Design of an Active Axis Wind Turbine (AAWT) That Can Balance Centrifugal and Aerodynamic Forces to Reduce Support Infrastructure While Maintaining a Stable Flight Path

Abstract

This study introduces a novel approach to wind energy by investigating a novel Active Axis Wind Turbine design. The turbine is neither a horizontal nor vertical axis wind turbine but has an axis of operation that can actively change during operation. The design features a rotor with a single blade capable of dynamic pitch and tilt control during a single rotor rotation. This study examines the potential to balance the centrifugal and aerodynamic lift forces acting on the rotor blade assembly, significantly reducing blade, tower, foundation and infrastructure costs in larger-scale devices and decreasing the levelised cost of energy for wind energy. The design of a laboratory prototype rotor assembly is optimised by varying the masses and lengths in a lumped mass model to achieve equilibrium between centrifugal and lift forces acting on the turbine’s rotor assembly. The method involves an investigation of the variation of blade pitch angle to provide a balance between centrifugal and aerodynamic forces, thereby facilitating the cost advantages and opening the opportunity to improve the turbine efficiency across a range of operation conditions. The implication of this study extends to different applications of wind turbines, both onshore and offshore, introducing insight into innovation for sustainable energy and cost-effective solutions.

1. Introduction

Owing to concern over global warming and climate change, power generation from renewable resources has obtained much interest in recent decades [1]. Wind energy is one of the most promising renewable energy sources for energy production for several reasons, including the availability of sustainable resources, decreasing cost and advanced technology, which has minimal environmental effects [2,3,4]. There have been a variety of designs over the years, but all wind turbines can be classified as either horizontal-axis wind turbines (HAWTs) or vertical-axis wind turbines (VAWTs). Of these, the HAWT has been by far the most commercially successful. Figure 1a,b show the main components of a HAWT and a VAWT, respectively.

HAWTs have higher powertrain efficiency than VAWTs, making HAWTs more efficient in energy conversion and more competitive in the market for both onshore and offshore applications during the last 30 to 40 years [7]. The aerodynamic performance of HAWTs is scalable, as proved in aerodynamic theory. However, the greatly increased mass at the top of the rotor deteriorates the upscaling behaviour of the turbine system because the increased weight due to the larger rotor size is greater than the increased aerodynamic load [7]. Due to this, the upscale limit of a HAWT is around 30 MW [6]. Cyclic gravitational loading causes stress problems on blades and other parts of the turbine structure. On a wind turbine rotor, thrust is the main source of aerodynamic force and is directly proportional to the square of the hub-height wind speed. Modern wind turbines require larger concrete and steel foundations to withstand the high overturning moments computed by multiplying the thrust force by the hub height, which, in modern turbine towers, can be over 100 m. The supporting infrastructure in the form of these foundations may significantly impact the total installed cost of the turbine. Blair Loftis [8] states that utility-scale wind farms cost approximately around $2 million for each installed turbine. About seventeen percent of the utility-scale cost is related to the turbine foundation. The foundation expenses for offshore wind turbines are much greater, ranging from around 20% for water depths of 10–20 m to 36% for ocean depths of 40–50 m [9]. Using floating platforms can lower the cost of an offshore foundation. Still, the high centre of mass of heavy-nacelle HAWTs atop tall towers necessitates using large area floating structures for stability.

VAWTs have some special advantages because of their inherent characteristics. For example, the operation of a VAWT is independent of wind direction (omnidirectional), and VAWTs operate without a yawing system, unlike HAWTs, which need to track the changing wind direction continuously. Further, VAWTs can have a simple blade design with low noise, and the mechanical and electrical equipment, such as generators and drivetrains, is easily accessible because they are ground-based and easy to maintain [10]. An important distinction between offshore VAWTs and HAWTs is that due to the ground-based location of heavy machinery components, VAWTs have a lower centre of gravity and wind load, enhancing structural stability. For offshore wind turbines on floating platforms, this lessens the likelihood of the platform overturning, leading to the possibility of smaller platform sizes and spacing. Consequently, a smaller mooring system will be needed for the reduced foundation. Any decrease in the hull and mooring system size, and thereby cost, is significant because the cost of the floating foundation is a major contributor to the total cost of floating wind turbines [11]. For these reasons, a recent surge has emerged in VAWT research concerning commercial offshore designs [12,13,14].

Despite the advantages mentioned above, VAWTs encounter several challenges. For example, Darrieus VAWTs face disturbed airflow in the downwind pass of the rotor blade because of turbulence created in the upstream pass and shed downstream [15,16]. This decreases the aerodynamic efficiency of the turbines and makes them less effective than HAWTs. Also, VAWTs are associated with a weak self-starting capability. Finally, there is high cyclic loading in Darrieus VAWTs because of the fluctuating acting forces on the blades. These fluctuations generate fatigue and increase capital costs associated with turbine design [17]. On the upwind pass, the centrifugal and the aerodynamic forces act in opposite directions, whereas on the downwind pass, these forces act in the same direction. Due to this imbalance, VAWTs need additional support structures, such as struts, to stabilise the rotor. This adds more materials and costs to the turbine [18].

