Design Process for Scaled Model Wind Turbines Using Field Measurements

The validation of a wind turbine design requires several prototype tests, including scaled model and full-size testing. The most effective scaled-model testing is in wind tunnels; nevertheless, extrapolating scaled data to full-size machines has several limitations due to the model's inability to reproduce all the operating conditions. The objective of a scaled-model test is to obtain data under controlled conditions, but the scaled-model behavior varies concerning the full-size prototype. These variations depend upon the specific analysis and validation. Thus, to define the scaled-model dimensions and operating conditions, several considerations must be taken, namely, the similarity parameters. The scaled model is designed to validate a full-size wind turbine, but the reference data for defining the similarity parameters are limited. The best procedure for validating a scaled model is the use of reference data obtained from a field turbine, and with these data, determine the dimensions, dynamic properties, and wind tunnel parameters. The scaling process is obtained for the dimensioning by applying the dimensional analysis [1]. Bolster and coauthors described the concepts and their historical evolution. The Buckingham method is one of the most used scaling procedures. It creates dimensionless groups (π groups) that simplify complex physical relationships, making analyzing, interpreting, and applying scaled test results to full-size prototypes more accessible. It is widely used in fluid mechanics, aerospace engineering, heat transfer, structural mechanics, and chemical engineering. According to this method, the number of dimensionless groups to define a problem equals the total number of variables, n (like density or viscosity) minus the fundamental dimensions, p (like length or time). When scaling a full-size prototype, it is necessary to define the type of similarities. The similarity between a full-size system and the scaled model had different definitions throughout history. Initially, only geometric similarities were considered, meaning the model and prototype have the same shape and proportions. In this way, both systems' flow path and boundary conditions were equivalent. The kinematic similarities imply that the flow field is similar between the model and the prototype; these similarities ensure that the flow analysis will be similar regardless of the time interval, which is critical for analyzing fluid behavior over time. To achieve dynamic similarity, it is often necessary to adjust the physical properties of the fluid, the flow speed, or other parameters in the model. Suppose all relevant dimensionless numbers match between the model and the prototype. In that case, the model is said to be dynamically similar to the prototype, and the results from the model can be confidently scaled up to predict the behavior in the real-world scenario. Finally, for a wind turbine, it is essential to include other similarities, such as the rotor's inertia, generated power, and structural dynamics, to represent the actual machine completely. Before defining the methods for scaling a wind turbine, it is important to remember that a wind turbine is a structure interacting with a fluid, similarly as a wing aircraft; thus, they have similar limitations.

Bolster et al., for the airflow around a solid (similar to a wind blade), recommended that the ratio of the acting forces on the surface be constant, and both systems must have similar dynamic responses [1]. They concluded that it is possible to reproduce the physics of an actual system if the dynamic similarities are fully considered. It has been demonstrated that thick airfoils' lift, drag, and stall conditions are susceptible to the Reynolds number, particularly when it is closed to 10 × 10⁴. This condition will depend on the actual wind tunnel characteristics. The profile must be adequate using the Froude number to achieve the same thrust and drag coefficients using power law for the upstream flow conditions. He et al. test a scaled model for finding the dynamic similarities in a flying wing aircraft [2]. They investigated a flying wing's flutter and modal behavior and used the dynamic model to study the entire wing structure. Shah reported the wind tunnel analysis of a damaged aircraft [3]. Korzun and Cassel reported scaling a single nozzle supersonic in a wind tunnel [4].

The scaling of large wind turbines presents a significant challenge due to the limiting dimensions of wind tunnels and the reproduction of wind conditions at the laboratory scale, for example, the test of a 60 m rotor (diameter) requires a geometrical scale of at least 20:1. Bottasso et al. presented work on wind tunnel tests of scaled wind turbines [5]. They used the Buckingham theorem, starting with the wind speed ratio, Reynolds number, Froude number, Mach number, Lock number, and the nondimensional natural frequency. They set equal values for the dimensionless numbers between the actual wind turbine and the scaled model. For the Reynolds number and the time to speed factor (wind tunnel), they proposed to minimize the sum of the two rations with a pondering constant k. Hao et al. analyzed several approaches for determining the correct scaling method for the aerodynamic and hydrodynamic loads on a floating offshore wind turbine [6].

