Original research paper  Open  Published:
Multidimensional optimization of small wind turbine blades
Renewables: Wind, Water, and Solarvolume 2, Article number: 9 (2015)
Abstract
This paper describes a computer method to allow the design of small wind turbine blades for the multiple objectives of rapid starting, efficient power extraction, low noise, and minimal mass. For the sake of brevity, only the first two and the last objectives are considered in this paper. The optimization aimed to study a range of blade materials, from traditional fibreglass through sustainable alternatives to rapid prototyping plastic. Because starting performance depends on blade inertia, there is a complex interaction between the material properties and the aerodynamics. Example blades of 1.1 m length were designed to match a permanent magnet generator with a rated power of 750 W at 550 rpm. The materials considered were (a) traditional Eglass and polyester resin; (b) flax and polyester resin; (c) a typical rapid prototyping plastic, ABSM30; and (d) timber. Except for (d), hollow blades were used to reduce the rotor inertia to help minimize starting time. Two airfoils are considered: the 10% thick SG6043 which has excellent lift:drag performance at low Reynolds number and the SD7062 whose extra thickness (14%) has some structural advantages, particularly for the weaker material (c). All blade materials gave feasible designs with material (d) the only one that required a blade shell thickness greater than the specified minimum value of 1% of the blade chord. Generally, the blade chord and twist increased as starting was given greater importance. In all cases, the associated increase in blade inertia was outweighed by the larger aerodynamic torque. Materials (a), (b), and (d) were better suited to the SG6043 airfoil whereas ABSM30 benefitted from the thicker SD7062 section.
Background
Small wind turbines are under strong competitive pressure from photovoltaics (PV) which have decreased significantly in price over the past decade. A similar reduction in the manufacturing cost of small wind turbines is probably not achievable, but significant efforts can be made in this direction. For example, the recent increase in the use of monopole towers may well be reversed and the cheaper lattice tower gain renewed application (Adhikari et al. 2014a).
In addition, there are two aspects of small wind turbines that will remain attractive. The first is that remote renewable energy systems generally should use multiple resources, such as sun and wind, to provide the most costeffective system despite any cost difference between the technologies (e.g. Kaldellis 2010). Second, most developing countries must import PV equipment, whereas there is scope for significant local manufacture of many wind turbine components, often using indigenous and sustainable materials, such as bamboo for the tower (Adhikari et al. 2014b) and timber or biocomposites and bioresins for the blades (Peterson and Clausen 2004, Holmes et al. 2009, Astle et al. 2013, and Shah et al. 2013). In addition, the recent advent of rapid prototyping brings the promise of cheap manufacture of the complex threedimensional blade shape and the potential to design blades for specific applications, such as rapidly starting blades for lowwind areas or especially quiet blades for building mounting.
In order to exploit these opportunities, the small wind turbine blade design must become increasingly multidimensional. Starting is often a critical issue for small blades as most do not have pitch adjustment. Recent twodimensional optimizations considered starting and power extraction (Wood 2011, Pourrajabian and Mirzaei 2014, and Shah et al. 2014). CliftonSmith (2010) added noise. The common finding of these studies is that a multidimensional design is actually a tradeoff. Typically, the most powerful blade is the slowest to start and CliftonSmith (2010) found that for blades of nearly equal power extraction, the quietest blade was the slowest to start whereas the fastest starting blade was the noisiest.
Of the papers just cited, only Pourrajabian and Mirzaei (2014) considered structural optimization which is important for largeblade design partly because blade manufacturing cost correlates with blade weight (e.g. Sessarego et al. 2014). Structure is important for small blades in another way: starting depends on the aerodynamic torque generated by the blade shape and is improved by a low blade inertia, J _{B}, which depends on shape, material, and manufacturing technique. It is to be expected that including structure in smallblade optimization will introduce complex interactions between the objective functions. To fully exploit the possibilities of structural optimization, it is necessary to design hollow blades, whose shell thickness can be minimized to reduce J _{B} while remaining sufficiently strong. Examples of hollow composite small turbine blades are given in Clausen et al. (2013).
