Cost of WT (M¤) Table IV. Optimization results: influence of the number of blades 0.72 0.34 0.38 Cost of project (M¤) 0.84 0.43 0.49	0.52 0.63
	Parameter Cost of kWh (¤cent) Cost reduction (%)	6.3	Reference WT 3	Optimized 4.0 p D 2 37	4.1 Optimized p D 3 35	4.6 27
		p					3		2	3
		D R (m)				48		35	35
		H hub (m)				70		35	35
		N (rpm)				28		40	40
		V des (m s 1 )				9		14	14
		P n (kW)			600		900	1000
		Type of regulation			Pitch		Pitch	Pitch
		Type of generator		Var. speed	Var. speed	Var. speed
		Energy (kWh year 1 )		1 .9 ð 10 6	1.5 ð 10 6	1.6 ð 10 6
		Cost of WT (M¤)			0 Ð72	0Ð34	0Ð37
		Cost of project (M¤)			0 Ð84	0Ð43	0Ð47
		Cost of kWh (¤cent)			6Ð3		4 Ð0	4 Ð11
		Cost reduction (%)					37	35
		Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12
		at 30 m.

Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m. Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m.

Table VII .

 VII Optimization results: influence of the rotation speed

	Parameter	Reference		Optimized	Optimized	Optimized
		WT 3	N D 20 rpm	WT N D 50 rpm
	p	3		2	2	2
	D R (m)	48		50	35	30
	H hub (m)	70		45	35	35
	N (rpm)	28		20	40	50
	V des (m s 1 )	9		13	14	14
	P n (kW)	600		1500	1000	700
	Type of regulation	Pitch		Pitch	Pitch	Pitch
	Type of generator	Var. speed	Var. speed	Var. speed	Var. speed
	Energy (kWh year 1 )	1 .9 ð 10 6		3.0 ð 10 6	1.5 ð 10 6	1.2 ð 10 6
	Cost of WT (M¤)	0 Ð72		0Ð81	0Ð34	0Ð24
	Cost of project (M¤)	0 Ð84		0Ð96	0Ð43	0Ð34
	Cost of kWh (¤cent)	6Ð3		4 Ð6	4 Ð0	4 Ð1
	Cost reduction (%)			28	37	35
	Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m.
	Table VIII. Optimization results: influence of the design speed
	Parameter	Reference		Optimized	Optimized	Optimized
		WT 3		V des D 6 m s 1	V des D 10 m s 1	WT
	p	3		2	2	2
	D R (m)	48		55	35	35
	H hub (m)	70		45	35	35
	N (rpm)	28		15	40	40
	V des (m s 1 )	9		6	1 0	1 4
	P n (kW)	600		800	700	1000
	Type of regulation	Pitch		Pitch	Pitch	Pitch
	Type of generator	Var. speed		Var. speed	Var. speed	Var. speed
	Energy (kWh year 1 )	1 .9 ð 10 6		2.0 ð 10 6	1.3 ð 10 6	1.5 ð 10 6
	Cost of WT (M¤)	0 Ð72		0Ð69	0Ð29	0Ð34
	Cost of project (M¤)	0 Ð84		0Ð84	0Ð40	0Ð43
	Cost of kWh (¤cent)	6Ð3		6 Ð1	4 Ð2	4 Ð0
	Cost reduction (%)			3	33	37

Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m.

Table IX .

 IX Optimization results: influence of the nominal power 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m.

	Parameter		Reference		Optimized	Optimized	Optimized
			WT 1/3	P n D 300 kW	WT P n D 1500 kW
	p		3		3	2	2
	D R (m)		43/48		25	35	35
	H hub (m)		50/70		35	35	35
	N (rpm)		27/28		55	40	40
	V des (m s 1 )		8/9		10	14	15
	P n (kW)		600		300	900	1500
	Type of regulation		Stall/pitch		Stall	Pitch	Pitch
	Type of generator		Const./var. speed	Const. speed	Var. speed	Var. speed
	Energy (kWh year 1 )	1.6 ð 10 6 /1.9 ð 10 6	6.5 ð 10 5	1.5 ð 10 6	1.8 ð 10 6
	Cost of WT (M¤)		0.5/0.72		0Ð15	0Ð34	0Ð43
	Cost of project (M¤)	0.61/0.84		0Ð21	0Ð43	0Ð55
	Cost of kWh (¤cent)	5.5/6.3		4Ð7	4 Ð0	4 Ð2
	Cost reduction (%)				13	37	33
	Wind characteristics: k D Table X. Optimization results: influence of the regulation
	Parameter	Reference	Optimized	Optimized	Optimized
			WT 1/2	Pitch (N var.)	Stall (N const.)	Pitch (N const.)
	p		3		2	3	2
	D R (m)		43/47		35	35	35
	H hub (m)		50/60		35	35	35
	N (rpm)		27/26		40	40	40
	V des (m s 1 )		8		14	10	11
	P n (kW)		600/660		900	700	900
	Type of regulation	Stall/pitch		Pitch	Stall	Pitch
	Type of generator		ConstÐ speed		VarÐ speed	ConstÐ speed	ConstÐ speed
	Energy (kWh year 1 )	1.6 ð 10 6 /1.6 ð 10 6		1.5 ð 10 6	1.3 ð 10 6	1.4 ð 10 6
	Cost of WT (M¤)		0 Ð5/0		0Ð34	0Ð32	0Ð34
	Cost of project (M¤)		0 Ð61/0		0Ð43	0Ð41	0Ð44
	Cost of kWh (¤cent)		5Ð5/6		4Ð0	4 Ð4	4 Ð6
	Cost reduction (%)				27/38	20/32	16/28

Wind characteristics: k D 1Ð2, c D 8 m s 1 . Site characteristics: ˛D 0Ð12 at 30 m.

