Experimental investigation and multi-objective optimization of savonius wind turbine based on modified non-dominated sorting genetic algorithm-II

A numerical data set will be developed to assist in the design of Savonius wind turbines. The main objective of study is to improve Savonius turbine blade designs to increase torque coefficients, rotational speeds, and pressure coefficients. Simulating 3D models and validating them with wind tunnel data were part of the experimental design methodology. Multi-objective optimization is used to optimize turbine performance. Twist angle, aspect ratio, and overlap ratio are all important factors in determining the optimal torque and power coefficients. Data-driven objective functions were modeled using the group method of data handling (GMDH). Using an evolutionary Pareto-based optimization approach, polynomial models were used to plot Pareto fronts and TOPSIS to calculate optimal commercial points. The torque coefficient, rotational speed, and power coefficient are all improved by 13.74%, 0.071%, and 5.32%, respectively. As a result of the multi-objective optimization of the turbine, some significant characteristics of objective functions were discovered.


Introduction
Due to environmental pollution, increased energy demand, and the depletion of fossil fuel resources, wind turbines have been gaining a great deal of attention.Since 2000, wind energy has contributed to an annual increase of 24% in electricity generation (Brenden et al., 2009).For the generation of electricity, there are two types of wind turbines: the horizontal axis wind turbine (HAWT) and the vertical axis wind turbine (VAWT) (Zare et al., 2023a(Zare et al., , 2023b)).This article focuses on the VAWT, and in particular, the Savonius turbine.There are a variety of applications for the Savonius wind turbine, including water pumping, ventilation of buildings, electricity generation, and hybrid renewable energy systems (Chong et al., 2013;Goodarzi and Keimanesh, 2015;Maleki et al., 2016aMaleki et al., , 2016b;;Roy and Saha, 2014).There have been numerous studies conducted on the geometric parameters of VAWTs (Dhamotharan et al., 2018;Naseem et al., 2020;Su et al., 2020;Tian et al., 2019;Xu et al., 2019).An experimental study was conducted by Damak et al. (2018) to optimize the rotor of a helical Savonius wind turbine.Based on their results, the modified twisted rotor displayed a higher power coefficient and static torque than the conventional twisted rotor.Montelpare et al. (2018) studied and reported experimental data on a modified Savonius system generating street lighting at different twist angles.To increase the rotor's aerodynamic performance, a conveyor and a deflector are incorporated into their design.The results of Shaheen et al. (2015) were compared with experimental data for the two-bucket Savonius wind turbine cluster in parallel and oblique positions.A numerical analysis of VAWT farms' performance parameters was also performed by Shaheen and Abdallah (2016).An experimental and numerical study was conducted by Lee et al. (2016) to determine the performance and geometric specifications of a Savonius wind turbine under different twist angles.A constant torque coefficient was observed at twist angles .90°whennumerical results obtained in an unsteady state were compared with experimental data.For the purpose of ventilating buildings, Tahani et al. (2017) designed and constructed a Savonius VAWT with a direct discharge flow.Savonius wind turbines have been designed with a twist angle that reduces negative torque and improves performance.Kamoji et al. (2009) examined the effects of overlap ratios, aspect ratios, and Reynolds numbers on the performance of the Savonius blade.It was found that the modified Savonius blade with an overlap ratio of 0 and an aspect ratio of 0.7 had the highest power coefficient of 0.21 at a given wind speed, achieving an overall power coefficient of 0.21.By comparing different blade shapes, Saha et al. (2008) concluded that when two blades were arranged adjacently, the performance was better than one, two, or three blades.Additionally, the twisted blades performed better than the cylindrical blades.
A new, innovative method of optimizing the blade geometry and guide gap flow of Savonius wind turbines was introduced by Al-Gburi et al. (2023).Increased overlap distances result in less efficiency since the returning flow becomes turbulent and interferes with the directed flow from fully connecting with the forward blade, thus reducing efficiency.It is reported that Alexander and Holownia (1978) added end plates to the turbine design in order to examine the effects of the aspect, overlap, and gap ratios on the performance of the turbine in a wind tunnel.Savonius turbines with two blades outperformed 3-and 4-bladed designs when aspect and overlap ratios were increased.Torres et al. (2022) recently reported on the performance of Savonius vertical-axis wind turbines.A numerical model with four key parameters will be used in this paper to generate 340 turbine configurations in order to propose a novel design optimization method.In a wind speed of 12 m/s, the optimized turbine exhibits superior aerodynamic performance with a