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Models for characterizing Darrieus turbines and synthesizing wind speed scenarios

With the evolving global energy dynamics, new and innovative ways to integrate renewable energy technologies into residential and commercial complexes are being adopted. Techno-economic models such as NREL-SAM and HOMER are handy for evaluating such integration of renewables. However, the unavailability of libraries for vertical axis wind turbines (VAWT) and wind speed data limitations have been identified in the two platforms (NREL-SAM and HOMER). Accordingly, this project looks into developing models for evaluating power curve tables for Darrieus turbines and synthesizing wind speed scenarios. Analytical and numerical models for Darrieus turbines have been presented, and the ability to evaluate power curve tables from both models has been demonstrated. Double-Multiple Streamtube model was implemented for the analytical modelling, and 2-D CFD simulations on OpenFOAM were developed for the numerical modelling. The analytical model was found to fail for small-sized turbines, likely due to poor lift and drag coefficient values at high local angles of attack, but matched sufficiently with experimental results for large-sized turbines. Numerical simulations for Darrieus turbines were of two types - torque evaluation at constant wind speed and angular velocity, and starting characteristics for constant wind speed. Auto-regressive moving average models were implemented to characterize historical wind speed measurements, thus synthesizing wind speed scenarios. For this, the wind speeds or wind speed residuals (following detrending) were first normalized using the cumulative distribution functions, following which an ARMA model was fit.


INTRODUCTION
With an increasing popularization of renewable energy, more and more residential and commercial complexes are looking into developing microgrids and smart grids. As a result, multiple techno-economic models and platforms have been developed to facilitate the decision-making process behind the integration of these renewables. These include the likes of NREL's System Advisor Model (SAM) [19], URBANopt and Renewable Energy Integration and Optimization (REopt), and Hybrid Optimization of Multiple Energy Resources (HOMER) [6]. NREL SAM and HOMER are popular among such platforms in academia, with SAM having over 130,000 users and applications including grid integration studies, policy and utility rate design, and estimate Levelized Cost of Energy for systems. However, while SAM is open-source, HOMER is a paid software and has many subtle additional benefits. In this project, models to complement these platforms have been explored, particularly in wind power performance estimations.
SAM and HOMER have models to predict the hourly electrical outputs from wind farms or stand-alone wind turbines using the location's weather data and the turbine's power curve and wake model. For a turbine's power curve, the platforms have libraries of common horizontal axis wind turbines, which contain pre-defined tables of wind speed vs turbine output.
Uploading user-defined power curves is also possible by editing these tables. Figure 1.1 shows screenshots of the power curve tables in NREL SAM and HOMER.
All the pre-defined tables available in the library are specifically for horizontal axis wind turbines (HAWT), and the alternate vertical axis wind turbines (VAWT) are not included in the library. This is due to the sidelining of VAWT's by the wind energy industry over the years because of the superior efficiencies offered by HAWT's. However, factors including ease of manufacturing, simplicity of design, and the omnidirectional operation of the turbines have led to smaller sized VAWT's gaining popularity, specifically for residential use cases (like rooftop applications). But with limited commercial VAWTs deployed, power curve tables for these turbines are limited. With the intention of accommodating VAWT integrated systems on these platforms, generating power curves using analytical and numerical models have been explored in Chapter 2 and 3, respectively.
The most common design of VAWT's are Darrieus (lift-based) and Savonius (drag-based) turbines, and hybrid turbines (combination of the two) 9. Darrieus turbines consist of multiple aerofoil blades attached to a shaft, and as fluid flows past it, a lift is generated, thus producing torque. These turbines are known to achieve higher power coefficients than other designs, and have thus been chosen for modelling in this project. However, Darrieus turbines are also known to stall at low speeds leading to poor starting torques [9,10]. Thus, the build-up of angular velocity starting from a stationary state has also been modelled in Chapter 3.
Abundant weather data is also a necessary component for evaluating the wind resource potential for a system. Both platforms (SAM & HOMER) require a weather text file that contains at least one year's worth of data in an hourly or sub-hourly time step for their evaluation and optimization processes. Typical Meteorological Year (TMY), a database maintained by NSRDB that contains yearly data for many locations, is the default weather data loaded on both systems. Besides that, obtaining hourly weather data for a location, particularly for multiple years, is typically complex. HOMER has an additional feature of generating synthetic wind speed data using specific user-defined parameters. However, it does not offer the user features to determine these parameters using available historical data. In Chapter 4, models have been developed for generating hourly wind speeds for a location using historical wind speed data for the same location or a location with a similar climate.     Momentum or BEM (Blade Element/Momentum) models is based on calculating the flow velocity through the turbine by equating the aerodynamic force with the change in momentum of air flow. These models are however known to fail at large values of solidity, σ.
The first and simplest BEM model developed was the single streamtube model by Templin (1974). In this, the entire turbine is assumed to be enclosed within a single streamtube, and a uniform flow velocity within the swept area equal to the average of the free-stream and wake velocity is considered. This model is applicable for only lightly loaded turbines, and often over-predicts the produced torque. The multiple streamtube model was first proposed by Wilson and Lissaman (1974) where the swept area was divided into a series of adjacent and aerodynamically independent streamtubes. It was further developed by Strickland (1977) incorporating the wind shear effects.
The double-multiple streamtube model was proposed by Paraschivoiu (1977), where the turbine rotation was divided into upstream and downstream half-cycles, thus "doubling" the number of streamtubes. This is the model chosen for the project, and is explained in the following section.
Vortex models are potential flow models based on calculating the velocity field about the turbine through the influence of vorticity in blade wakes. It was first proposed for 2-D analysis of VAWT's by Larsen (1975). The turbine blades are represented by bound or lifting-line vortices whose strengths are determined using the airfoil coefficient for the calculated relative flow velocity and angle of attack. These models are accurate for only small local angles of attack, and are also known to take the most computational time.
Cascade models are commonly used for other turbo-machines, and was first applied to Darrieus turbines by Hirsch and Mandal (1987). In the model, the blades are rolled-out onto a cascade (a plane normal to the turbine axis) with the interspace equal to the circumferential distance between two adjacent blades. After calculating the local angles of attack and flow relative velocity, the blades are arranged in the cascade and the aerodynamic forces are evaluated for a revolution of the reference blade. This model is known to arrive at solutions much faster than the vortex model and is applicable for a wider range of solidity than the BEM models.

