Particle System Modelling and Dynamic Simulation of a Tethered Rigid Wing Kite for Power Generation

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Master of Science Thesis

Mustafa Can Karadayı

September 21, 2016

Faculty of Aerospace Engineering Delft University of Technology ·

Particle System Modelling and Dynamic Simulation of a Tethered Rigid Wing Kite for Power Generation

Master of Science Thesis

For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology

Mustafa Can Karadayı

September 21, 2016

Thesis Registration Number: 088#16#MT#FPP

An electronic version of this thesis is available at http://repository.tudelft.nl

Faculty of Aerospace Engineering Delft University of Technology · Copyright c Mustafa Can Karadayı

The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled “Particle System Modelling and Dynamic Simulation of a Tethered Rigid Wing Kite for Power Gener- ation” by Mustafa Can Karadayı in partial fulﬁllment of the requirements for the degree of Master of Science.

Dated: September 21, 2016

Chairperson of Committee: Dr.-Ing. Roland Schmehl

Reader: Dr. Axelle C. Vir´e

Reader: Dr.ir. Mark Voskuijl

Daily Supervisor: Uwe Fechner, MSc

Abstract

Kite Power research group at TU Delft developed the KiteSim framework for dynamic simulation of crosswind kite power systems. Currently, KiteSim has the capability to simulate leading edge inflatable kites. The use of rigid wing kites in crosswind kite power systems is becoming widespread. Accordingly, KiteSim framework is planned to be augmented to simulate the rigid wing kite power systems as well. Therefore, objective of this thesis is set to enhance the KiteSim framework in order to include the capability of the dynamic simulation of a rigid wing pumping crosswind kite power system by developing a particle system model to represent the tethered rigid wing. The particle system model of rigid wing consists of 6 point masses and 13 spring-damper elements. Positions and masses of the discrete particles are calculated in order to represent the rigid wing properties accurately. The spring-damper elements are interconnecting the particles. Lifting line theory and Kirke’s post stall correlation methods are used to create the full angle of attack aerodynamic model for wings and tails. Available atmospheric, tether and winch models of the KiteSim framework are also used for rigid wing dynamic simulations. Equation of the motion of the particle system model is formulated as an implicit problem, which is simulated by implicit Runge-Kutta method of fifth-order. Validation simulations are conducted with the particle system model of NASA SGS 1-36 flight test sailplane. Validation cases show satisfactory results with the flight test data. For the remaining simulations, AP2 PowerPlane of Ampyx Power is modelled as particle system and manually flown in KiteSim. Mass properties of the model is found to be highly accurate throughout the simulations. Power generation capabilities of the model is checked by flying figure-of-eight trajectories. Reel-out phases are investigated, a peak mechanical power of 45 kW and a mean power of 10 kW are obtained. Further comparison of the reel-out dynamic simulations with the quasi-steady theoretical calculations is done. Moreover, gliding and stalling manoeuvres are simulated for a plausibility check study. The developed particle system model for rigid wings shows satisfactory results and a potential for further development.

v vi Abstract Acknowledgements

My enthusiasm for airborne wind energy was born when I started taking kite power lectures back in 2013 from Dr.-Ing. Roland Schmehl. I want to thank him for introducing me to this fascinating topic and leading me to choose this thesis project. I would like to thank my daily supervisor Uwe Fechner, MSc. who has been always ready to help me with my countless questions. I want to thank Ampyx Power for sharing the data of their previous prototype with me. I am also thankful to Ted Spinders for the help and feedback on piloting the simulation. I am grateful to all past and present members of Kite Power team and Room 6:08 who have been great mates not only in office but also outside. Thank you Tassos, Lukas, Jo, Felix, Chris, Pietro, Lorenzo, Julius, Mike, Apurva, Rachel, Andres, Bas, Viktor, Pranav. I want to thank my friends from FPP; Ashwin, Phillip, Martin, Akshay, Malcom and Roderick for being such a great company. In particular, I feel so happy to meet my companion Dino; our endless talks from science to politics shall never end. Thanks to all Happy Hour People, for sharing the life in Delft since the first day. Special thanks to the great panpas of GülizAbla who became my family in Delft: Sercan, for pre-/peri-/post-fitness adventures; Emre, for interesting topics on the mops; Yasemin, for great midnight dessert sessions; Argun, for good coffee and hazelnut taste; Onursal, for interesting philosophical conversations and even more interesting laptop wars; Alper, for random tricky dj skills; Cansın, for tasty sushi workshops; Burak, for famous one- hand-capable-water-skiing tricks; Ekin, for overnights and broodjes memories; Tilbe, for hilarious countryside trips; and Cansel, for good cheer and ’But, I!’ moments. Last but not the least, I am sending my greatest thanks to my parents for their trust and support during my whole life.

Delft, The Netherlands Mustafa Can Karadayı September 21, 2016

vii viii Acknowledgements Contents

Abstract v

Acknowledgements vii

List of Figures xv

List of Tables xvii

Nomenclature xix

1 Introduction1

2 Crosswind Kite Power Fundamentals3 2.1 Physics of Crosswind Kite Power Systems...... 3 2.1.1 Power formula...... 3 2.1.2 Power harvesting factor...... 4 2.2 Classification of Crosswind Kite Power Systems...... 4 2.2.1 Power extraction mode classification...... 4 2.2.2 Flexible wing vs. rigid wing classification...... 5 2.2.3 Other classifications...... 6 2.3 Working Principles of Crosswind Kite Power Systems...... 6

3 Literature Review9 3.1 Modelling and Simulation...... 9 3.2 Crosswind Kite Modelling...... 10 3.3 TU Delft Dynamic Kite Power System Simulator...... 12 3.3.1 Atmospheric model...... 12 3.3.2 Tether model...... 13 3.3.3 Kite models...... 13

ix x Contents

3.3.4 Winch model...... 14 3.3.5 Implementation...... 14 3.4 Particle System Modelling...... 15 3.5 Component Buildup Method...... 16 3.6 Aerodynamic Models for Dynamic Simulation...... 18

4 Research Objective 21

5 Methodology of the Simulation Modelling 23 5.1 Rigid Wing Particle System Model Structure...... 23 5.1.1 Discretization of the particles...... 24 5.1.2 Formation of the spring-damper elements...... 29 5.2 Rigid Wing Aerodynamic Model...... 29 5.2.1 XFLR5 software...... 31 5.2.2 Post-stall correlation method...... 32 5.2.3 Aerodynamic data approximation script...... 34 5.3 Rigid Wing Controller Implementation...... 34 5.4 Atmospheric Model...... 35 5.5 Tether Particle System Model...... 36 5.6 Rigid Wing Particle System Model...... 39 5.6.1 Calculation of the gravitational forces...... 40 5.6.2 Calculation of the spring-damper forces...... 40 5.6.3 Calculation of the aerodynamic forces...... 41 5.7 Equation of Motion of the Particle System Model...... 46 5.8 Formulation of the Implicit Problem...... 46 5.9 Simulation Implementation...... 47 5.9.1 Numeric solver...... 47 5.9.2 Initial and simulation parameters...... 48

6 Model Validation 49 6.1 Validation Aircraft...... 50 6.2 Validation Methodology...... 50 6.2.1 Simulation of non-tethered ﬂight in KiteSim...... 50 6.2.2 Fuselage drag estimation...... 50 6.3 Validation Results and Discussions...... 51 Contents xi

7 Simulation Results and Discussions 57 7.1 Mass Properties Results...... 57 7.1.1 Case 1: Tethered, non-maneuvering kite at very high atmospheric wind speed...... 57 7.1.2 Case 2: Tethered, maneuvering kite at high atmospheric wind speed 60 7.2 Reel-Out Phase Results...... 63 7.2.1 Case 3: Tethered, reel-out, figure-of-eight trajectory flight at strong wind...... 63 7.2.2 Case 4: Tethered, reel-out, figure-of-eight trajectory flight at moderate wind...... 67 7.2.3 Comparison of reel-out cases results and theoretical analysis.... 70 7.3 Plausibility Checking Results...... 71 7.3.1 Case 5: Non-tethered, gliding flight...... 71 7.3.2 Case 6: Non-tethered, stalling maneuver...... 73 7.4 Discussions...... 75 7.4.1 Discussion on the developed particle system model...... 75 7.4.2 Discussion on the controlling and piloting...... 76 7.4.3 Discussion on the mass properties...... 76 7.4.4 Discussion on the power generation...... 77

8 Conclusions and Recommendations 79

References 81

A Input File for the Initialization Script of AP2 PowerPlane 85

B XFLR5 Models for Wing, HT and VT of AP2 PowerPlane 87

C Correction Factor Plots for Post-Stall Aerodynamic Coeﬃcients 89

D Approximate AeroData RW Script 91

E Wing Axes System Creation for Wing Dihedral Eﬀects 95

F Schweizer SGS 1-36 Sailplane 97 F.1 Input ﬁle for SGS 1-36 Sailplane...... 97 F.2 General Data for SGS 1-36 Sailplane...... 99 F.3 XFLR5 models for wing, HT and VT of SGS 1-36 Sailplane...... 103

G Winch Model and Winch Controller Parameters 105 G.1 Winch Model Parameters...... 105 G.2 Winch Controller Parameters...... 106

H Additional Figures and Data for Quasi-Steady Analysis 107

I Additional Simulation Results 111 xii Contents List of Figures

2.1 Examples of flexible, hybrid and rigid wing kite designs...... 5 2.2 Different ground based CWKP steering and control concepts...... 6 2.3 Traction and retraction phases of the pumping cycle operation...... 7 2.4 3D representation of the figure-of-eight flight pattern...... 7

