Evaluation of a Blade Force Measurement System for a Vertical Axis Wind Turbine Using Load Cells

Abstract

Unique blade force measurements on an open site straight-bladed vertical axis wind turbine have been performed. This paper presents a method for measuring the tangential and normal forces on a 12-kW vertical axis wind turbine prototype with a three-bladed H-rotor. Four single-axis load cells were installed in-between the hub and the support arms on one of the blades. The experimental setup, the measurement principle, together with the necessary control and measurement system are described. The maximum errors of the forces and accompanying weather data that can be obtained with the system are carefully estimated. Measured forces from the four load cells are presented, as well as the normal and tangential forces derived from them and a comparison with theoretical data. The measured torque and bending moment are also provided. The influence of the load cells on the turbine dynamics has also been evaluated. For the aerodynamic normal force, the system provides periodic data in agreement with simulations. Unexpected mechanical oscillations are present in the tangential force, introduced by the turbine dynamics. The measurement errors are of an acceptable size and often depend on the measured variable. Equations are presented for the calculation of measurement errors.

1. Introduction

The vertical axis wind turbine (VAWT) research at the Division of Electricity at Uppsala University has been going on for more than ten years. VAWTs are omnidirectional, and therefore, the yaw mechanism is excluded. Additionally, the generator may be placed at ground level. Major challenges associated with the design of VAWTs are the varying forces on the turbine and drive train [1]. To perform accurate scale experiments for wind turbines, the Reynolds number should match. For small-scale experiments (wind tunnels and water towing tanks), it is hard to reach realistic flows, as the Reynolds number is too low to obtain turbulent boundary layers. Therefore, it is important to carry out experiments at a larger scale to reach force coefficients similar to those of full-scale VAWTs.

Sandia National Laboratories conducted studies on force measurements of large-scale VAWTs with curved blades (Darrieus turbines) in the 1980s [2–4]. These experiments provided data with a high Reynolds number and are still used for the validation of simulation models of VAWTs [5–8]. However, data from force measurements on straight-bladed VAWTs (H-rotor) at high Reynolds numbers are still limited, since these require an operating VAWT at an open site. This motivates the type of measurement described in this work.

The 12-kW H-rotor built by Uppsala University in 2006 is a small-scale prototype designed for experimental purposes. The design has fixed blades, and the power absorption is controlled by adjusting the rotational speed though the control of the power output. This type of turbine is not reliably self-starting; instead, the generator is used as a motor to start the turbine. The starting process consumes negligible power and is described in [9]. The development and construction of the turbine prototype is further described in [10–12]. The generator is a directly-driven permanent magnet synchronous generator (PMSG) placed on the ground, designed to handle the varying power and rotational speed of the turbine [13]. The power coefficient of the turbine was studied in [14] and was found to peak at a value of 0.29 with a tip speed ratio of 3.3.

This paper evaluates a system for measuring the blade forces on the 12-kW VAWT by the use of four load cells attached between the support arms and the hub. Additionally, the control and measurement system for this experiment is described. The details, method, experimental setup and error estimations are described in Section 3. Example results and a discussion of the experimental approach are presented in Section 4. The conclusions are summarized in Section 5. First, Section 2 presents the underlying theory behind the studied forces.

2. Theory

2.1. Aerodynamic Blade Forces

The forces acting on the blade of a VAWT are shown in Figure 1. The wind speed at the blade $\overrightarrow{U}$ is usually lower in magnitude than the asymptotic wind speed ${\overrightarrow{U}}_{\infty }$, since the turbine extracts energy from the wind. The speed of the blade during rotation is:

$${\overrightarrow{U}}_b={\Omega }_{tur}R\widehat{t}$$

(1)

