A d esign and performance prediction method for small horizontal axis wind turbines and its application

: The paper deals with small wind turbines for grid-independent or small smart grid wind turbine systems. Not all small turbine manufacturers worldwide have access to the engineering capacity for designing an efficient turbine. The objective of this work is to provide an easy-to-handle integrated design and performance prediction method for wind turbines and to show exemplary applications. The underlying model for the design and performance prediction method is based on an advanced version of the well-established blade-element-momentum theory, encoded in MATLAB™. Results are (i) the full geometry of the aerodynamically profiled and twisted blades which are designed to yield maximum power output at a given wind speed and (ii) the non-dimensional performance characteristics of the turbine in terms of power, torque and thrust coefficient as a function of tip speed ratio. The non-dimensional performance characteristics are the basis for the dimensional characteristics and the synthesis of the rotor to the electric generator with its load. Two parametric studies illustrate typical outcomes of the design and performance prediction method: A variation of the design tip speed ratio and a variation of the number of blades. The predicted impact of those parameters on the non-dimensional performance characteristics agrees well with common knowledge and experience.

research with the aim of maximizing power coefficient and minimizing the starting time of small wind turbines blades. Liu et al. [24], developed an optimization method for wind turbines blades in order to maximize the annual energy output. A 7.5% increase in the annual energy output was realized in the study. Meng-Hsien [25], undertook a different study both numerically and experimentally on a small wind turbine blade, where one blade was designed using the BEM theory and the other was non-twisted blade. It was found out that the blade designed with BEM theory had higher coefficient of performance compared to the non-twisted blade.
Most wind turbines optimizations are conducted for wind turbines working in on design conditions. Wind turbines are exposed to different working conditions. In addition, matching the electric generator to the turbine performance characteristics curves is fundamental in ensuring the components are protected against extreme design effects.  Hence, the objective of the paper is to report a procedure for designing blades for small HAWTs. The turbine rotor with these blades should yield a shaft power as close as possible to theoretical maximum and match best to a given electric generator. The underlying theory is an advanced version of the well-established blade-element-momentum (BEM) theory, encoded in our own in-house MATLAB™ code deapWind 1 and validated extensively Kaufmann [5]. Eventually, as an exemplary case study, this paper describes the design of a small generator/battery charger with a 1 m diameter turbine rotor. Starting point is a one-year histogram of measured wind speed at the candidate site at Dedan Kimathi University of Technology in rural central Kenya. Upon design and manufacturing, a concluding wind tunnel experiment demonstrates how well the methodology works.

Design and performance calculation procedure
The turbine design methodology comprises three fundamental steps: (i) The design of the HAWT rotor, especially its profiled and twisted blades, (ii) the prediction of its performance characteristics in terms of shaft power, shaft torque and axial thrust force on the complete rotor, (iii) the overall power produced when "loaded" by the torque from the electric generator. Upon this last steps, it is straightforward to determine the overall electric power output for a given-say annual-wind speed histogram. Eventually, measures to achieve low cut-in wind speed and switch-off features for high winds have to be designed. The latter steps are out of the scope of this paper.