The aim of this paper is to explore multidimensional small wind turbine blade optimization including structure, by designing hollow blades using (a) traditional Eglass and polyester; (b) flax and polyester; (c) a strong, rapid prototyping material; and (d) solid blades using timber. The 1.1mlong blades were designed to match a permanent magnet generator with a rated capacity of 750 W at 550 rpm and efficiency of 74%. Solid timber blades for this generator were designed in Chapter 7 of Wood (2011). The optimizations use the MATLAB code Small Windturbine Rotor Design Code (SWRDC) developed by the authors and available from them. The code combines the multiobjective problem in a single objective by using a scalarization function of all the objective functions, as explained below. Optimization is by a simple genetic algorithm.
Material (a) is typical of many current small blades. The use of flax rather than Eglass in material (b) is one way to reduce the environmental footprint of small blades (Shah et al. 2013). Timber, material (d), is commonly used for ultralight aircraft propellers and also may be environmentally preferable to material (a). Hoop pine is a native Australian tropical pine that grows well in plantation (Peterson and Clausen 2004) and has been used for many years for propeller manufacture^{a}. It is possible that the advent of rapid prototyping or threedimensional printing will revolutionize smallblade manufacture because it does not need expensive moulds and allows specific blade design for specific site conditions. For example, a windy site may not require rapidly starting blades, whereas a turbine to be installed next to a house may need to be extremely quiet. SWRDC is a very suitable tool to undertake the initial design of blades for different requirements. However, there is one major concern with rapid prototyping materials: an extensive search of the literature has found none whose fatigue properties have been established. Although it is not directly considered in the present optimization, adequate fatigue behaviour is essential for any blade to be safe.
It is common to use a single airfoil for small blades as the largeblade practice of increasing the airfoil thickness near the root will cause undesirable aerodynamic behaviour at the low chord Reynolds number, Re, encountered by small blades (e.g. Wood 2011). The first airfoil considered here is the SG6043 developed for small wind turbines by Giguere and Selig (1998). As seen in Figure 1, this airfoil has an outstanding lift to drag ratio which is necessary to maximize power output. The SG6043 has a maximum thickness of 10% which may cause high stress levels near the root, so the 14% thick SD7062 (Giguere and Selig 1997) was also considered. Figure 1 shows that it has a good lift to drag ratio but is likely to produce less power than the SG6043. The noise analysis is that used by CliftonSmith (2010) which was based on the semiempirical formulation of Zhu et al. (2005). This objective function will not be described in detail as it is not used in this study.
One of the prime requirements for multidimensional optimization involving the evaluation of thousands of objective functions is their rapid evaluation. Therefore, it is not possible to use, say, detailed computational fluid dynamics or finite element structural analysis. The optimization results from SWRDC cannot be considered as final designs until validated or modified by higherfidelity simulations. It is also necessary to emphasize that the optimization is of the aerodynamic part of the blade and excludes the structurally important connection to the turbine hub. This ‘attachment section’ typically does not contribute substantially to the blade inertia but is likely to carry the major blade load, the root bending moment. The assumption of hollow blades may also be a significant simplification. Some small blades use a foam core to separate the blade surfaces and prevent buckling (e.g. Clausen et al. 2013), and it is also possible for small blades to have a shear web or spar as do large blades. These additional structural features are not considered here.
The next section demonstrates the importance of starting and describes its aerodynamic modelling using a modified blade element analysis. This is followed by a description of the structural analysis. Power extraction uses standard blade element theory (e.g. Hansen 2008, Wood 2011, and Sessarego 2013). The last reference described the detailed validation of the code in reference to the NREL code Wt_Perf using the AWT27 turbine performance data (Buhl 2012).
Methods
Starting performance and starting time
Figure 2a shows a highspeed starting sequence of the twobladed 5mdiameter 5kW wind turbine at the University of Newcastle, Australia, depicted in Figure 2b. The blades were designed by the second author for good starting performance using the analysis that is described below. Starting is important for small turbine blades which have no pitch adjustment as the stationary or slowly rotating blades experience high angles of attack, α, which generate low aerodynamic torque and lead to long starting times in which no power is extracted. Figure 2a indicates that the starting time, T _{s}, to reach powerextracting angular velocity is about 13 s. Since T _{s} scales as the inverse of the wind speed, this welldesigned turbine takes around 40 s to start at a desirable cutin wind speed of 3.5 m/s.