Double-Multiple Streamtube Model
An analytical model for Darrieus wind turbine has been developed in Python following Paraschivoiu (1977). While [25] was developed keeping curved-blade turbines in mind, the model has been implemented for straight-blade turbines in this report. [25] also considers tall turbines with local free-stream velocities varying as a function of height -this has been ignored in this project. presented, the flow past the turbine has been divided into multiple "streamtubes", each having an upstream and a downstream section. Along the streamtube, different induced velocities have been identified (V u , V e and V d ) assuming two actuator discs in tandem [11].

Induced velocities and local relative velocities
V u and V d are averages of V ∞ (free-stream velocity) and V e , and V e and V w (wake velocity) respectively. Interference factors to account for the decrease in velocities at the upstream and downstream discs have been introduced -u 1 and u 2 , and their values are determined in an iterative manner as described later ahead. Thus the induced velocities become: The upstream half-cycle of the rotor is for Azimuth angles −π/2 < θ < π/2. In this region, the local relative velocity at the blade tip is given by: and the local angle of attack is given by: where α 0 is the blade's angle of attack w.r.t. the tangential direction. λ u is the upstream tip speed ratio given by: The downstream half-cycle of the rotor is for Azimuth angles π/2 < θ < 3π/2. The local angle of attack (α d ), relative flow velocity (W d ) and tip speed ratio (λ d ) can be determined from Equations 2.4, 2.5 and 2.6 using V d instead of V u .

Force Coefficients
Due to the flow past the blades, an elemental lift and drag force is generated. The tangential and normal coefficients for a blade is given by: C L and C D are the blade's lift and drag coefficients, and is a function of both -the local Reynolds Number (Re) and the local angle of attack (α). The local Re for a given streamtube can be evaluated as: The values for C L and C D have been interpolated from the values in [28]. Other sources for these values include Xfoil [4] and airfoiltools.com.

Interference factor evaluation
As stated, the interference factors, u 1 and u 2 are iteratively evaluated. The convergence criteria from BEM theory for them are: It is taken that 0 < u 2 < u 1 < 1, and their values decrease with increased tip speed ratio. By starting with an assumption of u 1 = 1, u 1 can be iteratively determined. Following it, starting with u 2 = u 1 , u 2 can also be determined iteratively.