where Ω_tur is the rotor angular velocity, R is the rotor radius and $\widehat{t}$ is the unit vector in the tangential direction. The flow at the blade, due to its motion, is $-{\overrightarrow{U}}_b$, and the total flow velocity at the blade is ${\overrightarrow{U}}_{rel}=\overrightarrow{U}-{\overrightarrow{U}}_b$; see Figure 1. The angle of attack α is the angle between the chord line and the ${\overrightarrow{U}}_{rel}$, and the blade pitch angle δ is the angle between ${\overrightarrow{U}}_b$ and the chord line. The angle between ${\overrightarrow{U}}_{rel}$ and ${\overrightarrow{U}}_b$ is referred to as the angle of the relative wind, φ = α + δ. The lift force F_L is orthogonal to the relative wind vector ${\overrightarrow{U}}_{rel}$, and the drag force F_D is aligned with ${\overrightarrow{U}}_{rel}$. The tip speed ratio $\lambda =\frac{{\Omega }_{tur}R}{{\overrightarrow{U}}_{\infty }}$ influences the magnitude of the variation of α, which, in turn, affects the lift and drag force.

The resultant of F_L and F_D in the direction of the blade motion is denoted as the tangential force F_T, and the resultant of F_L and F_D in the radial direction is referred to as the normal force F_N. The tangential force F_T is used to estimate the turbine power, and the normal force F_N reflects the loads on the blades. The forces F_T and F_N are measured in this study according to the method described in Section 3.1.

2.2. Turbine Torque

The torque provided by one blade is:

$${\tau }_B=F_{T}R$$

(2)

For constant wind speed, constant wind direction, non-turbulent wind and for a complete number of turbine revolutions, the average turbine torque can be expressed as:

$${\tau }_{tur}=n_B{\langle {\tau }_B\rangle }_{rev}$$

(3)

where n_B is the number of blades and 〈τ_B〉_rev is the average torque provided by one blade for a complete number of revolutions.

2.3. Aerodynamic Simulation Model

The double multiple streamtube model is used in this study to obtain theoretical values of F_N and F_T. The model is based on the conservation of momentum, and it is described and tested in [8]. The only difference to the implementation in [8] is the dynamic stall modeling for low magnitudes of α. For an angle of attack α lower than half of the static stall angle α_ss (i.e., for |α| < 0.5α_ss), the dynamic tangential force coefficient C_T,dyn is substituted with the static C_T,st, found in [15]. For 0.5α_ss < |α| < α_ss, the linear scaling from C_T,st to C_T,dyn is applied to ensure the smooth transition between the static and the dynamic value of C_T. This modification is applied to ensure that the tangential coefficient approaches static values for low angles of attack.

3. Method

3.1. Blade Forces

Four sensors are used to measure the forces on one blade of the turbine. These are single-axis load cells, which measure tension and compression. The two blades without load cells were fitted with spacers of equal weight and radial dimension as the load cell assembly. Figures 2 and 3 show the assembly together with dimensions, the measured forces and the spacers for the unmeasured blades. To minimize the bending moment on the sensors, nuts were installed at both ends of each load cell, as seen in Figure 3. The sum of the measured forces F₀, F₁, F₂ and F₃ represents the radial force, F_R. Assuming that the rotational speed is constant, the radial force is the sum of the centrifugal force and the normal force, F_N, acting on the blade and support arms. Additionally, the differences between F₀, F₁ and F₂, F₃ is related to the tangential force, F_T, acting on the blade and support arms. Furthermore, the placement of the load cells allows for estimation of the bending moment of the blade and support arms, τ_bend. All of the forces and the bending moment can be summarized as:

$$F_R=F_0+F_1+F_2+F_3-F_{N,zero}$$

(4)

$$F_N=F_R-m{L}_C{\Omega }_{tur}^2$$

(5)

$$F_T=\frac{L_1}{2L_B}(F_0+F_2-F_1-F_3-F_{T,zero})\equiv \frac{L_1}{2L_B}{F}_{T,LC}$$

(6)

$$F_{bend}=\frac{L_0}{2}(F_0+F_1-F_2-F_3-F_{B,zero})\equiv \frac{L_0}{2}{F}_{B,LC}$$

(7)

where F_T,LC and F_B,LC are introduced to simplify subsequent equations. The definition and values of the constant parameters (m, L_C, L₁, L_B, R) together with their estimated maximum errors are given in

Table 1. Turbine parameters and estimated maximum error.