3.1 Complexity vs computation time for diﬀerent kite modelling approaches. 10 3.2 Description of point mass, rigid body, particle system (lumped mass) and multi plate models...... 11 3.3 Representation of the rigid body model of the kite and the particle system model of the tether...... 12 3.4 Particle system representation of the tether (points P0 to PKCU) and the four point kite (points A to D)...... 13 3.5 Representation of the point mass and four-point model...... 14 3.6 Representation of a particle and the systems of particles formed by the particles along with local axis frames...... 15 3.7 Representation of the external forces that cause the translation (left) and the rotation (right) on a rigid body...... 16 3.8 The discretization of the kiteplane aerodynamic surfaces...... 17 3.9 Illustration of lift and drag forces on the discrete elements of the kiteplane 17 3.10 Lift and drag coeﬃcients versus angle of attack curves for LEI kites in KiteSim...... 18

5.1 Representation of the PS model on a conventional RW kite, with 6 discrete point masses and 13 spring-damper elements...... 24 5.2 RW kite particles of the PS model for the AP2 RW kite...... 28 5.3 The particles and the spring-damper elements of the PS model, a generic RW kite is drawn for comparison...... 30 5.4 3D aerodynamic coeﬃcients versus AoA obtained from XFLR5 software. 33 5.5 Aerodynamic coeﬃcients as a function of AoA...... 35

xiii xiv List of Figures

5.6 Power law, logarithmic law and fitted wind profiles along with three measured wind speeds at different altitudes...... 36

5.7 Discrete tether particles (points tp0 to tpn) of the tether model...... 37 5.8 Representation of the position and velocity vectors along with the segment vector and segment relative velocity for a segment...... 38 5.9 Representation of the position and velocity vectors along with the segment vector and segment relative velocity for a segment...... 40 5.10 Representation of the RW particles with the lift and drag forces acting on the particles m2, m3, m4 and m5 ...... 41 5.11 Apparent wind velocity and its components for the calculation of angle of attack and sideslip angle...... 43 5.12 Equivalent flat plate method used in the calculation of the control surface deflection effect on the AoAs...... 43 5.13 The aerodynamic moments and aerodynamic moment induced forces explained on the PS model of the RW kite...... 45

6.1 Modified Schweizer SGS 1-36 sailplane used by NASA...... 49 6.2 L/D versus airspeed plot obtained from the flight tests and the KiteSim. 53 6.3 Sink rate versus horizontal velocity plot obtained from the flight tests and the KiteSim...... 53 6.4 Rudder input and directional time history...... 55 6.5 Aileron input and lateral time history...... 55 6.6 Elevator input and longitudinal time history...... 56

7.1 Flight trajectory; case 1...... 58 7.2 Apparent wind speed versus time; case 1...... 58 7.3 Reel-out speed versus time; case 1...... 58 7.4 PS model mass properties versus time; case 1...... 59 7.5 Flight trajectory; case 2...... 61 7.6 Apparent wind speed versus time; case 2...... 61 7.7 Reel-out speed versus time; case 2...... 61 7.8 PS model mass properties versus time; case 2...... 62 7.9 Flight trajectory; case 3...... 63 7.10 Time history; case 3...... 65 7.11 Simulation results for the power related parameters; case 3...... 66 7.12 Flight trajectory; case 4...... 67 7.13 Time history; case 4...... 68 7.14 Simulation results for the power related parameters; case 4...... 69 7.15 Flight trajectory; case 5...... 71 7.16 Time history for gliding phase; case 5...... 72 7.17 Flight trajectory; case 6...... 73 7.18 Time history for stall maneuver; case 6...... 74 List of Figures xv

B.1 AP2 PowerPlane Wing...... 87 B.2 AP2 PowerPlane HT...... 88 B.3 AP2 PowerPlane VT...... 88

F.1 SGS 1-36 general data, from [33]...... 99 F.2 SGS 1-36 mass properties, from [33]...... 99 F.3 SGS 1-36 general data, from [19]...... 100 F.4 SGS 1-36 general data (continued), from [19]...... 100 F.5 SGS 1-36 mass properties, from [19]...... 100 F.6 SGS 1-36 three-view drawing , from [33]...... 101 F.7 SGS 1-36 side drawing , from [6]...... 102 F.8 Schweizer SGS 1-36 Wing...... 103 F.9 Schweizer SGS 1-36 HT...... 103 F.10 Schweizer SGS 1-36 VT...... 103

H.1 Representation of kite velocities, with azimuth angle φazi, polar angle θele and course angle χ shown; from [29]...... 107 H.2 Representation of aerodynamic forces and velocities, with the azimuth angle φazi and polar angle θele shown; from [29]...... 108 H.3 Representation of aerodynamic forces, gravity force and velocities, with the azimuth angle φazi and polar angle θele shown; from [29]...... 108

I.1 Kite lift and kite weight during the simulation case 1...... 111 I.2 Kite lift and kite weight during the simulation case 2...... 111 xvi List of Figures List of Tables

5.1 Initial guesses, lower bounds and upper bounds of the optimization variables in the optimization problem...... 27 5.2 Basin Hopping algorithm parameters used in the optimization...... 27 5.3 The masses and positions of the AP2 RW kite particles in PS model... 27 5.4 Comparison of the mass properties and the cg location of the AP2 RW kite and its PS model...... 28 5.5 XFoil Direct Analysis parameters for the AP2 RW kite...... 31 5.6 Wing & Plane Design tool advanced settings for wing, HT and VT of the AP2 RW kite...... 32 5.7 Control Surface Sign Convention...... 34 5.8 Tether spring stiﬀness in tension and compression...... 39 5.9 Radau5DAE solver parameters used in the simulation...... 48 5.10 Initial and simulation parameters...... 48

6.1 SGS 1-36 Sailplane fuselage drag calculation parameters...... 51

H.1 Averaged values of the system parameters during the approximate horizontal ﬂight state, along with theoretical traction force and power..... 109 H.2 Tether force and traction power comparison of the simulation cases 3 and 4 with the theoretical analysis...... 110

xvii xviii List of Tables Nomenclature

Latin Symbols

2 Aside Fuselage Side Projected Area [m ] 2 Atop Fuselage Top Projected Area [m ] AR Aspect Ratio [-] b Span [m] BL Buttock Line [m] c¯ Mean Aerodynamic Chord [m] c Chord Length [m] ccs Chord Length of the Control Surface [m]

CD,t,eq Tether Equivalent Drag Coeﬃcient [-]

CDt Tether Drag Coeﬃcient [-]

CD Drag Coeﬃcient(3D) [-]

Cd Drag Coefficient(2D) [-] ceff Effective Chord Length [m]

Cf,t Turbulent Flat-plate Skin Friction Drag Coeﬃcient [-]

CL Lift Coeﬃcient(3D) [-]

Cl Lift Coeﬃcient(2D) [-]

CM Quarter Chord Moment Coeﬃcient(3D) [-]

Cm Quarter Chord Moment Coeﬃcient(2D) [-] croot Root Chord [m] df Fuselage Maximum Diameter [m] dt Tether Diameter [m]

xix xx Nomenclature

FB Body-Fixed Reference Frame [-]

FW Wind Reference Frame [-] FF Fuselage Form Factor [-] FS Fuselage Station [m] iHT Horizontal Tail Incidence Angle [rad] or [deg] iw Wing Incidence Angle [rad] or [deg] 2 Ixx Moment of Inertia about xB axis [kgm ] 2 Ixy Product of Inertia with respect to xB and yB axes [kgm ] 2 Ixz Product of Inertia with respect to xB and zB axes [kgm ] 2 Iyy Moment of Inertia about yB axis [kgm ] 2 Iyz Product of Inertia with respect to yB and zB axes [kgm ] 2 Izz Moment of Inertia about zB axis [kgm ] −1 kd Damping Coeﬃcient [Nsm ] −1 ks Spring Constant [Nm ] lf Fuselage Length [m] ls0 Initial Segment Length [m] ls Length of Tether Spring-Damper Segment [m] lt Tether Length [m] M Aerodynamic Moment [Nm] th mi Mass of i Particle [kg] mkite Kite Mass [kg] mtp Mass of a Tether Particle [kg] N Number of Particles of the PS Model [-] n Number of Tether Spring-Damper Elements [-] S Reference Area [m2] 2 Swet,f Fuselage Wetted Area [m ] t Time [s] −1 va Apparent Wind Velocity [ms ] −1 vt,ro Tether Reel-Out Speed [ms ] −1 vw,exp Wind Speed at Altitude According to Power Law [ms ] −1 vw,log Wind Speed at Altitude According to Log Law [ms ] −1 vw,ref Reference Wind Speed [ms ] WL Water Line [m] th xi, yi, zi x, y, z Coordinates of the i Particle in FB [m] xB, yB, zB x, y, z Axes of Body-Fixed Reference Frame FB [-] xcg, ycg, zcg x, y, z Coordinates of the cg Position in FB [m] xW, yW, zW x, y, z Axes of Wind Reference Frame [-] Y¯ Spanwise Distance of the Mean Aerodynamic Chord [m] Nomenclature xxi

z Altitude [m] z0 Surface Roughness Length [m] zref Reference Altitude [m] p, q, r Roll, Pitch, Yaw Body Rates [deg s−1]

Vector Symbols

D Drag [N]