Torque and power coefficient evaluation
The average torque produces in one revolution is determined as: where, From the above, the over-all power coefficient for the turbine can be determined as: where, S is the swept area -which for a straight-bladed turbine is 2RH.

Results and Discussions
Results for large turbines Using the above procedure, double-multiple streamtube (DMST) model for a straight-bladed Darrieus turbine has been developed in Python. The turbine is considered to be of R = 2.5 m and σ = 0.22 with 3 NACA0015 blades, similar to SANDIA's 5-m curved-bladed turbine [27].
Experimental data for this turbine is available for ω = 150 rpm. Thus, for comparison purposes the results from the DMST model for the same angular speed have been considered.  experimental C P obtained from SANDIA's 5-m turbine [27]. It can be seen that the two graphs present similar characteristics. It must be noted that since the experimental data is for a curvebladed turbine, a higher C P and a shift in the peak is expected from the analytical solution.
We can thus say that the analytical model is able to predict the Darrieus turbine performance with sufficient accuracy for large turbines. Figure 2.3 (c) presents the Torque coefficient, C τ , obtained for the turbine using the DMST model. to determine the optimal power that can be harvested for a given wind speed -which is what is required for softwares such as HOMER and NREL SAM.
When small sized VAWT's are considered, the model was found to fail. A possible reason for this could be the high local angles of attacks encountered by small turbines. As stated, experimental data from [28] was used for the lift and drag coefficients in the model. These values were listed for discrete values of Reynolds number and angle of attack, and the model interpolates accordingly. Due to a high variation in C L/D and larger separation between consecutive α's, these interpolations are likely erroneous leading to the model's poor performance. Since analytical models require much less computational time than CFD models discussed in Chapter 3, these models are beneficial for parametric studies -though not relevant in the context of techno-economic models. Fig. 2.7 presents the variation in C P for varying solidity (σ) and blade angles of attack (α 0 ) as an example. The plots obtained are similar to those in [10]. Continuing, sudden drops in C P are observed in Fig. 2.7 (a). This is once again likely due to the erroneous interpolations for C L and C D , as was observed for small-sized turbines.   SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) [2] is an iterative numerical procedure for solving Navier-Stokes equations, and is meant for steady state flows alone.
In it, the discretized momentum equation and the pressure correction equation are solved approximate velocity field, then the pressure distribution, and finally the corrected velocity, and the processes is iterated until convergence [24]. PISO (Pressure Implicit with Splitting of Operators) [8] is an extension of the SIMPLE algorithm, and includes a predictor step followed by two corrector step, and is applicable for transient flows too. Implementation of the PIMPLE algorithm has been illustrated in Fig. 3.1. near-wall treatment, and is very sensitive to free-stream turbulence. In this project, the k − ω SST model [15], which is known to "combine best of worlds (k − and k − ω)" has been chosen for modeling the turbulence. The SST (shear stress transport) formulation emphasizes on the k − ω behaviour in the inner parts of the boundary layer and switches to the k − behaviour in the free-stream. The drawbacks of the model is over prediction of turbulence at stagnation regions, which are less pronounced for Darrieus turbines, and more computational requirements.

Geometry and Mesh Creation
The first step of a CFD simulation is the "pre-processing". This includes creating or deciding a geometry for the simulation, and following it generating a mesh system that can be used by the solver.

Geometry Generation
STL is a convenient and typical format used for 3-D geometries. In an STL file, a raw triangulated surface of the geometry is represented with a combination of the normal and vertices of these triangles. Generating the relevant STL files for the Darrieus turbine has been simplified using the Python library numpy-stl [26]. The Python file create_blade generates an STL file for a single NACA blade as per the inputs. These inputs include the blade profile, number of vertices, blade height (z-axis), chord length and angle of attack. Following it, the python file create_VAWT uses the generated blade.stl file to generate geometry.stl which represents the Darrieus turbine 1 . The numpy-stl library is also used to evaluate the created geometry's volume and inertia tensor, which proves useful for simulating the turbine's starting characteristics. Figure 3.2 presents a generated Darrieus turbine using 3 NACA0018 airfoils. The geometry.stl file is then converted to a 2-D mesh, as described ahead, allowing it to be used by the solver.