Table 1. Comparison between HAWTs and VAWTs.

Feature	HAWTs	VAWT
Direction	The rotor shaft is horizontal	The rotor shaft is vertical
Airflow disturbance	Less significant compared to VAWTs due to the rotor orientation	High disturbance on the downwind pass
Induction factor	Impacts efficiency once	Impacts efficiency twice
Aerodynamic efficiency	Better performance—more efficient than VAWTs by about 25%, making them more suitable for large-scale applications [19]	Lower performance due to aerodynamic issues, only extracting 30–40% of the kinetic energy in the wind and needing additional power to start the rotation [20]
Application	Suitable for open space and stable wind status	Suitable for turbulent and low wind areas, for example, in urban settings [21]
Infrastructure	Need a more substantial foundation but no other support	Need a less substantial foundation but additional support such as struts
Blade design	More complicated blades—twisted shape and sophisticated profile	Simpler blade design—especially H-rotors
Cyclic loading	Lower amplitude cyclic loading	Higher amplitude due to force imbalance through each rotation
Market penetration	Higher because of better efficiency and performance	Lower because of structural and efficiency issues
Thrust impact on rotor	High impact due to large hub heights and stronger winds	Moderate impact
Overturning forces	High—an expensive foundation is needed. Floating foundation for offshore application uneconomic	Lower for some designs. VAWTs may be more suitable for offshore floating applications
Centre of mass	High	Low for the same capacity
LCOE (USD/kWh)	0.027 to 0.074 for onshore applications and 0.09 to 0.191 for offshore applications for 5 MW HAWT [22]	0.157 to 0.190 for 5 MW VAWT [11]

summarises the main comparable aspects of HAWTs and VAWTs.

Moreover, pitching control is vital for optimising efficiency in both technologies. In HAWTs, the pitch control system limits power, protecting electrical components when the wind velocity exceeds the rated speed and eliminating the need to shut down the turbine. Samani et al. [23] conducted a study focusing on pitch-to-stall and pitch-to-feather approach control in HAWTs and showed that the pitch-to-stall method assists in load reduction and power regulation. An individual pitch control method based on a data-driven technique implemented by [24] to balance fatigue with reduced actuator damage illustrated improvement in employing the adaptive pitch method. HAWTs’ pitch control system involves complex mechanical and hydraulic system techniques.

Conversely, in VAWTs, the pitch control system enhances turbine performance and enables self-starting capabilities. Employing a control system that manipulates the angle of attack using pitching blades can improve VAWT performance [25]. El-Faham et al. [26] studied the implementation of active pitch control using a Fuzzy Logic Controller. They found it can enhance the performance and self-starting of VAWTs. Another study of variable pitch control on Darrieus VAWTs was conducted by [27], which found that variable pitch in VAWTs can improve power production and torque compared to fixed-pitch turbines. In their study, the aerodynamic behaviour of VAWTs optimised if an appropriate pitch control method was utilised. Another study introduced by [28], which was comparing VAWT-shaped motions and the sinusoidal pitch angle. Implementing pitch control in VAWTs is less complex than in HAWTs, as a simple mechanic adjustment is performed. However, because of their vertical rotation, different forces act on the blades as they rotate around the axis. Consistent and active design for pitch control is difficult to implement when compared with HAWTs.

Introduction to the Active Axis Wind Turbine (AAWT)

With these limitations in both HAWTs and VAWTs, there is a need for innovation to harness the advantages and address the disadvantages of both types and introduce a new type that could significantly reduce the levelised cost of energy (LCOE) for wind. The AAWT is a novel type of wind turbine that potentially achieves these goals by lowering the required support infrastructure and increasing aerodynamic efficiency. The AAWT consists of a rotor with a single blade that can be pitched and tilted synchronously as a function of the azimuth angle. Thus, the AAWT turbine can actively tilt the rotor’s rotation axis to an incline rather than fixing it horizontally or vertically. Also, the pitch angle is actively controlled, and the change in the angle of the blade about a longitudinal axis through the spar centreline, which is approximately aligned with the quarter chord point, occurs via a simplified and effective technique compared with the approaches used in HAWTs and VAWTs. By adjusting the pitch angle, the aerodynamic performance of the AAWT can be optimised at each azimuth angle, wind speed and direction. The design of this turbine is specifically directed toward the following:

• Reducing the substantial infrastructure required to cope with large overturning forces acting on HAWTs and VAWTs;
• Balancing the aerodynamic and centrifugal forces acting on the wind turbine blade to enable longer spans, reduced mass and lower cost;
• Minimising the blade flight through to a lower velocity area, with lower energy downwind or disturbed air, and specifically addressing the lower efficiency associated with VAWTs;
• Allowing cost-effective floatation and anchoring systems to be used in offshore floating wind turbine installations.