Df Fuselage Drag [N] exB , eyB , eyB Unit Vectors of Body-Fixed Reference Frame [-] Fd Tether Aerodynamic Drag Force [N]

Fg Gravitational Force [N]

Fsd Spring Damper Force [N] g Gravitational Acceleration [ms−2] L Lift [N] p Position [m] s Segment Vector [m] −1 sv Segment Relative Velocity [ms ] v Velocity [ms−1] −1 va Apparent Wind Velocity [ms ] −1 va,xz Apparent Wind Velocity Projected on XZ Plane [ms ] −1 va,xy Apparent Wind Velocity Projected on XY Plane [ms ] −1 vs Segment Velocity [ms ] Y State Vector [-] Y˙ State Derivative Vector [-]

Greek Symbols

α Angle of Attack [rad] or [deg]

αcs Additional Angle of Attack due to Control Surface Deﬂection [rad] or [deg]

αeﬀ Eﬀective Angle of Attack [rad] or [deg] β Sideslip Angle [rad] or [deg]

δcs Control Surface Deﬂection Angle [rad] or [deg] Γ Dihedral Angle [rad] or [deg]

κd Finite AR Correction Factor for Post-Stall Drag Coeﬃcient [-]

κl Finite AR Correction Factor for Post-Stall Lift Coeﬃcient [-] λ Taper Ratio [-] xxii Nomenclature

ΛLE Leading Edge Sweep Angle [rad] or [deg] ν Kinematic Viscosity [m2s−1] Ψ, Θ, Φ Tait-Bryan Roll, Pitch, Yaw Angles [deg] ρ Air Density [kg m−3] −3 ρt Tether Mass Density [kg m ] ξ Airfoil Camber [% of the chord] ζ Power Harvesting Factor [-]

Abbreviations ac Aerodynamic Center cg Center of Gravity 2D Two Dimensional 3D Three Dimensional AoA Angle of Attack AWE Airborne Wind Energy CFD Computational Fluid Dynamics CWKP Crosswind Kite Power DOF Degree(s) of Freedom EOM Equation(s) of Motion FoE Figure-of-Eight HAWP High Altitude Wind Power HT Horizontal Tail KiteSim Kite Power System Simulator L/D Lift-to-Drag Ratio LE Leading Edge LEI Leading Edge Inﬂatable LLT Lifting Line Theory PS Particle System RW Rigid Wing TU Delft Delft University of Technology UAV Unmanned Aerial Vehicle VLM Vortex Lattice Method VT Vertical Tail Chapter 1

Introduction

Our world is going through a change, the share of the renewable energy technologies in the energy generation is increasing. Airborne Wind Energy (AWE) is a recent and state- of-the-art concept in the field of the renewable energy technologies. The humankind had invented the kites in ancient times and flown them since then for various reasons [40]. However, the concept of using AWE for the electricity generation needed to wait until the 20th century, where the general investigations were made in the 1930s and the more detailed attempts to produce electricity by the airborne windmills were taken place in the 1960s [20]. High Altitude Wind Power (HAWP) and Crosswind Kite Power (CWKP) constitute two main concepts of the AWE systems. The content of this thesis concerns the latter. In 1980, Miles L. Loyd published his famous paper ”Crosswind Kite Power” [18], which is considered as the scientific foundation of the CWKP systems. However, the research and the development of the CWKP systems stayed inactive until the late 1990s. The improvements in tether and wing materials along with automatic control and navigation systems led to the development of the first CWKP concepts in 2000s, where different solutions gave birth to the different concepts in the CWKP systems [7]. Delft University of Technology (TU Delft) has started the research on the AWE systems with the visionary work of Professor Wubbo Ockels. Ockels’ solution was the Laddermill concept which was developed in the late 1990s. The concept includes a series of multiple kites which are linearly connected by a cable that drives a ground based generator [22]. Since then, TU Delft KitePower Research Group has been continuing to the research and development of the CWKP systems and currently has a system demonstrator where a tethered Leading Edge Inflatable (LEI) kite is used in a pumping mode to generate electricity using a ground generator [30]. Dynamic simulations are essential during the development process of the complicated systems such as a CWKP system. Realistic simulations help to test the system without requiring to manufacture different kite configurations and conducting the flight tests. Modelling of the system components and the environment is necessary for dynamic simulations. The modelling of kite, tether, generator and atmosphere is needed in order to

1 2 Introduction

simulate a CWKP system. The TU Delft KitePower Research Group currently has a simulation framework which is able to simulate the current pumping CWKP system demonstrator which has a LEI kite [10]. The LEI kites are represented either as a point mass model or as a particle system model in the framework. The current simulation is lacking a dynamic model to study Rigid Wing (RW) kites. Therefore, the purpose of this thesis is to develop a particle system model for a tethered RW kite for enhancing the current framework. By achieving this, the KitePower Research Group will have the ability to simulate RW kites which will be helpful for the development of the future CWKP systems. This thesis report is structured as followed. The Crosswind Kite Power Fundamentals chapter gives a background information about the CWKP technology aiming to give the reader an understanding on the physics behind the technology, the classiﬁcation of the CWKP systems and the working principles of the pumping cycle operation. The Litera- ture Review chapter presents a literature review on the modelling and simulation, crosswind kite modelling, KiteSim framework, particle system modelling, component buildup method and aerodynamic models for dynamic simulation. The Research Objective chapter gives the objective and the goals of the thesis. In the Methodology of the Simulation Modelling chapter, the methodology of the numeric simulation model and the simulation are explained. The Model Validation chapter presents the validation study of the particle system model for rigid wing kite. The Simulation Results and Discussions chapter presents and discusses the dynamic simulation results of the several test cases conducted with the developed particle system model. Finally, in the Conclusions and Recommenda- tions chapter, the conclusion of the thesis is presented along with the recommendations for future developments. Chapter 2

Crosswind Kite Power Fundamentals

Anyone, who flew a kite before knows the feeling that the kite is pulling stronger while flying in a crosswind direction rather than while staying static in the air. That is the very basis of the Crosswind Kite Power, which is the term that Miles Loyd created when describing and studying the idea of power generation with tethered wings in a crosswind flight [18].

2.1 Physics of Crosswind Kite Power Systems

Using the ambient wind flow, a kite can be flown in a crosswind direction with an apparent wind velocity va, which is way higher than the ambient wind velocity vw. Hence, in crosswind flight, a kite experiences higher apparent wind velocity when compared to the static flight. A crosswind flying kite extracts a significant amount of power from the wind and this can be used for power generation [7], using either directly the high apparent wind velocity or its result in high aerodynamic forces and therefore high tether tension.

2.1.1 Power formula

Using idealized assumptions, Loyd [18] estimates the maximum power that a crosswind kite can extract from the wind as:

2 2 3 CL Pmax = ρ vw SCL (2.1) 27 CD

It is seen that the maximum power Pmax is proportional to the ambient air density ρ and ambient wind speed vw which are atmospheric parameters. Also, it is dependant on kite parameters which are kite area S, as well as kite lift and drag coeﬃcients CL, CD. The 3 2 ratio CL/CD is an important kite parameter that eﬀects the performance of a CWKP system considerably.

3 4 Crosswind Kite Power Fundamentals

Equation 2.1 neglects the kite weight and the tether eﬀects. Moreover, it assumes a steady-state ﬂight for the kite at zero elevation angle and at a position which is directly crosswind that is at zero azimuth angle.

2.1.2 Power harvesting factor

The ratio of the useful power P that is extracted from the wind by a crosswind kite of a surface S, to the wind power Pwind which is ﬂowing through the cross sectional area of the same size with S is named as power harvesting factor ζ. It is important to note that the power harvesting factor is not an energy conversion eﬃciency; rather it is an performance parameter.

P P ζ = = 1 3 (2.2) Pwind 2 ρ vw S

Using Equation 2.1, the maximum power harvesting factor is obtained as in Equation 2.3. Diehl [7] states that the power harvesting factors of modern CWKP systems are around 5.5 and the existing highest experimentally obtained value is 8. Moreover, the rigid wings have higher power harvesting factor compared to ﬂexible wings as a result of their higher lift-to-drag ratios, as discussed in Section 2.2.

2 4 CL ζmax = CL (2.3) 27 CD

2.2 Classiﬁcation of Crosswind Kite Power Systems

The classiﬁcation of the CWKP systems is not solely dependant on one concept. There is a large variety of interesting concepts in the area of CWKP systems. The most common classiﬁcations of the CWKP systems will be given here as a background information, the more detailed ones can be obtained from [5] and[7].

2.2.1 Power extraction mode classiﬁcation

In CWKP systems, the power of the kite can be extracted using two diﬀerent power extraction modes that Loyd describes as drag mode and lift mode [18]. The drag mode makes use of the high apparent wind velocity to drive an on-board wind turbine and the lift mode makes use of the tension in the tether either to drive a ground generator or to propel a vehicle on the ground. With this division of the lift mode, Diehl states that the CWKP systems can be divided in three groups regarding their means of power production [7]; on-board power generation systems, ground based power generation systems and vehicle propulsion systems. On-board power generation systems carry on-board turbines to generate electricity directly from the high apparent wind velocity on the kite. Payne and McCutchen, described the idea of electricity generation with the on-board generators on a tethered sailplane in 2.2 Classiﬁcation of Crosswind Kite Power Systems 5

their patent ”Self-Erecting Windmill” [23] in 1976. Right now, 40 years later, Makani Power [39] is working on this concept. Ground based power generation systems have a ground station where a drum and an electric generator is present. The tether tension that is generated by the kite is used to unroll the tether from the drum. Hence, the drum rotates and drives the generator. This concept is wide spread in the ﬁeld of CWKP, for example, Ampyx Power, TwingTec and EnerKite are using this concept in their solutions [5]. Vehicle propulsive systems are not electricity generating systems. They make use of the tether tension directly to propel a vehicle on the ground. SkySails [11] is a company that is using this concept.

