# A Tensile Rotary Airborne Wind Energy System—Modelling, Analysis and Improved Design

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. A Brief History of Daisy-Kite AWE Rotary Kite Turbine

#### 1.2. Motivation and Main Work Organisation

## 2. Modelling Framework

#### 2.1. Overall System Configuration

#### 2.2. Power Extraction

#### 2.2.1. Rotor Aerodynamics

#### 2.2.2. Wing Characteristics

^{5}and $AR$ to be 5. The results presented in [23], along with the predictions from Xfoil [24], were used to define the lift and drag coefficients for the foam blades for angles of attack in the range from −10° to 110°. For values outside of this range, similar to the HQ kites, the coefficients were calculated using NREL’s AirfoilPrep.

#### 2.2.3. Lift-Kite Aerodynamics

#### 2.2.4. Wind Models

- (1)
- The first is the uniform and constant wind speed used to analyse steady-state performance.
- (2)
- The second wind model assumes that the wind speed varies with time but is uniform in the plane perpendicular to the wind’s direction. This model is used for the simulation of dynamic system responses.
- (3)
- The third wind shear model accounts for the variations in wind speed in both time and altitude. The variation in wind speed with altitude is calculated following the power law [25]. This wind shear model is used for the entire system to integrate all modules into the same modelling scheme.

#### 2.3. Ground Station—Power Take Off

^{2}and 0.019 kgm

^{2}, respectively. Other moments of inertia grouping the inertia due to the chain drive, disc brake and power meter, were calculated to be 0.002 kgm

^{2}in this work.

## 3. Power Transmission—TRPT Representations and Tether Drag Models

#### 3.1. Steady State TRPT Model

- Wind reference frame. It is defined as (${x}_{w},{y}_{w},{z}_{w}$), in which ${x}_{w}$ is parallel to the wind velocity vector, ${V}_{w}$, which is parallel to the ground; ${y}_{w}$ is perpendicular to the wind vector and also parallel to the ground; and ${z}_{w}$ is perpendicular to the ${x}_{w}-{y}_{w}$ plane.
- Rotating reference frame for the lower ring. It is defined as (${x}_{a},{y}_{a},{z}_{a}$), with the origin at ${O}_{1}$. ${x}_{a}$ lies on the system’s axis of rotation, ${y}_{a}$ and ${z}_{a}$ are in the plane of the lower ring, and ${z}_{a}$ is towards point A.
- Rotating reference frame for the upper ring. It is denoted by (${x}_{b},{y}_{b},{z}_{b}$), the origin is at ${O}_{2}$, ${x}_{b}$ lies on the axis of rotation, ${y}_{b}$ and ${z}_{b}$ are in the plane of the upper ring, and ${z}_{b}$ is towards point B.

#### 3.2. TRPT Dynamic Model 1: Spring—Disc Representation

^{3}) and the diameter of each ring with the inner and outer diameters of 2.5 mm and 4.5 mm, respectively.

#### 3.3. TRPT Dynamic Model 2: Multi-Spring Representation

#### 3.4. Tether Drag Models-Calculation of Torque Loss in TRPT

#### 3.4.1. Simple Tether-Drag Model for Steady-State TRPT Representation

#### 3.4.2. Improved Tether Drag Model for Dynamic TRPT Representations

## 4. Model Validation and Modifications

#### 4.1. Steady State Model

^{−4}. The increase in tether length from TRPT–3 to TRPT–4 is 3.4 m, these simulation results show that the increase in tether drag due to this additional length is minor. The effect of tether drag on the Daisy Kite system is analysed further in Section 5.3.

