Innovative multiobjective optimization of a curved‐blade vertical axis wind turbine modeled by modified double multiple streamtube method

The attractive specifications of vertical axis wind turbine (VAWT) are its operating with low noise and wind in various directions. To achieve a higher performance of these turbines, modeling of a curved‐blade VAWT by the modified double multiple streamtube (DMST) method and optimizing this wind turbine are performed. The power coefficient and the total normal force acting on blades were selected as two objective functions. Minimizing the normal force acting on turbine blades as the second objective function was not observed in the open literature. The amount of normal force is crucial in the structural design of VAWT because it generates both the bending moment on blades and forces on supporting arms of VAWT. Another innovation of this study is to consider the shape coefficient parameter that determines the shape of the rotor geometry as a design variable in optimization procedure. A 3 m height curved‐blade VAWT type was optimized by this multi‐objective optimization technique. The optimum values of the design variables obtained by the Genetic Algorithm were selecting an elliptical rotor geometry, pitch angle −1.4°, diameter/height 1.3, blade aspect ratio 19.8, tip speed ratio 4.3 and selecting three blades which could provide a power coefficient of 0.49 and a normal force of 158.7 N.


| INTRODUCTION
2][3] For harnessing the wind energy and converting the wind kinetic energy into mechanical or electrical energy, a wind turbine is typically employed. 4Vertical axis wind turbines (VAWT) and horizontal axis wind turbines (HAWT) are the two main groups of wind turbines.][14] Different approaches are applied to predict the performance of VAWTs.2][23] Templin in 1974 24 proposed single streamtube model as the simplest model to predict the performance of a wind turbine based on momentum theory.In this method, the whole rotor was surrounded by a single streamtube without considering two upstream and downstream turbine swept volumes.6][27] In another study, Strickland introduced multiple streamtube method 28 which contained a series of parallel independent streamtubes in the rotor domain and the momentum theory for each streamtube was used for each blade element.This model can be used for turbines with high tip speed ratios. 28,29he double actuator disk theory is a simplified mathematical model used to describe the flow of air through a vertical axis wind turbine (VAWT).In this model, the VAWT is represented by two actuator disks of equal diameter, one representing the upper half of the turbine and the other representing the lower half.The theory assumes that the whole rotor is surrounded by one streamtube. 30ouble multiple streamtube (DMST) is one of the most popular approaches which is often used in wind turbine design and optimization studies, as it enables designers to predict the performance of a wind turbine under different operating conditions.It combines the multiple streamtube and double actuator disc theory. 12,31,32In the DMST method, each pair of adjacent streamtubes features a double actuator disc in tandem which allows the velocity to change between the upstream and downstream of the rotor. 33,34treamtube expansion is a phenomenon that occurs in the streamtube model of a vertical axis wind turbine.As the air flows through the streamtubes, it undergoes a change in velocity and pressure due to the presence of the blades.This causes the streamtubes to expand radially outward, which reduces the velocity of the air and increases the pressure, Since the original DMST method (as well as other streamtube models) assumes zero expansion of the streamtubes (which is corrected in this study), and streamtube models assume that the flow is steady and incompressible, which means that the air density is constant and that there is no turbulence in the flow therefore it refuses to consider the wake blade interaction.and also, the effect of the downstream half of the rotor on the upstream half.Thus, DMST fails to accurately compute the aerodynamic loads on the rotor in complex situations.For these reasons, in some cases, there might be discrepancies between experimental data and computational results yet because of its mean low computational cost, high robustness, ease of implementation and also acceptable results in many cases, DMST remains a popular prediction model in VAWT design protocols. 35,36angga et al. 37 showed that a low solidity turbine (σ = Nc/R ≈ 0.2) is mainly influenced by the streamtube expansion effect (which is described in Section 2.1).They illustrated that using a modified DMST model, which is the combination of the standard DMST and streamtube expansion model, yielded results closer to experimental data.
Different authors presented several optimization routines to maximize the power output of vertical axis wind turbines as an objective function.Important contributions in this regard are cited below: Chen et al. 38 coupled DMST with covariance matrix adaptation evolutionary strategy (CMAES) optimization method (a heuristic search algorithm) to maximize power coefficient (Cp).Using three geometric design variables, including radius, the ratio of radius over height and number of blades, they improved Cp for 12.5%.
Tahani et al. 39 proposed a novel heuristic optimization method (combined continuous and discrete optimization routines) with the use of DMST.In their study, the chord and diameter of turbines were continuous design variables, however, the type of airfoil and blade number were discrete design variables.The average value of Cp among tip speed ratios of 1.5-9.5 is selected as the objective function.Optimization results showed 44% improvement of power coefficient in comparison with that for the original expressed turbine in their study.
Baghdadi et al. 40 applied a dynamic shape optimization based on computational fluid dynamics (CFD), utilizing fluid form deformation (FFD) algorithm and a mesh morphed coupled with Torczon optimizer for a three-straight blade VAWT with NACA 0021 airfoil.The objective of their study was to maximize the power generated by that turbine.The design variables were blade profile shape as a function of azimuthal position.Their results showed a significant improvement in power output.
Ning Ma et al. 41 used CFD coupled with Genetic Algorithm to optimize the airfoil shape of a VAWT with high solidity factor (σ = Ncl = S, the ratio of the total area of blades over the swept area of the turbine).The results