This research is part of a larger project that involves developing the AAWT. The project’s first phase is to construct a small laboratory prototype, a proof-of-concept AAWT, with a blade that is 920 mm in span, 120 mm in chord and weighs 115 g.

The research question of this paper is whether the AAWT can be designed so that the centrifugal and normal aerodynamic forces can be balanced sufficiently to achieve a satisfactory fight path during each rotation. If this can be achieved, the required support infrastructure could be reduced by reducing the size of foundations and removing items such as towers and supporting stays. The blade chord and span can potentially increase.

Based on the literature, no studies directly discuss balancing centrifugal and normal aerodynamic forces to decrease load and costs in VAWT operation. However, in HAWT designs, the rotor is sometimes coned downwind to decrease the flapwise bending moment, using the centrifugal force on the blade to counter the normal aerodynamic force [29,30]. Further investigations examining how to achieve equilibrium between these forces and the impact on an AAWT’s design would be beneficial.

This paper presents a design study that constructs an aerodynamic model for the AAWT, which neglects any external wind for simplification. Aerodynamic force is generated on the blade via pitching the rotating blade. Because the scale of the blade is small, it needs to be lightweight for the centrifugal force to balance the small aerodynamic force. A lumped mass analysis is used to aid in designing the required weight and locations of the AAWT’s various components and examines their contribution to the net centrifugal moment. An optimisation approach using Python code version 3.10.12 was developed employing the Sequential Least Squares Programming (SLSQP) method to provide optimum component masses and distances from the centre of rotation. This optimisation seeks to achieve the necessary equilibrium, balancing the centrifugal and aerodynamic moments under various blade pitch angles and at different tilt angles.

The greatest contribution is specifying the balancing mechanism of the newly introduced AAWT of the centrifugal and aerodynamic lift forces. This equilibrium is achieved by optimising the underdesigned prototype’s lumped mass and identifying the pitch and tilt angles that can match to provide the balance, which decreases structural stress, making a stable and cost-effective design. This approach is unique for this wind turbine model and can initiate a more efficient and economical installation in wind turbines.

The other contributions of this study can be listed as follows:

• Introducing the new concept of the AAWT prototype within laboratory availability and safety constraints;
• Developing a lumped mass model for the rotor components and optimising the mass and length of each component to achieve equilibrium in operation;
• Identifying the pitch angles’ range with tilt angles to balance centrifugal and lift forces;
• Validating the concept of AAWTs in a small-scale laboratory prototype.

The structure of this paper is as follows: Section Two provides a theoretical explanation of VAWT aerodynamics, which is based on the literature (as background for the forces on the AAWT), while Section Three discusses the methodology of the study, including the materials used for the designed prototype and method of operation design and construction followed by the developed lumped mass model of the AAWT and the optimisation of the components for this model. The fourth section covers the results and discussion of the prototype’s optimised rotor components and the relationship between the optimal pitch and tilt angles to maintain the equilibrium during the operation, summarised in the conclusion in Section Five.

2. Theoretical Background

This theory relates to an H-rotor VAWT but gives a suitable background for the AAWT described in Section 3.

Figure 2 illustrates the velocity vectors’ aerodynamic notation and the forces on the rotor blades of an H-rotor VAWT. The wind velocity at the blade, Vind, is less than the incoming freestream flow velocity, V, due to the slowdown on the freestream by the blade. As the rotor extracts energy from the wind, the tangential velocity of the blade is as follows:

$$V_{rot}=\omega R$$

(1)

where ω is the rotor’s angular velocity and R is the radius of the rotor. The relative velocity, V_rel, results from the vector addition of the wind velocity on the turbine and tangential velocity. This study calculates the relative wind velocity based on the cosine rule of the angle opposite to the relative wind, θop, calculated per Equation (2) and highlighted in Figure 2a.

$$V_{rel}=\sqrt{V_{ind}^2+{\left(\omega R\right)}^2-2V_{ind}\omega R\cos {\theta }_{op}}$$

(2)

The angle between the relative velocity and the tangential velocity is known as the flow angle, $\mathsf{\phi },$ and it can be calculated from the geometry as per Equation (3), based on the sine rule in Figure 2a.

$$\mathsf{\phi }={\sin }^{-1}(\sin ({\theta }_{op})\ast {\displaystyle \frac{V_{ind}}{V_{rel}}})$$

(3)

The flow angle is the summation of the angle of attack, AOA or α; the angle between the relative wind and the chord line of the blade’s airfoil; and the pitch angle, β, the angle between the chord line of the airfoil and tangential velocity vector [31].

$$\phi =\alpha +\beta$$

(4)

As the blade rotates, the rotation is driven by the lift force perpendicular to the relative wind, which helps to generate torque to rotate the turbine. The generated power on the wind turbine increases with increasing lift and decreases with increasing drag. The lift force on the blade F_L and the drag force F_D can be calculated as per the following [32]:

$$F_L={\displaystyle \frac{1}{2}}\rho V_{rel}^2{C}_{L}cl$$

(5)

$$F_D={\displaystyle \frac{1}{2}}\rho V_{rel}^2{C}_{D}cl$$

(6)

C_L represents the lift coefficient, C_D is the drag coefficient, ρ is air density, V_rel is the relative wind striking the blade, c is the chord length and l is the blade length.