maximum power coefficient of 0.21.Simultaneously, Marinic´-Kragic´et al. (2022a) focus on robust Savonius wind turbine deflector blade optimization, achieving significant power coefficient improvements, particularly with Scooplet-based designs.In another study, Marinic´-Kragic´et al. (2022b) combined a genetic algorithm with computational fluid dynamics (CFD) modeling to enhance the Savonius wind turbine power coefficient.In addition to developing a novel two-blade design that achieves 12% improvements in power efficiency, they refine Scooplet-based designs for four and six-blade configurations with a power efficiency coefficient of 0.32 and 0.34, respectively.Roy and Saha (2014) and Naseem et al. (2020) adopted the Bach-type blades for the Savonius turbine and studied the effects of geometrical parameters on the performance.Their wind tunnel and numerical simulations indicated that the power coefficient was increased by up to 16% for the modified Savonius turbine given the optimal geometrical parameters, compared with that of the conventional turbine with semicircular blades.
In order to improve the performance of the Savonius turbine, Damak et al. (2013) conducted an experimental study on the Savonius rotor with a 180°twist angle.A Savonius rotor with a twist angle of 180°outperformed a conventional Savonius rotor, according to their results.To improve the pressure coefficient of the Savonius turbine, Chan et al. (2018) conducted studies on the optimization of the blade shape of the turbine.A new geometry was proposed and analyzed using genetic algorithm (GA) and computational fluid dynamics.Based on the results, the optimized blade performed better than the previous model.For the design of vertical-axis wind turbines, Bedon et al. (2013) developed a database generation technique that was validated for symmetric profiles.There was an improvement in the performance of wind turbines as a result of the results.Jafaryar et al. (2016) conducted a numerical study based on the central composite design to obtain an optimal design for VAWT blades with an asymmetric geometry.According to their results, the maximum torque coefficient was obtained at a rotational speed of 450 rpm.Mohamed et al. (2011) proposed the use of symmetric airfoil blades based on coupling optimization and CFD algorithms to increase the tangential force generated by the Wells monoplane turbine.Their optimization of the process resulted in an increase in the output power and efficiency of the machine.
According to the literature review, there are no studies that apply the TOPSIS and modified NSGA-II algorithms to the multi-objective optimization of Savonius wind turbines.The NSGA-II has been extensively studied as a multi-objective optimization algorithm (Deb et al., 2002;Hosseini and Keshmiri, 2022;Shojaeefard et al., 2018).Although this method has been presented previously based on GA, it has been widely used in the literature (Dehghani et al., 2019;Li and Zhang, 2009;Lotfan et al., 2016;Wei et al., 2017).Savonius wind turbines have been studied experimentally and numerically, but little information is available regarding the role of geometrical and operational parameters.Specifically, based on the literature review presented above, there is a clear lack of research on multi-objective optimization of Savonius wind turbines using TOPSIS and modified NSGA-II which is fundamentally different from NSGA-II; this is a first-of-its-kind application of multi-objective optimization to a Savonius wind turbine.
An evaluation of the performance of a Savonius helical wind turbine is performed through experimental testing and numerical simulation in this study.Using available experimental data in the literature, the measured experimental data were validated.Afterward, the simulation results are validated against the experimental data.Based on the twist angle, aspect ratio, and overlap ratio as design variables, a series of simulations is conducted in order to obtain the polynomial of the models using GMDH.The objective functions selected are torque coefficient, rotational speed, and power coefficient.Using polynomial models, we then conducted a multi-objective optimization to optimize the turbine performance.TOPSIS is then used to determine the best trade-off points.

Test sample and apparatus
Figure 1 shows a helical Savonius wind turbine that was tested in a wind tunnel at Radman San'at Co.This is an open-circuit suction tunnel with a test section area of 600 3 600 mm.For aerodynamic testing, it is capable of reaching a maximum speed of 60 m/s.Four different days were scheduled for the tests in order to ensure accuracy and repeatability.The rotational speed of the turbine was calculated using a tachometer.To detect the rotation of the shaft, axis, blade, or any other rotating object, the optical tachometer emits a red beam on a luminous label.RPM is then calculated and displayed as the rotational speed.The setup operates in the range of 2.5-99,999 rpm.At speeds higher than 1000 rpm, the precision is 0.1 and 1 rpm, respectively.In front of the wind turbine, a standing hotwire was used to measure the airspeed.An S-shaped load cell was used to calculate weights up to 1000 N with a precision of 0.001.