#### 4.2. Spring-Disc Representation Compared to Field-Testing Data

#### 4.2.1. Steady-State Response Testing

#### 4.2.2. Dynamic Response Testing

#### 4.3. Multi-Spring Representation Compared to Field-Testing Data

#### 4.3.1. Improving Computational Efficiency with Assumption of Rigid Wings

#### 4.3.2. Multi-Spring Model Compared to Experimental Data

#### 4.4. Comparison of Spring-Disc and Multi-Spring TRPT Models

#### 4.4.1. Response to Short-Term Step Changes in Torque and Tension

#### 4.4.2. Impact of TRPT Length

#### 4.4.3. A Few Remarks

## 5. System Analysis and Improved/Optimised Design

#### 5.1. TRPT Design Analysis

#### 5.2. Rotor Design Analysis

#### 5.2.1. System Elevation Angle

#### 5.2.2. Blade Pitch Angle

#### 5.2.3. Blade Length

#### 5.3. Tether-Drag Analysis

#### 5.3.1. Analysis with Simple Tether-Drag Model

#### 5.3.2. Analysis with Improved Tether-Drag Model

#### 5.4. Optimised/Improved Design

#### 5.4.1. Optimised Rotor Design

#### 5.4.2. Optimised TRPT Design

#### 5.4.3. Optimised Elevation Angle and Tether Length

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AWE(S) | Airborne wind energy (system) | BEM | Blade element momentum |

DoF | Degree of freedom | EOM | Equation of motion |

RMSE | Root mean square error | TRPT | Tensile rotary power Transmission |

## Appendix A. Pseudo Codes of Model Development

#### Appendix A.1. Spring–Disc TRPT Modelling

**Table A1.**Pseudo code for the spring-disc TRPT representation. $\mathbf{R}$ is the vector for the discs radii, ${\mathbf{l}}_{s}$ is the vector containing the TRPT section lengths, ${\mathbf{l}}_{t}$ the vector for tether lengths in each TRPT section, $\mathbf{J}$ is the inertia matrix, $AeroQ$ the rotor torque, $AeroT$ the rotor thrust, $AeroP$ the rotor power, $l{k}_{T}$ the lift kite line tension, ${F}_{x}$ the axial force, ${\mathbf{f}}_{D}$ the torque loss due to tether drag, $\mathbf{Q}$ the torque applied to each disc, $\mathbf{k}$ the stiffness matrix, $\mathsf{\Delta}t$ the time step length and $\mathit{\theta},\dot{\mathit{\theta}},\ddot{\mathit{\theta}}$ the vectors containing the discs rotational positions, velocities and accelerations, respectively. $\mathbf{a},\mathbf{b},$ and $\mathbf{c}$ are variables used within the algorithm.

Inputs | Wind speed ${\mathit{V}}_{\mathit{w}}$, TRPT geometry R and ${\mathbf{l}}_{s}$, elevation angle $\mathit{\beta}$, initial conditions ${\mathit{\theta}}_{2}$ and $\dot{{\mathit{\theta}}_{2}}$, and generator torque $\mathit{G}\mathit{e}\mathit{n}\mathit{Q}$ |

Line 1 | Find ${\mathbf{l}}_{t}$ and $\mathbf{J}$ |

Line 2 | Find $Aero{Q}_{2}$, $Aero{T}_{2}$ and $Aero{P}_{2}$ |

Line 3 | Find $l{k}_{T}$, ${F}_{x,2}$ and $\mathbf{k}$ |

Line 4 | Find ${\mathbf{f}}_{D,2}$ and ${\mathbf{Q}}_{2}$ |

Line 5 | $\ddot{{\mathit{\theta}}_{\mathbf{2}}}={\displaystyle \frac{\mathbf{J}}{{\mathbf{Q}}_{2}}}-\mathbf{k}{\mathit{\theta}}_{\mathbf{2}}$ |

Line 6 | ${\mathit{\theta}}_{1}={\mathit{\theta}}_{2}-\mathsf{\Delta}t\dot{{\mathit{\theta}}_{2}}+{\displaystyle \frac{\mathsf{\Delta}{t}^{2}}{2}}\ddot{{\mathit{\theta}}_{2}}$ |

Line 7 | $\mathbf{a}={\displaystyle \frac{\mathbf{J}}{\mathsf{\Delta}{t}^{2}}}$, $\mathbf{b}={\displaystyle \frac{2\mathbf{J}}{\mathsf{\Delta}{t}^{2}}}$ |