The AAWT uses a NACA0018 airfoil, and the calculated Reynolds number of the model in this study is around 125,000. This study re-plotted the lift and drag coefficients from previously calculated wind tunnel experimental results for an NACA0018 airfoil with a Reynolds number of 150,000 [33,34], the closest Reynolds number to these test data for the NACA0018 found in the literature. Figure 3a and Equation (7) show the curve fit regression for the lift coefficient graph.

$$C_L=0.000344{\alpha }^4-0.0083{\alpha }^3+0.058{\alpha }^2+0.0056\alpha$$

(7)

The drag coefficient is shown in Figure 3b, represented in Equation (8).

$$C_D=5.89\ast {10}^{-5}\ast {\alpha }^2-6.62\ast {10}^{-4}\ast \alpha +0.0168$$

(8)

After α reaches 10 degrees and beyond, the NACA0018 airfoil stalls, according to the experiment, when Reynolds number equals 150,000.

The lift and drag forces are decomposed into normal F_N and tangential F_T forces based on the inflow angle. These forces can be calculated in Equations (9) and (10).

$$F_N=F_L\cos \left(\phi \right)+F_D\sin \left(\phi \right)$$

(9)

$$F_T=F_L\sin \left(\phi \right)-F_D\cos \left(\phi \right)$$

(10)

The normal force contributes to the structural load on the blade, while the tangential force provides the estimated torque from the rotor. The generated torque, T, from the rotor is calculated by multiplying the tangential force by the radius of the rotor, R, as follows:

$$T=F_{T}R$$

(11)

The generated power is defined as the work rate over a specified distance. The average power generated from the rotor over one revolution can be calculated by multiplying the average torque with the rotor rotation ω.

$$P={\displaystyle \frac{1}{\pi }}{\int }_0^{2\pi }T\left(\theta \right)\omega d\theta$$

(12)

The power coefficient is the power generated by the turbine over the power available in the wind.

$$C_P={\displaystyle \frac{P}{0.5\rho V^{3}2Rl}}$$

(13)

3. Materials and Methods

3.1. Prototype Components

The laboratory prototype AAWT consists of the following:

(1)

A polystyrene foam and balsa wood blade assembly with a carbon fibre spar running through the length of the blade to provide stiffness and avoid buckling or structural failure. The blade is designed to be as light as possible to reduce the magnitude of the centrifugal forces so that they are similar to the lift forces, which is challenging on a small prototype of this scale.

(2)

A 3D-printed tee assembly which couples the blade to the wheel and counterweight assembly. The tee assembly contains mechanisms for pitching the blade and setting the wheel assembly’s height.

(3)

An adjustable wheel assembly to allow a tilt on the cross-arm. Before take-off, the wheel runs on a table as the AAWT rotates. When the blade pitch is zero, and centrifugal loads dominate, the wheel acts as a device to limit the tilt angle and prevent the blade from contacting the table. Once the blade is pitched, the wheel floats free above the table to generate sufficient lift. Pitching the blade further generates more lift and changes the tilt of the AAWT. The blade can be “flown” at a range of tilt angles that can be set in flight by changing the blade pitch.

(4)

A 3D-printed assembly which provides a counterweight to the other assemblies via an adjustable lead weight in a carrier. The assembly also contains a servo motor, which activates rods in the cross-arm that connect to the blade pitch mechanism in the tee assembly, and a battery to power the servo motor and flight controller.

(5)

A cross-arm (or balance arm) that connects the counterweight assembly to the tee assembly and sits atop the motor/generator assembly.

Figure 4a shows an image of the laboratory prototype AAWT with the cross-arm in the horizontal position (zero tilt). This image identifies each of the assemblies to be used in the lumped mass analysis, the centre of rotation and the centre of rotation for tilt. At the same time, Figure 4b shows AAWT assemblies and initial design dimensions.

The wheel assembly can be adjusted manually between 80 mm and 155 mm to set or change the maximum tilt angle on the cross-arm. When the tilt angle is changed, the geometry of the AAWT will be changed, as in Figure 5.