SANAYE and HOSSEINKHANI
| 2119 showed a 27% increase in the maximum power coefficient at a tip speed ratio of 0.9.
Bedon et al. 42 employed airfoil shape optimization with a genetic algorithm to increase the aerodynamic performance of a Darrieus wind turbine.The objective function in their study was the rotor torque (computed as an average value during one complete rotation).The optimal airfoil with the highest torque was named "WUP 1615".
Nguyen and Metzger 43 investigated the effect of different design parameters on power coefficient of a VAWT to find an optimal turbine design with applications in urban/suburban area.They considered thirteen design configurations (design #1 to #13) consisting of various turbine parameters such as height to diameter ratio (H/D), blade airfoil (symmetrical NACA 4-digit), blade chord (c) and turbine solidity (σ).The results showed that the turbine with H/D = 1.2, σ = 12%, c = 8 mm and blade airfoil shape of NACA 0015 (design #10) had the highest power coefficient (Cp = 0.15).In this study, the normal force (Fn) acting on turbine blades was defined as a component of aerodynamic load which might cause turbine blade failure.This normal force acting on blades is also computed for different turbine designs.As the result, the turbine with the selected design parameters (design #10) also experienced the lowest normal force (Fn=77.5 N).
Chattopadhyay et al. 44 introduced augmentation techniques that significantly enhanced the power coefficient and self-start ability of both lift (Darrius) and drag (Savonius) type vertical axis wind turbines (VAWTs).Devices like deflectors, guide vanes, converging ducts, diffusers, and stators which are located at the inlet airflow of VAWTs, resulted in a venturi effect that increased the positive torque (in the direction of rotation of rotor) which led to create power.Augmented devices for lift type VAWTs (Darrius type) also generated a greater lift force on airfoil-shaped blades by directing wind flow at an improved angle of attack and minimized the negative torque during the second half of the revolution.Research has demonstrated substantial increases in power coefficient (CP) and output power for various VAWTs by incorporating augmentation systems.

| Innovations in this research are as follows
Based on a thorough review of existing literature, there appears to be a gap in research as no study has specifically addressed the optimization of both the power coefficient of vertical axis wind turbines and the normal force applied on blades simultaneously.In this research, the modified double multiple streamtube (DMST) method is applied to model a Darrieus curved-blade vertical axis wind turbine.Then a multi-objective optimization method is applied to maximize Cp (to its possible highest value) and to minimize the normal force acting on blades Fn (to its lowest possible value).This systematic multi-objective optimization of a VAWT is not observed in open literature.
The reason for selecting the normal force as an objective function is that this force which is caused by wind flow on blades (aerodynamic force) is the main cause of bending and flatwise stresses acting on blades of a VAWT which leads to damaging and/or deflecting of blades.Furthermore, higher normal force dictates stronger supporting arms, its construction and higher cost.
Six design variables, including diameter to height ratio (β), blade aspect ratio (μ), number of blades (N), pitch angle (α p ), tip speed ratio (λ) and a shape coefficient, are also selected.Selecting Cp and Fn as objective functions, as well as achieving a new curvedblade turbine rotor geometry by optimizing, are innovations discussed in this paper.Furthermore, the blade airfoil shape also has a great effect on the power coefficient, normal force and torque distribution.In this study, the effects of three symmetrical NACA 4-digit airfoils (NACA 0012, NACA 0015, and NACA 0018) on power performance, normal force acting on blades and toque distribution of optimal turbine are investigated.Finally, the power coefficient of an asymmetric NACA airfoil (NACA 7512) is compared with NACA 0018 airfoil.