Measured Parameter	Value	Maximum Error
Turbine radius, R	3.24 m	±0.01 m
Center of mass, LC	1.83 m	±0.01 m
Distance sensor to blade, LB	2.99 m	±0.01 m
Vertical distance between sensors, L0	0.500 m	±0.005 m
Horizontal distance between sensors, L1	0.200 m	±0.0005 m
Mass of blade and support arms, m	35.79 kg	±0.05 kg
Turbine speed, Ωtur		±0.05 rpm

and Figures 2 and 3. The weight of the structure and the tension in the assembly add offsets to the measured forces F₀, F₁, F₂ and F₃. The offsets can be identified as the no-load forces or zero values of the turbine. These zero values, F_N,zero, F_T,zero and F_B,zero, are determined by:

$$F_{N,zero}=F_{0,zero}+F_{1,zero}+F_{2,zero}+F_{3,zero}$$

(8)

$$F_{T,zero}=F_{0,zero}+F_{2,zero}-F_{1,zero}-F_{3,zero}$$

(9)

$$F_{B,zero}=F_{0,zero}+F_{1,zero}-F_{2,zero}-F_{3,zero}$$

(10)

where F₀,_zero, F_1,zero, F_2,zero and F₃,_zero are the no-load forces measured by the load cells.

The output of the sensors is measured by a National Instruments (NI) NI 9237 module, which interfaces directly with the load cells. The NI 9237 is mounted to a NI 9191 cDAQ WiFi chassis to transfer the signals to the control and data acquisition system (described in Section 3.6). Data are sampled at 2 kHz. The setup requires about 10 W of electric power, which is supplied through weather-protected brushes from the turbine tower. There is also a small buffer composed of batteries and capacitors connected to the power terminals inside the signal conditioning box.

The NI 9237 was calibrated together with the load cells. During the calibration procedure, each load cell was subjected to known tension and compression from weights. For each sensor, a linear curve fit was made of the calibration values. For all measurement values in this work, the curve fits are used to determine F₀, F₁, F₂ and F₃.

As a verification of the system, the support arms and the load cells were assembled and mounted on a wall in the laboratory. Known forces were applied at the center of the blade to imitate tangential and normal forces. The method of obtaining F_R according to Equation (4) was verified with deviations of less than 5 N for forces of 15 N to 140 N. The measurement of F_T according to Equation (6) was verified with deviations of less than ±3 N for tangential forces of 25 N to 105 N. Additionally, attempts were made to use the load cells to verify the position of the center of mass. This was unsuccessful, since the no-load forces caused by tension in the assembly were difficult to separate from the mass of the assembly.

3.2. Load Cells

The load cells are sealed single-axis 4-wire cells with a capacity of ±5000 N. The maximum capacity of the load cells was chosen to give good resolution at moderate wind speeds. Therefore, the load cells were intentionally chosen too weak to be used at the rated wind speed, as this wind speed is relatively uncommon at the site. It was considered preferable to have high accuracy at the more common wind speeds. The horizontal distance between the load cells was chosen as relatively low, as the radial forces are much stronger than the tangential forces. A short distance is required to get a sufficient difference in forces between the load cells to be able to calculate the torque. Hence, the distance was chosen to get a difference in forces that has a similar magnitude as the mean force.

3.3. Turbine

The turbine is a straight-bladed Darrieus turbine, also known as an H-rotor. There are two support arms for each blade, and the construction is reinforced by an interconnecting cross; see Figure 4. For this experiment, the original design of the turbine was modified to hold the load cells. The turbine radius has been increased and, therefore, also the swept area. The turbine power and rotational speed are limited with installed load cells, due to the capacity of the load cells, as described in Section 3.2. The turbine specifications are listed in

Table 2. The turbine characteristics.

Characteristic	Original	With Load Cells
Power	12 kW
Rotational speed	127 rpm
Rated blade tip speed	40 m/s
Rated wind speed	12 m/s
Number of blades	3
Hub height	6 m
Swept area	30 m2	32 m2
Turbine radius, R	3 m	3.2 m
Blade length	5 m
Blade airfoil	NACA0021
Chord length	0.25 m
Tapering, linear	1 m (from tip)
Tip chord length	0.15 m
Blade pitch, δ	2°	2°

.