Aerodynamic design of rotor blades
The blade design is based on the standard GLAUERT/SCHMITZ blade element momentum (BEM) theory as described for instance Gasch and Twele [6] or Gerhard et al. [7]). It is summarized only briefly; some more details are presented in the following section. The turbine design process starts with the specification of:  the design wind speed far upstream of the turbine c0,design,  the design rotor diameter Dtip (= 2rtip),  the density of the air , the turbine will operate with, and  the design tip speed ratio design, of the turbine s 0, 0, with the circumferential velocity of the blade tip due to its rotation utip and the rotor speed n.
Essentially, c0,design and Dtip determine the power harvested by the turbine, and design, its rotational speed for a given c0,design. Each blade is thought as a number of blade elements (BE) with a small radial size r, stacked from root to tip, Figure 3 left. The BEs in a coaxial strip form themselves into an elementary blade cascade (CA). The most relevant geometrical parameters of each local BE are, Figure 3 right,  the chord length, l,  the pitch angle, ,  the angular distance, t, between two circumferentially adjacent BE's,  the number of blades, B, and  the solidity, , of the CA (2) Figure 4 left shows schematically a wind turbine with a boundary stream tube. The free stream wind speed far upstream, c0 is decelerated to c3 far down stream in the wake. c3 comprises a circumferential velocity component cu3 (i.e., the swirl) and an axial (= meridional) cm3. The spatially averaged velocities at the entrance and exit plane of a CA are depicted in Figure 4 right. c's indicates absolute frame of reference, w's rotating frame of reference flow velocities; u is the circumferential velocity of the CA due to the rotation of the rotor. Again, c2 comprises a circumferential and a meridional velocity component cu2 and cm2, respectively. Instead of carrying on with dimensional quantities, the standard GLAUERT/SCHMITZ theory deals with the non-dimensional axial and tangential induction factors  For maximum power extraction from the wind the classic BETZ theory [8] requires which is assumed to hold true, also within the GLAUERT/SCHMITZ theory. The outcome of the combined application of the axial and angular momentum conservation are design values of a and a' or in other words, the kinematics of the flow, each local BE ensures in order to provide maximum power output at a specified design wind speed, c0, design. The design values of a and a' are depicted in   The effect of any flow deflecting elementary blade cascade CA is attributed to lift and drag forces L and D from all BE's in the CA which are considered to be isolated infinite span airfoil sections aligned to the free stream incoming flow velocity w  with an angle of attack , Figure 6 left. Key parameters of an airfoil sections are its lift coefficient CL and its drag/lift ratio , defined as 2 , 1 2 The airfoil shape is chosen by the designer, who also has to identify the CL () and CD () from wind tunnel data or other sources. The inflow velocity to each BE corresponds to the vector mean of the inlet and outlet relative velocities w1 and w2; its magnitude is and the flow angle Eventually, this results in the solidity of each elementary blade cascade CA.
and the pitch angle In our implementation the public domain software XFOIL (Drela, [9,10]) is integrated, which yields the airfoil lift and drag coefficients as a function of the chosen airfoil shape, angle of attack and the Reynolds number  being the kinematic viscosity of the air. Finally, the geometrical parameters design, hence l for the chosen number B of rotor blades, and design are sufficient to draw the blades and the complete rotor.