By assuming that starting is quasisteady and using the generic flat plate equations for high angle lift and drag, Wood (2011) determined the aerodynamic torque, Q, for a turbine with N blades of tip radius R, generated by a wind speed U as
ρ is the air density, r _{h} is the normalized hub radius, and λ _{r} is the local speed ratio, which is the product of, r, the radius normalized by R, and λ, the tip speed ratio. Ө _{ p } is the blade twist angle, measured from the plane of rotation. If the resistive torque of the drivetrain and generator is Q _{r}, then the tip speed ratio, λ, obeys
where J is the rotor inertia. For turbines of all sizes, J is dominated by the blade inertia, J _{B} (Wood 2011), so that J ~ NJ _{B}. If Q _{r} can be neglected, as in the example shown in Figure 2, then starting becomes independent of the number of blades. Figure 2a shows the solution of Equations 1 and 2 obtained using the AdamsMoulton method and the measured U. The equations are a good fit to the measurements for the period of slow acceleration, up to around 11 s, that dominates the starting sequence. Soon after that point, the angles of attack have reduced significantly and the rotor accelerates rapidly. In general, Q _{r} cannot be neglected, unless it is less than about 1% of the rated torque (Wood 2011). An example of its importance is the analysis of altitude effects on smallturbine performance by Pourajjabian et al. (2014). The reduction in air density with altitude can require significant blade redesign to achieve rapid starting as the altitude increases if Q _{r} is not affected by altitude. Q _{r} is assumed to be zero for the present calculations. SWRDC solves Equations 1 and 2 for a userspecified number of blade elements; this was 30 for the present calculations. The same number of identical elements is used to find the power output.
Structural analysis
Even though the blade surface shape is determined largely by aerodynamic requirements, the structure is very important. The chosen blade materials must be sufficiently strong and resistant to fatigue, and the resulting blade inertia is a key element of the starting performance. This interaction of material properties and aerodynamics is unique to small blades and is one reason to include structural analysis in the optimization. The structural model is based on simple EulerBernoulli beam theory (e.g. Hansen 2008) and is a simpler version of the one used for largeblade optimization by Sessarego (2013) and Sessarego et al. (2014). They assumed a layup consisting of multiple laminate shells, two shear webs, fore and aft panels, and two spar caps of equal thickness. Since small turbine blades often are shells of thickness, s (Clausen et al. 2013), the structural model is shown in Figure 3. SWRDC assumes s is constant for each blade element but can vary between elements. A minimum value of s/c is a necessary user input. Actual wind turbine blades are made from composite anisotropic materials, and classical laminate theory or finiteelementbased approaches must be used to predict accurately the blade structural performance (e.g. Chen et al. 2010 and Lin et al. 2010). For the current study, the complexity of the structural analysis is reduced by treating all materials as isotropic using the material properties listed in Table 1. These materials range from conventional Eglass/polyester through sustainable materials to rapid prototyping plastic. In all cases, the first two material properties were taken directly from the reference. The maximum allowable strain which must be an input to SWDRC was calculated as follows, using Eglass/polyester as an example. Table 2 of Shah et al. (2013) gives the composite strength as 567 MPa. We assume that this is the ultimate tensile strength (UTS) as is assumed for all other materials. Taking a generous ‘safety factor’ of 2 between the UTS and yield strength gives the latter as 283.5 MPa. The resulting maximum strain is found from Hooke’s law: 283.5 × 10^{6} = 36.9 × 10^{9} ε, where ε is the strain. Thus, ε = 7,684 μstrain. Table 2 of Shah et al. (2013) gives the composite fibre failure strain as 1.9% which is greater than 7,684 μstrain = 0.77%, so the latter value is used.
The strain is determined around the contour of the crosssection of each blade element in SWRDC to find the maximum value. The blade can be solid or hollow; in the latter case, s is calculated to ensure that the allowable maximum strain is not exceeded.