The servo motor of the pitch actuation system and the battery that powers it are included within the counterweight assembly. The battery and cable weigh 18 g, the servo motor and cables 53 g and the counterweight carrier 66 g. Once the lead weight is inserted in the carrier, the total weight of the counterweight assembly is 350 g and the counterweight length from the centre of rotation is increased from 250 mm, as per Figure 5, to 350 mm. The design of this counterweight system will be optimised as a part of this study.

In designing the AAWT, several safety measures were taken into consideration, including the following:

• The rotation speed of the rotor does not exceed 300 RPM to avoid overstressing lightweight components;
• The average aspect ratio of the blade AR = Length/Chord is not greater than 10;
• For the laboratory scale, the radius should not exceed 0.5 metres.

Based on the above limitations, the characteristics of the AAWT turbine prototype physical design were derived in this study, as shown in

Table 2. Characteristics of the turbine.

Parameter	Value
Nominal rotational speed (rpm)	300
Number of blades	1
Blade airfoil	NACA0018
Rotor radius [m]	<0.5
Chord length [m]	0.12
Span [m]	0.92
Aspect ratio	7.7
Air density [kg/m3]	1.2
Minimum manufacturing mass per unit area [kg/cm2]	1

.

3.2. Lumped Mass Model

Most research on HAWTs and VAWTs is focused on modelling aerodynamics and power generation, and limited studies are available on the issues of the balance of forces that act on the turbine. The lumped mass model can simplify complex wind turbine mechanical system design analysis by focusing on dynamic structure and rotor balance. Lorenzo et al. [35] studied the application of lumped mass simulation for damage in offshore HAWTs. This technique provided progressive mass addition for the simulation, such as ice formation, to examine the structure response and evaluate real-time damage detection. Another study [36] utilised lumped mass analysis of VAWTs, particularly to analyse the rotor imbalances that affect vibration and the interaction with the turbine support structure. This method assisted in ensuring the developed design is more stable and has resilient masts with dynamic loads.

For this study, a lumped mass model was developed using Python code to find the assembly’s optimum height, radius and mass to ensure flight stability. As noted above, the freestream wind velocity, i.e., V_ind = 0. From Equation (2), this means that the relative wind velocity equals the tangential velocity, ωR. It is also assumed that the rotor’s angular velocity, ω, is constant. In this scenario, the angular momentum rate of change can be deemed zero, and the applied centrifugal force on the blade is constant. Although the freestream wind velocity is zero, there is still aerodynamic lift on the blade if it is pitched at an angle so that there is an angle of attack between the airfoil’s chord line and the airfoil’s direction of motion. For balance, the blade needs to be pitched so that the normal component of the aerodynamic force acts radially inwards to counter the centrifugal force, which acts radially outwards and is independent of wind flow. In practice, the blade’s centrifugal force is also counteracted by the structural components that connect the blade to the rest of the turbine to prevent the blade from flying outward.

A lumped mass analysis was conducted to design the weight and position of the rotating components of the AAWT. For simplicity in modelling, the blade, tee, wheel and counterweight (with adjustable lead weight) assemblies and the adjustable lead weight are each assumed to act as a lumped mass at a point, i.e., the combined mass of each assembly is considered to be concentrated at the centre of mass of the assembly. Figure 6 shows a lumped mass diagram for the AAWT.

Moment arms are measured from the horizontal position. The blade’s lumped mass is above the horizontal position, and tilting reduces the length of the moment arm.

$$L_{b,t}=R_b\sin {(\theta }_{b,0}-{\theta }_t)$$

(14)

In all other cases, the tilt increases the length of the moment arm:

$$L_{T,t}=R_T\sin {(\theta }_{T,0}+{\theta }_t)$$

(15)

$$L_{l,t}=R_l\sin {(\theta }_{l,0}+{\theta }_t)$$

(16)

$$L_{w,t}=R_w\sin {(\theta }_{w,0}+{\theta }_t)$$

(17)

$$L_{c,t}=R_c\sin {(\theta }_{c,0}+{\theta }_t)$$

(18)

Adopting the convention that a clockwise moment is positive, the blade assembly has a positive moment, whereas the other assemblies have negative moments. For balance, the sum of the moments of the assemblies must be zero. Assuming the angle of attack produces no lift on the blade for the time being, we obtain the following equation:

$$F_b{L}_{b,t}+{F_T{L}_{T,t}+F}_l{L}_{l,t}+F_w{L}_{w,t}{+F}_c{L}_{c,t}=0$$

(19)

where F_b, F_T, F_l, F_w and F_c are the centrifugal forces (=mω²r) for the blade, tee, lead weight (the adjustable part in the counterweight), wheel and counterweight, respectively, and L_b,t, L_T,t, L_l,t, L_w,t and L_c,t are calculated from Equations (14)–(18).