Prediction of aerodynamic rotor performance
The goal of this analysis is the aerodynamic performance of the rotor with its geometry as found in the previous section. Among others the shaft power, the shaft torque and the axial thrust the rotor The method chosen for estimating the performance characteristics is again based on the blade element momentum (BEM) theory, although refined for reasons of accuracy. In contrast to the design method, the flow velocities, or synonymously a and a', and the angle of attack, , hence lift and drag at each BE need to be computed for arbitrary values of . For the sake of completeness, some more steps, which have been omitted in section 2.1 are presented in this section. Full details can be found in Kaufmann [5].
Starting point is the conservation of axial momentum applied to the fluid in a coaxial strip from 0 to 3 as in Figure 4 left. It yields the axial force acting on the elementary blade cascade CA The mass flow rate through the CA is The theoretical shaft power is obtained via EULER's turbine equation (i.e., conservation of angular momentum, friction ignored) as Eventually, with the definitions of the induction factors Eq (3a,b), the CA thrust and power coefficients become; respectively. For any given value of , the two Eq (14a,b) bear the four unknown quantities CT,CA, CP,th,CA, a and a'. Additional equations are obtained by considering each CA as a set of isolated BE's experiencing a lift and a drag force. Referring to Figure 6 right the force components on one BE in the CA are Eventually, the local thrust and power coefficient of the complete CA (consisting of B BE's) are obtained as respectively. Eventually, equating CT,CA from Eq (14a) and (16a) and CP,th,CA = CP,CA from Eq (14b) and (16b) yields two non-linear equations for the two unknowns a and a' for any  considered: So far the analysis refers to one CA at r/rtip. The overall turbine thrust and power coefficients, i.e., the targeted characteristics of the complete rotor, are obtained by summation over all CA's: A number of supplementary semi-empirical models are required to take into account effects that had been neglected so far. For instance, friction was partly excluded resulting in a theoretical CP,th. or: At the blade tip a radial flow component develops causing a tip vortex. And the root region of the blade in the vicinity of the hub is affected by vortex type secondary flows.
A customary model for the flow in the blade root and tip region is due to BETZ/PRANDTL and Glauert [11] (see e.g., a summary in Sorensen [12], Gasch and Twele [6], Moriaty and Hansen [13]). The thrust and power coefficients derived from momentum conservation (Eq (15a,b)) are modified by a factor F1 that gradually reduces the axial and tangential forces from those CA's which are located close to the blade tip and the hub: Since the flow "leaks" around the blade tip from pressure to suction side, lift and drag of airfoil sections in the tip region are different from what the infinite span polar data suggest. Shen In practice, as the axial induction factor a exceeds a critical value acrit ≈ 1/3, complicated flow patterns result in much larger thrust coefficients as compared to the prediction by the basic BEM. Hence, Eq (14a) is replaced according to Buhl [15] by; As in the design part, the polar data of the infinite span airfoil sections are obtained via XFOIL. Post stall data, however, are obtained from Viterna et al. [16].
Some literature suggests solving the non-linear Eq (18a,b) with all their hidden supplementary model equations for a and a' explicitly (see e.g., the textbooks of Hansen [17] or Sørensen [12]). Since the authors encountered stability problems, an implicit scheme was set up [5]. It is encoded in our own inhouse MATLAB code, deapWind, which turned out to be very stable and reliable.

Converting non-dimensional into dimensional performance characteristics
Once the non-dimensional performance characteristics are known, it is trivial converting them into dimensional: It is worth noting that the value of Re stays the same only, if the size of the turbine Dtip is preserved and c0 equals the design wind speed c0,design. If the dimensional performance curves are required for other than the original design parameters, for instance for various wind speeds c0, two options are possible: Either eqs. (24a,b,c,d) are used neglecting the effect of Re, or running the performance prediction scheme as described in section 2.2 for each c0. The first option is faster and often sufficient for most practical purposes.

Exemplary parametric studies
The results of two parametric studies targeting at small wind turbines illustrate the outcome of the design and performance prediction method. The first study shows the effect of the choice of design, as "a family of quiet, thick, natural-laminar-flow airfoils for 1 to 3 m-diameter turbines" (SOMERS [18]).
In particular, the root, middle and outer part of the blades are made from S835, S833 and S834 airfoils with a thickness/chord ratio of 21, 18 and 15%, respectively. Computing airfoil lift and drag polars with XFOIL requires an assumption how the initially laminar boundary layer becomes turbulent in chord wise direction on the airfoil pressure and suction side. Particularly at low Re, which are typical for small wind turbines, the chord wise position of the laminar/turbulent transition depends on many details. This has a substantial effect on the predicted lift and drag. XFOIL offers the option "natural transition", where the position of laminar/turbulent transition is calculated using the e N -method with a chosen threshold amplification ratio of Ncrit. Ncrit is a measure of the quality of the flow incident to the airfoil with, 15 for very clean inflow, 1 for very "dirty" (corresponding to a free stream turbulence level of 2%) with the consequence of transition very close to the airfoil leading edge; in a second option, positions x/l (= 0: leading edge, = 1: trailing edge) at the pressure and suction side are specified where transition is forced at the latest, Drela [10].
In these two parametric studies Ncrit is set to 1 with natural transition accordingly. The XFOILcomputed polars for the three BE's at hub, mid-span and blade tip are depicted in Figure 7. At each BE design is chosen close to the most favorable minimum drag/lift ratio, . Table 1 shows the results for three-bladed rotors. Increasing values of design lead to slimmer blades. As expected CP always reaches its peak close to design but stays with approx. 0.42 the same. Note that the theoretical (BETZ) limit is 0.59. CM is affected severely by the choice of design: The lower design, the larger CM. The effect of design on CT is mainly at   design. The dimensional characteristics in the lower row throw further light on the properties of the different designs: Increasing design shifts the peak power to larger rotor speeds, resulting in lower shaft torques. Nevertheless, the peak powers are very similar.  Table 2. The performance characteristics are very similar, irrespective of the number of blades chosen. It must be noted that the spanwise distribution of solidity and blade pitch of all three rotors is kept the same as obtained by Eqs (9,10). The minor effect of B stems from the impact of Re and those loss mechanisms that are a function of B. Hence, a criterion for the choice of B-among others-is the technical practicability of many slim or a few more massive blades, but not the power output.