Table XI .

 XI Wind turbines optimized for k D 2 and c D 6-12 m s 1 (k and c are defined at 30 m height)

	Parameter	k D 2,	k D 2,	k D 2,	k D 2,
		c D 6 m s 1	c D 8 m s 1	c D 10 m s 1	c D 12 m s 1
	p	2	2	2	2
	D R (m)	55	35	35	30
	H hub (m)	45	35	35	35
	N (rpm)	15	40	40	50
	V des (m s 1 )	9	11	13	15
	P n (kW)	600	500	800	800
	Type of regulation	Pitch	Pitch	Pitch	Pitch
	Type of generator	Var. speed	Var. speed	Var. speed	Var. speed
	Cost of kWh (¤cent)	7Ð77	4Ð42	2Ð90	2Ð29

Table XII .

 XII Wind turbines optimized for k D 1Ð6 and c D 6-12 m s 1 (k and c are defined at 30 m height)

	Parameter	k D 1Ð6,	k D 1Ð6,	k D 1Ð6,	k D 1Ð6,
		c D 6 m s 1	c D 8 m s 1	c D 10 m s 1	c D 12 m s 1
	p	2	2	2	2
	D R (m)	60	35	30	35
	H hub (m)	45	35	35	35
	N (rpm)	15	40	50	40
	V des (m s 1 )	1 0	1 3	1 5	1 5
	P n (kW)	1100	700	800	1200
	Type of regulation	Pitch	Pitch	Pitch	Pitch
	Type of generator	Var. speed	Var. speed	Var. speed	Var. speed
	Cost of kWh (¤cent)	7Ð01	4Ð12	3Ð05	2Ð44

Table XIII .

 XIII Wind turbines optimized for k D 1Ð2 and c D 6-12 m s 1 (k and c are defined at 30 m height)

	Parameter		k D 1Ð2,		k D 1Ð2,		k D 1Ð2,		k D 1Ð2,
		c D 6 m s 1	c D 8 m s 1	c D 10 m s 1	c D 12 m s 1
	p		2		2		2		2
	D R (m)		60		35		35		35
	H hub (m)		45		35		35		35
	N (rpm)		15		40		40		40
	V des (m s 1 )		1 1		1 4		1 5		1 5
	P n (kW)		1500		900		1200		1300
	Type of regulation		Pitch		Pitch		Pitch		Pitch
	Type of generator		Var. speed	Var. speed		Var. speed	Var. speed
	Cost of kWh (¤cent)		6Ð10		3Ð96		3Ð20		2Ð90
		1.6
		1.4				Mediterranean site
		1.2
	P n /S R (kW m -2 )	0.8 1			k=1.2 k=1.6
					k=2
		0.6
		0.4			Northern Europe site
		0.2	6	7	8	9	10	11	12
					c (m s -1 )

Table 6

 6

		2	35	35	40	14	900	11Ð13
	Reference WT 3	3	48	70	28	9	600	8Ð69
	Optimized WT	2	55	45	15	9	600	7Ð77

Acknowledgements

The authors wish to acknowledge the agency ADEME (Paris, France) and the company Valorem (Bégles, France) for their financial support (thesis of the first author) and consultation.

Contract/grant sponsor: ADEME (Paris, France). Contract/grant sponsor: Valorem (Bègles, France).


Yoanna Álvarez Ginarte
email: yoanna@fq.uh.cu

Roy González Aleman

Taher Yacoub

Fabrice Leclerc
email: fabrice.leclerc@u-psud.fr

Luis A Montero Cabrera

Roy González

Montero Cabrera

Yoanna M Álvarez Ginarte

Identification of selective inhibitors against BACE 1 over BACE 2 in Alzheimer's disease by Quantitative Structure-Activity Relationship (QSAR)

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cluster analysis is used in QSAR models to build the training and test sets as well as to determine the structural diversity of the dataset

üØ

  Compounds with biochemical activity below 1 μM were selected for further inhibition assay ü The IC 50 value is the concentration of compound which inhibits BACE 1 and BACE 2 binding by 50 % ü In our study, the negative logarithm of the biological activity, pIC 50 , was used as the dependent variable to determine QSAR correlation equations ü Active molecule (pIC50 greater or similar than 6) and inactive molecule (pIC50 lower than 6) (hetero)aromatic ring, B: an (hetero)aromatic ring and an amide linker is installed between A-and B-rings allowing the B-ring to enter deep into the pocket. X, Y, Z: Substituent groups, Position 4: C or N atom The two-dimensional (2D) chemical structures of the inhibitors were generated using ChemDraw Ultra and saved in mol format.