Line 8 | For each time step, i |

Line 9 | Find $Aero{Q}_{i}$, $Aero{T}_{i}$ and $Aero{P}_{i}$ |

Line 10 | Find $l{k}_{T}$, ${F}_{x,i}$ and update $\mathbf{k}$ |

Line 11 | Find ${\mathbf{f}}_{D,i}$ and ${\mathbf{Q}}_{i}$ |

Line 12 | $\mathbf{c}={\mathbf{Q}}_{i}-\mathbf{a}{\mathit{\theta}}_{i-1}-\mathbf{k}{\mathit{\theta}}_{i}+\mathbf{b}{\mathit{\theta}}_{i}$ |

Line 13 | ${\mathit{\theta}}_{i+1}={\displaystyle \frac{\mathbf{a}}{\mathbf{c}}}$ |

Line 14 | $\dot{{\mathit{\theta}}_{i}}={\displaystyle \frac{{\mathit{\theta}}_{i+1}-{\mathit{\theta}}_{i-1}}{2\mathsf{\Delta}t}}$, $\ddot{{\mathit{\theta}}_{i}}={\displaystyle \frac{{\mathit{\theta}}_{i+1}-2{\mathit{\theta}}_{i}+{\mathit{\theta}}_{i-1}}{\mathsf{\Delta}{t}^{2}}}$ |

Line 15 | End For |

Outputs | $\mathit{\theta}$, $\dot{\mathit{\theta}}$, $\ddot{\mathit{\theta}}$, $\mathbf{AeroQ}$, $\mathbf{AeroT}$, $\mathbf{AeroP}$, $\mathbf{Q}$, ${\mathbf{F}}_{x}$ |

#### Appendix A.2. Multi-Spring TRPT Modelling

**Table A2.**Pseudo code for the multi-spring TRPT representation. $\mathbf{R}$ is the vector for the discs radii, ${\mathbf{l}}_{s}$ is the vector containing the TRPT section lengths, ${\mathbf{l}}_{t}$ the vector for tether lengths in each TRPT section, $\mathbf{M}$ the mass and inertia matrix, $AeroQ$ the rotor torque, $AeroT$ the rotor thrust, $AeroP$ the rotor power, $l{k}_{T}$ the lift kite line tension, ${\mathbf{f}}_{S}$ the spring forces, ${\mathbf{f}}_{D}$ the aerodynamic forces on the tethers, $\mathbf{p}$ the force applied to each point mass, $\mathsf{\Delta}t$ the time step length and $\mathbf{u},\dot{\mathbf{u}},\ddot{\mathbf{u}}$ the vectors containing the masses positions, velocities and accelerations respectively. $\mathbf{a},\mathbf{b},\mathbf{c}$ are variables used within the algorithm.

Inputs | Wind speed ${\mathit{V}}_{\mathit{w}}$, TRPT geometry R and ${\mathbf{l}}_{\mathit{s}}$, elevation angle $\mathit{\beta}$, initial conditions ${\mathbf{u}}_{2}$ and $\dot{{\mathbf{u}}_{2}}$, generator torque $\mathit{G}\mathit{e}\mathit{n}\mathit{Q}$, and number of tethers ${\mathit{N}}_{\mathit{t}}$. |

Line 1 | Find ${\mathbf{l}}_{t}$ and $\mathbf{M}$ |

Line 2 | Find $Aero{Q}_{2}$, $Aero{T}_{2}$ and $Aero{P}_{2}$ |

Line 3 | Find $l{k}_{T}$, ${\mathbf{f}}_{S,2}$ and ${\mathbf{f}}_{D,2}$ |

Line 4 | Find ${\mathbf{p}}_{2}$ |

Line 5 | ${\ddot{\mathbf{u}}}_{2}={\displaystyle \frac{\mathbf{M}}{{\mathbf{p}}_{2}-{\mathbf{f}}_{S,2}}}$ |

Line 6 | ${\mathbf{u}}_{1}={\mathbf{u}}_{2}-\mathsf{\Delta}t{\dot{\mathbf{u}}}_{2}+{\displaystyle \frac{\mathsf{\Delta}{t}^{2}}{2}}{\ddot{\mathbf{u}}}_{2}$ |