Case study: Micro wind turbine with battery charger
The objective of this exemplary case study is the design of a HAWT for a small generator/battery charger. The analysis involves matching the wind turbine rotor performance curves to the torque characteristic of the battery charger. Upon manufacturing, a concluding wind tunnel experiment validates the theoretical analysis.

Available wind resource and turbine specification
Starting point is a one-year histogram of measured wind speed data collected at the candidate site, which is the campus of Dedan Kimathi University of Technology, Nyeri in central Kenya. The wind data were monitored from December 2016 to December 2017. The data includes the measured wind speeds and directions. Figure 8 visualizes the hours of occurrence of the ten (10) minutes averages of prevailing wind speeds at this site. Obviously, at this site wind speeds in the 4, 5 and 6 m/s wind speed bins are most prevalent. For the micro turbine to be designed c0,design = 6 m/s is chosen. The specified air density is  = 1.058 kg/m 3 , which corresponds to the altitude above sea level of the planned site for the turbine.  At any rotor speed n, the shaft torque provided by the turbine rotor must equal the torque required by the electric generator:

MS,Turb=MS,Gen
The generator/charger unit characteristic MS,Gen(n), is determined experimentally by coupling the shaft of the electric generator to an auxiliary electric motor. The torque required by the generator is measured for any set speed via a torque measurement shaft in-between. The measured generator/charger characteristic for the controller status "Battery empty-charge" is the red curve in the middle graph of Figure 10.
It is worth to note that for maximum energy yield the generator/charger characteristic should be a quadratic parabola. Once parameterized correctly this would enable to operate the turbine at maximum CP for any c0. This strategy is provided by more advanced controllers or grid-connected turbines. In contrast, objective of this case study is the aerodynamic design of a rotor to the given generator/charger characteristic that ensures unambiguous rotor speed for any wind speed, keeps the rotor diameter as small as possible and provides a maximum annual energy output for the given wind resource.

Electric generator/battery charger
The electric generator/battery charger unit consists of three main components:  The electric generator; the turbine rotor is attached to the shaft of the generator without gearbox  The battery to be charged  The control unit, essentially comprising a 3-phase rectifier, a charge controller, a voltage threshold switch acting on a power relay which connects the generator to a load resistor rather to the charge controller, and a supply point to consume e.g., USB ports for charging wireless phones. Its layout is depicted in Figure 9. The charge controller distinguishes between the bulk mode "Battery empty-charge with a current up to 2 A" and the float mode "Battery full-trickle charge with 0.6 A". Power provided by the turbine rotor but not absorbed by the battery is dumped via the load resistor. Voltage and current at the charge controller can be monitored via Bluetooth ® on a smart phone.