PICSignificant

  50 = -0.44 F03[C-C] + 0.60 MR99+ 0.19 TPSA(Tot) + 0.45 LOGP + 2.78 LLS_02-3.13 Training sets : n=55; R=0.90; R 2 = 0.81; s=0.26; F=42.71; Q2=0.76; P=0.00 Test sets: n=14; R=0.80; R 2 =0.64; s=0.70; -2.05 SCBO +1.29 D/Dtr06 +8.94 DLS_cons +7.07 LLS_01 -10.85 QEDu + 9.53 Training sets: n=28; R=0.91; R 2 = 0.83; s=0.42; F=22.47; Q2=0.75; P=0.00 Test sets: n=6; R=0.92; R 2 = 0.85; s=0.48; F=22.98 Ken-ichi Kusakabe , Chien-Chi Hsiao & Harrie J. M. Gijsen (2020): Smallmolecule BACE1 inhibitors: a patent literature review (2011 to 2020), Expert Opinion on Therapeutic Patents, DOI: 10.1080/13543776.2021.1832463 Ø Challenges for BACE1 inhibitors, notably achieving central penetration and avoiding toxicity, been progressively overcome in the past 10 years Ø The most recent progress was made in getting high BACE1/BACE2 selectivity, which further reduces the toxicity risk.

Estimation of molecular properties (descriptors)

	BLOCK	TOTAL	SELECTION	EXCLUDE
	1. Constitutional indices	50	14	36
	2. Ring descriptors	35	5	30
	3. Geometrical descriptors	38	0	38
	4. Randic molecular profiles	41	0	41
	5. Functional group counts	154	1	153
	6. Pharmacophore descriptors	165	3	162
	7. 2D Atom Pairs	1596	4	1592
	8 3D Atom Pairs	36	0	36
	9. Charge descriptors	15	0	15
	10. Molecular properties	27	11	16
	11. Drug-like indices	30	6	24
	12. Chirality descriptors	70	0	70
	TOTAL	2257	44	2213
	Ø The 3D structure of each inhibitors were obtained by the geometrical
	optimization of each molecule using Kohn-Sham's DFT B3LYP/6-
	31G method included in Gaussian 09 program routines

Exclusion criteria: Ø At least one value as zero Ø Constant values Ø Not

applicable Selection of training and test sets using Cluster Analysis 69 BACE 1

		Training Set
		55 molecules
		80 %
	hierarchical
	Cluster Analysis	Test Set
		14 molecules
		20 %
		Training Set
		26 molecules
		80 %
	hierarchical
	33 BACE 2
	Cluster Analysis	Test Set
		7 molecules
		20 %

Journal of Computational Chemistry 2016, DOI: 10.1002/jcc.24307

2) Hydrophobic, Steric and Electronic significant molecular descriptors included in the predictive QSAR model of BACE 1 and BACE 2 inhibitors allow the structural interpretation of the biological process, evidencing the main role of the shape of molecules, its hydrophobicity and its electronic properties in the transport and the Ligand-Receptor interaction.

3) QSAR models allowed the identification of five modified nucleotides as selective inhibitors against BACE 1/ BACE 2 in Alzheimer's disease. Phosphate group in the selective modified nucleotides have a positive impact in the inhibitor's activity. According the value of the significant molecular descriptors include in the QSAR models of BACE 1, a Phosphate group increases the MR, TPSA and LOG P values with a positive contribution to the BACE 1 inhibitor activity.

Conclusions


Q Clémenc ¸ot

P.-L Delafin

T Maître

Numerical Study of Bi-axial Cross-flow Turbine

Keywords: bi-axial turbine, CFD, cross-flow turbine, marine renewable energy, OpenFOAM, overset mesh

A bi-axial cross-flow water turbine is numerically simulated to evaluate its performance with a view to carrying out experiments on a prototype. Two axes of rotation and a chain-pulley system allow for an innovative blade kinematics. Indeed, the angle of attack of the blades is constant over a portion of their trajectory. This is a sought-after characteristics but difficult to obtain for conventional cross-flow turbines. 2D URANS k-ω SST simulations are carried out. Experimental data from an oscillating airfoil case are used to validate the numerical simulation process. The impact of solidity for one-blade turbine and the impact of the number of blades at isosolidity are studied. The unbalance performance between the upstream and downstream parts of the turbine are also investigated.

I. INTRODUCTION

D ARRIEUS turbines are named after their inven- tor, Georges Darrieus, a French engineer who patented them in 1931 [START_REF] Darrieus | Turbine having its rotatings shaft transverse to the flow of the current[END_REF]. This type of turbine has been widely studied in recent decades [START_REF] Worstell | Aerodynamic Performance of the DOE/Sandia 17-m Diameter Vertical Axis Wind Turbine[END_REF]- [START_REF] Clary | An optimally-controlled RANS Actuator force model for efficient computations of tidal turbine arrays[END_REF]. Their main particularity is to have an axis of rotation perpendicular to the current. This allows the turbine to have a rectangular swept area, making it particularly suitable for use on rivers, for example. But this particularity brings two obstacles to improving their performance: a non-constant angle of attack and half a revolution where blades travel in the wake of the upstream part of the turbine.