| Double multiple streamtube
Streamtube models are based on conservation of momentum theory and forces acting on rotor blades which change the flow momentum in a streamtube through turbine.In double multiple streamtube, flow is divided into a series of streamtubes.For VAWT operating analysis by the use of DMST, the main rotor is replaced with two independent halves in the upwind and downwind sections. 45,46enerally, the air free stream velocity profile changes with height as following relation 31,47,48 : cc where V ∞ i refers to the local ambient wind velocity in the vertical direction measured at a height Z i from the bottom of the wind turbine, Z EQ , is the distance from the bottom of the wind turbine structure to the equator plane (midpoint between the upper and lower surfaces of the wind turbine's airfoils) V ∞ i and Z EQ are shown in Figure 2. Also, α w , is the atmospheric wind shear exponent also known as the power law exponent, is a parameter used to describe the rate at which wind speed changes with height in the atmosphere.that varies from 0.1 to 0.4.
It is a function of the local roughness of ground and air temperature gradient.In this study α w is con- sidered to be 0.1. 49With defining interference factor (u), the upstream half of the rotor is: The equilibrium-induced velocity is: And finally for the downwind side of rotor:

| Upstream of the rotor
For the upstream side of the rotor   π θ π − /2 /2 the local relative velocity is: where X = rω/V and δ is angle between the blade normal and the equatorial plane.The expression for the local flow angle derived as: The local flow angle (φ) is defined as angle between relative velocity (W) and tangential velocity (V t = Rω).The pitch angle (α p ), the local angle of attack (α), the flow angle and also aerodynamic forces for an azimuthal angle θ are shown in Figure 1.
where α p (the pitch angle) is the angle between the chord line and tangential velocity and α is the local angle of attack.Due to the wind flow, aerodynamic lift and drag forces (F L, F D ) act on blades of a Darrieus VAWT.With the definition of angle of attack, α, resultant forces of F L and F D in directions of tangential velocity (F T ) and perpendicular to the blade (F N ) are also obtained.The normal force gives the structural force on blades which cause bending and flatwise stresses on turbine blades.Tangential force provides the generated torque of blades which is used to compute the power coefficient of turbine rotor. 50,51he upwind function (f up ) can be derived in terms of interference factor (u): Parameter η r R = / is defined and the upwind function is then given by equation: where: C L and C D for various NACA airfoils at various angles of attack are estimated by the use of available experimental data.The local Reynolds number is 52 : For a streamtube with a given velocity, with u the interference factor (equal to unity) and ω angular velocity, a value for local tip-speed ratio X is selected.Then, in this study values of Re l and α evaluated in the former try and values of the airfoil characteristics C L and C D were obtained from Q-blade software.This data is necessary for estimation of normal and tangential force F I G U R E 2 Curved-blade VAWT rotor geometry. 48efficients for blade sections from Equation ( 9) and (Equation 10).The parameter f up computed from Equation 8 and the new interference factor obtained from Equation 7. Iterations continued until successive sets of interference factor (u) was less than 10 .−4 46,48 Conventional models of curved-blade Darrieus VAWT are known as guy-wired phi-type in which the top of rotor shaft is connected to the ground by wire.The drawback of these turbines is the high axial force acting on bearings due to the weight of rotor as well as the connecting wire downward force. 53he modeled turbine in this study (Figures 2 and 3), has blades connected to the rotor shaft by supporting arms, while there is a space between the rotor shaft and blades at the top and bottom of the rotor.Furthermore, less axial force acts to the bottom of support bearing which reduces the need for high-capacity bearings and reduces the capital cost of the turbine.As shown in Figure 3, the parameter ζ (z/H) is considered to be between −0.9 to 0.9.The normal and tangential forces for each blade upstream are given by: C T are the blade normal and tangential force coefficients for an element at upwind where: The total blade torque for a complete blade is then obtained as a function of the blade position θ: The average value for half a cycle, that is, N /2 blades is thus given by: Then the average value for torque coefficient of half a cycle is also expressed as 46 : For specified rotor geometry, η = r/R is function of ζ = z/H as bellow, also the swept area of turbine is defined by Equation ( 20): −0.9 +0.9 (20)   Finally, the power coefficient is:

| Downstream of the rotor
The governing equations for downwind side of the rotor 3 /2 are: where: The same procedure of upwind estimation of power coefficient is applied for downwind part of the rotor, by considering approximately the first value of u′ as u.The normal and tangential forces for each blade in downstream are given by:

F′ C N and F′
C T are the blade normal and tange- ntial force coefficients for an element at downwind, where: SANAYE and HOSSEINKHANI The average value of torque for N/2 of blades, the average torque coefficient and the power coefficient for downwind half of the rotor are as: Thus, the power coefficient for a full cycle of the rotor is sum of two upwind and downwind power coefficients: (36)

| Streamtube expansion correction model
As previously mentioned, turbines with low solidity that can operate at high tip speed ratios are primarily affected by the expansion of the streamtube.This phenomenon occurs when the wind speed decreases, causing the flow to diverge towards the downwind section of the turbine, resulting in the expansion of the streamtubes.To address this issue, it is essential to enlarge the streamtube both upstream and downstream of the rotor.This requires correcting the interference factors in the standard DMST.As illustrates in Figure 4, by employing the continuity equation for each section, for expansion effects 54 : where e with u which is a new interference factor in upstream ( ) e is expanded length.As stated in Section 2.1, with defining u 0 as interference factor with no streamtube expansion, the modified inference factor for upstream is: For obtaining u in Equation ( 38), u n−1 was computed from previous iteration.Similarly, for downstream region 54 : To access the advantage of Modified DMST (M-DMST) model in comparison with that for Simple F I G U R E 4 Schematic of the streamtube expansion model. 54MST (S-DMST) model, these two models were developed and their results were obtained and compared in Figure 5 with experimental data for SANDIA 17 m.55 This comparison showed that M-DMST results had better agreement with the experimental results.

| Genetic algorithm (GA)
GA is used in recent years by many researchers [39][40][41] for optimizing the performance of VAWT.GA is a part of evolutionary process in which good characteristics dominate in the population because they are straight relevant to the survival of a member in a population.Due to GA high accuracy and reliability in simple and complex cases and due to GAs ability to avoid getting stuck in local optima, which is a common issue with other optimization methods, it is the most used among other evolution algorithms. 49,56The structure of this algorithm is such that it first creates an initial population with predetermined bound limits by using a combination of individuals.Then after the algorithm initialized, the fittest individuals which are defined based on the fitness function, are selected using the selection function.The fittest individuals exchange their genetic information by cross over function (mated together).Then set of new offspring is generated which inherit characteristic from both parents.All procedures are repeated until a convergence condition satisfied.Some of the individuals which have good genetic characteristics might have lost through the selection and cross over procedures.In this case mutation function is applied to reproduction that genetic characteristics.Figure 6 shows the processes associated with the genetic algorithm.

| Multi-objective genetic algorithm (MOGA)
Many engineering problems require more than one objective function.Genetic algorithm optimization technique can be used in these cases.In multi objectives, every point on Pareto front is a potential design point.Thus, in this research two decision making algorithms are used to select a point on Pareto front.Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) and Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) are two methods for decision-making process. 54In LINMAP method, the target point on Pareto front was selected based on the minimum distance of the points on the Pareto curve from the ideal point and in TOPSIS method, the final optimal point is obtained based on the minimum distance to the ideal point and the maximum distance to the nonideal point (the definition of ideal and nonideal points for this study is given in Section 3.2).
In the next step, all objective functions should be dimensionless before decision-making process.There are several methods for making objective functions Nondimensional.In Euclidian approach, the objective matrix is considered to be F ij , where i and j specify each point on Pareto curve and each objective function, respectively.Then, the following Nondimensional objective function (F ij n ) gained.

| Defining objective functions and design parameters
The optimization process in this study is performed by means of multi objective genetic algorithm (MOGA).The power coefficient of turbine determines the ability of a turbine to extract the kinetic energy of the wind and the normal force applied to the blades of turbine in a complete rotation leads to a bending and flatwise stresses on blades and supporting arms.Therefore, the purpose of optimization in this study is to maximize the power coefficient (objective function Ι) and to minimize total normal force acting on blades (objective function ΙΙ) of a curved-blade VAWT with three meters height and symmetric NACA 0018 airfoil.Design variables including diameter/height ratio, blade aspect ratio, tip speed ratio, Comparison of experimental results from ref. 55 with simple DMST (S-DMST) and modified DMST (M-DMST).
number of blades and pitch angle with their range of variation are given in Table 1.
The parameter η = r/R defines the rotor geometry.The torque in the upstream and downstream of the turbine is influenced by η, as indicated by Equations ( 16) and (32).Furthermore, the turbine power coefficient is directly related to torque, as is illustrated by Equations ( 18) and ( 21) also by Equations ( 34) and (35).Thus, the rotor geometry significantly impacts the turbine power coefficient.Additionally, the rotor geometry affects the normal force acting on the turbine blades, as shown in Equations ( 14) and ( 16) also in Equations ( 28) and (31).Consequently, the rotor geometry plays a critical role in optimizing both objective functions.
The last design variable here is j which shows the geometry of a curved-blade.For this purpose, the general shape of VAWT rotor for the present study is specified by defining a relation between Nondimensional radius and Nondimensional turbine height (Equation 41). 58In this equation (j), is the shape coefficient parameter with values in the range of 0.5 to 1.As shown in Figure 7 for j = 0.5, the blade geometry is elliptic and for j = 1, it is parabolic.
As stated in Equation ( 19), the Nondimensional local radius of the turbine (η = r/R) is a function of the Nondimensional local height (ζ = z/H), which is defined by the shape coefficient parameter (j) in Equation ( 41).This parameter determines the relationship between the local radius and height of the turbine, ultimately specifying its overall geometry.Based on Equations ( 16) and ( 32), the torque in the upstream and downstream of The flowchart of an optimization process using a genetic algorithm. 57[61] Variables Range the turbine has a direct relationship with η.Therefore, the value of j affects the torque.Additionally, as shown in Equations ( 18), ( 21), ( 34) and ( 35), the power coefficient of the turbine has a direct relationship with torque.Consequently, the power coefficient changes with the variation of j.