There are two contradicting scaling models in a floating wind turbine: one is for the aerodynamic scaling model and the other is for the hydrodynamic case. Froude similarity is used for aerodynamic scaling but produces smaller Reynolds numbers. Some parameters they analyzed were the steady-state force, the drag disk, geometrical similarities (Froude scale model), redesigned blades, and the real-time hybrid model. The commitment is between the aerodynamic loads, the actual power, and their equivalent trust. The hybrid method modifies the scaled test conditions with a sophisticated control within the wind tunnel. The challenge is determining the appropriate scaled model to maximize the wind tunnel results complying with the aerodynamic forces and the power generated (speed and torque). A particular condition not considered is the tower's dynamic response and the excitation frequencies associated with the blades and rotor's dynamics. They propose a similar thrust and torque scaling model by redesigning the blade's profile; it is not necessarily a scaled version of the original design. The scaled blade has a different profile with similar torque and drag forces. The forces did not necessarily have similar behavior within the velocity range. Li et al. presented a method for designing the test procedure for scaled wind turbines. They kept the same tip ratio and Strouhal number. For the dynamic similarity, they considered the Jensen's wake model. Miller et al. studied the dynamic similarities of vertical wind turbines in a wind tunnel [7]. They used highly compressed air to enable high Reynolds numbers, tip speed ratio (TSR), and Mach number. Their experiments had a good correlation with field data. Vardaroglu et al. used another approach that considered the frequency domain of a six degree-of-freedom for the entire system when considering wave wind variations [8]. Yang et al. [9] find the similarities between the wake characteristics of a full-scale wind turbine and a scaled model [9]. They constructed the scaled model with the blade element momentum (BEM) theory and tested it in a wind tunnel. They found that the model had lower thrust coefficients than the wind turbine of the actual size. They assumed the scaled model should have the same tip speed ratio and Reynolds number as the original design. This assumption is because the tip vortex of both turbines has the same spiral structure.

Other applications described similar limitations as scaled wind turbines tested in wind tunnels. Baker et al. presented a paper related to scale analysis for predicting the interaction between passing trains and fixed structures [10]. Kurabuchi and Goto presented a method for evaluating the dynamic similarities in wind tunnel tests for building ventilation [11]. Ohba et al. analyzed the dynamic similarity of a scaled model for predicting cross-ventilation in buildings [12]. Sunny studied the drag coefficient of a scaled vehicle model compared to a full-scale computational model [13].

This paper is an actualization of a previous work [14] for designing and testing wind turbine-scaled models based on field data. Scaling models requires careful consideration of the studied system's physical laws and similarity criteria. The design of scaled models is crucial for predicting wind turbine operating conditions and validating new turbine designs. Thus, the method presented here considers most similarity recommendations, including the system's dynamic modeling.

According to the type of similarities described before, the three similarity parameters—geometric, kinematic, and dynamic—must be set for the design of a scale wind turbine to be tested in a specific wind tunnel [15]. The structural dynamic response and the rotor's dynamic behavior must also be included since these considerations modify the scaled model's mass and inertia.

where D is the diameter of the rotor, and the subscripts denote m for the model and p for the prototype.

The tip ratio was selected considering the wind speed range based on the field measurements, the wind tunnel conditions, and the optimal TSR = 5.5 of the full-scale design. This coefficient was obtained considering the starting torque and calculated during the actual machine's design process.

The Reynolds and Froude numbers are also verified. The goal is to maintain the same Reynolds and Froude numbers. It is difficult to control the wind tunnel to have the desired wind velocity; therefore, a commitment between these two numbers must be achieved.