Even at constant rotational speed and constant upstream fluid velocity, the angle of attack of the blades (α) with respect to the relative speed of the fluid varies over a revolution (fig. 1). In order to simplify, α is classically calculated with the undisturbed upstream velocity. Under this assumption, α is a function of the blade angular position θ and λ = V /U ∞ , the ratio between the speed of the blades with respect to the fixed reference frame V , and the fluid speed far upstream U ∞ . The speed of a blade is given by the product of the turbine rotational speed ω and the turbine radius r: V = ω × r. Equation [START_REF] Darrieus | Turbine having its rotatings shaft transverse to the flow of the current[END_REF] gives an approximation of the relation between these parameters. α = arctan sin θ cos θ + λ

However, this equation that can be found in the literature [START_REF] Shiono | An Experimental Study of the Characteristics of a Darrieus Turbine for Tidal Power Generation[END_REF] is based on a very strong simplification so that the predicted angle of attack can be quite Q. Clémenc ¸ot (e-mail: quentin.clemencot@univ-grenoble-alpes.fr), P.-L. Delafin and T. Maître are with Univ Grenoble Alpes, CNRS, Grenoble INP, LEGI, 38000 Grenoble, France. different from the actual angle of attack. Firstly, the main velocity component decreases when approaching the turbine, leading to an angle of attack smaller than the one calculated using equation [START_REF] Darrieus | Turbine having its rotatings shaft transverse to the flow of the current[END_REF]. Secondly, to have a better approximation of the angle of attack, it is important to take into account the effects related to the dynamics of the blade. Migliore [START_REF] Migliore | Flow Curvature Effects on Darrieus Turbine Blade Aerodynamics[END_REF] showed that the flow curvature effects on a Darrieus turbine create in particular a virtual incidence that shifts to the left the lift curve of blade. Moreover, in the downstream part of their trajectory, i.e. θ ∈ [180°, 360°], the blades evolve in the wake created by their passage in the upstream part, i.e. θ ∈ [0°, 180°]. The variation of angle of attack in the upstream part may cause it to exceed the stall angle, leading to eddies creation and their advection in the wake of the blades. Therefore, the angle of attack is hard to predict in the downstream part of the trajectory and the power coefficient in this zone is lower than in the upstream part. These difficulties are not encountered by horizontal axis turbines (HAT), so their performances is still higher than cross-flow turbines [START_REF] Hau | Wind Turbines Fundamentals, Technologies, Application[END_REF]. In an attempt to overcome the limitations of classic cross-flow turbines, several possibilities are being explored. Passive or active variable pitch control technologies, for example, have led to an increase in efficiency. The blades are no longer fixed in the turbine frame but have a degree of freedom in their angular position. Delafin et al. [START_REF] Delafin | Performance Improvement of a Darrieus Tidal Turbine with Active Variable Pitch[END_REF] find an active pitching law allowing a 40% increase in the power coefficient together with a better balance in torque generation between the upstream and downstream parts. Even more disruptive innovations, moving away from the Darrieus turbines, are also envisaged in the world of cross-flow turbines. In the early 1980s, McKinney [START_REF] Mckinney | The Wingmill: An Oscillating-Wing Windmill[END_REF] described and experimentally tested a single oscillating wing system with simultaneous hammering and pitching motions, capable of extracting energy from the flow. In this configuration, the blade see only unperturbed upstream flow and never work in its own wake. Kinsey [START_REF] Kinsey | Computational Fluid Dynamics Analysis of a Hydrokinetic Turbine Based on Oscillating Hydrofoils[END_REF] used CFD tools to improve the performance of this turbine and achieve attractive efficiency.

The turbine presented here is considered to partially overcome the non-constant angle of attack issue of cross-flow turbines by adopting two axes of rotation and a chain-pulley system (fig. 2). This design allows a perpendicular translation of the blades with respect to the upstream velocity on two portions of the trajectory. Naccache [START_REF] Naccache | Development of the Dual Vertical Axis Wind Turbine Using Computational Fluid Dynamics[END_REF], [START_REF]Parametric study of the dual vertical axis wind turbine using CFD[END_REF] performed a 3D Unsteady-RANS study on a single-blade turbine of this kind and obtained a promising power coefficient: Cp = 0.4. The aim of this study is to better understand the influence of both solidity and number of blades on the performance of the turbine.

II. TURBINE AND FLOW CHARACTERISTICS

A. General description

As represented in fig. 2, the blades are arranged at regular intervals along two chains, themselves connected to pulleys of the same radius r. The pulleys are placed so as to form two rotation axes orthogonal to the upstream flow velocity. Parameter h designates the distance between the axes. The total turbine height is therefore d = h + 2r and the distance travelled by the blades during one revolution is l = 2h + 2πr. Considering b the blade span, the projected area of the turbine is a rectangle of area S = b × (h + 2r). All turbines simulated in this study have an aspect ratio h/r = 4. The dimensions of the turbine are summarized in Table I.