| Modeling validation
For validation, two cases are considered from references 47,48 ; (1) a 17 m curved-blade turbine with two NACA 0015 blades and parabolic rotor geometry, and (2) a 6 m straight-blade VAWT.The characteristics of these turbines are listed in Tables 2 and 3 respectively.The aerodynamic data of airfoils (NACA 0012, NACA 0015, NACA 0018 and NACA 7512) are computed for different local Reynolds number using Q-Blade software and then the obtained data used in our model for evaluating power coefficients, normal force and torque distributions.As are shown in Figures 8 and 9, the value of the power coefficient is computed in terms of different tip speed ratios for two turbine blades with two above mentioned airfoils.These figures illustrate that our modified DMST results are consistent with findings at references. 47,48herefore, based on the above results, the modified DMST model is used for further study.As Figures 8 and 9 show, power coefficient increases with rising in λ, due to the fact that with increasing λ, the relative velocity (W) increases, Equations 16 and 32 result in an increase in the torque value acting on blades in upstream and downstream sections which raises the power coefficient.This process continues until C p reaches its maximum value after which the power coefficient decreases with rising λ value.At this situation, the angle of attack decreases to a point where the air flow passes around the turbine instead of acting force on blades and making them to rotate.

| Optimization results
Optimization process is performed to maximize the power coefficient (objective function Ι) and to minimize the total normal force (objective function ΙΙ) acting on blades of the wind turbine with NACA 0018 airfoil and with six design parameters which mentioned in Table 1 in a complete rotating cycle.Optimization results converged and finalized by selecting the tuning parameters which are given in Table 4.
The results showed that for about 100 iterations or less, the convergence reached.The Pareto curve of the multi-objective optimization is shown in Figure 10.According to this figure, the maximum power coefficient is at design point A (Cp = 0.59), at which the normal force was at its maximum value (Fn = 623.7 N).Furthermore, the minimum value of the normal force was at point B (Fn = 30.6N), at which the power coefficient was at its lowest value (Cp = 0.19).Thus, if only the power coefficient is considered as one objective function, then design point A is selected as the optimal design point.However, point B can be selected as the optimal design point with the normal force as single objective function.
The power coefficient (objective function Ι) is a dimensionless parameter and the total normal force acting on blades of turbine (objective function ΙΙ) has Newton dimension.To obtain the final optimal design point, objective functions became Nondimensional by Euclidian method (described in Section 3.2).Pareto front curve in terms of Nondimensional objectives is shown in Figure 11.The ideal and nonideal points are marked in this figure.The ideal point here is the point that has the highest power coefficient and the lowest normal force and the nonideal point is the point with the lowest power coefficient and the highest normal force.
To find the final optimal design point, LINMAP and TOPSIS methods are used.As mentioned in Section 3.1 in LINMAP method, the decision-making process is based on the minimum distance of the points on the Pareto curve from the ideal point.In TOPSIS method, the final optimal point is obtained based Equation 39.
where Cl i is closeness coefficient, d i− is the distance from point i to the nonideal point and d i+ is the distance from point i to the ideal point.The point with the highest closeness coefficient is selected as the final optimal design point.As is shown in Figure 11 by both decision-making methods, an optimal design point is obtained.The values of the first objective function (power coefficient) and the second objective function (normal force acting on blades of turbine in a complete rotation) are 0.49 and 158.7 N respectively (Table 5).Table 6 lists optimum values of design variables.In this table the shape coefficient parameter (j) of the optimized turbine is 0.5.Thus, the turbine had the elliptic rotor geometry.The shape of 3 m curved-blade optimal turbine with five design parameters (mentioned in Table 6) is shown in Figure 12.
Considering the fact that this wind turbine is going to be used for residential buildings and regions with low and medium wind velocity, the wind velocity was F I G E 8 Comparison power coefficient for = 1.5-10 obtained from DMST modeling for parabolic rotor geometry with the results reported at reference. 48I G U E 9 Comparison power coefficient obtained from DMST modeling in terms of tip speed ratio with results reported at reference. 47A B L E 4 Tuning parameters for multiobjective genetic algorithm optimization applied in the present study.selected to be about 6 m/s.Then, by the optimum value of tip speed ratio (λ opt = 4.3), the value of rotational speed is 120 rpm which is in the typical range of 100-150 for a medium scale VAWT.The values of total power coefficient and normal force obtained for cases in which the power coefficient or the normal force are one objective function as well as when both parameters are two-objective functions are given in Table 7.

