Adaptive surface distortions for torque ripple control in vertical-axis wind turbines

Abstract

Vertical-axis wind turbines have been confined to small-scale generation in urban environments where their omnidirectional capability offers them an advantage over the more ubiquitous horizontal-axis wind turbine. With a drive towards renewable energy, more opportunities exist for the implementation of wind turbines in a multitude of environments. Based on its inherent operational drawbacks, the vertical-axis wind turbine has not undergone extensive investigation. Recently, there has been a resurgence of interest in the technology. This article addresses the torque ripple, a variation in torque produced by the turbine, present during operation. The variation in torque generated by a vertical-axis wind turbine increases the likelihood of failure due to fatigue. Current treatment is symptomatic and addresses the result of the torque fluctuation and not the cause. A novel blade design, capable of altering the lift and drag response through shape alteration, is presented as a solution. The blade design and operation is achieved through genetic algorithm optimization and computational fluid dynamic simulations. Comparisons with previous work show the novel blade presented here surpasses the reduction seen with pitching solutions. A 25% reduction in torque ripple was demonstrated for a 17% reduction in performance coefficient using the surface distortion approach. This surpasses the foil pitching approach which achieved a 15% torque ripple reduction for the same loss in performance coefficient.

Keywords

VAWT optimization airfoil BGA power quality CFD turbulence model

Introduction

The horizontal-axis wind turbine (HAWT) and the vertical-axis wind turbine (VAWT) represent the two categories of wind turbine used for power generation. HAWTs are the turbine of choice for commercial-scale wind energy farms. The VAWT configuration has seen less deployment and similarly less development in recent years (Eriksson et al., 2008). However, in the early years of investigation the two configurations saw equal interest.

One of the limitations of VAWTs is their struggle to ramp up to operational speed without an external aid. The design tip speed ratio (TSR), where the turbines achieve the maximum performance coefficient $(C_{P})$ , is beyond the turbines’‘dead zone’. This region is so termed because it has a negative $C_{P}$ value; thus, the turbine needs external power to rotate (Bogateanu et al., 2013). Figure 1 shows the effect a fixed pitch blade has on the $C_{P}$ as a function of TSR.

Figure 1.

Power coefficients for NACA 0012 at Re 360,000 with various blade pitch angles, recreated from Bogateanu et al. (2013).

As the turbine starts rotating, the $C_{P}$ increases but invariably experiences a dip prior to achieving its maximum value. A VAWT can generate power with less wind but its maximum achievable $C_{P}$ value is less than a HAWT and thus costs more per kW (Pope et al., 2010). The lower return on investment can be viewed as a reason for the slow development of VAWT technology.

Torque ripple is not only a VAWT problem, but its effect is more substantial when compared with the ripple seen in HAWTs. The severity is due to the constantly changing angle of attack in a VAWT, which causes a significant yet predictable variation in load. The HAWT on the other hand sees a fluctuation in torque because of wind shear and the tower wind shadow effect. These fluctuations reduce the life cycle of the equipment due to fatigue and add to poor power quality (flicker and harmonics) (Reuter and Worstell, 1978).

Systems with blades that can adjust their pitch have been proposed to overcome some of the VAWT issues. Miau et al. (2012) bypassed the ‘dead zone’ while Chougule and Nielsen (2014) increased the C_P of VAWTs through variable pitching approaches. Paraschivoiu et al. (2009) showed an increase in C_P while Kirke (2011) and Miau et al. (2012) showed it could improve the self-starting characteristics. Erfort et al. (2019b) showed that pitching the blade during operation could also reduce the torque ripple with a related drop in C_P. In Tjiu et al. (2015), it was shown that VAWTs with helical blades reduce vibrations during operation, but they have a lower running $C_{P}$ than the straight bladed configuration. A troposkein design also offers a reduction in torque ripple but with an increased cost to manufacturing and installation.