One-blade, two-blade and three-blade turbines are simulated. Three different chord lengths are used: c/r = 0.8, c/r = 0.4 and c/r = 0.266. This makes it possible, in particular, to study the influence of the number of blades with a constant solidity (σ = N ×c/l). In order to get closer to the experimental test conditions, the simulations are not carried out at iso-Reynolds. The kinematic viscosity of the fluid is that of water, i.e. ν = 10 -6 m 2 .s -1 . Taking the relative velocity of the fluid during the upstream vertical translation, W = U ∞ × √ 1 + λ 2 , as the reference velocity, the Reynold number varies between Re = 8.43 × 10 4 (the shortest chord length and the lowest λ) and Re = 4.07 × 10 5 (the longest chord length and the highest λ).

B. Blade kinematics

The blades are NACA 0012 symmetrical profiles. There are two kinematic connection points (A and B) between a blade and the chain (fig. 3). These points are located at x A /c = 1/8 and x B /c = 7/8 of the leading edge respectively. The kinematic link at point A, close to the leading edge, is a pivot of axis (A, e z ).

The kinematic link at point B is closer to the trailing edge. It allows z-axis rotation and translation along the chord line. Points A and B will pass through the same points in space during a turbine revolution. During one revolution, a blade will thus successively have an upstream translation movement directed downwards, a rotation around the lower pulley (semicircle), a downstream translation directed upwards and finally a rotation around the upper pulley (semicircle). The blades are therefore either in translation, in rotation or in a more complex transition movement between these two states (fig. 3). The transition takes place when the attachment points A and B are not on the same rectilinear or circular portion. This transition avoids a theoretical discontinuity in the blade rotational speed. Indeed, with a rigid attachment at a single point, the blade rotation speed would be a step function of time passing instantaneously from 0 to ω. And according to the mechanics of non-deformable solids, this infinite angular acceleration would imply an infinite torque. In practice, the blade would be subject to a strong mechanical stress 4 times per revolution. A two-point fixing solution seems more interesting for the durability of the turbine.

C. Hydraulic power coefficient

The instantaneous power transmitted by the fluid to the blade is equal to the product of the mechanical action tensor by the kinematic tensor expressed at the same point. When writing this relation at point A comes equation [START_REF] Worstell | Aerodynamic Performance of the DOE/Sandia 17-m Diameter Vertical Axis Wind Turbine[END_REF]. belonging to the blade with respect to the fixed frame of reference in m.s -1 .

P (t) = - → F f →b (t). - → V A∈b/R0 (t) + -→ M A,f →b (t). - → ω b/R0 (t) (2
-→ M A,f →b is the moment exerted by the fluid on the blade calculated at point A in N.m. Finally, -→ ω b/R0 is the rotation speed of the blade with respect to the fixed reference frame in rad.s -1 .

To lighten the notations, the value taken by a time function φ at time t will be noted φ(t) = φ. Its temporal average will be noted φ. The instantaneous power coefficient received by the blade i is equal to the ratio between the captured power and the kinetic power of the fluid passing through the projected surface in the absence of the turbine, as in equation [START_REF] Dabiri | Potential order-of-magnitude enhancement of wind farm power density via counter-rotating vertical-axis wind turbine arrays[END_REF].

C p,i = P i 1 2 ρU 3 ∞ db (3)
In the case of a multi-blade turbine, the total instantaneous power coefficient is given by the sum of the contributions of the blades, as in equation ( 4). modeling for regions close to a wall to kmodeling for the rest of the simulation domain [START_REF] Menter | Two-Equation Eddy-Viscosity Turbulence Models for Engineering Application[END_REF]. This model is widely used in research and in industry [START_REF] Menter | Ten Years of Industrial Experience with the SST Turbulence Model[END_REF]. It allows in particular a good prediction of the flow in the presence of an adverse pressure gradient. One of its limitations is that it assumes a totally turbulent boundary layer, contrary to a transition model of type k -kl -ω for example.

C p,tot = i C p,i (4)

B. Domain & mesh

The rectangular simulation domain extends over a length of 55d and a height of 60d (fig. 4). Symmetry plane boundary conditions are applied on the upper and lower borders. At the entry of the domain, a uniform velocity is imposed U ∞ = 1m.s -1 with a turbulent intensity I = 8%. A zero-gradient condition is imposed for the pressure. At the exit of the domain, a uniform pressure is imposed.