(a) TU Delft LEI kite de- (b) TU Delft Kiteplane (c) Makani Power design sign design from [36]

Figure 2.1: Examples of ﬂexible, hybrid and rigid wing kite designs

2.2.2 Flexible wing vs. rigid wing classiﬁcation

Diehl [7] gives another classification in the CWKP systems which is between the flexible wings like LEI kites or ram-air kites and the rigid wing like sailplanes or aircrafts. Flexible wings are lighter kites due to their structures; they maintain their shapes using the airflow around them. They do not fly at very high speeds; thus, they are easily controllable by pilots. All these aspects also result in the fact that the flexible kites are safer for the operations near humans. Rigid wings, contrary to flexible wings, have heavier structures, which is due to the fact that they maintain their shapes without the need of the airflow around them. The rigid structure allows the use of more efficient profiles in the design, resulting in higher lift-to- drag ratios; thus, can reach higher speeds compared to the flexible wings. These aspects make the rigid wings more useful for power production; however, also more dangerous in case of a crash. There also exist some kites that have hybrid designs, making use of both flexible and rigid elements in their structures in order to have a design which can have the benefits of the flexible and rigid wing designs. 6 Crosswind Kite Power Fundamentals

2.2.3 Other classiﬁcations

There exist more classifications that can be made for CWKP systems. Diehl describes the use of multiple wing system in [7]. Although this concept is not built yet, the concepts can be classified considering the number of kites of the system. Cherubini classifies the ground based CWKP systems with respect to their means of steering and control [5]; such as systems which are using on-board actuators or airborne control pods or multiple tether lines as seen in Figure 2.2.

Figure 2.2: Diﬀerent ground based CWKP steering and control concepts, from [5]

2.3 Working Principles of Crosswind Kite Power Systems

As mentioned in the classification of CWKP systems section, there are on-board power generation, ground based power generation and vehicle propulsion systems. The TU Delft KitePower research group is currently using a ground based power generation system with pumping cycle operation. Therefore, in this section, pumping cycle power generation is explained. The operation of the pumping cycle systems do share some common principles with the other CWKP systems. However, there exists differences, for which Diehl gives the necessary explanation in [7]. The main components in a CWKP system using pumping cycle operation are ground station, tether and kite. The pumping cycle operation can separated into two phases; namely traction (reel-out) and retraction (reel-in) phases as shown in Figure 2.3. The traction phase is where the kite is flying a crosswind maneuver such as a figure-of- eight (FoE) pattern. Due to the crosswind flight, the high aerodynamic load generated by the kite increases the tether tension and the tether reels-out from the drum in the ground station and rotates it. Consequently, the drum drives the generator. Once the maximum design tether length is reached the traction phase ends and kite goes through a fast transition phase in order to start the retraction phase. In the retraction phase, the kite is de-powered where its aerodynamic load is minimized and the tether tension is decreased. In this stage, the de-powered kite is reeled-in by an electric motor. De-powering can be done in different ways such as by decreasing the angle-of-attack, or by stopping the crosswind maneuver, or else by flying in the sides of the wind window. Once the retraction phase is over, the kite starts again with the traction phase. 2.3 Working Principles of Crosswind Kite Power Systems 7

Wind

Reel-out (traction) phase: Reel-in (retraction) phase: energy generation energy consumption

Figure 2.3: Traction and retraction phases of the pumping cycle operation, from [38]

In the pumping cycle operation, the system generates power by driving the generator during the traction phase and consumes power for reeling-in the kite in the retraction phase. Due to the de-powering of the kite, the required power for the retraction is signiﬁcantly lower than the power generated in the traction phase. Therefore, a net power production occurs over the pumping cycle operation.

Figure 2.4: 3D representation of the ﬁgure-of-eight ﬂight pattern, from [25] 8 Crosswind Kite Power Fundamentals Chapter 3

Literature Review

3.1 Modelling and Simulation

Modelling and simulation is a fundamental discipline for the development of complex engineering systems. A simulation framework can be helpful for the design, optimization and operation of the aerospace systems. As a matter of fact, the simulations are based on the mathematical models. Therefore, model building is necessary for every simulation. The modelling process is stated as an iterative process in [34]; a mathematical model is established on the laws of physics and further it is refined by the experimental results until a satisfying model is developed which is good enough for the scopes but with the accepted limitations. The simulations generate simulated data for a real system using the mathematical models. On the other hand, experiments make use of the physical models and collect observed data for the real system. The observed data coming from the experiments is also used in the iterative development process of the mathematical models for the verification and validation of the models. Yun and Li describe simulations as a method to obtain information through the use of assumed system models which presents techniques for analysis, decision-making, design- ing and training [42]. Moreover, they state that the simulations can decrease the flight periods, cost and risk. Stevens et al. express that the high cost of building and flight testing of a real aircraft leads to the development of the modelling and simulation discipline [34]. They indicate that the use of mathematical models together with computer simulations provides a relatively cheaper way for the performance evaluation and design improvisation, also creates the means to investigate the accidents and develop training simulators. At the moment, the modelling and simulation technology is an important discipline which is widely used in the aerospace industry, for the development of unmanned aerial vehicles (UAV), general aviation aircrafts, passenger aircrafts, military aircrafts, rotorcrafts, wind turbines, etc.

9 10 Literature Review

Comparatively, CWKP systems are complex systems using aerial and ground components. Therefore, the modelling and simulation technology is helpful for the development of the CWKP systems in following aspects:

Testing new design conﬁgurations • Checking ﬂight characteristics • Conducting performance analyses • Implementing automatic control systems • Training kite pilots •

3.2 Crosswind Kite Modelling

As the simulation technology plays an important role for the development of CWKP systems, modelling of the kite is required in order to have a CWKP system simulation. Many kite modelling approaches can be found in the literature such as point mass, particle system, rigid body, multi plate, multi body and ﬁnite element models. A comparison of these modelling approaches is shown in Figure 3.1 considering the complexity and the computation time. The literature review on crosswind kite modelling is done for point mass, particle system, rigid body and multi plate approaches.

Figure 3.1: Complexity vs computation time for diﬀerent kite modelling approaches, taken from [37]

Williams et al. investigate point mass model, rigid body model and ﬂexible multi plate model for surf kites in [41]. They state that the point mass model is the simplest model representing the kite. However, the lack of rotational inertia in the point mass model makes it not useful for high level control algorithms. On the other hand, it is useful enough 3.2 Crosswind Kite Modelling 11

for basic flight trajectory and performance analysis. The rigid body model represent the kite with six degrees of freedom (DOF), which is required for the fully investigation of the flight dynamics of the kite. Hence, rigid body models give the position, velocity, attitude and rotational rate outputs for the kite by making use of the forces and moments acting on the body. The multi plate flexible model is built up as a series of hinged plates representing the kite. Williams et al. state that this model is useful for the representation of the deformation of the kite which is a common case for flexible kites [41]. Ruppert gives a detailed list of the CWKP system models available in the literature [26], along with an overview for the different modelling approaches as can be seen in Figure 3.2.

Figure 3.2: Description of point mass, rigid body, particle system (lumped mass) and multi plate models, taken from [26]

The rigid body modelling is a common modelling approach for the kites, especially for RW kites. There is an enormous amount of literature and research on the rigid body models for aircraft flight dynamics [8], [34]. A RW kite is basically an aircraft, more specifically an unmanned glider with the only difference that it is connected to a ground station by a tether. Hence, the previous literature and research on aircraft rigid body models are helpful for the CWKP modelling, with the only difference that when getting the equations of motion (EOM) of a rigid body kite model, the contribution of the tether forces and moments needs to be included. Figure 3.3 shows a model of the kite and a tether of a CWKP system [12], where a rigid body model is used for the kite and a particle system model for the tether, the forces f and moments m acting on the rigid body with mass M and inertia tensor I are also shown. Terink et al. use a rigid body modelling approach in the simulation of a tethered inflatable kiteplane [36], where the kiteplane is connected to a single tether with two bridle lines. 12 Literature Review

Figure 3.3: Representation of the rigid body model of the kite and the particle system model of the tether, from [12]

In order to include the kinematic constraints coming from the bridle, the Lagrange’s equations are used to obtain the EOM of the rigid body model. Winch launching of gliders and CWKP operations show some similarities. The glider is connected to a winch via a tow cable which is reeled during take-oﬀ and climb. This is similar where a CWKP kite is connected to a ground station via a tether which is reeled in and out during a pumping cycle. Thus the research on the glider launch modelling can be helpful for the crosswind kite modelling. Santel investigates glider winch launch using a numeric simulation, where the glider is modelled as a rigid body and the EOM of the rigid body model is obtained using Newton’s second law [27], [28].

3.3 TU Delft Dynamic Kite Power System Simulator

Fechner et al. describe a dynamic model of the pumping kite power system that is used in the Kite Power System Simulator (KiteSim) framework in [10], where the models of the atmosphere, tether, winch and kites are explained in detail. In this thesis, the KiteSim models for atmosphere, tether and winch are used directly; only slight modiﬁcations are done when required. In this section, these models and the implementation of the KiteSim is brieﬂy explained; for further details reader should refer to [10].