Line 7 | $\mathbf{a}={\displaystyle \frac{\mathbf{M}}{\mathsf{\Delta}{t}^{2}}}$, $\mathbf{b}={\displaystyle \frac{2\mathbf{M}}{\mathsf{\Delta}{t}^{2}}}$ |

Line 8 | For each time step, i |

Line 9 | Find $Aero{Q}_{i}$, $Aero{T}_{i}$ and $Aero{P}_{i}$ |

Line 10 | Find $l{k}_{T}$, ${\mathbf{f}}_{S,i}$ and ${\mathbf{f}}_{D,i}$ |

Line 11 | Find ${\mathbf{p}}_{i}$ |

Line 12 | $\mathbf{c}={\mathbf{p}}_{i}-\mathbf{a}{\mathbf{u}}_{i-1}-{\mathbf{f}}_{S,i}+\mathbf{b}{\mathbf{u}}_{i}$ |

Line 13 | ${\mathbf{u}}_{i+1}={\displaystyle \frac{\mathbf{a}}{\mathbf{c}}}$ |

Line 14 | $\dot{{\mathbf{u}}_{i}}={\displaystyle \frac{{\mathbf{u}}_{i+1}-{\mathbf{u}}_{i-1}}{2\mathsf{\Delta}t}}$, $\ddot{{\mathbf{u}}_{i}}={\displaystyle \frac{{\mathbf{u}}_{i+1}-2{\mathbf{u}}_{i}+{\mathbf{u}}_{i-1}}{\mathsf{\Delta}{t}^{2}}}$ |

Line 15 | End For |

Outputs | $\mathbf{u}$, $\dot{\mathbf{u}}$, $\ddot{\mathbf{u}}$, $\mathbf{AeroQ}$, $\mathbf{AeroT}$, $\mathbf{AeroP}$, ${\mathbf{f}}_{D}$, ${\mathbf{f}}_{S}$, $\mathbf{p}$ |

## Appendix B. Four TRPT Configurations

**Figure A1.**Diagrams of TRPT iterations 1, 2, 3, and 4 used throughout this work. (

**a**) TRPT−1; (

**b**) TRPT−2; (

**c**) TRPT−3; (

**d**) TRPT−4.

## Appendix C. Comparison of Multi-Spring Model and Experimental Data

**Figure A2.**Comparison of the ground station angular velocity between the multi-spring model and experimental data. (

**a**) 20 September 2018: rigid wing, TRPT−3; (

**b**) 27 August 2018: rigid wing, TRPT−3; (

**c**) 5 June 2018: soft wing, TRPT−2; (

**d**) 18 June 2017: soft wing, TRPT−1.

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**Figure 3.**Representation of a single TRPT section within the Daisy Kite system [28].

**Figure 4.**Schematic of the spring-disc TRPT representation. Each ring is described by a moment of inertia, J, and the multiple tethers in each section are replaced by a single torsional spring of stiffness k. Disc 6, on the left hand side of the schematic, represents the ground-station end of the TRPT.

**Figure 5.**Schematic of the multi-spring TRPT representation. Each ring is represented by ${N}_{t}$ (${N}_{t}=6$ in this design) linear springs with stiffness $kr$ and the tethers by linear springs with stiffness $kt$. The number of degrees of freedom for a single ring is ${N}_{t}+1$.

**Figure 6.**Diagram showing the change in tether length, $\mathsf{\Delta}{l}_{{t}_{i,j}}$, of the j-th tether within the i-th TRPT section.

**Figure 9.**TRPT laboratory test set-up and results, 30 kg axial load case. (

**a**) TRPT section lab testing with two wheels; (

**b**) Lab testing results compared to calculated torsional deformation.

**Figure 10.**Steady-state power coefficient against tip speed ratio, Daisy Kite configuration comparisons using spring-disc model. (

**a**) Rigid and soft wings; (

**b**) 4° & 0° pitched rigid wings; (

**c**) TRPT–3 and 4; (

**d**) 3 and 6 blades.

**Figure 11.**Power coefficient against tip speed ratio for various Daisy Kite configurations using the steady-state spring-disc model compared to experimental data. (

**a**) Soft wing TRPT−1; (

**b**) Soft wing TRPT−2; (

**c**) Soft wing TRPT−3; (

**d**) Rigid wing TRPT−3; (

**e**) Rigid wing TRPT−4; (

**f**) Rigid 6−wing TRPT−5.

**Figure 12.**Power spectral density of the ground station angular velocity for the spring-disc model with and without a low pass filter applied.