For constant angular velocity, the net moment is as follows:

$$m_b{r}_{b,t}{L}_{b,t}-{m_T{r}_{T,t}{L}_{l,t}-m}_l{r}_{l,t}{L}_{l,t}-m_w{r}_{w,t}{L}_{w,t}-m_c{r}_{c,t}{L}_{c,t}=0$$

(20)

where m_b, m_T, m_l, m_w and m_c are the masses of the blade, tee, wheel and lead weight with counterweight assemblies, respectively. The symbols r_b,t, r_T,t, r_l,t, r_w,t, and r_c,t are the horizontal distances between the centre of rotation and the tilted blade, tee, lead weight, wheel and counterweight, respectively.

3.2.1. Lumped Mass Model for Maximum Tilt Angle

The lumped masses of the initial design of the blade, cross-arm and wheel are shown in

Table 3. AAWT initial lumped masses and their height and radius.

Lumped Masses
	Blade	Tee	Wheel	Counterweight
Mass (grams)	115	136	77	350
Zero X (mm)	380	380	380	350
Zero Y (mm)	507	8.9	116	0
Radius to the centre of mass (mm)	633.60	380.10	397.31	353.55
Angle to centre of mass (deg)	53.15	1.34	16.98	0

. Zero X is the horizontal distance (radius) between the centre of rotation and the centre of mass of the assembly piece at zero tilt angle. Zero Y is the vertical distance (height) between the cross-arm pivot and the centre of mass of the sub-assembly at zero tilt angle (see Figure 6). A lumped mass model with the initial suggested values derived based on the physical limitation design, as shown in Table 3, was used to calculate the moment at a 12-degree tilt angle. With the initial parameters, the net moment could not achieve a zero value, i.e., maintain equilibrium as per Equation (20). Therefore, an optimisation algorithm using Python code was implemented to obtain the optimum value of each parameter in the assemblies that achieve equilibrium and make the net moment zero, as described in Section 3.3.

3.2.2. Lumped Mass Model Optimisation

This study employed a fine-tuned optimisation procedure for the lumped mass model to find the wheel’s optimum height, radius, mass and counterweight assemblies in the AAWT model. The mass and position of the blade and tee were considered fixed. The system’s main objective was balancing the moments generated by the distributed lumped masses. This section illustrates the method, and the computation procedure implemented to achieve this target.

Constant and Initial Values

Predefined initial values and constants were considered when carrying out the optimisation. The constants included the parameters defined in Table 2. The initial values of masses and lengths of various assembly components were provided, as shown in Table 3.

Moment Calculation

A custom function calculated the moment based on the mass and length parameters. This function computes the moment based on Equations (21) to (26).

The distance from the centre of rotation to the centre of mass (DCM) for each assembly is calculated as

$$DCM=\sqrt{{\mathrm{Z}\mathrm{e}\mathrm{r}\mathrm{o}\_\mathrm{x}}^2+{\mathrm{Z}\mathrm{e}\mathrm{r}\mathrm{o}\_\mathrm{y}}^2}$$

(21)

while the angle to the centre of mass (ACM) of each assembly at zero tilt angle is calculated as

$$ACM={\tan }^{-1}({\displaystyle \frac{\mathrm{Z}\mathrm{e}\mathrm{r}\mathrm{o}\_\mathrm{y}}{\mathrm{Z}\mathrm{e}\mathrm{r}\mathrm{o}\_\mathrm{x}}})\ast 180/\pi$$

(22)

At zero tilt angle, the resultant angle is equal to the angle to the centre of mass (ACM) since the resultant angle (RA) is equal to the angle to the centre of mass (ACM) minus the tilt angle (see Figure 6).

The horizontal radius from the centre of rotation to the centre of mass (RCM) of each assembly is calculated as follows:

$$RCM=DCM\ast \mathit{cos}\left(RA\ast {\displaystyle \frac{\pi }{180}}\right)$$

(23)

The centrifugal force for each assembly is calculated as

$$F_c=m\ast {\left(RPM\ast 2\ast {\displaystyle \frac{\pi }{60}}\right)}^2\ast RCM$$

(24)

The length of the moment arm from the pivot to the line of action of the centrifugal force of the assemblies can be found in the following equation:

$$M_{pivot}=DCM\ast \mathit{sin}\left(RA\ast {\displaystyle \frac{\pi }{180}}\right)$$

(25)

The moment of the assembly is calculated as

$$M_f=F_c\ast Mpivot$$

(26)

The moment is a crucial value in the model because it influences the stability and performance of the system.

Objective Function

The objective function in the optimisation methodology was implemented to minimise the square of the moment. In this way, the optimisation ensures the net moment is reduced to close to zero, thereby making the system stable at a 12-degree tilt angle. The formulated objective function as follows:

$$Minimizef(x)={\left(\sum _{i=1}^n{F}_{ci}.M_{pivoti}\right)}^2$$

(27)

where F_ci is the centrifugal force for each component i and $M_{pivoti}$ is the moment arm length that corresponds to that component.