Optimization
The present calculations considered only starting and power extraction. The optimization seeks to
for blade i in the current population. C _{p} is the conventional power coefficient at the design wind speed, taken to be 10.5 m/s. The weight w (0 ≤ w ≤ 1) determines the relative significance of power extraction and starting. When w = 1, the optimization is purely for power. T _{s} in seconds is the starting time required to reach a tip speed ratio at a wind speed specified by the user. Since the calculations cover only the slow part of starting as shown in Figure 2, the actual values of T _{s} are not significant, but the relative magnitude of them is critical.
The blade chord and twist are determined using Bézier curves with a userspecified degree number. The number of control points is one greater than the degree of the curves, i.e. nine control points were used for all the present calculations. Limitations on maximum and minimum chord and twist can also be specified. For all calculations, the chord was constrained between 0.04 and 0.02 m which can be viewed as realistic limitations of the manufacturing techniques. Since the minimum chord occurs at the tip, Re is approximately 1.7 × 10^{5} and approximately constant along much of the blade. Similarly, the allowable twist range is 30° to −5°.
SWRDC uses a genetic algorithm (GA) for optimization, which was based on the NSGAII C program available from the Kanpur Genetic Algorithms Laboratory^{b}. The relevant inputs are entered in the Genetic Algorithm Parameters panel shown in Figure 4. A reasonable population size is needed to search the design space adequately. Numerous trials have indicated that a minimum size of 100 is necessary. This value was used in all the present calculations. The optimization stops at the maximum number of generations which was set as 100. The GA utilizes crossover and mutation operators set by the user. The crossover operator exchanges traits  the chord and twist  (called genes) between two or more solutions (parents) in the hope of producing superior solutions (children). Mutation ensures that each child is unique by altering their genes slightly. The crossover and mutation settings can significantly influence the performance of the GA. The Crossover Type can be either SBX or Uniform. SBX crossover is performed between two parents according to a scheme described in Deb (2001) and requires a SBX index which controls the probability of creating nearparent solutions or distant solutions as children. When Uniform is selected, a random binary vector is created and selects the genes from the first parent when the value is 1 and from the second when the value is 0, and combines the genes to form the child. The Crossover Probability controls whether the parents will be subjected to crossover or not and is a parameter for both the SBX and Uniform crossover types. The values shown in the Genetic Algorithm Parameters box in Figure 4 were used for all the present calculations.
Mutation causes small random changes to the children, promoting the diversity of the population. Only polynomial mutation as described in Deb (2001) is available in SWRDC, which depends on the Mutation Index and Mutation Probability parameters. The Mutation Index controls the intensity of the mutation (or the randomness) when altering the genes of a child. The Mutation Probability is the probability that each gene from a child will be mutated. A commonly used heuristic for Mutation Probability is 1/(number of variables) = 1/(2N + 2) in SWRDC, where N is the degree of the Bezier curve.
The structural analysis is combined with the aerodynamic analysis that gives the power output and starting time. For hollow blades, the shell thickness is determined by reducing the shell thickness iteratively starting from the maximum thickness value (solid blade) to the point where the specified minimum thickness (e.g. 0.01 of the chord, see subsequent paragraph) or the structural limit, defined by the maximum strain, is reached. The code gives two alternate ways of assessing structural loads, both based on the simplified load model (SLM) of the international safety standard for small turbines (International Electrotechnical Commission IEC 61400–2 ed3.0 (2013)). The SLM defines eight load cases that determine the loads mainly in the blade root and main shaft. An example of SLM analysis is given in Wood (2011) for a 0.94mlong timber blade. It is shown that load case E ‘Maximum Rotational Speed’ and case H ‘Parked Wind Loads’ gave the greatest blade stresses. SWRDC allows the user to choose between these two cases: all the examples considered here used case H. The IEC also mandates load and material safety factors. For ‘full characterization’ of the blade material, the material safety factor is 1.1 and the safety factor for ‘aeroelastic modelling’ of the loads is 1.35. This is lower than the value of 3.0 that is mandated for SLM calculations. All the present cases used an extreme (50year) wind speed of 52.5 m/s for load case H making the design suitable for an IEC Class III wind turbine.