The motion of the blades in the simulation domain is made possible by the use of an overset mesh. Two types of meshes are used. A first fixed background mesh extends over the entire simulation domain and at the boundaries of which boundary conditions are imposed. Each blade is surrounded by an overset mesh which is superimposed on the background mesh and follows the blade motion (fig. 4). Both types of meshes exchange information by interpolation. The cells at the border of the overset calculate the values of their field by interpolation with the closest background cells (fig. 5). Some background mesh cells that are under an overset mesh are disabled. A background cell adjacent to a disabled cell calculates the value of its fields by interpolation with the overset mesh. The inverse distance method is used for interpolation. The behaviour of this type of mesh is summarized for a 1D case in fig. 6. This meshing method is more flexible to use than an Arbitrary Mesh Interface (AMI) since it allows to impose any trajectory. It is also well adapted to multiblade cases as it is possible to manage overset mesh trajectories independently of each other. Finally, this approach is potentially less costly in computing time and more robust than a dynamic remeshing.

C. Verification 1) Space discretization convergence:

The mesh convergence study was carried out for a classical non-overset mesh on a fixed blade at Re = 7 × 10 5 . The cells density according to the chord and according to the direction normal to the blade surface were multiplied until reaching a plateau on the value of the quantities of interest, namely the lift and drag coefficients. The retained mesh size divides the blade into 364 cells. The thickness of the cells attached to the profile ensures a y + ∼ 2 in their centre. Thus, at least one cell of the mesh is located in the viscous sublayer (fig. 7).

The background mesh is composed of 136 × 10 3 cells and each overset mesh of 28 × 10 3 cells. The total number of cells for each simulation is summarised in Table II. The smallest cells in the background mesh are located in the area where the blades are moving. The ratio between the size of the background mesh cells in this area and the size of the cells at the overset boundary is ∆x background /∆x overset = 1.58. The closer this ratio is to 1, the more accurate the interpolation between the meshes is. Thus, the mesh size ratio employed allows a correct interpolation while limiting the number of cells in the background mesh.

The simulations are run sequentially. Indeed, the implementation of solvers with overset meshes in OpenFOAM is relatively recent. The parallelization of this type of simulation was observed in this study to bring little or no time savings, at least on the cluster configuration used. 2) Time step convergence: To ensure that convergence in time step is achieved, the average power coefficient during the 12th blade revolution is calculated on a oneblade turbine case with c/r = 0.8 and λ = 4 with different time steps. The results are presented in Table III. For the rest of the study, the time step is chosen so as to have 1200 time steps per revolution.

To evaluate the cycle-to-cycle convergence of the simulations, the evolution of the average power coefficient over each cycle is considered. After 25 turbine revolutions, a variation of less than 1% is obtained for most of the simulations, with the exception of some low efficiency cases. The worst convergence is 3% variation in C p between the last two cycles.

D. Validation

The validation of the numerical simulation results cannot be done by direct comparison with experimental results of the bi-axial turbine as it has not yet been experimentally tested. Therefore, we choose to validate the simulations by comparison with a more classical case of a pitching airfoil. In this configuration a NACA 0012 airfoil oscillates with respect to an axis located at c/4 from its leading edge. The angle of attack during its sinusoidal oscillation is given by equation [START_REF] Clary | An optimally-controlled RANS Actuator force model for efficient computations of tidal turbine arrays[END_REF].

α(t) = α 0 + ∆ α sin(ωt) (5)
This time varying angle of attack is a common feature with the bi-axial turbine. It is a case that has been studied both experimentally by Lee [START_REF] Lee | Investigation of flow over an oscillating airfoil[END_REF] and numerically in particular by Gharali [START_REF] Gharali | Dynamic stall simulation of a pitching airfoil under unsteady freestream velocity[END_REF] and Delafin [START_REF] Delafin | Comparison of low-order aerodynamic models and rans cfd for fullscale 3d vertical axis wind turbines[END_REF]. The overset mesh method was used in the present study with very similar characteristics to the turbine simulations and a slightly smaller background mesh. The overset mesh in this study and those in the bi-axial turbine simulations have the same O-grid topology. In both cases, the blade is discretized into 364 elements and the thickness of the first cell on the airfoil is defined in order to have y + max ∼ 1. The numerical parameters are the same as in the turbine simulations (turbulence model, numerical schemes). A comparison of the instantaneous lift and drag coefficients obtained is shown in fig. 8. In the present study, the lift coefficient (C L ) is well predicted from 0 • to 20 • during the upstroke phase. The peak of C L is predicted with a similar accuracy to the other simulations. In general, the simulation results presented in fig. 8 do not capture accurately the C L in the downstroke phase and our results tend to underestimate a little more C L in this phase. Regarding the drag coefficient (C D ), the present study gives results similar to the experiment close to the peak of C D . Away from the peak, the present study predicts C D with a similar accuracy to the other simulations (none of them being perfect). The simulations of Delafin [START_REF] Delafin | Comparison of low-order aerodynamic models and rans cfd for fullscale 3d vertical axis wind turbines[END_REF] as well as those of the present study use a standard k-ω SST turbulence model. It assumes a fully turbulent boundary layer which is not the case in the experiment due to the low Reynolds number. This may explain some of the discrepancies between the experimental data and the simulation results (like the hysteresis at low AoA). The simulations of Gharali [START_REF] Gharali | Dynamic stall simulation of a pitching airfoil under unsteady freestream velocity[END_REF] use a k-ω SST model with a low-Reynolds number correction which accommodates the transitional regime and therefore may explain the slight hysteresis observed at low AoA.