This article looks at adjusting the characteristic lift and drag curves of a blade while in operation. Inspired by nature, a blade with a morphing surface distortion capable of changing height during operation was modelled numerically to investigate its effect on the torque ripple. An analytical model using the double multiple stream tube (Paraschivoiu and Delclaux, 1983) approach was implemented in Python (Software Foundation, 2019) to obtain the VAWT characteristics. The coefficients of lift $(C_{l})$ and drag $(C_{d})$ used in this model were sourced from computational fluid dynamic (CFD) simulations. The best height and location of the distortion were obtained through genetic optimization.

Blade design

The blade design presented here is inspired by drag adaptation techniques in nature, in particular the mechanism used by swimming dolphins to improve their efficiency. In Fish and Hui (1991), the various myths and proposals surrounding these techniques are discussed. In Gray (1936), the theory of a turbulence-free boundary layer was postulated as the reason for dolphin swimming efficiency. Attempts to verify the laminar theory included viscous damping through compliant surfaces, namely, that the skin acts to passively damp the Tollmein-Schlicting waves that add turbulent fluctuations (Fish and Battle, 1995). The attempt was ultimately unsuccessful but did manage to reduce the skin friction coefficient $(C_{f})$ in some instances.

This laminar theory was disputed by Fish and Rohr (1999) through experiments on flat plates and surrogate rigid bodies. Figure 2 shows that drag values over a flat plate for turbulent boundary flow more closely match the experimental results for rigid body models of dolphins and other cetaceans. In experiments involving a trip wire, there was no reported difference in drag coefficients for the surrogate bodies. This indicated the presence of a turbulent boundary layer. Such layers are noted to delay separation which, in turn, minimises drag. In Fish and Battle (1995), the pectoral fins of humpback whales showed that leading edge tubercles affected lift and drag characteristics during swimming. By inducing a turbulent boundary layer, the separation point can be delayed further downstream along an airfoil surface. The mechanism for tripping the boundary layer has been shown to be small adjustments to the surface of a rigid body. Finally, placement of these distortions should be near the leading edge to effect change along the surface of the airfoil further downstream.

Figure 2.

Annotated image from Fish and Rohr (1999) showing turbulent versus laminar $C_{d}$ values.

Manufactured distortions

The use of surface distortions for flow manipulation was identified in the design of golf balls. Alam et al. (2010) concluded that the dimpled surface of the ball significantly affects the drag characteristics of the ball. These effects allow the ball to travel farther due to its reduced drag at low speeds. Figure 3 shows the reduced drag for various golf balls in comparison with the smooth surface of a squash ball. Since each manufacturer has a specific pattern and arrangement, the effects on the ball characteristics also vary. Transition from laminar to turbulent flow occurs earlier in golf balls than it does for the smooth squash ball.

Figure 3.

Partially recreated image from Alam et al. (2010) for drag of various golf balls.

Proposed blade

Studies have shown dolphins make use of a turbulent boundary layer to reduce their drag while swimming. Cutaneous ridges have been shown to effect the dynamics of vortex filaments resulting in a reduction of the skin friction coefficient. The humpback whale has tubercles on its pectoral fins that positively influence their swimming ability. Finally, the golf ball design includes a tripping mechanism to effect the boundary layer and thus maximise its flight potential. These factors influenced the proposed blade design as follows:

Implementation of active surface distortion, capable of changing height, based on the control seen in dolphins.

Placement of the distortion is envisaged for the upper surface of a foil, near the leading edge after investigating the pectoral fins on humpback whales.

The distortion is shaped as a half circle protruding from the foil surface. The golf ball studies indicated different shapes have different effects. The simple half circle provides a good baseline geometry.