Geometric model
Two twisted blades are attached to the central shaft for torque transmission.The turbine blades are made of PLA (Polylactic Acid) and have a diameter of 115 mm and a height of 162 mm.By keeping the extruded plastic warm, the 3D structure of the turbine shape can be manufactured using a 3D printer's robotic heat bed (Quantum Generous Pro).
The Quantum Generous Pro printer has an accuracy of 50 microns, and the largest dimensions are 300, 250, and 300 mm on the x, y, and z axes, respectively.As well, the 3D printing layer thickness has been set to 200 microns.The dimensions and operating parameters of the model, as well as the three-dimensional wind turbine model, are used in the simulations presented in Table 1 and Figure 1.Initially, the wind speed was adjusted to 7 m/s by positioning the turbine in the middle of the test section.For accurate measurement of turbine performance at different torques, an optical tachometer was preferred over a dynamic tachometer.Dynamic calculations were used to calculate the turbine torque.

Governing equations
The governing equations for numerical simulation are based on the assumptions of incompressible fluids.The time-averaged Navier-Stokes equations used for determining the mass and momentum are as follows: where the stress tensor, t, is related to the strain rate by: where m is the fluid viscosity.The Boussinesq (1877) concept which relates the Reynolds stresses to the mean velocity gradients with the turbulent eddy viscosity (m turb ) as the proportionality factor is used to model the stress tensor ( À ru 0 i u 0 j ).
where k is the kinetic energy of turbulence.Finally, based on the available data in the literature (Almohammadi et al., 2013;Shaheen et al., 2015), the shear stress transport (SST) model was selected as the most appropriate model.The transport equations for the kinetic energy of turbulence (k) and its turbulent frequency (v) are obtained from the following equation (Rodi, 2017): Where u 0 i is fluctuating velocity.The kinetic energy of turbulence deformation tensor is determined by: P k is production of turbulent kinetic energy due to interaction between relative flow and flow field: In which b Ã rvk is turbulent kinetic depreciation.The turbulent frequency deformation tensor is given by: F 1 is a blending function composition which is given by: Also: where y is a distance to the nearest wall, CD kv is the positive portion of the cross-diffusion term of equation ( 12): The turbulent viscosity is obtained using a limiter.Menter (1994) has argued that the use of this limiter does not increase the turbulent viscosity in regions close to the stagnation point: where O is the absolute value of vorticity given by O The blending function F 2 can be expressed by: F1 and F2 are the blending functions which are based on distance from nearest wall to blend the near-wall k-v model with the away from-wall k-e closure \ (recast into k-v variables), this being a fundamental attribute of the SST model.In the present work, these two functions redefined in a form that replaces wall distance with a local representation.

Solver settings
The governing equations were numerically simulated using ANSYS-CFX (Version 16.1).A high-resolution scheme, a SIMPLE scheme was used to discretize the advection term.In addition, 10 26 convergence criterion was applied.Figure 2(a) presents a rectangular duct with a cross-section of 3 3 3.5 m and a length of 7 m, comprising a rotating domain and a stationary zone.Wind speed is specified at the inlet, pressure is specified at the outlet, and all other boundaries are defined as non-slip.
Due to the complexity of the domain, ANSYS ICEM (Version 16.1) was used to generate unstructured tetrahedral and prismatic elements.The small gap between the blade and axis zones also required mesh refinements to capture flow details more accurately.Figure 2 illustrates the schematic of the mesh used in this study.It is necessary to solve the governing equations for different numbers of grid points to ensure mesh-independent results, and Table 3 presents the results of numerical simulation for wind turbine pressure coefficient, torque coefficient, and rotational speed.According to Table 3, no changes can be observed in these parameters when considering 8,510,325 cells.