The results presented show that the numerical setup used in our study gives a similar level of accuracy to other published simulation results. The relatively poor agreement with the experimental data observed in the downstroke phase indicates that the turbine simulations at low tip speed ratios will have to be analyzed carefully in the semicircular motions. However, this validation process gives confidence in the results obtained in the linear regions and everywhere at high tip speed ratio, when angles of attack are below the stall angle.

IV. RESULTS

A. Effect of solidity and blade number on performance

For each turbine configuration, a mean power coefficient curve as a function of the λ is plotted. This bell-shaped curve gives the optimum λ (λ opt ) for each configuration. A prototype of turbine equipped with a pulley speed control system would allow the turbine to always operate at this λ opt . Conversely, if the turbine is designed with a constant drive speed, the hydraulic efficiencies achieved as a function of the upstream fluid speed can be deduced from a curve of mean C p versus λ.

A first set of simulations of single-blade turbines with different chord sizes allows to study the influence of the solidity parameter on the performances (fig. 9). The dashed lines correspond to the values obtained in the penultimate cycle, the solid lines correspond to the values of the last cycle. The fact that both continuous and dashed lines are almost perfectly superimposed indicates that the convergence in number of turbine cycles is reached. It can be seen that λ opt decreases as the solidity increases. This is an expected result as this phenomenon occurs for other types of turbine with horizontal or vertical axes [START_REF] Paraschivoiu | Wind Turbine Design: With Emphasis on Darrieus Concept[END_REF], [START_REF] Delafin | Effect of the number of blades and solidity on the performance of a vertical axis wind turbine[END_REF]. It also appears that σ = 0.028 leads to a higher peak of C p than the two other cases. A maximum efficiency C p = 0.47 is then achieved with a single-blade turbine. It is noted that a lower solidity will give a lower maximum C p but the performance curve as a function of λ will be flatter. These solidity-induced effects on turbine performance curve have been observed in the case of the Darrieus turbine by Paraschivoiu [START_REF] Paraschivoiu | Wind Turbine Design: With Emphasis on Darrieus Concept[END_REF].

A second simulation set is launched to study the effect of the number of blades at iso-solidity (σ = 0.056) on the efficiency (fig. 10). The 2-and 3-blade cases give very close and even perfectly similar power coefficients for λ ≥ 4. Again, this is an expected result that can be found for example in the case of Darrieus turbines [START_REF] Delafin | Effect of the number of blades and solidity on the performance of a vertical axis wind turbine[END_REF]. The difference observed at lower λ could be linked to Reynolds effects. On the other hand, the performance obtained with an 8cm chord does not coincide with the other two configurations. The difference could be explained by the effect of the c/r ratio on the dynamics of the flow in the turns. Indeed, in the turns the blade sees a curved flow. This induces a greater camber and virtual incidence as the c/r ratio is greater. With c = 8cm, this ratio c/r = 0.8 is a rather extreme case and rarely encountered in the literature.

B. Influence of the number of blades on the instantaneous power coefficient

In this section, the evolution of the instantaneous power coefficient during a blade revolution is studied. To this purpose, a position parameter, noted "s" and varying from 0 to 1, is introduced (fig. 11). It allows to locate the blade along its trajectory. By convention, s = 0 when the link-point A close to the leading edge starts its downward vertical translation.

The instantaneous power coefficient of a single-blade turbine with chord c/r = 0.4 and λ=5, corresponding to the maximum C p , is shown in fig. 12. At t/T = 0, the blade position parameter is s = 0. The downstream translation is the part of the trajectory where the blade receives the most power from the flow. During a part of the lower turn, the Cp becomes momentarily negative before becoming positive and gradually increasing again during the upward downstream translation. However, the power captured during the downstream translation remains significantly lower than the power captured during the upstream translation. This is due to the decrease in the flow velocity of the fluid as a result of wake effects (fig. 13).

Due to the large variation of the instantaneous C p during a cycle, two turbine configurations with the same mean C p can hide quite different behaviours. For example, fig. 14 shows the instantaneous C p of a twoblade turbine with c/r = 0.4 and λ = 4. At t/T = 0, the position parameter of blade 1 is s 1 = 0, and the position parameter of blade 2 is s 2 = 0.5. Therefore, both blades will be in translation or rotation simultaneously. So the variation of the total C p is more important than the variation of C p received by each blade. The instantaneous C p of a case with the same overall efficiency but with a chord such that c/r = 0.266 and in a 3-blade configuration is plotted in fig. 15. At t/T = 0, the position parameter of blade 1 is s 1 = 0, it is s 2 = 1/3 for blade 2 and s 3 = 2/3 for blade 3. This time, the phase shift between the blades results in an instantaneous C p that is always positive and varies much less around its mean value. The instantaneous C p of a single blade in both 2-and 3-blade configurations (fig. 14 field near the blade at different times in the cycle are represented in fig. 17 and fig. 18 for λ = 3 and λ = 4, respectively. The areas of overpressure on the blade are shown in red, the areas of depression in blue. The black arrows correspond to the forces transmitted by the blade to the chain at points A and B. If the arrows representing the forces are not visible it is because they are so small that they are hidden by the points of application of these forces (red point for A and blue point for B). This is notably the case for λ = 3 and s = 0.4. The scales of pressures and forces are the same in fig. 17 and fig. 18. At λ = 4, the angle of attack between the relative speed of the fluid and the blade is inferior to the stall angle during most of its trajectory. On the other hand, the blade stalls over most of its trajectory at λ = 3, which produces eddies that disturb the flow and decrease the performance.