Hours of occurence t [h]
Energy output @ shaft [kWh] Mainly for low manufacturing cost, the rotor is two-bladed. Laminar/turbulent transition was fixed to x/l = 0.02 on each BE suction side and 0.05 on the pressure side, assuming that the foreseen method of manufacture does not yield to perfect smooth and precise blade surfaces, with the consequence of laminar/turbulent transition close to the blade leading edge. Relevant parameters of some of the rotor/charger assemblies are compiled in Table 3. One criterion for selection is the annual energy output at the shaft of the rotor 2 3 , 0 , all bins 2 4 which translates into the electric energy output when subtracting all electric losses. Other criterions are the rotor speed at the most prevalent wind speed 6 m/s and at high winds, here 12 m/s. Low rotor speeds are easier to handle and produce less noise. Also, the axial thrust is a criterion. Large axial thrust forces require more expensive structures to hold the turbine.  The rotor ultimately selected has the parameters Dtip = 1 m and design = 4.0. Important geometrical parameters of each BE are compiled in Table 4. Figure 10 illustrates the synthesis of the measured generator characteristic ("mode Battery empty-charge", red line with red crosses in the middle diagram) and this rotor at different wind speeds; the black circles indicate the operating points of the complete assembly. Figure 8 right shows the predicted shaft energy output at each wind speed bin of the turbine/generator assembly.  The software deapWind yields the BE coordinates of 15 BE's along the span. From that a 3D CAD model and 2D technical drawings are established. For manufacturing a traditional technique is chosen that requires a standard carpenter's workshop only: The blades are carved from a high quality wood block with the help of a grid of drill holes, Figure 11. The perpendicular distance from each of the grid point on the blade surface to the surface of the block is measured in the 3D CAD model. This is performed for both sides of the blade. This distance is the depth of the 8 mm holes to be drilled. The drilling of the holes is to ensure that the correct depth of material removal is achieved. It acts as a guide on the maximum depth of material removal obtain the blade profile. The material was removed manually to almost the end of the drilled holes that mark the surface of the blades. The finishing was done using rough and fine sand papers. In order to achieve the finest surface finish, the blades were applied with wood filler, allowed to dry up, then filed using the finest sand papers. At the hub, the material removal and smoothening was achieved by use of rough and fine wood files.

Wind tunnel validation test
For the experiments, the rotor was mounted upstream on the drive train, comprising a steel shaft, separate bearings and the electric generator. The battery charger was connected to the generator. The complete assembly was tested in the University of Siegen wind tunnel. This wind tunnel is a closedcircuit type allowing a maximum wind speed of the wind tunnel of 70 m/s at a turbulence intensity less than 0.35%. Figure 12 shows the experimental set up in the wind tunnel (all dimensions are in millimeters). The wind speed was measured using a propeller anemometer, the rotational speed with an optical laser tachometer. The wind speed was varied in steps. Upon steady-state operation, the rotational speed of the turbine was recorded. It is important to note that the air density in the wind tunnel lab was approx.  = 1.17 kg/m 3 , resulting in different dimensional turbine performance characteristics as compared to those in the previous sections. This has been taken into account. Ideally, the shaft power of the rotor would have been measured. This was not possible. The electric power was measured instead. A comparison of the rotational speed of the complete rotor/charger assembly in bulk mode ("Battery empty-charge") at different wind speeds is depicted in Figure 13 left. Obviously, the "loading" of the rotor by the electric generator is very close to what was expected from the previous analysis. This partly validates the aerodynamic rotor design. The difference of predicted rotor shaft power and the measured electric power, Figure 13 of the complete assembly. A major loss is attributed to the electric generator. Its electric efficiency characteristic was measured separately-similarly as the complete generator/charger/battery characteristics before, and is depicted for n = 560 rpm in Figure 14. At PS = 29 W (corresponding to c0 = 6 m/s) the generator's electric efficiency is 47%. Other electric losses stem from the battery charger, and-up from c0 = 7.5 m/s-then active load resistor for protecting the battery. Eventually, fundamental errors due to blockage of the wind tunnel test section by the rotor and a potentially reduced turbine efficiency due to the manual manufacture may account for a lower than predicted power coefficient of the rotor. Note, that the turbine shaft power could not be measured directly during the wind tunnel measurement campaign.