The Strouhal number must be considered alongside the aerodynamic loads, the actual power, and the equivalent trust and drag forces. The construction of the scaled model neglects the kinematic viscosity. According to Gasch and Twele, achieving similar nondimensional curves of C_P, C_T, and moment coefficient C_M indicates identical flux conditions in a wind turbine [16]. Therefore, the scaling methodology must adhere to the fundamental principle of maintaining geometric and kinematic similarity, in order to achieve similar performances in terms of their dimensionless dynamic numbers (to have similar curves).

where $KMmM$ is the natural frequency of the scaled model, and $KPmP$ is the natural frequency of the full size turbine.

where $JM$ is the moment of inertia of the scaled model and $JP$ is the moment of inertia of the full size rotor.

Once all similarities are considered, it's crucial that the scaled model's blade design is adequate. It should have at least the first natural frequencies along the chord and perpendicular to the chord, matching those of the full-size blade. This is a key factor in replicating the behavior of the full-sized system.

The methodology for designing wind turbine scaled models is validated using an actual wind turbine (Fig. 1). The machine is installed at the Autonomous University of Queretaro, where the wind velocity varies between 3 and 19 m/s. The rotor has a 12 m diameter, a nominal generating power of 13.5 kW, a nominal speed of 60 rpm, and a constant pitch (the pitch angle can be adjusted but remains constant once the rotor starts). The main shaft is connected to a permanent magnet generator through a 2:1 speed increaser. Further details for this wind turbine are given in Table 1.

The aerodynamic design was obtained with Qblade™. This comprehensive open-source package not only simulates HAWT (Horizontal Axis Wind Turbines) but also models the entire rotor. Based on the BEM (blade element momentum) theory, it divides the blade into small segments to calculate the aerodynamic forces acting on each segment. QBlade determines the Lift (C_l), Drag (C_d), Moment (C_m), Axial Force (C_ax), Tangential Force (C_t), Power (C_p), Thrust (C_t) coefficients, and TSR, and uses them to simulate and optimize the aerodynamic performance, allowing to evaluate different blade geometries and operating conditions. The software then computes the coefficients and operating conditions for the wind range and the different pitch angles. Table 2 shows the distribution of profiles, chords, and twist angles according to the design of the 13.5 kW full-size model.

The analytical C_p coefficient, calculated with the BEM method, is plotted in Fig. 2.

The site's wind velocity varies widely, from 0 to 19 m/s. Figure 3 shows the histogram of the measured data.

The actual power curve was determined from test field data (Fig. 4). There is a difference between the experimental curve and the analytical results. Estimating the actual resistance torque was necessary to improve the BEM model. This torque was determined from the rotor's angular acceleration and its moment of inertia, which was calculated from the computer aided design (CAD). The angular acceleration was calculated from the spectrogram (Fig. 5) obtained from the vibration measurements rechorded at the gearbox. The maximum angular acceleration rechorded was 0.087 (rad/s²). Figure 5 shows two dominant frequencies around 1 Hz (60 rpm) and 2 Hz. The first dominant frequency corresponds to the rotor's speed and the other corresponds to the generator's speed. The difference between the measured data and the BEM's results is due to the starting torque and the wind speed variations that cannot be easily modeled.

The equivalent radius of gyration was estimated from the starting torque. It was calculated from the experimental data and compared with the CAD model. The blade's natural frequencies were determined with an experimental test. The blade was forced to vibrate with a gravimetric shaker, increasing the excitation frequency following a ramp-up and ramp-down path. Two tests were conducted, one with the applied force parallel to the chord and the other perpendicular to it. The results are shown in Figs. 6 and 7, and the natural frequencies are parallel to the chord 13 and 24 Hz, and perpendicular to the chord 16.8 and 30.5 Hz. These are the reference values for determining the scaled model blade stiffness.