Experimental data and numerical results validation
It was first necessary to examine the accuracy of the measurement system to evaluate the power and torque coefficients of the turbine as well as the rotational speed.Using a 3D printer, the blade was fabricated according to the specifications in Table 1. Figure 3 compares the power and torque coefficients with those reported in Damak et al. (2018).The Savonius wind turbine performance is characterized by the pressure coefficient (C p ) and torque coefficient (C T ) as follows: where P w represents the power generated by the turbine, r is the air density, V is the wind speed, h represents its height, and D represents its diameter of the turbine.
In view of the strong dependence of the Savonius wind turbine performance on wind speed, the effect of wind speed is examined for wind speeds of 4.9, 7, 8, and 8.8 m/s.The results of this study differ slightly from those of the reference study (Damak et al., 2018) due to friction, different power and torque measurement systems, and  different test section specifications.As part of these differences, oscillations are included along with the diagram as well as within the tip speed ratio (l = vd V ) interval of the turbine performance.Therefore, the measurement error of power and torque coefficients at the tip speed ratios between 0.7 and 0.9 is less significant than at the other tip speed ratios, possibly because of bearing friction embedded beneath the turbine.Further, when the speed ratio is low, the effect of wind speed on the pressure coefficient is less pronounced than when the speed ratio is high.By reducing the tip speed ratio, it is evident that the torque coefficient increases.Specifically, the torque coefficient occurs at C T = 0.15 and 0.35.The maximum error percentage for the power and torque coefficients is 3% and 4%, respectively, for tip speed ratio values of 0.8 and 0.61, for a wind speed of 7 m/s.
CFD simulations and wind tunnel experiments were kept consistent in terms of dimensions and aerodynamics to ensure a strong correlation.To achieve geometric similarity, a 1:3 scale Savonius wind turbine physical model was used in the wind tunnel experiments.We used a scaling factor of 3 for CFD simulations.Atmospheric parameters were precisely replicated in the wind tunnel experiments.In addition, turbulence models were selected to match experimental observations closely, with boundary conditions reflecting the wind tunnel's constraints.Through this meticulous approach, we were able to develop a comprehensive understanding of Savonius turbine performance under real-world conditions.
Figure 4 presents a comparison between the numerical and experimental results.As shown in Figure 4(a), as the load increases, the power coefficient reaches its maximum value of 0.16 at a tip speed ratio of 0.74, exhibiting a linear behavior up to the tip speed ratio of 0.47.The power coefficient significantly decreases when the turbine is excessively loaded and when friction is excessive in the transmission system as suggested in earlier studies (Hassanshahi and Kharati-Koopaee, 2023;Hosseini Imeni et al., 2022).Numerical results are in very good agreement with experimental data, which demonstrates the accuracy of the numerical method and assumptions used.It is also observed that the difference between the experimental and numerical results is the highest, with an error percentage of approximately 6%.The torque coefficient of the turbine is shown in Figure 4(b).As the applied load is increased, the torque coefficient increases up to 0.27 and then decreases slightly after that, which is consistent with the results from Fatahian et al. (2022).According to the aforementioned relationships, Figure 4(c) illustrates a linear relationship between the tip speed ratio and the rotational speed.Maximum and average relative errors are 4.68% and 2.85%, respectively.
As a result of the air flowing through the gap between the blades, there were significant improvements in wind power and torque coefficient, thanks to the pressure drag that was introduced on the suction side of the return blade due to the air flowing through the gap.The overlap ratio of the blades allows for the flow of air to pass between the progressive blades and the opposite blades when rotating blades are separated by an overlap ratio between them.As a result, this type of rotor is able to generate a greater amount of torque than other types.It is important to note that gravity may result in a higher output rate from the lower part of the rotor than from the upper part.Consequently, wind turbine shafts are designed with the upper part of the rotor closed and the central shaft sloped.This is so that wind flows downward into the rotor.