3.4. Generator and Shaft

The permanent magnet generator is located at the ground level and is directly connected to the turbine via a stiff shaft and a coupling, i.e., there is no gearbox. The speed of the generator was measured using three Hall sensors mounted in the air gap of the generator. The sensors register the passing of the magnets. The speed was estimated from the switching times of these sensors. Due to imperfections in the geometry of the rotor, the estimation of the rotational speed can acquire some errors. To even out these variations during the rotation, a moving average of the speed estimations from ten electric periods ( $\frac{10}{16}$ of one revolution) was used as the current speed value. For fixed speed operation and for a complete number of revolutions, these errors will cancel out. Therefore, the total error of the speed estimation is assumed to be low, as seen in Table 1.

3.5. Weather and Site Conditions

The location of the turbine is the Marsta Meteorological Observatory (59°55′32″ N, 17°35′12″ E) close to Uppsala, Sweden. The site has well-known weather conditions and a convenient location, rather than the potential of high wind energy production [16]. The turbine is positioned fairly close to the ground and to some obstacles that cause turbulence. Therefore, good knowledge of the wind direction will make it possible to choose data when the wind is less affected by the obstacles. The wind speed, wind direction, air temperature, air pressure and air humidity were measured with a Vaisala WXT520. The WXT520 is placed 15 m north of the turbine at hub height. The position is chosen to limit the disturbances from the turbine from the dominating wind direction, which also is the direction of lowest turbulence due to the site obstacles. The distance, 15 m or 2.5 turbine diameters, is based on the recommendations in [17]. The WXT520 has a serial digital RS-485 output protocol, which limits all of the weather measurement errors to the WXT520 device. All weather data were sampled at 1 Hz.

The wind speed and air density are important for the interpretation of the collected forces. The air density can be determined as:

$$\rho =\frac{1}{T}\left(\frac{p}{287.1}-2.700\cdot {10}^{-10}h\exp (0.06318(T+273.15))\right)$$

(11)

where p is the air pressure in Pa, h is the relative humidity in percent (%RH) and T is the temperature in °C [17]. The maximum error of ρ can be estimated in the same manner as described in Section 3.8. All maximum errors from weather measurements are specified in

Table 3. Maximum error of the weather measurements.

Measured Parameter	Max Error		Comments
Wind speed	ΔU∞ =	±0.3 m/s	at 0 m/s to 10 m/s
		±3 %	at 10 m/s to 35 m/s
Wind direction	ΔWdir =	±3.0°
Barometric pressure	Δp =	±0.05 Pa	at 0 °C to 30 °C
		±0.1 Pa	at −52 °C to 60 °C
Air temperature	ΔT =	±0.3 °C
Air humidity	Δh =	±3 %RH	at 0 %RH to 90 %RH
		±5 %RH	at 90 %RH to 100 %RH
Air density	Δρ =	±0.0016 kg/m3	calculated from Δp, ΔT, Δh
			valid for T < 15 °C, h < 90 %RH

.

3.6. Control and Data Acquisition

The control and logging software was developed in National Instruments LabVIEW. A total of three executables were running simultaneously, one on each platform; FPGA, real-time target (RT) and Windows Server 2008. A simplified illustration of the data and power flow can be seen in Figure 5. The AC output from the generator varies in frequency, and the voltage is diode-rectified into a DC-voltage. The DC-voltage is fed into a two-level (2L) inverter. To regulate the turbine speed, a PI-controller was implemented on the FPGA. The turbine speed was used as input and the modulation level of the inverter as output. The controller loop speed was 10 Hz, and a conditional anti-windup was used. The conditional anti-windup provides acceptable performance without adding more parameters to the controller [18]. The controller parameters were found from a simulation of the turbine and then fine-tuned by tests on site.

The power, rotational speed, controller output and weather conditions were sampled by the FPGA, transferred to the RT-controller and then transferred to the server for calibration and synchronization with the forces F₀, F₁, F₂ and F₃. All data were stored on the server hard drive. The system has additional components for start, safety and control that are outside the scope of this paper.