The procedure for designing the scaled model considered the field data described in full size wind turbine (description) section and the wind tunnel dimensions (Fig. 8). The flow conditions were scaled based on the actual wind profile registered at the site.

Figure 9 show the designed experimental setup, featuring a wind tunnel with an effective area of 1.2 m × 1.2 m and a testing length of 3 m. The wind tunnel is equipped with a sophisticated flow controller, ensuring the regulation of wind speed from 0 to 25 m/s with utmost accuracy. The scaled model, complete with a scaled rotor and a DC 60 W generator, is designed to simulate the load with precision. The instrumentation, including a wind velocity probe, temperature, and pressure sensors, and the voltage and current sensors, further enhances the precision of the experiment.

The scaling factors are summarized in Table 3 according to the geometric, kinetic, and dynamic similarity conditions discussed in full size wind turbine (description) sections.

The blade's scaled profile must be modified to achieve the same dynamic similarities between the scaled model and the full-size turbine, meaning the scaled model must have a different aerodynamic profile. The length of the scaled blade is computed with the dimension scaling factor, but the chord and aerodynamic profile are different. Since there are many possible solutions, it was decided to reduce the search to two types of profiles: the original NACA profiles (see Table 2) and a high-performing low Reynolds airfoil SD7003. It was a decision to maintain the same twist angle as the original design to reduce the number of input variables, reducing the decision-making process.

The procedure consisted of determining the power coefficient C_p with the BEM theory, a process that involved using Qblade™ to calculate the aerodynamic performance of the blade. This was done by modifying the chord's length of each section. For example, the model Chord 20 (Table 4) has the same profiles as the original design, but the chord's length is 20% longer. The model Chord 30–20 has two different chord increments; half of the blade has a 30% increment, and the other half has a 20% increment.

According to the wind tunnel dimensions and kinematic similarity factors, the best geometric scale factor was 15:1. Then, the other scale factors were calculated as described in Table 5. This parameter was implemented in QBlade™ to determine the performance of each proposed scaled model.

The position within the wind tunnel is set considering the measured wind profile at the site. Since the prototype was installed on-site, the wind has been continuously measured. (Figure 3 shows the histogram of the site's wind velocities.)

Figure 10 shows the calculated C_p coefficient of each proposed profile. In all cases, the twist angle was kept fixed, and the differences were the percentage of chord length variation from the full-size design (10–30%) plus an extra profile, the high-performance SD7003 profile.

The first step in the design process was to scale the full-size profiles using only geometric similarity. This initial design, which showed lower C_p values for all the TSR ratios, led to changes in the chord dimension and the blade profile. Each proposed design was then simulated with Qblade, and the results revealed a range of aerodynamic responses. The wind conditions for the simulations were determined using kinematic similarities and wind tunnel operating conditions. The model Chord 30, which was significantly different from the prototype, and Chord 14SD and Chord 20SD, which had similar C_p values at lower TSR ratios but not at higher ratios, all contributed to our understanding of the design process. The model SD7003 exhibited a completely different behavior, suggesting that modifications should be limited to similar airfoil designs. These findings are significant as they provide valuable insights into the aerodynamic performance of different wind turbine blade profiles.

It's essential to understand that the wind turbine is located in an area where the wind speed can change suddenly from low to very high within a short time. Therefore, the blade's design is a critical factor in ensuring the wind turbine's overall performance across a wide range of wind conditions, including gusty winds.

Figure 10 shows that using the SD7003 profile did not provide good results for the scaling objectives of the present work. The scaled model Model Chord 30–20 was the one that obtained the best performance for similarity with the three dimensionless parameters: C_P, C_T, and C_M. This profile has a 30% chord increase near the blade root and a 20% chord length change at the tip (Fig. 11). Table 6 shows the data for the final scaled design.

As discussed above, the procedure for determining the best similitude was integrating the cumulative difference between the scaled model and the prototype along the TSR range. This method gives an absolute value of the cumulative difference. The results are plotted in Fig. 12.