The examples discussed earlier all have multiple distortions over the base geometry. This work is a preliminary investigation into the effects of a single adaptive distortion. The use of multiple distortions of fixed height is documented, and this study is more concerned with the ability of the distortion to vary in height. CFD simulations, carried out in the open source software OpenFOAM (Weller et al., 1998), were used to obtain lift and drag information for the various blade designs. The results of such a simulation on a smooth blade showed the transition point to be around the apex, that is a non-dimensional distance along the chord length of $(X / c)$ 0.3, for 0° angle of attack at the lowest predicted blade Reynolds number using the analytical model. The transition point moves forward as the angle of attack increases and since a larger effect was assumed for distortions nearer the leading edge (LE), the distortion was bound between 0.05 and 0.2 $X / c$ . Figure 4 illustrates the size of a distortion relative to the airfoil. A NACA 0012 airfoil with a distortion located at 0.15 $X / c$ from the LE is shown. The magenta box illustrates the bounds in which the distortion could be located.

Figure 4.

The distortion to be implemented on a NACA 0012 foil.

Howell et al. (2010) showed that surface roughness could actually improve the $C_{P}$ of small VAWTs at low TSRs. The effect on the performance coefficient curve for both rough and smooth surfaces is presented in Figure 5.

Figure 5.

Improvement in $C_{p}$ due to surface roughness of a two-bladed VAWT, recreated from Howell et al. (2010).

The proposed distortions in this study are close to the surface roughness scale, being less than an order of magnitude larger. Their inclusion in the foil geometry lowers the achievable $C_{P}$ as the TSR being investigated is greater than 3; therefore, a relationship between $C_{P}$ decrease and corresponding torque ripple reduction was investigated. A single semi-circular distortion would be placed between the leading edge and 0.2 $X / c$ on the upper surface, and the height varied as a function of azimuthal position. The exact location was determined through optimization. The variation in height as related to azimuth angle conformed to a sine wave function, whose phase, frequency, and amplitude are also determined through optimization.

Numerical work

DMST model

The double multiple stream tube (DMST) model has been shown as a fast and effective tool for predicting the $C_{P}$ value of a VAWT for various TSR (Islam et al., 2008). The pitching study by Paraschivoiu et al. (2009) was used to validate the DMST model implemented in Python. The remainder of this study uses the parameters listed in Table 1 as a base line model (BSL). Using this configuration, the blades see a variation in Reynolds number between 480,000 and 710,000. The angle of attack changes between 28 and 4 degrees.

Table 1.

Parameters for DMST model input.

Parameter	Value
Rotor diameter	6 m
Chord	0.2 m
Airfoil	NACA0012
Number of blades	2
Stream tubes	51
Tip speed ratio	5.3

Lift and drag data

Accurate lift and drag data are important inputs for the DMST model. CFD was used to obtain these curves over the range of Reynolds numbers already mentioned. Extensive effort was put into turbulence model selection as the distortion proposed directly effects tripping in the boundary layer. A transitional turbulence model was selected for its ability to capture both the laminar and turbulent boundary layers along the airfoil surface. The $γ$ – $\tilde{R} e_{θ_{t}}$ model from Erfort et al. (2019a) was used for its increased computational speed when compared with the original model of Langtry and Menter (2005). Figures 6 and 7 highlight the benefit of using a transitional model over a fully turbulent model.

Figure 6.

$C_{d}$ at Re 360,000 for NACA 0012.

Figure 7.

$C_{d}$ at Re 700,000 for NACA 0012.

The drag predictions from the $γ$ – $\tilde{R} e_{θ_{t}}$ more closely match the experimental data of Sheldahl and Klimas (1981) over the working range for the VAWT. As the Reynolds number increases, the difference in model selection becomes less important, with the predictions of both models converging to a similar value. CFD runs are computationally expensive, so a surrogate model based on the CFD results was built to be used during optimization.

Methodology

The surface distortion specifics (location, width and height variation) were undefined at the outset. These values were determined during the optimisation stage in conjunction with torque ripple reduction. Reproducing experimental data for the possible blade configurations would have been time-consuming. Instead a virtual laboratory approach was adopted. Figure 8 highlights the work flow for both approaches.

Figure 8.

Work flow for virtual laboratory.