3.7. Blade Position

The position θ of the blade (often referred to as the “azimuth angle”) is calculated through the response of the normal force F_N. The normal force on the blade is close to zero when the blade passes from upwind to the downwind side of the rotor; see Figure 6. It is assumed that during constant rotational speed, the blade is at θ = 90° when passing the middle of the upwind disk during one revolution; Figure 6. The start position of the blade θ is updated on every revolution.

3.8. Error Estimation Method

The error of a value Z derived from measured values x, y,… can be determined by the mean error:

$$\langle \Delta Z\rangle =\sqrt{{\left(\frac{\partial Z}{\partial x}\right)}^2{(\Delta x)}^2+{\left(\frac{\partial Z}{\partial y}\right)}^2{(\Delta y)}^2+\dots }$$

(12)

or the maximum error:

$$\left|\Delta Z\right|=\left|\frac{\partial Z}{\partial x}\Delta x\right|+\left|\frac{\partial Z}{\partial y}\Delta y\right|+\dots$$

(13)

where Δx and Δy are the maximum errors of the measured values. The mean error estimation method is only valid if all contributing errors can be considered independent of each other. As can be seen by Equations (3) to (7), the F_R, F_N, F_T and τ_bend depend on several different measurements. The errors of some of the measurements have some correlation. To handle such partly correlated measurements, the method of maximum error was chosen; Equation (13). This approach gives a large error span, and it is important to note that all errors given in this paper are maximum errors, unless otherwise stated.

For the turbine parameters, maximum errors were estimated based on the tools and the measurement technique; see Table 1. The maximum errors for the individual load cells ΔF₀, ΔF₁, ΔF₂ and ΔF₃ were estimated as the sum of the maximum deviation from the linear calibration curve (see Section 3.1), the maximum error of the weight measurement (for calibration) and the highest error in the load cell specification.

The no-load zero values F_N,zero, F_T,zero and F_B,zero had to be found as an average during low wind speeds, since aerodynamic forces on the turbine were always present just before a measurement. To estimate the zero values, the load cell output values were grouped in 200 s periods when the wind speed was below 1 m/s. The averages of those periods were used to get a baseline of the deviation in the zero values over time. The most deviating temperatures (>15 °C) were discarded from the zero value estimations. The remaining data resulted in 1300 datasets of F₀, F₁, F₂ and F₃. The zero values F_N,zero, F_T,zero and F_B,zero, which all vary with time and temperature, were calculated for each set. The maximum errors, ΔF_N,zero, ΔF_T,zero and ΔF_B,zero were taken as the maximum deviation found in all of the sets. Any additional errors in the load cell measurements caused by temperature or the measurement electronics were assumed to be included in ΔF₀, ΔF₁, ΔF₂, ΔF₃, ΔF_N,zero, ΔF_T,zero and ΔF_B,zero.

Finally, the estimated maximum errors are determined by applying Equation (13) on Equations (3) to (7). Equations (16) and (17) can be simplified to depend on the actual measured value,

$$\Delta F_R=\left|\Delta F_0\right|+\left|\Delta F_1\right|+\left|\Delta F_2\right|+\left|\Delta F_3\right|+\left|\Delta F_{N,zero}\right|$$

(14)

$$\Delta F_N=\left|\Delta F_R\right|+\left|L_C{\Omega }_{tur}^2\Delta m\right|+\left|m{\Omega }_{tur}^2\Delta L_C\right|+\left|2{\Omega }_{tur}m{L}_C\Delta {\Omega }_{tur}\right|$$

(15)

$$\Delta F_T=\frac{L_1}{2L_B}\left(\left|\Delta F_0\right|+\left|\Delta F_1\right|+\left|\Delta F_2\right|+\left|\Delta F_3\right|+\left|\Delta F_{T,zero}\right|+\left|\frac{F_{T,LC}}{L_1}\Delta L_1\right|+\left|\frac{F_{T,LC}}{L_B}\Delta L_B\right|\right)$$