Population type
Results show that the highest power coefficient corresponds to when power coefficient is the objective function (Cp = 0.59 which is 20.4% bigger than that for two objective functions, Cp = 0.49).Also, in this case, the maximum total normal force is acting on turbine blades (Fn = 623.7 N which is 293% bigger than that for two objective functions, Fn = 158.7 N).On the other hand, the lowest normal force applied to the turbine blades occurs when only the total normal force is selected as the objective function (Fn = 30.6N which is 80.7% less than that for two objective functions, Fn = 158.7 N).In this case, the turbine has the lowest power coefficient (Cp = 0.19 which is 61.2% less than that for two objective functions, Cp = 0.49), which is not a suitable value for turbine power performance.Finally, in case of two-objective optimization, the optimum values of objective functions were power coefficient Cp = 0.49 and the normal force acting on turbine blades Fn = 158.7 N.
We are looking for a wind turbine with a maximum power coefficient (Cp) that closely matches the value obtained from a single-objective optimization with Cp as the objective function (Cp-objective).Additionally, we are interested in a wind turbine with a minimum total normal force that closely matches the value obtained from a single-objective optimization with Fn as the objective function (Fn-objective).As shown in Table 7, the wind turbine obtained through multi-objective optimization in this study meets our criteria.
Figures 13 and 14 provide power coefficient and total normal force curves for various tip speed ratios (2 ≤ λ ≤ 8) and highlight the values obtained from both two-objective and one-objective optimizations.These figures demonstrate that the turbine obtained via the Cp-objective optimization has the highest power coefficient, while the turbine obtained via the Fn-objective optimization has the lowest total normal force for different tip speed ratios.However, our goal is to find an optimal turbine that achieves both a high-power coefficient and a low normal force for various tip speed ratios.Based on Figures 13 and 14, the turbine obtained through multiobjective optimization has a power coefficient curve that closely matches the Cp-objective and a normal force curve that closely matches the Fn-objective.(Darrius type).The shape of 4-digit NACA airfoils is determined by three parameters.The first digit, define the maximum camber, the second digit define the location of the maximum camber and the last two digits are the maximum thickness of airfoil divided by chord length. 62Camber line, mean camber, thickness, and other parameters of airfoils are shown in Figure 15.NACA 0012, 0015, and 0018 are most frequently used in symmetrical NACA 4-digits due to their desirable aerodynamic properties, including their relatively low thickness distribution, good balance between lift and drag at low to moderate angles of attack, and established performance characteristics.