Modeling using GMDH
In simplest terms, artificial neural networks (ANNs) are new computational systems and methods for machine learning and knowledge representation that aim at applying the acquired knowledge for predicting the output of complex systems.These networks are partly based on the performance of biological neural networks when it comes to processing data and information for learning and knowledge creation.This idea focuses on creating new structures for the information processing system.Through electromagnetic communication, neurons, which are highly interconnected processing elements, solve problems in a coordinated manner.
Machine learning algorithms combine inputs with multilayered models, GMDH.This algorithm is considered advantageous due to its ability to adapt to different types of data, its capability for combinatorial learning, and its ability to reduce model complexity.To identify fundamental features in data, GMDH reduces model complexity.In addition, the algorithm also performs well in prediction tasks, especially when analyzing time series and modeling systems.Also, it can eliminate non-essential features using feature selection techniques.
There is no doubt that the GMDH algorithm is one of the most comprehensive and widely used neural networks in use today.Utilizing a variety of statistical parameters, it has been demonstrated that GMDH polynomials are highly accurate (Park et al., 2020;Roohi et al., 2021).As a result, polynomials were used for multi objective optimization of wind turbine parameters and extraction of the Pareto front using the modified NSGAII algorithm.
Table 4 lists the main design parameters for artificial neural networks.The turbine construction takes into account all operating conditions of the design points.This resulted in 60 different geometries being created using the specifications reported in Table 5.From our CFD analysis, we have obtained these different geometries separately.Blade twist angle, aspect ratio, and overlap ratio are three of the most important design parameters affecting turbine performance.A modified NSGA-II algorithm will be used for Pareto-based multi-objective optimization of the turbine.
To use the neural network, the data is divided into three categories.There are three categories of input parameters in the first category: twist angle, aspect ratio, and overlap ratio, and torque coefficient as the objective function.In the second and third categories, the input data are identical to the first category, but the rotational speed and power coefficient are considered the objective functions.The power, torque, and rotational speed of the wind turbine should be maximized to increase the efficiency of the wind turbine and optimize its performance.It is important to note that the ANN minimizes the functions.As a result, the torque and power coefficients, as well as the rotational speed, are inverted before being applied to the neural network.1/pressure coefficient, 1/torque coefficient, and 1/rotational speed are modeled as outputs using Ivankhenko's GMDH algorithm (Ivakhnenko, 1971).
Figure 5 displays the structure of the ANN for the objective functions, namely the power and torque coefficients and the rotational speed.The structure is the same for all three cases.
The ANN polynomials for the power and torque coefficients and the rotational speed of the turbine are as equations ( 17)to (19).Y 12 = 25:664 À 0:017u À 41:463a + 0:001u 2 + 22:735a 2 À 0:069ua Y 23 = 25:709 À 40:395a À 9:284d + 20:834a 2 + 41:163d 2 À 0:445ad Y 12 =À 0:00003 + 0:0002u + 0:058a À 0:0000008u 2 À 0:0293a 2 À 0:0001ua Y 23 = 0:023 + 0:012a À 0:018d À 0:006a 2 + 0:231d 2 À 0:005ad According to Figure 6, the power and torque coefficients and rotational speed predicted by the GMDH are compared to those predicted by the CFD.The results of the ANN are consistent with those of the CFD.Using the evolved group method of data handling type neural networks to obtain polynomial equations for ANNs, we applied multi-objective optimization to simultaneously vary blade twist angle, aspect ratio, and overlap ratio functions.Figure 6 clearly demonstrates that the polynomial equations predict the targets for the testing data that have not been utilized during training.These evolved GMDH-type neural networks are tested on two separate sets of data, one for training and one for testing, in order to demonstrate their prediction ability.In order to train the neural network models, 42 out of 60 input-output data sets are used.In the training process, the test set consists of 18 unpredictable input-output data samples, which are used only for testing the predictive ability of the evolved GMDH-type neural network models.
The performance of the ANN is evaluated using the absolute fraction of variance (R 2 ), root-mean-square error (RMSE), and mean relative error (MRE), which are defined as follows: where t is the target value, o is the output value and P is the pattern number.The results show that the model confirms the observations (Table 6).It is worth mentioning that the minimum of the bias for the power and torque coefficients and the rotational speed are 0.08, 0.002, and 0.056, respectively.