(16)

$$\Delta {\tau }_{bend}=\frac{L_0}{2}(|\Delta F_0|+|\Delta F_1|+|\Delta F_2|+|\Delta F_3|+|\Delta F_{B,zero}|)+\left|\frac{F_{B,LC}}{2}\Delta L_0\right|$$

(17)

$$\Delta {\tau }_{tur}=n_B(\langle |F_T\Delta R|\rangle +|R\Delta F_T|)$$

(18)

3.9. Experimental Procedure

The measurement campaign was from 13 September to 4 December 2014. For baseline purposes (see Section 3.8), the load cell data was being sampled at a large portion of that time with the turbine off. Actual running field tests were limited to times of suitable conditions. The weather conditions during the measurements divided the data into steady and turbulent conditions.

During the field tests, the speed controller was manually set to a speed that corresponded to reasonable power extraction from the turbine based on actual wind conditions. Since conditions at the site varied and the rotational speed was kept constant for some time, the resulting datasets had a wide spread in tip speed ratios. Data at different tip speed ratios are valuable for comparison with models.

4. Results and Discussion

4.1. Influence of Load Cells on the Turbine Dynamics

As described in Section 3.1, the load cell assembly on Blade 1 alters the dynamics of the turbine. The spacers used on Blades 2 and 3 are assumed to be fairly stiff, and the properties of Blades 2 and 3 should be similar to the original turbine design (with the modified radius). To evaluate the extent of the change on Blade 1, the natural frequencies of all of the blades were analyzed using an accelerometer. The test was carried out on site on the fully-assembled modified turbine. The accelerometer was temporarily attached to each blade at 400 mm from the lower end. The turbine was excited by impulse forces. The blades were tested one by one. During measurement of Blade 1, the load cells were activated to validate the accelerometer measurements and to confirm the mode shapes. The measurement discovered that the blades had the same set of modes for frequencies below 40 Hz.

The measured frequencies for the blades are presented in

Table 4. Measured natural frequencies of the blade assemblies. The excitations and modes are illustrated in Figure 7.

Mode	Blade 11 (Hz)	Blade 2 (Hz)	Blade 32 (Hz)	Excitation
Twisting	5.6	6.7	7.3	X1
Tangential 1	8.0	9.1	9.1	X2
Tangential 23	28.8	29.6	36.1	X2
Bending 1	3.2	3.2	3.2	Y1
Bending 2	4.6	4.6	4.6	X1
Normal 14	13.5	13.2	12.6	Z2
Normal 24	20.7	20.7	18.7	Z1

¹Notes: Blade with the load cells;

²Blade reinforced with carbon fibre;

³Clear in accelerometer spectrum, weak in load cell spectrum;

⁴Only visible in accelerometer spectrum.

. How the different modes were excited is listed in the table and illustrated in Figure 7. The bending modes have the same frequency for all blades. The twisting frequency of Blade 1 has dropped by 16% compared to Blade 2 and 23% compared to Blade 3. The tangential mode at 8.0 Hz of Blade 1 represents a 12% drop compared to Blades 2 and 3.

The normal modes are the same for Blades 1 and 2, and there is a small difference for Blade 3. A possible explanation is that Blades 1 and 2 are reinforced with fiberglass, while Blade 3 has carbon fiber reinforcements. The weights are the same, but the stiffness properties are different. The two natural frequencies in the normal direction were only present in the accelerometer spectrum. The frequencies are therefore assumed to represent the natural frequencies of the blade in the normal direction. This is supported by the difference in frequency for Blade 3 in the normal modes. The absence of other natural frequencies in normal mode indicates that the support arms are very stiff in the normal direction.

The lowered natural frequency in tangential mode indicates that the load cell blade assembly is a softer system in the tangential direction than the other blades. Since the dynamics in the bending direction (up and down) is unchanged, the lowered twisting mode frequency is presumably also a result of the increased softness in the tangential direction. For the tangential mode around 30 Hz, the spectral amplitude was high in the accelerometer data, but hardly visible in the load cell spectrum. This frequency is therefore assumed to be the natural frequency of the blade in the tangential direction. This is also supported by the higher frequency of Blade 3 for this mode.

4.2. Accuracy

The resulting maximum errors of the calibrations and zero values are listed in

Table 5. Maximum errors of zero and calibration values.