During the lower turn, power coefficients for λ = 4 and λ = 5 vary similarly (fig. 16). In the two cases the flow is nearly attached (not shown here for λ = 5). The greater negative peak for λ = 5 could come from a greater friction due to a higher blade velocity. During the upper turn, the C p variations for λ = 4 and λ = 5 are no longer similar. This could come from the flow detachment at λ = 4 (fig. 18), which is not present for λ = 5 (not shown here). In the last case, the negative peak is higher than for the lower turn probably because the relative velocity is higher.

V. CONCLUSION

The bi-axial turbine concept seems to be able to achieve interesting efficiencies. 2D overset mesh simulations of one-blade, two-blade and three-blade turbines with different chord lengths have been performed using the OpenFOAM solver. This allowed to highlight recurrent behaviour that can be found with other types of turbines, notably the Darrieus turbines. For example, it appears that, all other things being equal, increasing the solidity causes a decrease in the optimal λ. On the other hand, turbines with different blade numbers but the same solidity will have very similar power coefficient versus λ curves. This behaviour is observed for c/r ≤ 0.4 but it ceases to be true when the c/r ratio becomes too high (here c/r = 0.8), perhaps due to the effects of virtual camber in turns.

Finally, the observation of the instantaneous power coefficient over a revolution cycle shows a difference in behaviour between two-blade turbines and threeblade turbines. Indeed, with the same solidity and the same average efficiency, the arrangement of the blades gives a momentary negative global coefficient for the two-blade turbines, whereas it always remains positive and is less fluctuating for a three-blade turbine.

Fig. 1 .

 1 Fig. 1. Scheme of a 3-blades Darrieus turbine and evolution of the geometrical angle of attack.

Fig. 2 .

 2 Fig. 2. Scheme of a 3-blades bi-axial turbine

Fig. 3 .

 3 Fig. 3. Point A and B of kinematic connections between the blade and the chain and their dimensionless distance from the leading edge (a). Examples of three kinematic states of the blade: a pure translation (b), a pure rotation around the lower pulley axis (c) and a composite kinematics of translation and rotation (d).

IIIFig. 4 .

 4 Fig. 4. Simulation domain.

Fig. 5 .

 5 Fig. 5. Overset mesh with interpolated border cells in green.

Fig. 6 .

 6 Fig. 6. 1D representation of the superimposition of an overset mesh (top) with the background mesh (bottom). The blue cells are calculated conventionally, the yellow cells are interpolated and the red cells are deactivated. The arrows represent the exchange of information by interpolation.

3 Fig. 7 .

 37 Fig. 7. Velocity profile scaled on the top surface of a blade at 0.3c from the leading edge at an angle of attack α=15°and a Re = 7×10 5 (blue dots correspond to u + values at the cell centroid).

Fig. 8 .

 8 Fig. 8. Aerodynamic force coefficients comparison for α(t)=10°+15°sin(ωt) with k = ωc/2U∞ = 0.1 and Re = 1.35 × 10 5 . The continuous line corresponds to the increasing Angle of Attack (AoA) and the dashed line corresponds to the decreasing AoA.

Fig. 9 .

 9 Fig. 9. Mean Cp as a function of λ for one-blade turbines with different solidity values.

Fig. 10 .

 10 Fig. 10. Mean Cp as a function of λ for iso-solidity σ=0.056 turbine with different numbers of blades.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Position parameter s for different blade positions and associated coordinates of point A.

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Evolution of the mean streamwise velocity for the one-blade turbine with c/r = 0.4 and λ = 5.

Fig. 15 . 4 .Fig. 16 .

 15416 Fig. 15. Instantaneous Cp for 3-blade turbine with c/r = 0.266 and λ = 4.

  5+h/l s = (2h+1.5πr)/l

Fig. 17 .

 17 Fig. 17. Pressure distribution over the blade surface plus vectors of the force exerted by the blade on the chain (left) and vorticity field (right) for a one-blade turbine with c/r = 0.4 and λ = 3. The scales of force vectors and pressure are the same as in fig. 18.

  5+h/l s = (2h+1.5πr)/l

Fig. 18 .

 18 Fig. 18. Pressure distribution over the blade surface plus vectors of the force exerted by the blade on the chain (left) and vorticity field (right) for a one-blade turbine with c/r = 0.4 and λ = 4. The scales of force vectors and pressure are the same as in fig.17.

TABLE II MESHES SIZE

 IISIZE