| NACA airfoil shapes and its effects on wind turbine operating parameters
In this study, the effects of three symmetrical NACA 4-digit airfoils (NACA 0012, NACA 0015, and NACA 0018) on power performance, normal force acting on blades and toque distribution of optimal turbine are investigated.Then the power coefficient of an asymmetric NACA airfoil (NACA 7512) is compared with NACA 0018 airfoil from previous section.
NACA 7512 airfoil is a commonly used asymmetric airfoil due to its favorable combination of lift-to-drag ratio and good structural strength.It has a relatively high camber, which allows it to generate lift at low angles of attack. 63Symmetrical airfoils are commonly used in designing VAWT for their low manufacturing cost, producing equal aerodynamic lift from both sides of airfoil during a full revolution of turbine and generating lift with change in wind direction. 59,62he Torque ripple parameter (γ) that determines stability and smoothness of rotor operation is defined by the use of the maximum torque (T max ) and the average torque (T avg ) from Equation 43.The larger γ means the larger discrete circumferential spacing among blades, which leads to higher torque variations.Thus, a smaller γ is desirable to avoid turbine vibration and noise. 64 Optimization with a one objective function (Cp or Fn) and a multi-objective optimization (power coefficient and total normal force as two objective functions) are also used to compute the values of power coefficient and total normal force acting on the blades of turbine for different airfoil shapes and results are shown in Figures 16 and 17 (as studied for NACA 0018 in Section 3.2). Figure 16 shows that for optimization with Cp as objective, NACA 0012, NACA 0015 and NACA 7512 has power coefficient of 0.51, 0.55 and 0.54 and total normal force of 469.4 N, 640 N and 652.3 N respectively.Figure 17 shows that for optimization with Fn as objective, NACA 0012, NACA 0015 and NACA 7512 has total normal force of 72.28 N, 31 N and 225 N and power coefficient of 0.15, 0.18 and 0.39.Figures 16 and 17 show that for multi-objective optimization the values of power coefficient are 0.42, 0.46, 0.49 and 0.43 and the values total normal force are 167.1 N, 162.3 N, 157.8 N and 314.7 N for NACA 0012, NACA 0015, NACA 0018 and NACA 7512, respectively.Therefore, if both turbine power coefficient and total normal force acting on turbine blades have to be considered, multiobjective optimization are approximately between the maximum power coefficient obtained from Cp-objective optimization and the minimum normal force obtained from Fn-objective optimization as shown in Figures 16 and 17.
With the use of multi-objective optimization process for NACA 0012, NACA 0015 and NACA 7512 the optimum values of tip speed ratio (λ opt ) are obtained 5.1, 4.2 and 2.7 for NACA 0012, NACA 0015 and NACA 7512, respectively.The effect of symmetrical airfoils NACA 0015, NACA 0018 and NACA 0012 on power coefficient and the normal force acting on blades of turbine in a complete rotation are given in Table 8.According to this table, the optimum airfoil with NACA 0018 has the highest power coefficient (Cp = 0.49) and lowest total normal force (Fn = 158.7 N).
Torque distribution curves acting on blades for one complete turbine rotation are given in Figure 18.The curves illustrate the variation of torque experienced by each blade during one complete revolution of the turbine.This lower number of peaks in the xtorque distribution curves indicates that VAWTs operate more smoothly, as the fluctuation in torque experienced by the blades is less frequent and intense.This smooth operation can result in reduced stresses on the blades and other components of the turbine, leading to increase the reliability and longevity of the system.
The torque ripple parameter (Equation 43) also relates to the same above issue.As Figure 18 shows, NACA 0018 has a lower torque variation.Furthermore, the torque ripple parameter computed for different airfoils showed that this parameter is 1.84, 1.96 and 3.3 for NACA 0018, NACA 0015 and NACA 0012 airfoils respectively.Therefore, NACA 0018 has lower torque variation, lower noise and vibration and higher overall power coefficient than other airfoils, which leads to a better performance for a VAWT with this airfoil.
The estimations for asymmetric airfoils are similar to symmetrical airfoils.In the present study the aerodynamic characteristics of lift and drag coefficients (C L , C D ) of asymmetric NACA 7512 are obtained by open-source software Q-blade.To better understand the difference in appearance of NACA 7512 and NACA 0018, the schematic of their sections is shown in Figure 19.Table 9 compares the power coefficient and normal force of the optimum turbine obtained by multi-objective optimization for asymmetric NACA 7512 airfoil (at its λ opt ) and NACA 0018 airfoil (at its λ opt ).Results show that the turbine with NACA 0018 airfoil has higher power coefficient and lower normal force than that for NACA 7512. ) has negative torque, therefore, at its (λ opt = 2.7) optimum tip speed ratio, the power coefficient of this airfoil is lower.