Mesh Generation
Creating a background mesh is a key component of the pre-processing stage. A simple cuboid of appropriate dimensions (sufficiently larger than the turbine radius) and divisions is created using blockMeshDict for this purpose. The left face is declared as an inlet patch and right the outlet patch, while the rest are simple walls. Since we are proceeding with a 2-D simulation, the z-direction is one cell/division thick, and a thickness of 0.05m is arbitrarily chosen.    Since the simulation has a moving component (the turbine), a dynamic mesh is required.
For this purpose, the arbitrary mesh interface (AMI) feature of OpenFOAM is used along with a dynamicMeshDict file. The circular region in Fig 3.5 is the AMI region, and is generated using an AMI.stl file through a process similar to geometry.stl. The size of the AMI region is chosen as roughly equidistant from the turbine geometry and the background mesh's boundary. The rate at which the AMI region, thus the turbine, rotates is determined by the dynamicMeshDict file. For the constant λ simulation, a constant angular speed is assigned, whereas for starting characteristics, the angular speed is determined using the 6-dof library.
The top-view of the final mesh as displayed on paraView has been presented in Fig. 3.6. The left face of the outer cuboid is the inlet patch for the system, and the right face is the system's outlet patch. Following the mesh generation, the pimpleFoam solver is called and the system is

Evaluation of Torque at constant (W sp , Ω)
In order to evaluate the C p and power curve for the given Darrieus turbine, it is important to evaluate the torques at various values of wind speeds (W sp ) and angular speeds (Ω). An OpenFOAM case for this purpose has been included in the CO_VAWT folder. A VAWT of radius 0.3 m with solidity 0.3 and 3 NACA0018 blades has been selected for the following simulations. In order to modify it for different situations, the following can be done: • To change the geometry of the VAWT, the geometry.stl file in constant/triSurface/ must be replaced.
• Should the new turbine geometry be too big or small, the background mesh and AMI region too must be modified. This can be done by modifying the blockMeshDict file in system/ and resizing AMI.stl in constant/triSurface/ respectively.
• omega in constant/dynamicMeshDict file must be changed.
• 0.orig/U must be modified with the appropriate wind speed.
Using the forces library, the forces and torque experienced by the VAWT mesh is tabulated for each time-step. The forces and moments due to pressure, viscous and porous effects across the 3 directions is tabulated in forces.dat. The sum of moments due to pressure and viscous effects about the z-axis gives us the torque required for our calculations. It must be noted however, that these torques need to be multiplied by 20H to find the actual torque, as    The power and torque coefficients for the turbine can be calculated as:  Upon comparing with experimental and numerical results of torque and power for VAWT's of similar sizes in [9], it can be stated that the results are in similar range and the plots retain the characteristics expected from a straight-bladed Darrieus turbine. The C P and C τ curves also retain the expected characteristics, but have much smaller magnitudes than [9]. This anomaly hasn't been resolved yet.

Analysing the Starting Characteristics
While starting characteristics of a VAWT play no role in energy analysis softwares, it is a useful analysis to have handy -particularly when Darrieus turbines are known to have poorer performance than Savonios and Hybrid turbines at low wind speeds [9,10]. The torque produced by the turbine at low speeds can be determined from the C P/τ curves generated from the previous section. This section studies the build up of angular velocity, Ω, from a stationary state for varying wind speeds and number of blades.
The motion solver sixDoFRigidBodyMotion [23] has been used for the simulations in    Table 3.1.  Table 3.1) with identical wind speeds of 7 m/s. Due to the time taken by simulations of sixDoFRigidBodyMotion solver, none of the graphs were able to reach terminal angular velocity. However, a comparison between the angular accelerations can be seen from the plots.
It can also be noted that for N = 2 blades, with lower solidity and greater separation between adjacent blades than for N = 3 or 4, the torque produced and the resulting angular acceleration aren't as uniform as for N = 3 or 4. Rabiti (2017) are examples of such methods. In [16], the hourly wind speeds, which resemble a Weibel distribution, is first normalized following which an ARMA model is fit. Using the parameters obtained from the fitting process, new scenarios are generated. In [3], the seasonality and trend is first removed from the wind speed data before normalizing and fitting the ARMA model. In order to remove seasonality, each month's data is addressed separately, and a Fourier series is used to characterize the seasonal trend. The "de-trended" data is then used and auto-correlation in the obtained residue is captured. The two studies are used as the reference in this project, and hourly wind speeds are synthesized with an ARMA model using historical data from NOAA 1 [18] and TMY2 [21].