The numerical approach replaced the iterative experimental steps through parallel computing, drastically reducing the time needed to produce results. The DMST model provides a $C_{P}$ value and the torque curve for a single rotation. These outputs are dictated by the parameters in Table 1 and the three coefficients in the following equation

H = A \sin (B θ + C)

(1)

The amplitude $(A)$ , frequency $(B)$ and phase $(C)$ controlled the height $(H)$ of the distortion as a function of azimuthal position. The $C_{P}$ value was set as a constraint, and the optimizer was set to only reduce the amplitude of the swings in torque. In setting $C_{P}$ as a constraint and explicit trade-off study between $C_{P}$ and torque ripple reduction was made possible.

A breeder genetic algorithm (BGA) described by Mhlenbein et al. (1993) was used in this work. This particular variant of a genetic algorithm is based on selection as done by breeders in agriculture. The algorithm selects the best solutions and breeds a new generation of solutions with desirable traits from the best performers. The BGA reduced the number of random starts typically required for a gradient-based optimization approach.

Breeder genetic algorithm

This study implements a BGA in Python based on the approach described earlier where the function call is

B G A () = (f (x), \vec{x}, \vec{ζ}, G_{N}, P_{N}, E, ▹ ◃_{m}, ▹ ◃_{r}, η, Ω)

(2)

The objective function

f (x) = | \bar{T} - T_{m a x} |

(3)

was the difference between the average $(\bar{T})$ and peak torque $(T_{m a x})$ experienced by the VAWT during a single rotation, shown in Figure 9, as provided by the DMST model. The vector describing a member of the population

\vec{x} = [A, B, C, W, L]

(4)

consisted of the coefficients for equation (1), a distortion width $(W)$ and location $(L)$ . $\vec{ζ}$ represents the upper and lower boundaries for the genetic make-up of each individual. This ensured that $\vec{x} [i]$ , for example, remained within the design space by enforcing the boundaries on each member of the population. $G_{N}$ and $P_{N}$ are the maximum number of generations and individuals in a population as stipulated by the user. $▹ ◃_{(r, m)}$ represents the recombination and mutation rates, respectively. $η$ provides the number of elites to select from each population and $Ω$ is the termination criterion.

Figure 9.

The BSL torque ripple for a two-bladed H-rotor.

The code used the seed vector and created $P_{N} - 1$ number of individuals through random mutation of the initial population. Each member of the population is then confirmed to be within $\vec{ζ}$ ; if they are not, the offending component in the vector is limited to the boundary value. A function evaluation is carried out for every member of the population and the results stored for inspection. The top E candidates are selected for breeding; they, along with their offspring, populate the next generation. Portioned recombination was carried out by $▹ ◃_{r}$ . The offspring in the new generation are then mutated by $▹ ◃_{m}$ . In this particular code $▹ ◃_{m}$ was an adaptive mutation factor based on the population’s general fitness level. The optimisation loop is repeated while the number of generations is less than $G_{N}$ and the improvement between the population as a collective of two consecutive generations is greater than $Ω$ . The following conditions were used in this study:

Random seed vector containing five values; amplitude, phase and frequency of the sine wave controlling the height of the distortion and finally the width and location of the distortion

All variables were scaled between 0 and 1 prior to use in the BGA

The BGA was allowed a maximum of 50 generations $(G_{N})$

Within each generation there were 50 individuals $(P_{N})$

The mutation and selection rate were 0.5 and 0.15, respectively $(▹ ◃_{m}$ , $▹ ◃_{r})$

The top five members of any generation formed the elite class $(η)$ and breeding was limited to just these members $(E)$

The improvement between generations had to be less than $1 e^{- 4}$ to terminate early $(Ω)$

Results

Figure 9 shows the torque coefficient experienced by the VAWT during a single rotation. The smooth surface baseline model showed a peak value for the torque coefficient to be 30% larger than the average.