Multi-objective optimization
As multi-objective evolutionary optimization algorithms have been applied to diverse sciences, they have also become highly effective modeling tools for solving multi-objective problems.Every field of knowledge faces optimization problems with multiple objectives.Problems with conflicting objectives cannot have a single solution.Instead, the best possible compromise is sought between the objectives by applying good trade-off solutions.The improved VAWT blade shape (Figure 1(b)) was modeled using the modified multi-objective NSGA-II.The modified NSGA-II is one of the most widely used and most well-known multi-objective optimization algorithms.Multi-objective evolutionary optimization algorithms consider it one of their most fundamental members.The objective of this optimization is to obtain the best combination of design parameters (Shojaeefard and Zare, 2016).Nevertheless, there are currently different proposals of NSGA-II reported in the specialized literatures (Damavandi et al., 2017;Das and Hiremath, 2023;Safikhani, 2016).
A multi-objective optimization algorithm can be mathematically defined as the process of searching for the vector of design variables (X Ã = ½x 1 Ã , x 2 Ã ,:::, x n Ã , X Ã 2 < n ), such that the limitations imposed by m inequalities and p equalities are met: To optimize the vector of objective functions: In other words, all objective functions in the problem should be simultaneously optimized.Since these objective functions are conflicting so that and an increase in one affects the others.Hence, instead of a single optimal solution with respect to all the objective functions, a set of solutions known as the Pareto front exists (Coello, 1999).The modified NSGA-II is highly parallelizable, making it suitable for optimization problems involving large datasets or extensive computations.The algorithm can be executed simultaneously on multiple processors, resulting in reduced computational time and increased efficiency.Additionally, the flexibility of modified NSGA-II enables users to tune its parameters properly, improving their performance and enhancing the results.The algorithm's population adaptation, diverse exploration of solution spaces, robust convergence behavior, and excellent parameter tuning make it superior to other optimization algorithms.By combining these features, modified NSGA-II provides optimal solutions to complex problems.
This study uses the modified NSGA-II to optimize VAWTs.This modified NSGA-II is similar to the standard NSGA-II, with the exception that the -eliminate diversity method is replaced by the crowding distance division method in the modified NSGA-II.Using this approach, all clones and/or individuals with the same unique values will be removed from the current population.The elimination threshold in the present study is set at 0.001; hence, all members of the Pareto front within a certain distance of the individual are eliminated.Similarly, in order to prevent different individuals from being eliminated in the design variable space when they have similar values in the space of objective functions, similar values must be applied both within the objective and design variable spaces.A random selection of individuals from the population is then used to replace the eliminated individuals.As a result of this method, a more effective investigation of the search space is possible (Jamali et al., 2009;Shojaeefard et al., 2019).
According to the results and the relations used for wind turbine performance, the torque coefficient, rotational speed, and consequently power coefficient are affected by the twist angle, aspect ratio, and overlap ratio.
The flowchart shown in Figure 7 illustrates the process of establishing an artificial neural network and optimizing the wind turbine.Following the formulation of the problem, the design of the experiments, and the CFD simulations, the output data is used to construct the ANN structure.Datasets for training and testing are separated from the sample data.Using the least-squares method, the coefficients of the partial models are estimated.Based on the test set, the external criteria for each partial model are calculated, and the partial models with lower criterion values are used as input for the next layer or the final polynomials.
During the first stage of the optimization code, an initial population of N individuals is generated, and the objective functions are then assessed using polynomials.A non-dominant approach is used to classify the population using the e-eliminate diversity approach.The offspring population is created using a genetic algorithm that includes selection, cross-over, and mutation operators.A new population is created by reevaluating the objective functions, combining offspring and current populations.After the new population has been sorted again, N individuals are selected.An individual may be considered non-dominant when they are superior to others based on comparison of different objectives.Individuals in Pareto are entirely non-dominant as their cost function is assigned, since individuals in the second front are dominated only by individuals in the first front.Priority is given to individuals from the first Pareto.If the population size is not reached, the second Pareto individuals will be considered until the number of individuals is reached in order to start the next iteration.
In such a case, the GA operators are reapplied to produce the next generation, in order to get the Pareto optimal solution.Multi-objective optimization of turbine performance was carried out using the modified NSGA-II based on non-dominated sorting, taking into account the geometric and evaluation parameters mentioned above.The performance was optimized using polynomial ANNs.Three objective functions are optimized based on design variables: torque coefficient, rotational speed, and power coefficient.NSGA-II is modified to perform Pareto evolutionary optimization with multiple objectives.The following formulation is used to evaluate the problem: Objective Functions Design Variables Note that instead of maximizing the torque coefficient, rotational speed, and power coefficient, the inverse functions, namely 1/C T , 1/v, and 1/C p , are minimized.
For the three-objective optimization problem in 1000 generations, a population size of 200 is assumed with cross-over and mutation probabilities of 0.75 and 0.075, respectively.It is possible to plot individuals in different objective function planes as a result of solving this three-objective optimization problem.Using the same design variables, two-objective and multi-objective optimization problems are compared.