3.3.1 Atmospheric model

KiteSim includes three atmospheric models for the calculation of the wind speed at the altitude; the power law profile, the logarithmic law profile and the fitted profile which is a linear combination of the power and logarithmic law profiles [10]. 3.3 TU Delft Dynamic Kite Power System Simulator 13

The air density at the altitude is calculated assuming the standard average sea-level air density as 1.225 kg/m3 which decreases exponentially with the altitude.

3.3.2 Tether model

The tether is modeled as a particle system with a ﬁxed number of point masses1 connected with spring-damper elements2. The point masses representing the tether are equally placed on the tether creating n number of segments. Figure 3.4 shows the particle system representation of the tether with 6 segments, where PKCU represents last tether particle. The reel-out and reel-in of the tether is also simulated by varying the tether length as a function of tether reel-out speed. The forces acting on a tether particle are consisting of the weight of that particle, the spring-damper forces exerted on that particle by adjacent segments and the half of the aerodynamic drag of the adjacent segments.

Figure 3.4: Particle system representation of the tether (points P0 to PKCU) and the four point kite (points A to D), Pc does not represent a particle, taken from [10]

3.3.3 Kite models

KiteSim framework has two diﬀerent kite models representing the LEI kites that are used in the kite power system demonstrator of the research team: a point mass model and a particle system model which is a four-point model3. The point mass model is showed in Figure 3.5a. This is the simplest modeling approach where the kite is represented as one point mass placed at the end of the tether. All of the external forces acting on the kite are directly applied at the point mass; speciﬁcally these forces are lift, drag, side and gravitational forces. This model does not include the

1Point masses of a particle system model are also denoted as particles in this thesis 2Spring-damper elements are also denoted as segments in this thesis 3The particle system model that is used in [10] is denoted as four-point model in this thesis. 14 Literature Review

(a) Point mass model (b) Four-point model

Figure 3.5: Representation of the point mass and four-point model, from [10] rotational inertia of the kite, it only includes the three translational DOF. Thus, the accuracy of the model is not suﬃcient to use in the development of control algorithms. This model is used for the ﬂight path simulation and optimization, and for the initialization of the four-point model. The four-point model is the simplest particle system model that can include the rotational inertia of the kite in the simulation. It represents the kite with four discrete point masses (A, B, C and D) connected to each other by spring-damper elements as shown in Figure 3.5b. Total kite mass is distributed to the four particles using empirical equations. The kite geometry is parametrized using the kite width, kite height and bridle height of the LEI kite. Moreover, the positioning of the particles is done by empirical equations. This model attaches three aerodynamic surfaces to the top and side particles (B, C and D in Fig. 3.5b) for the aerodynamic model. The accuracy of the four-point model enables the development and optimization of control algorithms.

3.3.4 Winch model

The winch is modelled as assembly of an asynchronous generator, a gearbox and a drum. During the operation the tether can be reeled-in, reeled-out or braked. The winch is modelled by including the inertial dynamics of the winch and the torque-speed proﬁle of the generator.

3.3.5 Implementation

The diﬀerential equations of the particle system consisting of the tether and kite particles together are formulated as an implicit problem. Furthermore, the problem is extended to include the winch states and residuals. The obtained implicit problem is solved by the Radau5DAE solver of Assimulo simulation package. 3.4 Particle System Modelling 15

KiteSim presents a soft real-time capable model, written in a general purpose program- ming language with possibility to adapt for diﬀerent CWKP systems.

3.4 Particle System Modelling

The particle system modelling approach models the bodies as a finite number of point masses connected by spring-damper elements. Particles are defined as objects that have a mass, position and velocity. They react to the applied forces, but do not have spatial extent. A particle is represented by a point in space and it has the translational degrees of freedom in x, y and z directions. As being a point, it cannot have an orientation; thus, it does not have rotational inertia. However, multiple particles can be interconnected to form lines, areas and volumes leading to obtain rotational inertia as a system of particles. Figure 3.6 shows a particle, a line formed out of two particles and two triangles formed out of five particles. Here, the line has rotational inertia about y and z axes; it still does not have rotational inertia about the x axis. Likewise, the triangles have the rotational inertia about all axes.

Figure 3.6: Representation of a particle and the systems of particles formed by the particles along with local axis frames, from [37]

A particle system consists of the particles and the entities that apply forces to these particles. Knaap explains that a body can be subjected to the body and surface forces and these forces will result in internal forces [37]. Thus, in particle system modelling, the body is modeled by a number of particles connected by forces that represent the internal forces in the body. Hence, in a particle system where the particles are interconnected by spring-damper elements, these forces will be the spring-damper forces. The calculation of the spring-damper forces is done via Hooke’s Law and the damping force equation, using the positions and relative velocities of the connected particles. The motion of a particle system can be described using the Newton’s second law:

F(x, v, t) = Ax¨ (3.1) where A is the matrix of the particle masses, F is the vector function of the net force acting on the particles which is dependent on the states of the system (position vector x 16 Literature Review

and velocity vector v of the particles) and time. Introducing v = x˙ , the motion of the particle system can be written as a system of ﬁrst-order diﬀerential equations as given below, from [37].

d x d x v = = (3.2) dt ˙x dt v A−1F(x, v, t)

This system of differential equations can be numerically integrated in time to solve and simulate the motion, using implicit and/or explicit methods for the numerical integration. The dynamic behavior of a rigid body can be accurately represented by its particle system model [37]. However, there are some concerns that are needed to be taken into account when representing a rigid element in a particle system. The first concern is the rigid body rotation and translation. Rigid bodies can be exposed to forces which are applied anywhere on them. When a force lies on a line which crosses the center of gravity of the rigid body, this force will cause the translation of the rigid body as seen on the left in the Figure 3.7. When a pair of parallel forces of the same magnitude but in opposite direction is present, these forces will cause the rotation of the rigid body as seen on the right in the Figure 3.7. Therefore, the representation of the loads applied to a rigid body can easily be determined by finding the net force through the center of gravity and the net moment which can be represented by two opposing forces as stated.

Figure 3.7: Representation of the external forces that cause the translation (left) and the rotation (right) on a rigid body, from [37]

The second concern is the representation of the center of gravity, the mass moments of inertia and the products of inertia of the rigid element. The particle system model should have equivalent mass properties along the local axis frame as the real body in order to represent the physics of the real body.

3.5 Component Buildup Method

In the component buildup method, the aircraft is divided into the discrete components representing the basic elements of the aircraft such as wings, horizontal tails, vertical tails, fuselage, propellers. The component forces and moments are locally calculated at 3.5 Component Buildup Method 17

each component and further the total forces and moments acting on the aircraft center of gravity (cg) are obtained using the contribution of each component. This method is widely used in aircraft modeling and simulation, with some examples seen in [31], [35], [16]. Selig models the full-envelope aerodynamics of the small UAVs for real-time simulations using a component buildup approach in [31], where the aircraft is split into wing, horizontal tail, vertical tail, fuselage and propeller components. For each component, a different model is developed to calculate the component forces and moments. The component forces and moments are calculated using the lookup tables that cover the full flight envelope of the aircraft. The components are superposed and the component forces and moments are summed to obtain the total forces and moments acting on the aircraft cg; while doing this the interaction effects between the components are included. The local apparent flow at each component is determined using aircraft speed and rotation, wind speed, turbulence and the interference effects. For the wing and tail components, the aerodynamic coefficients and the induced angle of attack of the components are found using the lookup tables as a function of control surface deflection and angle of attack. Terink also uses a component buildup method in the simulation of tethered kiteplanes [35], the aerodynamic forces and moments of the components are locally calculated at the right wing, left wing, horizontal tail and vertical tail. Then, the contribution of these aerodynamic surfaces is used to calculate the total aerodynamic forces and moments at the kiteplane cg. The discretization can be seen in Figures 3.8 and 3.9.

Figure 3.8: The discretization of the kiteplane aerodynamic surfaces, from [35]

Figure 3.9: Illustration of lift and drag forces on the discrete elements of the kiteplane, from [36]

Khan and Nahon use the component buildup method for the dynamic modeling of an UAV with high maneuvering capabilities [16]. They state that the component buildup 18 Literature Review

method ensures that real aircraft phenomena such as induced roll or adverse yaw are directly included into the model, by calculating the aerodynamic forces and moments at the separate components instead of getting the total forces and moments of the aircraft as a whole. The aerodynamic forces are assumed to be acting at the aerodynamic center (ac) of the components; hence, the ac is chosen as the reference point for the calculations of the forces.

3.6 Aerodynamic Models for Dynamic Simulation

An aerodynamic model is required in order to calculate the aerodynamic loads acting on the particle system model representing the RW kite. The selection of the appropriate aerodynamic model can be critical for the dynamic simulation; as the complexity of the aerodynamic model increases, the required computing time also increases. The aerodynamic model that is used in KiteSim [10] consists of lift and drag coefficient curves that are functions of angle of attack as shown in Figure 3.10. These curves are obtained by using a model for unstalled and stalled airfoils and further modifications are made to better fit the curves for the LEI kites. In the simulation, the lift and drag coefficients are obtained as a function of angle of attack, which is calculated using the apparent wind velocity, steering settings and depowering settings. Further the aerodynamic lift and drag forces acting on the model are calculated, the aerodynamic moment is not taken into account. The control of the current LEI kite is being done with the depowering and steering lines. The former is changing the angle of attack of the whole LEI kite resulting in the pitch control, the latter is creating an angle of attack differential between the left and right sides of the kite resulting in steering control. Hence, the control of the LEI kite is modeled with the angle of attack changes at the aerodynamic surfaces.