Theoretical Background
An auto-regressive moving average (ARMA) model provides a framework to characterize a stationary (probability density function doesn't vary with time) series by establishing a linear dependence with the past values and a corresponding series of white noise [14]. A generic ARMA (p, q) model of a process {Z t } is given by: where φ i and θ j are the AR and MA parameters, and t denotes the white noise associated with that time instant. When p = 0, the model becomes an MA(q) model where dependence exists on the previous q noise terms alone. When q = 0, the model becomes an AR(p) model where dependence exists on the previous p values alone, with no dependence on the past noises.
HOMER Pro uses an AR(1) model to synthesize values for daily wind speeds for a user defined φ 1 and white noise standard deviation σ .

De-trending and normalizing
An almost stationary time series can be considered to have 4 components -level, trend, seasonality and noise. While level (average) and noise is what is reflected in an ARMA model, the trend and seasonality are unwanted components. A series without trend and seasonality would be "flat-looking" and without periodic fluctuations. To arrive at a stationary process, [3] analyses the wind speed data one month at a time to avoid seasonality, and fits a Fourier series to remove any additional trend and seasonality.
Data obtained from NOAA is for a continuous year and some deviation from a stationary process is observed. Figure 1.1 presents a comparison between plots for 45 days of data with and without de-trending using Fourier series. An improvement in characterization by the model can be seen. Data from TMY2 is, however, not for a continuous year. It instead considers monthly data from 1961 − 1990 and concatenates weather data that "best presents" the weather phenomenon of that location. Due to this, the need for de-trending the data like for NOAA isn't there.
While analysing the NOAA data, the trend and seasonality in the wind speed is represented using a Fourier series as presented in Eqn. 4.2.
{f k } is the set of frequencies used, and has been taken as { 1 500 , 2 500 , 3 500 , 4 500 , 5 500 , 7 500 }. For finding the optimal {A k , B k }, least square regression has been implemented -which leads to a set of 12 equations and 12 variables. The rest of the model fitting continues as is with the wind speed residuals, W r (= W sp − W F ), considered instead of wind speeds.
Finally, to obtain a stationary process, the data is mapped to a normal distribution as described in Eqn. 4.3 where Ω W and Ω N are cumulative distribution functions for W sp orW r and normal distribution respectively. The CDF for both distributions is evaluated numerically by implementing bins over the range.
The fitting is done through an iterative process starting with a guess value for Θ. The white noise at each instant, t , is estimated using the known process values, Z t , and the guess value for the ARMA parameters. For the first q instants, we assume that the process has no white noise. Meaning i = 0. Following it, for t > q, we estimate the white noise as: In the programs, a grid search was implemented to determine the optimal Θ which leads to the minimum σ .

Generating Scenarios
To generate wind speed scenarios, a set of {Z t } is first generated using Θ -which is then mapped back to a wind speed or wind speed residual using Eqn 4.5.
While generating {Z t }, the first p points are retained as it is with the corresponding t = 0, and the subsequent { t , Z t } are sequentially generated and evaluated respectively using Equations 4.1 and 4.6. To generate the white noise, the Box-Muller transform is used: where v 1 and v 2 are random variables in the range [0,1].
A flow diagram summarizing the series of steps involved in the generating synthetic wind speeds is presented in Figure 4.1. For models without the need for removing trends and seasonality, "Computing Fourier series coefficient" in Step 1 and "Add Fourier series" in Step 3 have to be ignored.

Plots and Discussions
The model fitting and scenario generation for hourly wind speeds have been programmed in C. Improvement in computational speed through parallel computing using OpenMP has also been explored as part of the course ID5130. The study was for ARMA(1,1) model with Fourier series de-trending, and a negligible improvement for 2 threads was found. More about it can be found in ID5130Report.pdf.