**Figure 13.**Comparison between the spring-disc model with a low pass filter and experimental data. (

**a**) Power output comparison; (

**b**) Angular velocity comparison.

**Figure 14.**Results from the multi-spring representation comparing rigid and flexible carbon fibre rings within the TRPT.

**Figure 15.**Power spectral density of the ground-station rotational speed for the multi-spring model with and without a low pass filter applied compared to the experimental data.

**Figure 16.**Comparison between experimental data and multi-spring model with a low pass filter. (

**a**) Power output comparison; (

**b**) Angular velocity comparison.

**Figure 17.**Response of angular velocity of the rotor to changes in torque and axial tension for the spring-disc and multi-spring models, TRPT length 30 m, wind speed of 8 m/s. (

**a**) Response to a change in torque; (

**b**) Response to a change in axial tension.

**Figure 18.**Amount of torque transmitted against the torsional deformation for a single TRPT section of the Daisy Kite.

**Figure 19.**Torsional stiffness variation with torsional deformation for a single TRPT section of the Daisy Kite.

**Figure 25.**Effect of blade length on rotor power output and ${C}_{p,max}$. (

**a**) ${C}_{p,max}$; (

**b**) Power output.

**Figure 27.**Optimised rotor design compared to the rotor used in prototype configuration 8. (

**a**) ${C}_{p}$ vs. $\lambda $; (

**b**) Power output.

Case | Test Date | Wing | TRPT | Wind Speed (m/s) | Power Output (w) | 1st Natural Frequency (Hz) |
---|---|---|---|---|---|---|

1 | 8 September 2019 | Rigid | 4 | 5.3 | 35 | 0.74 |

2 | 20 September 2018 | Rigid | 3 | 6.1 | 50 | 1.43 |

3 | 27 August 2018 | Rigid | 3 | 2.7 | 10 | 0.73 |

4 | 6 May 2018 | Soft | 2 | 5.8 | 10 | 1.47 |

5 | 18 June 2017 | Soft | 1 | 5.5 | 15 | 1.52 |

Wind Speed (m/s) | Change in Torque RMSE | Change in Tension RMSE |
---|---|---|

6 | 0.056 | 0.332 |

8 | 0.038 | 0.271 |

10 | 0.019 | 0.186 |

12 | 0.019 | 0.119 |

Models | Torque Loss (Nm) | TRPT–4 Efficiency (%) |
---|---|---|

Simple tether-drag model | 7.6 | 83.2 |

Improved tether-drag model ($\delta $ neglected) | 4.9 | 89.2 |

Improved tether-drag model | 5.1 | 88.6 |

Rotor Radius | Blade Length | TRPT Radius | TRPT Section Length | Elevation Angle | TRPT Total Length | Tip Speed Ratio |
---|---|---|---|---|---|---|

2.22 m | 1.4 m | 0.5 m | 1.25 m | 18.5${}^{\circ}$ | 190 m | 3.5 |

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## Share and Cite

**MDPI and ACS Style**

Tulloch, O.; Yue, H.; Kazemi Amiri, A.M.; Read, R.
A Tensile Rotary Airborne Wind Energy System—Modelling, Analysis and Improved Design. *Energies* **2023**, *16*, 2610.
https://doi.org/10.3390/en16062610

**AMA Style**

Tulloch O, Yue H, Kazemi Amiri AM, Read R.
A Tensile Rotary Airborne Wind Energy System—Modelling, Analysis and Improved Design. *Energies*. 2023; 16(6):2610.
https://doi.org/10.3390/en16062610

**Chicago/Turabian Style**

Tulloch, Oliver, Hong Yue, Abbas Mehrad Kazemi Amiri, and Roderick Read.
2023. "A Tensile Rotary Airborne Wind Energy System—Modelling, Analysis and Improved Design" *Energies* 16, no. 6: 2610.
https://doi.org/10.3390/en16062610