| CONCLUSIONS
For optimum design of a curved-blade VAWT, an innovative multi-objective optimization method is used to find VAWT optimum design variables.By maximizing power coefficient (objective function Ι) and by minimizing the total normal force (objective function ΙΙ) acting on blades, turbine design variables including diameter/ height (β), blade aspect ratio (μ), number of blades (N), pitch angle (α p ), tip speed ratio (λ) and shape coefficient (j) are obtained.The effects of symmetric and asymmetric NACA airfoil shapes (NACA 0018, NACA 0015, NACA 0012 and NACA 7512) on power coefficient, total normal force and torque distribution of the optimum turbine are also investigated.The main results are as below: VAWT, the parameter j is defined as the shape coefficient.This value (0.5) is obtained by the optimization procedure, thus, the optimal turbine has elliptical rotor geometry which is one of innovative findings.5. Comparison of optimal values for power coefficient and the total normal force acting on blades of VAWT with different airfoil shapes showed that multiobjective optimization provides better results than that for one objective optimization with either Cp or Fn as the objective.
Results for power coefficient and total normal force of VAWT showed that the symmetric NACA 0018 airfoil at the optimal point (λ opt ) has the highest power coefficient (Cp = 0.49) and the lowest total normal force (Fn = 158.7 N) acting on blades.6. Torque distribution curves of the optimal turbine with airfoil shapes (NACA 0018, NACA 0015 and NACA 0012) in a complete cycle shows that, NACA 0018 has the lowest torque variation and the lowest torque ripple (γ = 1.84).Thus, the optimal turbine has a smoother power performance with NACA 0018.7. Cp and Fn values of optimum turbine obtained by the use of multi-objective optimization with asymmetric NACA 7512 and symmetric NACA 0018 showed that, NACA 7512 had the lower power coefficient (Cp = 0.43) and the higher normal force (Fn = 314.7)acting on blades in comparison with that for NACA 0018 (Cp = 0.49 and Fn = 158.7 N).The torque curves also showed that NACA 7512 generates negative torque downstream of the rotor (   π θ π /2 3 /2).Thus, the use of NACA 0018 is suggested.

Abolfazl Hosseinkhani
https://orcid.org/0000-0002-4055-3959 Schematic of the flow angle, pitch angle, angle of attack, velocity vectors and aerodynamics forces (lift and drag as well as tangential and normal forces) acting at a desired position θ on the blade.

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I G U R E 7 Schematic of the rotor geometry for different shape coefficient (j).T A B L E 2The characteristic of VAWT turbine from reference.

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I G U R E 10 Pareto front curve of multi-objective optimization.F G U R E Nondimensional Pareto front curve of the multiobjective optimization.T A B L E 5 The optimum values of power coefficient and total normal force obtained by TOPSIS and LINMAP decision-making methods for the optimized wind turbine with NACA 0018 airfoil.

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4-digits NACA airfoils are the most popular and commonly used airfoils in design of lift type VAWT T A B L E 6 Results for optimum values of design variables of the turbine by multiobjective genetic algorithm for the optimized wind turbine with NACA 0018 airfoil.I G U R E 12 3D schematic of the optimal turbine.

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I G U R 14 Total normal force acting on blades of turbines curves in terms of tip speed ratio for optimization with one objective function Cp, one objective function Fn, and multiobjective functions Cp-Fn.F I G U R E 15 Airfoil nomenclature. 64SANAYE and HOSSEINKHANI | 2131 Figure 20  shows the torque distribution curves of optimum turbine obtained by multi-objective optimization in a complete rotation for the NACA 0018 and NACA 7512 airfoil.The turbine blade with NACA 7512 airfoil in the downstream part of rotor (

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I G U R E 18 Torque distribution curve of the optimum turbine obtained by multi-objective optimization for different symmetric NACA airfoils.F I G R E 19 Schematic of the symmetric NACA 0015 the asymmetric 7512 airfoils.T A B L E 9 The values of the power coefficient and the normal force of the optimum turbine obtained by multiobjective optimization for airfoil NACA 0018 and NACA 7512.I G U E 20 Toque distribution curve of a turbine for NACA 0018 and NACA 7512.

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With maximizing the power coefficient (Cp) as an objective function, then Cp max = 0.59 is obtained.With minimizing the normal force (Fn) acting on blades as an objective function, Fn min = 30.6N is obtained.With multi-objective optimization, both LINAMP and TOPSIS decision-making methods determined similar final optimal point on Pareto front curve.At the design point in this case, the values of the objective function І (power coefficient) and the objective function ІІ (normal force) are 0.49 and 158.7 N respectively.3. The optimum values of turbine design variables obtained from multi-objective optimization are: β = 1.3, N = 3, μ = 19.8,λ = 4.298, α p = −1.4°andj = 0.5.4. In the equation of rotor geometry for a curved-blade T A B L E 7 Values of power coefficient and total normal force acting on blades for optimization with one objective function Cp, one objective function Fn, and multiobjective functions Cp-Fn.
Values of power coefficient for optimization with one objective function Cp, one objective function Fn, and multi-objective functions Cp-Fn.I G U R 17 Values of total normal force acting on blades of turbine for optimization with one objective function Cp, one objective function Fn, and multi-objective Cp-Fn.Values of the power coefficient and the normal force of the optimum turbine obtained by multiobjective optimization for different symmetric NACA airfoils.