Constraints

A nonlinear constraint ensured the moment was equal or close to zero. For this functionality, the NonlinearConstraints function from the SiPy library was employed. This function enforces the constraints to make the resultant moment zero. The vital constraint to make sure the calculated total moment is equal to zero is as follows:

$$Constraint:g\left(x\right)=\sum _{i=1}^n{F}_{ci}.M_{pivoti}$$

(28)

where x represents the mass and length of the components. Each variable is defined with specific bounds as follows:

x = [m_b, m_T, m_w, m_c, L_b[x], L_T[x], …, L_c[y]

and

105  ≤  m_b … ≤  115, …130   ≤  m_T   ≤  145,  …, 0   ≤  L_c[y]   ≤  60

The symbols L_b[x] and L_T[x] are the horizontal length of the blade and tee, and L_c[y] is the vertical length of counterweight.

Optimisation Procedure

The optimisation method used in this procedure was solved using the SLSQP method. This optimisation method is widely utilised in different wind energy application applications because of its capability to deal with equality and inequality constraints. That makes an ideal optimisation technique for handling complex engineering requirements such as structural limits [37,38]. The initial values for this method considered the initial values for the mass and length of the assemblies’ parameters in Table 3. The bounds for the under-focus variables were constrained to ±10 (absolute value) of the initial values to limit the search boundaries and provide realistic physical parameters.

Results and Verification

The output values from the optimisation were the optimum masses and lengths for each component. The force moments were calculated considering the provided values from the optimisation algorithm, and the net moment was found to be close to zero, justifying the effectiveness of the optimisation procedure. The optimisation process proved a systematic approach to reduce the net moment in the lumped mass model to zero, which enhances the system’s stability. The employed SLSQP method and considerable constraints and bounds led to practical solutions.

3.3. Flow Chart

Figure 7 displays the decision diagram for this study’s procedure and the Python code’s functioning. When the AAWT rotates with the wheel fully retracted and touches the table’s surface, the tilt angle is 12 degrees. The optimisation is performed to find the optimum masses and lengths in the lumped mass model. The obtained optimised values were then used to compute the pitch angle for any tilt angle that would balance the centrifugal and lift forces.

4. Results and Discussion

4.1. Lumped Mass Model with 12-Degree Tilt Angle

The results from optimising the lumped mass model for the masses and lengths are presented in

Table 4. AAWT optimised lumped masses at 12 tilt angles.

Lumped Mass
	Blade	Tee	Wheel	Counterweight
Mass (grams)	115	136	76	301
Zero X (mm)	380	380	380	349
Zero Y (mm)	507	8.9	80	50

.

It can be seen that the optimisation process suggested a significant change to the Zero X (radius) and Y dimension (height) for the counterweight centre of mass, and this was accommodated with the design change shown in Figure 8 below.

The moment for each assembly was then calculated for different tilt angles within the range of movement, and then the total moment from the four assemblies was compared to the lift moment, where the lift moment M_l is calculated as

$$M_l=F_l\ast Zero\_y\left[Blade\right]$$

(29)

The lift force F_l is calculated as

$$F_l=0.5\ast \rho \ast Chord\ast Span\ast {V_{rot}}^2\ast C_L$$

(30)

where V_rot is the tangential velocity at 300 RPM, taken as a basis for the design of the rotor and calculated in the following equation:

$$V_{rot}=RCM\left[Blade\right]\ast RPM\ast 2\ast \pi /60$$

(31)

4.2. Lumped Mass Model with Different Tilt Angles

The change of moment with the tilt angle for each assembly is shown in Figure 9. The centrifugal moment for each assembly is calculated using Equation (26).

As per Equation (20), the equilibrium of moments on both sides of the centre of rotation is the summation of the moments of each assembly, considering the sign of each moment is positive where the movement is clockwise and negative if anticlockwise. Using the parameters in Table 4, the resultant moment when the tilt angle is 12 degrees is 0 Nm. To balance the moment at different tilt angles, the blade should generate a lift moment to provide equilibrium, which will be considered in the next section.

4.3. Lumped Mass Model with Tilt—Adding Lift Moment

As this study considers zero wind speed, the blade needs to be pitched to create an angle of attack and generate a lift force. From Equation (9), the normal aerodynamic force equals the lift force since the relative wind angle is zero. The lift moment can be found in Equation (27).

The relationship between the tilt angle and pitch angle that provides the required lift moment to balance the net centrifugal moment is shown in Figure 10.

To make the AAWT take off, the blade’s pitch increases, and the tilt angle decreases (see Figure 10), maintaining aerodynamic and centrifugal balance. In practice, the lower range of the tilt angle would be limited to around 6 degrees from the horizontal position since any lower would result in the stalling of the blade. At this low Reynolds number, the airfoil NACA0018 stalls at angles greater than 10 degrees (see Figure 3). In summary, the rotor first rotates on the table at a 12-degree tilt angle and has zero attack of angle. As the pitch angle increases in magnitude, which its sign is opposite to the angle of attack, the tilt angle decreases to 6 degrees. The angle of attack angle is 10 degrees. If the blade is pitched further, the lift will decrease due to stalling, and the balance between the centrifugal and aerodynamic forces will be lost. The operating range of tilt angles is 6 degrees, which is more than sufficient to prove the concept and test the control system.