Figure 13 shows the C_P curves of the original design and the optimum scaled model (the blue line corresponds to the original design and the red line to the scaled model). The more significant differences are localized at some TPS ratios. Figure 14 shows the pitch moment coefficient of both designs; it is clear that the scaled model has lower values at low TPS ratios, but it has a similar behavior elsewhere. The trust coefficient (Fig. 15) is similar to TPS ratios under the optimum operating condition, and then the scaled model has lower values. As can be seen, the behavior of the aerodynamic coefficients was sufficiently similar.

The frequency and inertia similarities must be considered during the structural design. The frequency similarity defines the blade's weight and stiffness values, and the inertia similarity defines the mass distribution and center of gravity. The results of this part of the process are out of the scope of this paper.

This paper presented a method for designing a scaled wind turbine; the design process included geometric, kinematic, and dynamic similarities, plus two extra concepts that considered the mechanical behavior, namely, the frequency and inertia similarities. The similarity between field conditions and wind tunnel parameters must be as much as possible. The new scaled model had a similar aerodynamic behavior considering the geometric, kinematic, and dynamic similarities, and the rotor's moment of inertia and natural frequencies were kept as similar as possible. In this case, the power, trust, and momentum coefficients were similar between the full-size and the scaled model. The scaled model was designed to have similar dynamic behavior that will be verified in future wind tunnel tests, reinforcing the importance of maintaining dynamic similarities. The equivalent moment of inertia of the scaled model was determined from the starting torque of the prototype, keeping the same starting torque time. It was found that the methodology for designing the scaled blade profile assures the same dynamic performance as the original turbine. The most crucial aspect of this research was maintaining a similar starting torque, a factor that significantly influences the machine's behavior.

The disparity between the measured data and the BEM's results can be attributed to the challenge of accurately modeling the starting torque and wind speed variations. This disparity underscores the complexities of wind turbine design, where real-world conditions often deviate from theoretical models. The actual operating conditions played a pivotal role in determining other design parameters, such as the moment of inertia, the rotor mass, and the starting torque of the scaled design.

The design process was conducted thoroughly to minimize the cumulative difference of the C_p coefficients and find the best chord length modification. For this paper, only two types of profiles were evaluated, and it was found that, for small-scale factors, the increment in the chord's length is sufficient for achieving a well-scaled design. The other similarity values are used to fine-tune the wind tunnel conditions. The dynamic conditions of the other elements, such as the generator and tower design, were also meticulously considered for the scaled-model design.

The future work includes testing the scaled model in the wind tunnel, considering the kinematic similarities and the startup conditions. This testing will provide further validation of the design process and its potential for application in the field of wind energy.

The authors would like to thank Univesidad Autonoma de-Queretaro for the support on the field data, and Universidad Nacional Autónoma de-Mexico through PAPIIT TA100323².

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$CM$ =

pitch momentum coefficient

$CP$ =

power coefficient

$CT$ =

trust coefficient

$Dm$ =

diameter, scaled model

$Dp$ =

diameter, original prototype

$JM$ =

full-size blade's moment of inertia

$JP$ =

scaled model blade's moment of inertia

$KM$ =

scale model blade's stiffness

$KP$ =

full-size blade's stiffness

$mM$ =

scaled model blade's mass

$mP$ =

full-size blade's mass

$PotP$ =

full-size nominal power

$PotM$ =

scaled model nominal power

$TSRm$ =

tip speed ratio, scaled model

$TSRP$ =

tip speed ratio, original prototype

$Vm$ =

scaled-model's wind velocity

$Vp$ =

prototype's wind velocity

Greek Symbols

$λFreq$ =

frequency scale ratio

$λJ$ =

inertia scale ratio

$λL$ =

geometric scale ratio

$λV$ =

kinematic scale ratio

$ωm$ =

scaled-model's angular velocity

$ωP$ =

full-size's angular velocity