Results and discussion
The dominated optimal design points obtained from the three-and two-objective optimizations overlap, as shown in Figure 8(a).Design points are also shown in Figure 8(b) and (c) on other planes.Since front points in Pareto solutions are regarded as the best points, their corresponding design variables are regarded as the most desirable.The two-objective values corresponding to another set of design variables would rank lower than the Pareto front if another set of design variables were selected.The Pareto set provides the best combination of all three objectives when the design variables are selected according to the Pareto set.In each plane, all the Pareto front points dominate each other, but all are superior to the others.
As shown in Figure 8, the results of three-objective optimization in each plane include the results obtained from a two-objective problem, providing designers with more choices.Furthermore, the results of the two-objective optimization are located at the boundary of the three-objective problem, indicating the validity of the results.
According to Figure 8, the values of 1/C T obtained using the two-objective problem are the same as those obtained using the modified NSGA-II.As a result, it is evident that selecting 1/C p as an objective function gives the same results as choosing 1/C T ; therefore, they can be used interchangeably.1/rotational speed is the result of 1/C p and 1/C T drops.As 1/C p increases, the 1/C T increases, but pressure has a much lower contribution to rotational speed than torque.Therefore, in the wind turbine under consideration, torque dominates rotational speed.
A multi-criterion decision-making analysis method known as TOPSIS, which ranks the alternatives of Pareto solutions based on multiple criteria, was used in order to determine the optimal points at which all objective functions are compromised (Shojaeefard et al., 2019).Each diagram shows the optimal design points.A list of objective function values can be found in Table 7.The TOPSIS method was used to investigate the design variables of the obtained points.In this case, Point A is obtained by applying the TOPSIS algorithm to the results of a three-objective optimization problem.In the selected objective functions, the Pareto exponents obtained from two different objective functions are clearly visible.The use of TOPSIS for these Pareto fronts results in deviations from the optimal design points (B to D), compromising both objective functions.
Torque and power coefficients are improved by 13.83% and 5.30%, respectively, when considering both the torque and power coefficients as independent objective functions (Table 8).As can also be seen in Figure 8, single-objective optimization results (Points E to G) indicate the most optimal values for each objective.By considering 1/C T as a single-objective function, the lowest value of 1/C T (4.9) can be obtained.1/v and 1/C p are minimized to obtain the maximum rotational speed (36.1) and power coefficient (0.121).
Furthermore, the closest point to the ideal point method can be used to determine the optimal trade-off point of the optimal design.Using this method, the objective function values for all non-dominating points are mapped from 0 to 1, and the distance between each optimal point and the ideal point is calculated.The optimal design point 39 is determined by the minimum distance to the ideal point.Using the design point (Table 9) obtained from this method, the torque coefficient, rotational speed, and power coefficient are improved by 13.74%, 0.071%, and 5.32%, respectively.
The optimization process significantly improves the power coefficient regardless of the method used for selecting the trade-off points of the optimal design, as shown in Table 10.Thus, reducing undesirable forces can help reduce depreciation and related costs by determining the optimal operating point of the VAWT.On the basis of the three-objective optimization method, the torque coefficient was improved by 14.18% and 13.74%, respectively.As a result of all the optimized trade-off points obtained, the twist angle has been reduced as compared to its initial value.
Figure 9 compares the pressure and fluid velocity of the original and optimized turbines.In Figure 9(a), the original turbine has a larger high-pressure area than the optimized turbine, but the modified turbine has a higher pressure gradient across the driving blade, which results in higher torque.
Figure 9(b) illustrates backward flows toward retarded blades in both turbines, with the backflow being stronger for the optimized turbine, resulting in an improved torque.Additionally, the overlap ratio enhances the fluid's ability to enter and exit the turbine more smoothly, especially in the optimized turbine.Therefore, the optimized turbine will experience fewer losses and blockages, which will improve the performance of the turbine as a whole.
There is an increase in pressure on the suction side of the opposite blade as a result of the flow of air from the progressive blade to the return blade.As a result, the performance of the rotor is affected.Due to the low-pressure regions on ascending blades, the new design of rotors can be positively influenced by the rotation of the rotor and the power output.In light of the results, it is safe to conclude that the optimized blade can be used to reverse the process of large-scale wind rotors to achieve greater performance simultaneously.Considering the results obtained for both two-bladed wind turbines, it can be concluded that the design of the new turbine has a significant impact on the discharge flow rate of the wind turbine.However, it causes negligible reductions in the power coefficient of the turbine.
To provide a comprehensive view of the relationship between objective functions and design variables, Figures 10 to 12 illustrate optimal variations of the objective functions with respect to the design variables.The values of all design variables are distributed across their permissible domains.
According to Figures 10 to 12, the blade twist angle varies almost linearly; however, the aspect ratio and overlap ratio are not constant.This paper presents a multi-objective Pareto optimization procedure that discovers relationships that are indefeasible between wind turbine optimal design variables.By using this optimization method, designers can select the desired design parameters according to their requirements.
In each region, one design variable changes for all figures, while the other variables remain almost constant.Furthermore, in the middle region, all objectives are competing, so this region is expected to contain the final design point (Point A in Table 7).
Due to the absence of these points in the training and test sets, TOPSIS optimal points are reevaluated by CFD.In Table 10, the CFD results are compared with those obtained from the GMDH.According to the results of the GMDH, there is a good agreement between the CFD and GMDH results, indicating that the GMDH is an accurate model.
With the use of a multi-objective optimization method, it would be impossible to consider all the competition criteria in the design of the VAWT and also discover the relationship between the optimal design variables.