Figure 3.10: Lift and drag coeﬃcients versus angle of attack curves for LEI kites in KiteSim, from [10]

The control of the RW kite is done with the control surfaces; conventionally with ailerons, elevator and rudder for rolling, pitching and yawing. Thus, the control surfaces should also be included in the aerodynamic model to represent the eﬀect of the control surface de- ﬂections on the aerodynamics of the RW kite. Considering these aspects, a more detailed 3.6 Aerodynamic Models for Dynamic Simulation 19

aerodynamic model should be used for the RW kite simulations. Terink uses a strip theory aerodynamic model for the determination of the aerodynamic loads on a kiteplane [35], the aerodynamic forces and moments are locally calculated at the right wing, left wing, horizontal tail and vertical tail. Further, the contribution of these aerodynamic surfaces is used to calculate the aerodynamic force and moment vector of the kiteplane. The flight of a RW kite shows strong nonlinearities due to the large flight envelope. These nonlinearities should be included to a dynamic simulation in order to have realistic results. Abdallah et al. investigate the nonlinearity of an aircraft simulation model in [2], where the model uses approximately 50 lookup tables for aerodynamic and propulsive data for a wide range of angle of attack, sideslip angle, throttle deflection and control surface deflections. Thus, the nonlinearities in the aerodynamic and propulsive coefficients are directly represented in the simulation by means of these lookup tables. Selig models the full-envelope aerodynamics of the small UAVs for real-time simulations using a component buildup approach in [31], where the methods used to model the wing, fuselage and tail surfaces of conventional aircraft are presented in detail. In the component buildup method used in the simulation, the component forces and moments are calculated using the lookup tables that covers the full flight envelope of the aircraft. Further, these components are superposed and the component forces and moments are summed to obtain the total forces and moments acting on the aircraft cg; while doing this any interaction effects between the components are included. Selig states that the component buildup method is a regular method that is used in a wide range of modeling and simulation platforms. In the method, the aircraft is split into wing, horizontal tail, vertical tail and propeller. For each component, a different model is developed to calculate the component forces and moments. The local apparent flow is determined using aircraft speed and rotation, wind speed, turbulence and the interference effects. Selig states that the use of traditional stability derivative methods is inconvenient for the nonlinear aerodynamic behavior. For the RW kite model that will be developed for KiteSim, lookup tables can be used to calculate of the aerodynamic forces and the moments. Lookup tables can be generated using many different sources such as flight test data, wind tunnel measurements, analytical calculations, Computational Fluid Dynamics (CFD) results, empirical methods [31]. Sequeira et al. investigate different aerodynamic models (stability derivatives, strip theory, vortex lattice method and panel method models) for dynamic simulation of aircraft in [32]. The strip theory, the vortex lattice method (VLM) and the panel method as aerodynamic models are compared as follows. In the strip theory, the geometry of the aircraft is divided into discrete segments, and the aerodynamic loads are calculated on each of these segments. The total aerodynamic loads are calculated by summing the loads on each segment. In [32], Sequeira et al. state that the strip theory can be employed as a rapid development and behavioral estimation tool for different aircraft geometries, and any aerodynamic surfaces can be specified by this method. The model in the article implements rectangular wings with symmetrical airfoils and does not include the skin friction drag, downwash and wake modeling. The model calculates the local angle of attack and the local sideslip angle at the aerodynamic center of the each segment. Then, by using a reduced thin-airfoil theory, it calculates the lift and drag coefficients of the 20 Literature Review

segments. Further, the lift and drag of each segments are computed. Finally, these forces are accumulated at the center of gravity of the aircraft to get the total aerodynamic load of the aircraft at the cg. In VLM [32], the wing bound vorticity is represented by a lattice of constant dipole panels. Using the vortex ring elements, the radiation condition is satisfied. Moreover, the strengths of the vortex rings are satisfying the ”no normal flow through the mean surface” condition. The vorticity is represented by use of vortex wake filaments in a wake sheet lattice. The strength of the wake sheet satisfies the zero spanwise vorticity Kutta condition at the trailing edge. This method calculates the loads using the vortex strengths and the free stream velocity. The method does not include the induced drag. Sequeira et al. mention also about several drawbacks of this method. The use of a simple quasi-steady flat sheet wake model is a source of error. Also, the usage of a low order ring vortex model causes slow convergence with an increased panel discretization. The lack of body thickness can introduce additional errors. The panel method that is used in [32] is the FastAero panel method which achieves a fast solution for the unsteady potential flow around the bodies with thickness. Sequeira et al. mention that with the traditional panel method wake models, the interaction of the wake with the downstream surfaces creates problems and further states that the usage of FastAero eliminates these problems. Gohl and Luchsinger use an in-house developed VLM aerodynamic model in [12]. They state that this method calculates the aerodynamic forces fast and therefore real time dynamic simulation is possible. The VLM is used to calculate the lift and the induced drag. The viscous drag is calculated by using XFoil. In a CWKP system, a kite will be exposed to a large range of angle of attack during the operation. Hence, the simulation must include a post-stall aerodynamic data for the kite. Obtaining the post-stall aerodynamic data is not possible using linear aerodynamic models. Empirical methods, CFD analyses and experiments are some ways to obtain the post-stall aerodynamic data. Bianchini et al. investigate different models for the extrapolation of the post-stall lift and drag coefficient in [4]. They compared Viterna-Corrigan, Montgomerie, Beans&Jakubowski, Kirke and AERODAS models with CFD simulations and experiments. One of the findings from the comparison is that the lift coefficient has a good agreement in the models for the deep-stall region where the flow tend to be independent from the profile shape and the airfoil tend to act like a thin flat plate. They conclude that Kirke, Montgomerie and AERODAS models provide a more accurate extrapolation of post-stall data than the others. Chapter 4

Research Objective

The KiteSim framework developed by the TU Delft KitePower research team has the capability to simulate the LEI kites using the point mass model and the four-point particle system model. As mentioned in Section 2.2, the CWKP systems can have different designs; flexible, rigid or hybrid wings. The LEI kites are categorized under the flexible wing designs. RW kite designs are investigated and used by several companies and academic groups in the world. The current lack of simulation capability to simulate a RW kite in the KiteSim framework gives birth to the research objective and the fundamental research question of this thesis which are therefore formed as:

Enhance the KiteSim framework to include the capability of the dynamic simulation of a rigid wing pumping crosswind kite power system by implementing a particle system model to represent the tethered rigid wing kite.

Can a rigid wing kite be accurately modelled using the particle system modelling approach?

In reaching this research objective, two main research questions and the following sub- questions are needed to be answered:

1. How can a particle system model for a tethered rigid wing kite be added to the current modelling and simulation framework?

(a) How can the current four-point model for the LEI kites be modiﬁed to implement rigid wing kite dynamics? i. Is the four-point model a suﬃcient particle system model for representing a rigid wing kite or should the number of points representing the kite be increased? ii. How will the aerodynamic surfaces and the parts of the rigid wing kite be represented in the particle system model?

21 22 Research Objective

iii. How will the structure of the particle system model be created? iv. What is the eﬀect of the tether connection point(s)? (b) Which aerodynamic model for the calculation of the aerodynamic forces and moments should be chosen? i. Can the strip theory, vortex lattice method and/or panel method be used? ii. What are the available algorithms for implementing these models? iii. How can the post-stall aerodynamics be included? iv. How can the aerodynamic model be implemented to the simulation? v. How can the control surfaces be modelled?

2. What can be a validation process for the implemented rigid wing particle system models?

(a) Can a previously designed rigid wing kite be simulated? (b) Are there any previous tethered rigid wing kite data for the validation? (c) Can ﬂight test data of a similar aircraft be used to validate the rigid wing kite model?

The following sub-goals are needed to be successfully completed in order to reach the research objective stated earlier.

Getting familiar with the current modeling and simulation framework • Modeling the rigid wing kite as a particle system model • Using the appropriate aerodynamic model for the aerodynamic surfaces • Implementing the particle system model of the rigid wing kite to the framework • Validating the particle system model of the rigid wing kite • Evaluating the dynamic simulation of a rigid wing kite design • Chapter 5

Methodology of the Simulation Modelling

5.1 Rigid Wing Particle System Model Structure

The particle system (PS) modelling approach can represent the geometric shape and the mass property of any object if the structure of the PS model is formed accordingly. The PS model is created by the discretization of the particles and the formation of the spring- damper elements. The rigid wing PS model contains 6 discrete particles connected to each other by 13 spring-damper elements as shown in Figure 5.1. The chosen PS model structure presupposes the following limitations and assumptions on the RW kites that are to be simulated.

The RW kite has a conventional conﬁguration; with a wing and a conventional tail • (horizontal tail, vertical tail)

The fuselage is considered as a very thin rod and therefore negligible. • The RW kite has a single line tether and the tether connection point is at the cg • location of the kite.

The plane of symmetry of the RW kite is xBzB plane; Ixy = Iyz = 0 • The wings of the RW have small dihedral angles, and the horizontal tail has zero • dihedral angle.

The control surfaces of the RW are ailerons, elevator and rudder which are extended • full span of the wings, HT and VT.