The three composite blades used hollow sections and the timber blade was assumed solid. The minimum shell thickness was set as 0.01 of the chord. The thickness of the blade sections shown in Clausen et al. (2013) was in the range 0.02c to 0.24c, and it is assumed that the smaller thickness is achievable. No calculations were done for the bamboo composite, whose overall performance would be similar to the flaxpolyester blade.
Fatigue behaviour was not considered. The IEC SLM has only one load case (case A) for fatigue. This typically gives lower stress levels than cases E and H, but of course, the allowable stress will be much lower. According to Equation (48) of IEC 614002, the present blade will experience 1.73 × 10^{10} fatigue cycles in a 20year life and so fatigue behaviour must be assessed very carefully. Wood (2011) showed that hoop pine has excellent fatigue properties, and it is likely that the composites studied here would also have good fatigue properties. The lack of fatigue data for ABS M30 is a major impediment to its use for small blades.
Results and discussion
All the blade designs were found to be feasible in that the maximum strains listed in Table 1 were never exceeded in any blade. Table 2 for the SG6043 airfoil and Table 3 for the SD7062 give the power coefficient, starting time, blade mass, and the first flapwise natural frequency of the blade (see Hansen 2008) and the tip deflection for IEC load case H. As expected, the SG6043 airfoil produces more power but the approximately 2% difference is not as large as the difference in the lift to drag ratio shown in Figure 1 and may not be observable in practice. Note that for all materials, C _{p} and the chord and twist distributions should all be equal which is true for the thicker SD7062 (the results are not shown) but not for the SG6043 blade. The distributions in Figure 5 indicate that the ABS M30 blade is an outlier because it needs a thicker shell: the thickness is plotted in Figure 6 for w = 1.0 and 0.75. In the tip region, where structural demands are low, the thickness reduces to the minimum and the chord and twist become equal to those of the other materials. For w = 0.5, and for all other materials at all w, the minimum s = 0.01c was sufficient to ensure structural integrity.
The fundamental natural frequency should be well above the frequency corresponding to the design angular velocity of 57.6 rad/s. This is the case for all but one or two of the ABS M30 blades. The tip deflection for IEC load case H is highest for the weakest material, ABS M30, but in most cases is small enough to justify the use of simple beam theory which does not account for any change in shape due to the deflection.
For w = 1, T _{s} scales with blade mass as this in turn scales the inertia for constant shell thickness which results in geometrically similar blades. With decreasing w, the general features of blade performance are common to all four materials and the two airfoils: both C _{p} and T _{s} decrease. For all materials, the decrease in C _{p} is significant only for w = 0.5: for all other w, it is unlikely that the small differences would be noticeable in terms of actual power production of a real turbine using these blades. The initial decrease in C _{p} with w is less than that which has been found in the previous optimizations of Wood (2011) and Pourrajabian and Mirzaei (2014). The trend in the blade mass depends on the blade material but is never monotonic.
Figures 7 and 8 show the chord and twist distributions for Eglass/polyester and ABS30 respectively. The top part of each figure gives the distributions for the SG6043 and the bottom half for the SD7062. For all materials, the changes in twist and chord are greatest near the hub, as was found by Wood (2011) and Pourrajabian and Mirzaei (2014), because the hub region generates most of the starting torque but little powerextracting torque. The Eglass/polyester chord is maximized at w = 0.75 for both airfoils, which gives by far the heaviest blade. The drastic change in blade shape for w = 0.5 is typical of these optimizations and indicates excessive emphasis on starting; for the reasons given in Wood (2011), the fastest starting blade has an infinitely small chord. In terms of the Pareto front, which should be the plot of C _{p} vs T _{s} from the optimizations, reducing w eventually increases drastically the slope of the front. Practical blades are more likely to be taken from higher values of w where reduced power extraction is more than matched by a much larger reduction in T _{s}. In other words, the tradeoff leads to better blade designs.