Measured Parameter	Value	Maximum Error
FN,zero	15 N	ΔFN,zero = ±7 N
FT,zero	37 N	ΔFT,zero = ±18 N
FB,zero	1960 N	ΔFB,zero = ±63 N
F0		ΔF0 = ±2.2 N
F1		ΔF1 = ±5.9 N
F2		ΔF2 = ±4.2 N
F3		ΔF3 = ±4.1 N

. The accuracy of the data that can be obtained is an important result of the presented method. The maximum errors based on the parameters in Table 1, the measurements in Table 5 and Equations (14) to (18) give the resultant maximum measurement errors as:

$$\Delta F_R=\pm 23\mathrm{N}$$

(19)

$$\Delta F_N=\pm (0.0049{\Omega }_{tur,rpm}^2+0.072{\Omega }_{tur,rpm}+23)[\mathrm{N}]$$

(20)

$$\Delta F_T=\pm (0.0058|F_T|+1.1)[\mathrm{N}]$$

(21)

$$\Delta {\tau }_{bend}=\pm (0.010|{\tau }_{bend}|+20)[\mathrm{Nm}]$$

(22)

$$\Delta {\tau }_{tur}=\pm (0.087\langle |F_T|\rangle +11)[\mathrm{Nm}]$$

(23)

As was explained in Section 3.8, the maximum error was considered to be the most suitable method for the measurements. For comparison, the mean error has been calculated according to Equation (12), and it is about half of the maximum errors given in Equations (19) and (23).

The radial force has a constant maximum error ΔF_R =±23 N, of which the calibration error contributes to ±12 N, the no-load error to load ±7 N and the load cell hysteresis ±4 N; see Table 5. For the maximum error of the normal force, two more terms are added because of the centrifugal force. In the span 40 rpm to 90 rpm, the maximum error goes from ±34 N to ±70 N.

The tangential force has a weak dependence on its own value. At F_T = 100 N, the maximum error has only increased from ±1.1 N to ±1.6 N. The maximum error of torque has slightly higher dependence on F_T and peaks around ±20 Nm at F_T = 100 N. The maximum error of the bending moment of the blade is fairly close to constant and may be approximated to Δτ_bend ≈ ±25 Nm with |τ_bend| < 500 Nm.

4.3. System Output and Performance

An example force measurement for three complete revolutions is captured in Figure 8. Wind speed and wind direction are fairly constant, and the system is keeping the rotational speed fixed. The output from the load cells appears very oscillatory. The resulting forces from the load cell data in Figure 8 are shown in Figure 9. It is revealed that the mechanical oscillations are limited to the tangential force and the bending moment, leaving the normal force with a remarkably smooth appearance. Tests to reveal the operational limits discovered that the speed limit was about 90 rpm and the maximum power was about 4 kW. Additionally, during the measurement campaign, the weather conditions were a further limitation.

4.4. Radial and Normal Force

To illustrate the performance of the proposed method in the normal direction, the measured normal force has been determined for two additional occasions. Figure 10 shows that the shape of the measured F_N is close to the expected F_N obtained from the simulation model (Section 2.3). The maximum magnitude of F_N increases with higher rotational speed at a constant tip speed ratio. This is due to the higher velocities at the blades and higher Reynolds number, and it agrees with the model prediction. The oscillations of the measured F_N at 180° < θ < 360° are consistent and present during each revolution.

The maximum error of the normal force in Figure 9 is ∆F_N = ±49 N. Since the average of the rotational speed is used in determination of F_N, the only difference to F_R is actually a constant, depending on the rotational speed. Therefore, the maximum error of the shape of F_N is the same as the maximum error of F_R. That is a fair improvement, since ∆F_R = ±23 N, which is less than half of ∆F_N.