Conclusions
To evaluate the performance of a twisted Savonius wind turbine, both numerical simulations and experimental tests were conducted.At a wind speed of 7 m/s, the average deviation between numerical and experimental results was approximately 1.65%, indicating that the numerical method employed in this study was quite effective.In addition, a multi-objective optimization technique was used to optimize the performance of the turbine.To increase the power coefficient, torque coefficient, and rotational speed, twist angle, aspect ratio, and overlap ratio were considered to be design variables.Input-output data were used to model the objective functions using the GMDH neural network.The polynomial models derived from the evolutionary Pareto-based optimization approach (the modified NSGA-II) were used to plot Pareto fronts, and TOPSIS was used to determine the optimal commercial points.Comparison of three-and two-objective optimization data indicates that two-objective optimization data fall within the boundaries of a three-objective problem.Multi-objective optimization improved the torque coefficient, rotational speed, and power coefficient by 13.74%, 0.071%, and 5.32%, respectively.Several important characteristics of the objective functions were discovered during the multi-objective optimization of VAWT.Turbine design can benefit greatly from the combination of TOPSIS and NSGA-II.Based on our research, we believe there are four important points to be made: 1.With an overlap ratio of 0.16, an aspect ratio of 1, and a twist angle of 45°, the highest power coefficient of 0.136 was obtained.As a result, air is able to escape through the blades, preventing the formation of the drag force that resists the movement.2. The power coefficient initially decreased as the overlap ratio increased from 0 to 0.16, but then increased and reached its maximum value as the overlap ratio increased from 0 to 0.16.3.In response to an increase in the overlap ratio, the power coefficient again experienced a decreasing trend with a higher gradient.4. We find that the highest power coefficient is obtained with an aspect ratio of 1 and a twist angle of 45°.

Figure 1 .
Figure 1.(a) Wind turbine blades in the test section, (b) blade geometry, and (c) scheme of variables.

Figure 2 .
Figure 2. (a) Computational domain and framework, (b) the unstructured mesh of the whole computational domain, (c) the blade unstructured mesh.

Figure 3 .
Figure 3. (a) Variation of power coefficient and (b) variation of torque coefficient due to tip speed ratio in validation.

Table 4 .Figure 4 .
Figure 4. (a) Variation of power coefficient C p , (b) torque coefficient C T , and (c) rotational speed v versus tip speed ratio l.

Figure 5 .
Figure 5.The artificial neural network structure for objective functions.

Figure 6 .
Figure 6.(a) Comparison of the predicted results by artificial neural network with training and test data sets for 1/C T , (b) 1/v, and (c) 1/C p .

Figure 7 .
Figure 7.The flowchart of the wind turbine (a) artificial neural network and (b) optimization procedure.

Figure 8 .
Figure 8.(a) The non-dominated optimum design points in the plane of 1/C T and 1/C p , (b) plane of 1/C p and 1/v, and (c) plane of 1/ C T and 1/v.

Figure 9 .
Figure 9.Comparison between the original and optimized turbines (a) pressure distributions around turbines and (b) the fluid velocity vectors.

Figure 10 .
Figure 10.Variations of the 1/C T versus (a) the blade twist angle, (b) the aspect ratio, and (c) the overlap ratio.

Figure 11 .
Figure 11.Variations of the 1/v versus (a) the blade twist angle, (b) the aspect ratio, and (c) the overlap ratio.

Table 1 .
Geometric parameters of the helical rotor.

Table 5 .
Samples of numerical results using CFD.

Table 6 .
The values of the R 2 , RMSE, and MRE for evaluating the artificial neural network performance.

Table 7 .
The values of the design variables and objective functions of the optimum points specified using TOPSIS.

Table 9 .
The values of the design variables and objective functions of the optimum point specified using nearest to the ideal point method.

Table 10 .
Comparison between CFD and GMDH predicted values of the optimum points.

Table 8 .
The baseline values of objective functions.