The optimisation was run with four different constraints on $C_{P}$ separated into groups of seven runs. Each run used a different seed vector to aid in workspace sampling. Reduction in torque ripple versus $C_{P}$ is plotted in Figure 10. Looking at the best performers in each group, a polynomial fit can be made for the relationship between ripple reduction and power coefficient. Equation (5) describes an upper bound estimate for ripple reduction with a single distortion.

y = - 0.035 x^{2} + 2.145 x - 0.722

(5)

Figure 10.

Reduction in ripple versus reduction in power coefficient.

The dotted yellow line in Figure 10 shows the relationship between ripple reduction and $C_{P}$ achieved when pitching the foil, as discerned by Erfort et al. (2019b).

Table 2 provides the average results from all the runs per group. The optimiser sought to place the distortion as close to the leading edge as possible. The larger distortions gave a larger ripple reduction for an accompany reduction in $C_{P}$ .

Table 2.

Optimisation results.

Group	Width (mm)	Location $(X / c)$	Reduction (%)
			Ripple	$C_{P}$
1	5.3	0.06	5.63	3.01
2	9.3	0.065	10.2	6.94
3	15.8	0.05	13.1	9.23
4	11.9	0.07	15.5	12.1

Group four was limited to a 17% drop in $C_{P}$ and had the widest spread in results. Figure 11 shows the resulting torque curves for each run. The optimizer attempted to create a plateau where there once were peaks. In doing so, the area under the curve is reduced and thus the $C_{P}$ value decreases.

Figure 11.

Comparison between base line ripple and optimised solutions for group 4.

Run three is now scrutinised having achieved the largest reduction in torque ripple (25%) for a 17% drop in $C_{P}$ . The distortion was 10 mm wide and located at 0.11 $X / c$ . Figure 12 shows the optimised torque ripple as well as the mean torque coefficient decrease.

Figure 12.

Comparison between base line ripple and optimised result from run 3.

The change in ripple is due to the effect of the distortion on the lift and drag characteristics of the foil. Figures 13 and 14 show how these traits differ due to the inclusion of an adjustable distortion. In the upwind zone, minimal changes exist between the BSL and optimised result, with the optimiser actually choosing to lower the maximum achievable $C_{l}$ . However, the $C_{d}$ graph shows that the cause for reduction in power was the large increase in drag throughout the rotation. Back et al. (2012) showed that any alterations to the initially smooth surface increase the drag.

Figure 13.

$C_{l}$ for run 3.

Figure 14.

$C_{d}$ for run 3.

Figure 15 shows that distortions are maximised close to 100 and 270 degrees azimuth angle. This is where the original model had its largest peak. In creating the largest displacement at this rotational position, the maximum achievable lift is diminished. This coupled with the increased drag causes a reduction in torque.

Figure 15.

The change in distortion height for each run in group 4.

The optimizer did not reach the generational limit in any of the simulations with convergence being found within 35 generations. The optimiser tended towards placing the distortions close to the leading edge, with the majority of final solutions as reported in Table 2 showing the distortion location before 0.8 $x / C$ . Incoming flow sees the most disturbance closest to the LE and these effects are transported along the foil surface. Thus, a distortion in this region would have a larger impact.

Conclusion

In this work, a novel blade design showed its ability to reduce the torque ripple of a VAWT during operation. A distortion on the upper surface of the blade, with the ability to change in height, outperformed a pitching blade on the same problem. The surface distortion approach was able to achieve a 25% reduction in ripple for an associated loss in $C_{P}$ of 17%. The pitching foil approach could at best achieve 15% reduction for the same $C_{P}$ reduction. The variation in height was associated with an azimuthal position and this removed a key feature of the VAWT. All pitch variation techniques require a yaw system to ensure the pitch varied according to azimuthal position.

This proposed blade is simple to construct and envisioned to house the actuator within the body of the blade. Further studies are suggested to identify the inclusion of multiple distortions with different height variation schemes. This work has provided a framework for causal treatment of the torque ripple seen in VAWTs to improve machine life cycle and efficiency.

Footnotes

Acknowledgements

This work was made possible thanks to the use of the CHPC cluster Lenagu.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Gareth Erfort

Gerhard Venter

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