Impact of de-trending
From NOAA, hourly weather data, including wind speed, for a particular location from Feb 2016 to Oct 2017 was retrieved. From it, wind speeds across 45 days (1080 hours/points) was chosen for the following analysis.
For the scenario generation a total of 6 models have been developed -ARMA (1,1), (1,2) and (2,2), with and without de-trending.  We can thus state that the method is useful for synthesizing wind speed scenarios even when the historical data available doesn't represent a stationary process.  TMY provides weather data that best represents the weather phenomenon for 239 locations.
It is the standard data set used by NREL and is made available for analysis in both SAM and HOMER. As stated before, the data obtained from TMY2 was sufficiently close to a stationary process that further modification prior to normalization was found unnecessary.   This method has also been extended to generating wind speed scenarios for San Diego using SD's available daily average wind speeds and LA's deviation from the daily mean. However, with unequal standard deviations for the two locations, a correction/scaling factor (0.794) has also been included. Thus the wind speed is evaluated as: The scenarios generated using this has been presented in Fig 4.6, and compared with the actual hourly wind speeds for San Diego. This method can be particularly useful for running simulations for regions with limited historical weather data. Resources such as NASA POWER [17] makes it possible to obtain daily average wind speeds for any given location and date. This can be combined with historical hourly data of locations with similar climates, as presented above, to arrive at suitable hourly wind speeds for the given location -which is of interest to softwares such as SAM and HOMER.  A demonstration video [YT Link] by NREL for wind power system modelling on SAM is chosen for context for the following statements. In the demonstration, a wind farm has been modelled from scratch using the available libraries, and results such as power generated, investment and payback, and more were obtained. The parameters required for the simulation include wind resource, wind turbine specification, wind farm layout and wake characteristics, system costs and degradation factors, and financial parameters.
Ignoring wake characteristics and using default values for the degradation and financial parameters, a user is concerned with the first two input sections alone: hourly wind speed data and turbine specifications. a CSV file containing hourly wind speed data or provide parameters for a Weibel distribution, the latter of which provides results for average power alone. The library for hourly wind speed data is from the TMY dataset, which is centred around USA. Obtaining hourly weather data outside of it is typically tedious and often a paid service. The paid software "HOMER Pro" has features for synthesizing wind speed data (using AR(1) model) using user-defined parameters -which are often difficult to estimate. Features for learning from an input existing dataset to generate synthetic scenarios is absent. Chapter 4 of the report has addressed this limitation.
ARMA models have been developed with additional detrending features to characterize existing wind speed data and thus generate new scenarios. ARMA(1,1) models found to perform as good as ARMA(1,2) and (2,2) models. Thus, ARMA (1,1) models were found the most suitable model for wind speed characterization due to their relative simplicity. Two detrending methods -fitting a Fourier series and using daily mean speeds have been explored as well, and the characterization was found to improve. A comparison of the overall statistical characteristics between the synthetic and historical data and graphical comparisons between the two have been presented in the chapter. From it, we can conclude that the synthetic wind speeds sufficiently represents the historical data. The chapter also looks into synthesizing wind speeds for a location using the deviation of hourly wind speeds from daily mean speeds for a location with a similar climate, along with the daily mean speed of that location. This has been implemented for San Diego using the hourly deviation for Los Angeles, a city sufficiently close with similar climates, with TMY2 data. From figure 4.6 it can be said that the synthetic data is sufficiently close to the actual data and is a good representation. With daily mean wind speeds being a much more accessible property from datasets such as NASA POWER, this method is useful for generating data for remote locations. Thus, a new user of SAM with no access to weather data for the location of interest can use the ARMA models developed as an alternative. With the programs written in C, the results are easily reproducible and can also be easily modified. Thus, with the two sections, this project presents models for characterizing Darrieus turbines and generating synthetic wind speeds, thereby addressing the stated limitations a new user of SAM will have to concern with for wind power system modelling.

Scope for Future Work
• Flow field around a Darrieus turbine is three dimensional in nature [9], with considerable variation in velocity field across the height of the aerofoils. Thus, developing easy-toedit 3-D OpenFoam cases will be a desirable feature to be made available for the user. 3-D models will also allow the user to study and compare designs such as curved-bladed and helical-bladed Darrieus turbines. These simulations would however come with the draw-back of requiring far more computational time.
• In Chapter 2 it was observed that for high local angles of attack the interpolated values of C L and C D were likely erroneous. Programs such as XFOIL [4] 1 are known to have good interpolation algorithms to determine these coefficients at small Reynolds numbers. Thus, the developed analytical models can be integrated with XFOIL to see if the results improve.