23 24 Methodology of the Simulation Modelling

m0 m1 m5

m3 m4

Figure 5.1: Representation of the PS model on a conventional RW kite, with 6 discrete point masses and 13 spring-damper elements

5.1.1 Discretization of the particles

The number of the discrete point masses representing the RW kite in the PS model is chosen considering the tether connection point on the RW kite and the representation of the aerodynamic forces and moments acting on the RW kite. The tether force is an external force and in a PS model an external force can be applied only to the point masses of the model. Thus, the PS model must have a point mass at the location of the tether connection point of the RW kite. In CWKP systems, the tether connection point is generally chosen at the cg of the kite or very close to the cg. Hence, for the model, a point mass is placed at the cg location of the RW, shown as m0 particle in the Fig. 5.1. The aerodynamic forces (lift and drag forces) are also external forces and must be applied only to the point masses. The aerodynamic surfaces of the RW kite are broken down as the right wing, left wing, horizontal tail and vertical tail. Each aerodynamic surface has an associated point mass and the aerodynamic forces that are generated by that surface is to be applied to that associated point mass. The positions of these associated point masses are chosen as the mean aerodynamic center of the corresponding aerodynamic surfaces. m2, m3, m4 and m5 particles in Fig. 5.1 show the particles representing the right wing, left wing, horizontal tail and vertical tail respectively, and they are located at the mean aerodynamic centers of the aerodynamic surfaces. The aerodynamic moments are also external loading and needed to be applied only to the point masses. However, in particle dynamics, particles do not have rotational degrees of freedom; therefore, the aerodynamic moments cannot be directly applied to the particles. The aerodynamic moments are applied with an alternative method that is explained later in the report, in Subsection 5.6.3. In order to apply this method, m1 particle is placed at the mirror symmetry of the m4 particle with respect to the yBzB plane of the RW kite. Considering these aspects, PS model of the RW kite is consisting of 6 particles and the simulation requires the positions and the masses of these 6 particles. The calculation of the positions and the masses of the particles is done via an initialization script (RW init v2.py). The initialization script requires an input ﬁle containing the mass and geometric properties of the RW kite, an example of the input ﬁle can be seen in AppendixA which contains the data for AP2 RW kite of the Ampyx Power. Geometric 5.1 Rigid Wing Particle System Model Structure 25

input values that are span (b), taper ratio (λ), root chord (croot), dihedral angle (Γ), leading edge sweep angle (ΛLE), the positions of the root leading edge points of the aerodynamic surfaces and the location of the cg of the kite are used to determine the positions of the particles. Input mass properties are the kite mass (mkite), cg position, mass moments of inertia (Ixx,Iyy,Izz) and products of inertia (Ixy,Ixz,Iyz). They are used to determine the masses of the particles once the particle positions are calculated. The mean aerodynamic chord (¯c) and the spanwise location of the mean aerodynamic chord (Y¯ ) are calculated according to Equation 5.1a and 5.1b. They are geometrical parameters of the kite required for the calculation of the particle positions. The equation for the spanwise location of the mean aerodynamic chord is done with the assumption that the lift is proportional to chord [24]; moreover, for the vertical tail, the span of the vertical tail is considered twice of the vertical tail height; bVT = 2 hVT.

2 1 + λ + λ2 c¯ = croot (5.1a) 3 1 + λ b 1 + 2λ Y¯ = (5.1b) 6 1 + λ

The input positions of the root leading edge (LE) points and the cg of the RW kite are given in the aircraft station coordinate system. The origin of the Fuselage Station (FS) is at the nose and the positive direction is to the back. The origin of the Buttock Line (BL) is at the centerline and the positive direction is to the right. The origin of the Water Line (WL) is at the cg level and the positive direction is upwards. The particle positions are calculated in the body-ﬁxed reference frame as given below.

x0 = 0, y0 = 0, z0 = 0 (5.2a)

x2 = x3 = FScg FSwing,LE + Y¯wing tan Λwing,LE + 0.25c ¯wing (5.2b) − y2 = y3 = BLwing,LE + Y¯wing cos Γwing BLcg (5.2c) − − z2 = z3 = WLcg WLwing,LE + Y¯wing sin Γwing (5.2d) − x4 = FScg FSHT,LE + Y¯HT tan ΛHT,LE + 0.25c ¯HT (5.2e) −

y4 = BLHT,LE BLcg = 0 (5.2f) −

z4 = WLcg WLHT,LE (5.2g) − x5 = FScg FSVT,LE + Y¯VT tan ΛVT,LE + 0.25c ¯VT (5.2h) −

y5 = BLVT,LE BLcg = 0 (5.2i) − 26 Methodology of the Simulation Modelling

z5 = WLcg WLVT,LE + Y¯VT (5.2j) − Once the positions of the particles are determined, the masses of the particles need to be calculated in a way that the particle system has the same mass properties as the RW kite. Thus, the Eqns 5.3a-5.3j need to be satisﬁed.

5 X mi = mkite (5.3a) i=0 5 5 X 2 2 X mi(yi + zi ) = Ixx (5.3b) mi(xiyi) = Ixy (5.3e) i=0 i=0 5 5 X 2 2 X mi(xi + zi ) = Iyy (5.3c) mi(xizi) = Ixz (5.3f) i=0 i=0 5 5 X 2 2 X mi(xi + yi ) = Izz (5.3d) mi(yizi) = Iyz (5.3g) i=0 i=0

5 5 5 P P P mi xi mi yi mi zi i=0 i=0 i=0 5 = xcg (5.3h) 5 = ycg (5.3i) 5 = zcg (5.3j) P P P mi mi mi i=0 i=0 i=0

The distribution of the masses is done by forming an optimization problem; so that the total mass, the mass moments of inertia, the products of inertia and the cg location of the PS model matches the RW kite. The objective function J that is minimized by the optimization algorithm is given in Equation 5.4. In the objective function, all the terms with the (rw) subscript are related to the RW kite; their values are constant and given in the input ﬁle. The terms with the (psm) subscript are related to the PS model of the RW kite. Because the positions of the particles are pre-determined and ﬁxed, the values of the PS model terms are only dependent on the masses of the particles. The masses of the particles in the PS model create the optimization vector of the problem. The variables of the optimization vector and the initial guesses, lower bounds and upper bounds for the variables are shown in Table 5.1.

2 2 2 mkite mkite Ixx Ixx Iyy Iyy J = rw − psm + rw − psm + rw − psm mkiterw Ixxrw Iyyrw 2 2 Izzrw Izzpsm 2 Ixzrw Ixzpsm 2 (5.4) + − + Ixyrw Ixypsm + − + Iyzrw Iyzpsm Izzrw − Ixzrw − 2 2 2 +(xcg xcg ) + (ycg ycg ) + (zcg zcg ) rw − psm rw − psm rw − psm The optimization problem is solved using the Basin Hopping algorithm which is a global optimization algorithm and available in the SciPy library [15]. The set parameters for 5.1 Rigid Wing Particle System Model Structure 27

Nr. Optimization Variable Unit Initial Guess Lower Bound Upper Bound

1 m0 kg mkiterw /6 mkiterw /100 mkiterw

2 m1 kg mkiterw /6 mkiterw /100 mkiterw

3 m2 kg mkiterw /6 mkiterw /100 mkiterw

4 m3 kg mkiterw /6 mkiterw /100 mkiterw

5 m4 kg mkiterw /6 mkiterw /100 mkiterw

6 m5 kg mkiterw /6 mkiterw /100 mkiterw

Table 5.1: Initial guesses, lower bounds and upper bounds of the optimization variables in the optimization problem the Basin Hopping algorithm are given in Table 5.2. The optimization ﬁnds the masses of each particle satisfying the global minimum for the objective function in the given bounds with the given optimization parameters.

Parameter Description of the Parameter Value func Function to be optimized See Eqn. 5.4 x0 Initial guess See Table 5.1 niter Number of Basin Hopping iterations 100 Temperature parameter for the accept or re- T 1.0 ject criterion Initial step size for use in the random dis- stepsize 0.5 placement interval Interval for how often to update the stepsize 50 Extra keywords argument to be passed to the minimizer kwargs method, bounds minimizer method The minimization method, type of solver L-BFGS-B bounds Bounds for variables See Table 5.1

Table 5.2: Basin Hopping algorithm parameters used in the optimization

Particle mass [kg] xB [m] yB [m] zB [m]

m0 4.218 0.000 0.000 0.000

m1 2.713 1.908 0.000 0.000

m2 11.429 0.042 1.283 0.000

m3 11.429 0.042 -1.283 0.000

m4 2.416 -1.908 0.000 0.000

m5 0.807 -1.903 0.000 -0.306

Table 5.3: The masses and positions of the AP2 RW kite particles in PS model 28 Methodology of the Simulation Modelling

An example input ﬁle for the initialization script given in AppendixA, contains the data for the Ampyx Power AP2 RW kite. In Table 5.3, the mass distribution and the positions of the particles for the PS model of the AP2 RW kite is shown. Moreover, Table 5.4 shows the comparison of the mass properties and the cg location of the AP2 RW kite and its obtained PS model, using the optimization algorithm.