Employing pitch and tilt angles and optimising them to maintain equilibrium is unique to this study, as all previous research used pitch angle optimisation to improve thepower coefficient, especially for VAWTs. Paraschivoiu et al. [39] optimised a blade-pitching system for a straight-bladed VAWT by numerical simulation. The optimisation implemented for maximum aerodynamic performance suggested the cyclic pitch system. A low-cost model was developed by [40] to evaluate the performance design of a small-scale VAWT. The model was optimised with pitch angle configuration. Giromill and Cycloturbine implemented simple optimisation where the turbine used a sinusoidal pitching system. The optimisation operated the turbine for a range of different values for a specified offset, maximising the pitch amplitude as a sinusoidal function in relation to the azimuth angle [41]. Other research supports Erickson’s findings that variable pitch can enhance the self-starting capability of VAWTs [42]. They studied an optimal blade pitch function for high-solidity VAWTs by developing a control device to mechanise the pitch angle. Tilting the blade is an innovation for this study, and it can cooperate with pitch control to maintain the balance between the centrifugal and aerodynamic forces. It can also be controlled to improve the power coefficient.

Future research will consider using an alternative airfoil designed to operate at a lower Reynolds number to provide lift and drag values over a wider range of pitch (or attack) angles. This would give a greater range of pitch and tilt values for which the forces on the AAWT are balanced and would aim to include flying the AAWT with a zero-tilt angle.

5. Conclusions

This study introduces a new wind turbine design with an axis of rotation that is neither horizontal nor vertical but can actively change during turbine operation: the Active Axis Wind Turbine (AAWT). This study aimed to show that the AAWT can operate with minimal supporting infrastructure if the centrifugal and lift forces on the rotor could be balanced to stabilise the device. The study involved optimising the masses and lengths of the rotor components using a lumped mass analysis to gain insight into the interaction and distribution of the forces in the AAWT and find an equilibrium between the centrifugal and lift forces, which act on the turbine components. This equilibrium can reduce the structural loads on an AAWT. A large-scale installation could reduce the need for expensive foundations and support infrastructure, especially the tower and guy-guides, and open the possibility of longer turbine blades with improved power. The reduced foundation loads may enable applications such as offshore floating platforms where the reduced installation cost could lead to offshore wind turbines that are more competitive with onshore wind turbines with conventional foundations.

The significant key findings in this study were the optimisation of the mass and radius of the rotating component assemblies in a prototype small-scale AAWT design. This study also emphasised the importance of the blade pitch angle as a control parameter to balance the forces acting on the rotor. Appropriate manipulation of the blade pitch angle can control the lift force, enabling stable operation of the AAWT. The relationship between the tilt and pitch angles in the situation where the centrifugal and lift forces can achieve equilibrium has been presented, which is a key objective of this study.

It was shown that for a laboratory prototype AAWT where the small scale heavily compromises the geometry, stable operation can be achieved over a range of 6 degrees of tilt. This range is limited by geometry at higher tilt angles and the blade pitch at lower tilt angles, where the stall angle of attack of around 10 degrees becomes the limiting factor. The blade will stall and lose lift at pitch angles greater than 10 degrees at the very low Reynolds numbers where this small-scale prototype operates. The centrifugal and lift forces will no longer be in equilibrium above these pitch angles and corresponding tilt angles.

A small-scale prototype was the focus of this study. However, the principle and results can be scaled up for larger AAWTs to achieve cost-effectiveness and improved performance. It should be noted that at a larger scale, the force balance can be achieved without counterweights and counterweight arms, which have been the subject of this study. Further work will attempt to validate this study’s theoretical model experimentally. The laboratory prototype described will be tested at a fixed rotational speed and over the range of pitch angles identified in this research to demonstrate force equilibrium at a range of tilt angles. This prototype will operate indoors with zero wind initially, and a number of control, geometry and hardware alternatives will be explored. In particular, using a low-Reynolds-number airfoil for the blade would aid in increasing the range of pitch and tilt angles for the AAWT. This would give greater flexibility in the flight paths of the blade. Moreover, developing and testing an advanced control system would improve the performance and longevity of the turbine under research.

The AAWT is a promising development in wind energy technology, which aims to tackle some of the deficiencies of the HAWT and VAWT designs. This novel design can potentially improve efficiency and decrease the costs of wind energy infrastructure, thus facilitating the lowering of the LCOE of wind energy in general. Lower-cost wind turbines would be welcomed in the need to accelerate the energy transition towards net zero by 2050.