For Eglass and flax blades, the better performing airfoil gave better blade designs because the minimum shell thickness was sufficient to maintain structural integrity. This is also the case for the solid hoop pine blades which are also very strong. For ABS M30, however, better performance was achieved with the thicker airfoil as this moved the shell further from the neutral axis and increased stiffness. A further feature of the SG6043 blades is that Ө _{ p } near the tip is negative for w = 1. This is clearly needed to maximize power, but negative twist produces negative starting torque by Equation 1 and so is eliminated by the optimization process as starting becomes more important.
Conclusions
This paper describes a multiobjective optimization program for small wind turbine blade design in which the optimization can be of any combination of maximum power and minimum noise, starting time, and structure. Traditional blade element theory predicts the power extraction, and a modification of it gives the starting time. Noise is not used in the present calculations. Simple beam theory assuming isotropic material properties determines the structural strength on the basis of a choice between two load cases of the IEC 614002 SLM for small wind turbines. The blades are assumed hollow with a constant shell thickness. The optimization uses a GA with user input population and maximum number of generations, as well as crossover type and mutation index. Starting performance is a specific requirement of small blades that have no pitch adjustment. Since the starting time depends on the blade inertia, which in turn depends on material and density, this introduces a complex interaction between the objective functions that does not occur if design is based only on power production.
The example design is a 1.1m blade for a permanent magnet generator producing 754 W at 550 rpm. Two airfoils are considered: the SG6043 which has the best known lift to drag ratio at Reynolds numbers typical of small turbines but is only 10% thick, and the thicker SD7062 which has 14% maximum thickness. Four blade materials were investigated:

1.
Eglass and polyester resin as used commonly for wind turbine blades

2.
Flax and polyester, as a potential way to reduce the environmental footprint

3.
A relatively strong rapid prototyping plastic

4.
Timber
Optimal designs were found for power extraction only and then two cases of increasing importance of the starting time.
For materials, 1, 2, and 4, the better performing SG6043 airfoil produced slightly better power output and starting time, partly because the minimum specified shell thickness was not exceeded for any design, meaning that the materials were sufficiently strong. Generally, the chord and twist increased as more weight was given to starting. This increased the blade mass and hence the inertia, but the increase in starting torque was even greater. The changes were mostly in the hub region where more starting torque is generated, but the twist in the tip region increased from the small negative values required to maximize the power output only. This is because Equation 1 gives a negative torque for a negative twist.
It was noted that rapid prototyping has great potential for small blade manufacture as it does not require expensive moulds and can produce blades designed for specific conditions. If the technique fulfills its promise, then easily modified multidimensional optimization will become a routine part of the design process. Table 1 shows that the example plastic that was selected, ABS M30, while a relatively strong plastic, is considerably weaker than materials 1, 2, and 4. It was the only material for which the shell thickness exceeded the minimum, and the thicker SD7062 airfoil was a better choice. Nevertheless, the ABS M30 blades were not significantly inferior to the others. One crucial warning must be made about rapid prototyping: none of the materials found by the authors have been tested for fatigue which is important as the IEC 614002 procedure stipulates that the blades designed here will experience in excess of 10^{10} fatigue cycles over a 20year lifetime.
The results presented here demonstrate the complex interaction between the multiple aerodynamic requirements of small wind turbine blades and the strength and density of the materials from which they are made.
Endnotes
^{a}See for example, http://www.recreationalflying.com/tutorials/scratchbuilder/timber.html (accessed 13 September 2014).
^{b} http://www.iitk.ac.in/kangal/codes.shtml. Version 1.1.6 was downloaded.
^{c} http://www.stratasys.com/materials/fdm/absm30 (accessed June 6, 2015).
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Acknowledgements
This research is part of a programme of work on renewable energy funded by the Natural Science and Engineering Council (NSERC) and the ENMAX Corporation. All the codes used in this paper are available without charge from the corresponding author.
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Both authors declare that they have no competing interests.
Authors’ contributions
The SWRDC code was written by MS. The calculations presented here were done by DW and the paper was written jointly. The authorship is alphabetical. Both authors read and approved the final manuscript.
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Keywords
 Small wind turbines
 Blade design
 Multidimensional optimization
 Power
 Starting performance