Summary and conclusions
An analytical method for the aerodynamic design of horizontal of axis wind turbines including performance prediction was described. The underlying theory for design and shaft power and thrust force prediction is an advanced version of the well-established blade-element-momentum (BEM) theory, encoded in our own in-house MATLAB™ code deapWind. The procedure yields the geometry of the aerodynamically profiled and twisted blades which are designed to yield maximum power output for a given design wind speed.
Two parametric studies illustrated typical outcomes of the design and performance prediction method: (i) A variation of the design tip speed ratio design; the higher design the larger the rotational speed at a given wind speed and the smaller the torque. Hence, the choice of design allows to match the turbine to sites with either low or high mean wind speeds. Moreover, the higher design the slimmer the blades, which may have an impact on manufacture and structural health. Load resistor R1 activated Load resistor R1 activated taken into account in the advanced blade-element-momentum theory. Hence, a criterion for the choice of B-among others-is the technical practicability of many slim or a few more massive blades, but not the power output as occasionally naively proposed. Eventually, a more comprehensive case study dealt with a micro wind turbine for a small low cost generator/battery charger. A goal was to foresee turbine blades of high aerodynamic quality that can be manufactured with techniques available in less developed regions of the world. The analysis involved matching the turbine rotor performance curves to the torque characteristic of the battery charger, manufacture of the rotor and a concluding wind tunnel experiment for validation of the theoretical analysis. Starting point was a one-year histogram of measured wind speed data collected at the candidate site in rural central Kenya. Prior to the wind turbine design, the generator torque/speed characteristic was determined experimentally in a laboratory by coupling the shaft of the electric generator to an auxiliary electric motor. The torque was measured for any set speed via a torque measurement shaft in-between. The generator was connected to the charge controller and empty battery. The interplay of various turbine rotors and the drive train/battery charging was simulated. Criterions for selection of the rotor were the annual energy output at the shaft, the rotor speed at the most prevalent wind speed and at high winds, and the axial thrust exerted on the rotor by the wind. A blade manufacturing technique was chosen that requires a standard carpenter's workshop only. Compared to 3D printing and additive manufacturing, this technology is simple, cheap and accessible to less developed nations for developing wind turbines for providing energies to communities off the national grids. Finally, the complete rotor/drive train//battery charger assembly was tested in the University of Siegen wind tunnel. The rotational speed for various wind speeds are very close to what was expected from the previous analysis. This partly validates the aerodynamic rotor design. The difference of predicted rotor shaft power and measured electric generator power. i.e., the value of the overall electric efficiency, is mainly attributed to the electric losses in the generator and the battery charger circuit.
Upcoming efforts must provide for passive yaw and storm protection. A tail vane can act as a passive yaw system, or a downwind installation of the rotor. The predicted thrust may facilitate the design of any mechanical tilt mechanism and eventually dimensioning of the tower. Safe cut-out at high winds and early cut-in of the turbine can be assisted by an advanced electric control.

Author contributions
S. K. Musau designed and manufactured the micro turbine described in the case study. K. Stahl assisted in designing the turbine including the electric part and guided the wind tunnel experiments and data evaluation. K. Volkmer created the software deapWind by linking an extended version of an existing wind turbine design method with N. Kaufmann's turbine performance prediction procedure. Th. Carolus is the leader for the project and compiled the manuscript with the help of all co-authors.

Acknowledgment
The project was partly funded by the Erasmus+ Programme of the European Union (Key Action 107 Mobility with partner countries). Partner were the University of Siegen and Dedan Kimathi University of Technology, Kenya. The authors gratefully appreciate this financial support. The authors also want to thank S. Kreklow and T. Schönemann from the University of Siegen electronic workshop for the design of the electronic circuits, J. Janssens from the mechanical workshop for the assistance in