Multi-dimensional optimization of small wind turbine blades
- Matias Sessarego^{1, 2} and
- David Wood^{2}Email author
DOI: 10.1186/s40807-015-0009-x
© Sessarego and Wood. 2015
Received: 9 January 2015
Accepted: 30 March 2015
Published: 1 July 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 E-glass and polyester resin; (b) flax and polyester resin; (c) a typical rapid prototyping plastic, ABS-M30; 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 ABS-M30 benefitted from the thicker SD7062 section.
Keywords
Small wind turbines Blade design Multi-dimensional optimization Power Starting performanceBackground
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 cost-effective 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 bio-composites and bio-resins 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 three-dimensional blade shape and the potential to design blades for specific applications, such as rapidly starting blades for low-wind areas or especially quiet blades for building mounting.
In order to exploit these opportunities, the small wind turbine blade design must become increasingly multi-dimensional. Starting is often a critical issue for small blades as most do not have pitch adjustment. Recent two-dimensional optimizations considered starting and power extraction (Wood 2011, Pourrajabian and Mirzaei 2014, and Shah et al. 2014). Clifton-Smith (2010) added noise. The common finding of these studies is that a multi-dimensional design is actually a trade-off. Typically, the most powerful blade is the slowest to start and Clifton-Smith (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 large-blade 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 small-blade 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 multi-dimensional small wind turbine blade optimization including structure, by designing hollow blades using (a) traditional E-glass and polyester; (b) flax and polyester; (c) a strong, rapid prototyping material; and (d) solid blades using timber. The 1.1-m-long 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 Wind-turbine Rotor Design Code (SWRDC) developed by the authors and available from them. The code combines the multi-objective 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 E-glass 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 three-dimensional printing will revolutionize small-blade 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.
One of the prime requirements for multi-dimensional 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 higher-fidelity 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 AWT-27 turbine performance data (Buhl 2012).
Methods
Starting performance and starting time
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 Adams-Moulton 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 small-turbine 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 user-specified 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
Material properties used for blade optimization
Quantity (units) | E-glass-polyester | Flax-polyester | ABS M-30 | Hoop pine | Bamboo-petroleum resin |
---|---|---|---|---|---|
Elastic modulus (GPa) | 36.9 | 23.4 | 2.4 | 12.1 | 20.5 |
Blade density (kg/m^{3}) | 1,640 | 1,290 | 1,040 | 550 | Not given |
Max. strain (μstrain) | 7,684 | 5,420 | 7,500 | 3,250 | 4,270 |
The strain is determined around the contour of the cross-section 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
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 user-specified 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°.
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 0.94-m-long 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 ‘aero-elastic 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 (50-year) 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 flax-polyester 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 61400-2, the present blade will experience 1.73 × 10^{10} fatigue cycles in a 20-year 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 M-30 is a major impediment to its use for small blades.
Results and discussion
Blade optimization results for the SG6043 airfoil
Material | w | C _{ p } | T _{ s } (s) | Mass (kg) | Frequency (rad/s) | Tip deflection (m) |
---|---|---|---|---|---|---|
E-glass/polyester | 1.0 | 0.477 | 1.42 | 0.288 | 156 | 0.146 |
0.75 | 0.476 | 1.28 | 0.331 | 175 | 0.231 | |
0.5 | 0.427 | 0.87 | 0.262 | 149 | 0.169 | |
Flax/polyester | 1.0 | 0.477 | 1.18 | 0.203 | 135 | 0.245 |
0.75 | 0.476 | 1.04 | 0.259 | 146 | 0.202 | |
0.5 | 0.457 | 0.79 | 0.240 | 141 | 0.226 | |
ABS M-30 | 1.0 | 0.474 | 1.79 | 0.698 | 81 | 0.447 |
0.75 | 0.470 | 0.90 | 0.520 | 114 | 0.352 | |
0.5 | 0.464 | 0.84 | 0.523 | 115 | 0.346 | |
Hoop pine | 1.0 | 0.477 | 1.71 | 0.351 | 124 | 0.200 |
0.75 | 0.475 | 1.54 | 0.516 | 164 | 0.116 | |
0.5 | 0.391 | 1.00 | 0.525 | 149 | 0.136 |
Blade optimization results for the SD7062 airfoil
Material | w | C _{ p } | T _{ s } (s) | Mass (kg) | Frequency (rad/s) | Tip deflection (m) |
---|---|---|---|---|---|---|
E-glass/polyester | 1.0 | 0.468 | 1.67 | 0.482 | 268 | 0.036 |
0.75 | 0.465 | 1.48 | 0.505 | 326 | 0.031 | |
0.5 | 0.308 | 0.66 | 0.206 | 142 | 0.205 | |
Flax/polyester | 1.0 | 0.469 | 1.36 | 0.386 | 242 | 0.056 |
0.75 | 0.467 | 1.19 | 0.370 | 262 | 0.060 | |
0.5 | 0.425 | 0.81 | 0.22 | 201 | 0.129 | |
ABS M-30 | 1.0 | 0.468 | 1.52 | 0.528 | 107 | 0.331 |
0.75 | 0.468 | 1.00 | 0.474 | 131 | 0.274 | |
0.5 | 0.453 | 0.87 | 0.457 | 139 | 0.277 | |
Hoop pine | 1.0 | 0.469 | 2.51 | 0.881 | 207 | 0.038 |
0.75 | 0.467 | 2.30 | 0.931 | 247 | 0.033 | |
0.5 | 0.410 | 1.44 | 0.549 | 160 | 0.104 |
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 M-30 blades. The tip deflection for IEC load case H is highest for the weakest material, ABS M-30, 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.
For E-glass 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 M-30, 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 multi-objective 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 61400-2 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.
- 1.
E-glass 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 multi-dimensional optimization will become a routine part of the design process. Table 1 shows that the example plastic that was selected, ABS M-30, 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 M-30 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 61400-2 procedure stipulates that the blades designed here will experience in excess of 10^{10} fatigue cycles over a 20-year 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/abs-m30 (accessed June 6, 2015).
Declarations
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.
Authors’ Affiliations
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