4.5. Tangential Force and Torques

To further illustrate the behavior of the proposed method in the tangential direction, the measured tangential force has been determined for two additional occasions. The tangential force contains large oscillations of multiple frequencies; see Figures 9 and 11. These oscillations are dependent on the rotational speed and are not predicted by the model; see Figure 11. For almost identical tip speed ratios, the shape of F_T is different at different rotational speeds, which is contrary to the simulated values. In Figure 9, F_T has a span from −37 N to 77 N. The average value, which provides the actual torque, is only 15.6 N. As shown in Section 4.1, the load cells change the dynamics of the system. The indicated change is low, and therefore, the behavior seen may reflect the tangential forces on the other blade assemblies. Furthermore, it is important to note that the blade force is measured at the hub. Since the support arms may act as a damper, the force on the blade may be higher or lower. The other blades that are unmeasured are likely also to provide oscillating forces, which could affect the hub and the force sensors. Some of the mechanical oscillations seen may therefore actually be the result of all three blades. It should be noted that the operating power is quite low (< 10 % of the rated power) and that these mechanical oscillations are possibly less significant at higher operating power. Overall, the tangential force measurement can provide interesting data of the mechanical dynamics of the prototype, but limited knowledge of the aerodynamic tangential force acting on the blade. The mechanical oscillations require further analysis.

The maximum error of the tangential force presented in Figure 9 is ∆F_T = ±1.6 N. That is good compared to the peak values. However, it is a larger fraction of the average F_T. Higher operational power would improve the relative size of the maximum error since ∆F_T is fairly constant; see Equation (21). As described in Section 4.3, the operational power is limited.

With all blades providing equal torque, the average torque in Figure 9 is 152±13 Nm. The assumption that all blades provide equal torque is a possible source of more errors. By using the average of many complete revolutions, that assumption can be justified. For more accurate torque measurements, a rotational torque sensor is probably better or by estimation from the electric power output of the generator, as in [14].

The bending moment on the blade, τ_bend, also seems affected by oscillations; see Figures 9 and 12. The shape of τ_bend changes with different rotational speeds. Note that τ_bend is not calculated by the simulation model described in [8]. Due to the wind shear pushing the upper and lower part of the blade unequally, the value of τ_bend is expected to change from positive to negative during one revolution. The presented data, however, have a constant negative offset. The center of mass of the blade and support arms is shifted vertically because of the gravitational force. The centrifugal force acts on the shifted center of mass, causing the constant offset. The measured behavior should be representative for the movement of the other blades, since the dynamics of the measurement blade and that of the other blades are the same in the bending direction; see Section 4.1.

4.6. General Discussion

The accuracy of the load cells is fairly high compared to the other factors affecting the total accuracy of the measurement. The operating power could be increased with load cells of higher capacity, without significant loss in accuracy of the normal force. Additionally, the horizontal separation of the load cells can be wider to increase stability in the tangential direction, hopefully lowering the mechanical oscillations. However, this would decrease the tangential force accuracy. As has been shown, the presented method already has limitations on measuring the average tangential force with high accuracy, so that can be a fair trade off.

It can be realized that good steady conditions of a desirable nature can be rare for open site experiments. This means that the experimental setup must be operational and ready for long periods of time, waiting for perfect conditions. This is of course a large practical challenge with prototypes and measurement systems in open site experiments.

5. Conclusions

A method for analyzing the aerodynamic blade forces using load cells at the hub of a vertical axis wind turbine has been presented and experimentally evaluated. The method was intended for both tangential and normal blade force measurements. The method provides a normal force with good periodicity over turbine revolutions and good agreement with theoretical values.

For the tangential force, mechanical oscillations have been discovered. The peak amplitudes of the measured forces are of a similar size as in the simulations. The shape of the tangential force is clearly distorted by the dynamics of the turbine. The presented method is therefore unsuited to estimation of the tangential aerodynamic forces acting on the blade. However, the provided data can still be interesting for studying the mechanical dynamics of the turbine. It should be noted that the turbine power during the tests was 5 % to 10 % of nominal.

The accuracy of the measurements of forces and accompanying weather conditions has been investigated carefully. Many of the errors are dependent on the measured values, and equations have been derived to simplify the estimations of the different errors. The measurements can be considered to be fairly accurate.

A final conclusion from this study is that open site experiments might be time-consuming, since they suffer from being dependent on the current weather conditions and, in this case, also require steady wind conditions.