AP2 Rigid Wing Kite PS Model of AP2 Percent Error (%)

mkite 33.010 kg 33.012 kg 0.006 2 2 Ixx 37.750 kgm 37.727 kgm -0.061 2 2 Iyy 21.730 kgm 21.722 kgm -0.037 2 2 Izz 59.240 kgm 59.298 kgm 0.098 2 2 Ixy 0.000 kgm 0.000 kgm n/a 2 2 Ixz 0.470 kgm 0.470 kgm 0.0 2 2 Iyz 0.000 kgm 0.000 kgm n/a

xcg 0.0 mm 0.0 mm n/a

ycg 0.0 mm 0.0 mm n/a

zcg 0.0 mm -7.5 mm n/a

Table 5.4: Comparison of the mass properties and the cg location of the AP2 RW kite and its PS model

0.35 mk5

0.30

0.25

z 0.20

y 0.15

0.10 mk2 m 0.05 k4 mk0 0.00 mk1 m 0.05 k3 3 1.5 1.0 2 0.5 1 0.0 0 y 1 y bo 0.5 x bod dy 1.0 2 1.5 3

Figure 5.2: RW kite particles of the PS model for the AP2 RW kite 5.2 Rigid Wing Aerodynamic Model 29

5.1.2 Formation of the spring-damper elements

The spring-damper elements of the PS model for the RW kite are required to be initialized for the simulation. In order to do this first, the way that the particle are connected to each other is chosen. Second, the spring constants and damping coefficients of the spring- damper elements are chosen. A limitation of the PS model requires the conventional configuration for the RW. There- fore, all of the particles, except the particle corresponding to the vertical tail, are lying approximately in the same horizontal level as seen in Figure 5.2. It shows the six particles of the simulation of the AP2 RW kite model with the mass and geometric properties given in Table 5.3. Hence, the choice of selecting the particles m0, m1, m2, m3 and m4 as the particles forming a horizontal base and the particle m5 as an out-of-the-horizontal-base particle is made. Thus, the shape of the PS model is consisting of the four oblique pyramids having different triangular bases but the same apex. Figure 5.3 shows the top and side views of the PS model, where the bases of the four pyramids can be seen from the top view as ∆m0m1m2, ∆m0m1m3, ∆m0m2m4 and ∆m0m3m4. Moreover, the apex of the pyramids can be seen from the side view which is the particle m5. According to this, all of the edges of the pyramids represent the spring-damper elements of the PS model. Hence, there exist 13 spring-damper elements which connect the six particles of the PS model of the RW kite. The spring constant and the damping coefficient of the RW spring-damper elements should be chosen in a way that the PS model behaves as a rigid structure throughout of the simulations. If the spring constants are too low, then the PS model deforms considerably under the loading of the external forces, the geometry of the model changes and the model cannot be assumed as rigid. If the damping coefficients are too low, severe oscillations occur in the PS model which affect the rigidity as well. A way to check the rigidity is to evaluate the mass properties of the PS model throughout the simulations. If the changes of the cg position, moments of inertia and products of inertia of the PS model are kept within a small margin during the simulation, then the PS model can be assumed and considered as a rigid structure. The selection of the spring constant and the damping coefficients is done by trying different coefficients for the spring-damper elements in the dynamic simulation and evaluating the simulation results for the rigidity and the speed of the simulation. The spring constant of 5 107 N/m and the damping coefficient of 100 Ns/m are selected for the spring-damper × elements forming the PS model of the RW.

5.2 Rigid Wing Aerodynamic Model

The PS model requires an aerodynamic model for the calculation of the aerodynamic forces and moments acting on the RW kite during the simulation. The aerodynamic model of the RW kite is responsible for the determination of the aerodynamic coefficients of the discrete aerodynamic surfaces as a function of angle of attack (AoA). Given the limitations in Section 5.1, the discrete aerodynamic surfaces are the right wing, left wing, horizontal tail and vertical tail. The lift coefficient (CL), drag coefficient (CD) and moment coefficient (CM) of these surfaces are obtained from the aerodynamic model. They are 30 Methodology of the Simulation Modelling

m3 m2 m0

m1 m4 m0 m2, m3

Figure 5.3: The particles and the spring-damper elements of the PS model, a generic RW kite is drawn for comparison further used to calculate the aerodynamic forces acting on the associated RW particles of the PS model. The aerodynamic model is created using the XFLR5 software and Kirke’s post-stall correlation method, for the full AoA envelope of the RW kite. The aerodynamic model of the RW kite is created and used under the following limitations and the assumptions.

The wings have small dihedral and sweep angles. • The aerodynamic coefficients (CL, CD and CM) are a function of AoA only. • The Reynolds number effects on the aerodynamic coefficients are neglected. • Interaction effects between wing, tail and fuselage are neglected. • 5.2 Rigid Wing Aerodynamic Model 31

5.2.1 XFLR5 software

The XFLR5 software is an analysis tool for airfoils and wings in low Reynolds number operations. It includes XFoil analysis capabilities as well as three dimensional (3D) wing design and analysis capabilities based on lifting line theory (LLT), VLM and 3D panel method. The aerodynamic coeﬃcients of the four discrete aerodynamic surfaces, which are the right wing, the left wing, the horizontal tail and the vertical tail surfaces, are calculated using the XFoil Direct Analysis and Wing & Plane Design tools of the software. The XFoil Direct Analysis tool is used to get the two dimensional (2D) aerodynamic analysis of the airfoils that are used in the aerodynamic surfaces of the RW kite. The analysis parameters that are used for the AP2 RW kite are shown in Table 5.5.

Airfoils: FDD14 (used in wing) and NACA0012 (used in HT & VT) Analysis type: Type 1 Reynolds number min: 2 105 max: 3 106 increment: 4 105 (FDD14) × × × range: min: 1 105 max: 1.5 106 increment: 2 105 (NACA0012) × × × Mach number: M = 0.0 Laminar to turbulent n = 9.0 (for en method) transition criterion Forced transition locations on top & bottom (x/c)top = 1.0 (at the trailing edge) (x/c) = 1.0 (at the trailing edge) surfaces: bottom AoA range: min: -50◦ max: 50◦ increment: 0.5◦

Table 5.5: XFoil Direct Analysis parameters for the AP2 RW kite

Once the 2D polars are obtained using the Xfoil Direct Analysis tool, the Wing & Plane Design tool is used to get the 3D analysis of the discrete aerodynamic surfaces which are the right wing, the left wing, the HT and the VT. The Wing & Plane Design tool has three diﬀerent method: the LLT, the VLM and the 3D panel methods. The LLT method is proposed as the preferred method in the manual of the XFLR5 software [1], as long as the geometry of the surface is consistent with the limitations of the theory. According to the limitations of the method; the surfaces should not have low aspect ratio, should not have large sweep angle, and should have a low dihedral angle. The LLT method is used to get the aerodynamic data of the surfaces because it gives better viscous drag evaluation, has better estimation at high AoAs around stall and is better supported by theoretical published work [1], compared to the VLM and 3D panel methods. The wing, HT and VT are discretely analysed by using the geometry of the AP2 RW kite, as described in AppendixB. All the analyses are done in wing-only calculation. The surfaces are modelled symmetrically, with the xz plane being the plane of symmetry for the surfaces. The model of the wing simply contains both right and left wing; therefore, the obtained 3D aerodynamic coeﬃcients are valid for both the right and the left wing. The HT is modeled as it is; whereas, the model of the VT is a symmetric surface containing the symmetry of the original VT. 32 Methodology of the Simulation Modelling

The 3D analyses are conducted with the following analysis properties, along with the advanced settings shown at Table 5.6.

Polar type is Type 1 which is the ﬁxed speed analysis; the free stream speed is • chosen as 35 m/s for the AP2 RW kite. Analysis method is LLT. • Moment reference location is at the quarter mean aerodynamic chord. • Reference area for the calculation of the aerodynamic coeﬃcients is the planform • area. Air properties are chosen as: an air density of 1.225 kg/m3 and a kinematic viscosity • of 1.5 10−5 m2/s. ×

Advanced Settings Wing HT and VT Number of Spanwise Stations 15 5 Relaxation Factor 20 20 Alpha Precision 0.01 0.01 Maximum Iterations 300 300

Table 5.6: Wing & Plane Design tool advanced settings for wing, HT and VT of the AP2 RW kite

LLT analyses are done for AoA range from -25◦ to 25◦ with an increment of 0.5◦ for wing, HT and VT surfaces. Figure 5.4 shows the 3D aerodynamic coeﬃcients (CL, CD, CM) versus AoA curves obtained from XFLR5.

5.2.2 Post-stall correlation method

The XFLR5 software cannot give an estimation for the aerodynamic coeﬃcients at very high AoAs for the post-stall regime. Thus, Kirke’s Correlation method is used as a post- stall correlation method in the aerodynamic model. Kirke suggests the Equations 5.5-5.6, as an approximation for the 2D lift and drag coeﬃcients at very high and very low AoAs [17], where ξ is the percent camber of the airfoil.

Cl = (1 + 0.05 ξ) sin 2α (5.5)

3 Cd = (0.9 + 0.025 ξ)(1.5 sin α + 0.5 sin α + 0.05 ξ) (5.6)

Kirke states that there is a rule of thumb that the aerodynamic center of an airfoil shifts ◦ from 25% to 50% of the chord as the stall develops. Also, at AoAs near 90 , Cl has smaller value than Cd. Thus, the Cm is dominated by Cd and the equation 5.7 can be used as an approximation for the moment coeﬃcient at very high AoAs.

Cm = 0.25 Cd (5.7) − 5.2 Rigid Wing Aerodynamic Model 33

(a) CL-AoA (b) CD-AoA

Figure 5.4: 3D aerodynamic coeﬃcients versus AoA obtained from XFLR5 software with red line showing wing, green line showing HT and blue line showing VT

Kirke suggests a correction factor to include the effect of finite AR in order to obtain the 3D post-stall aerodynamic coefficient approximations. AppendixC shows the plots for CL and CD correction factors for the finite AR. The correction factor equations in 5.8 and 5.9 are obtained by curve fitting to the these plots in order to calculate the correction factor as a function of AR.

2 κl = 0.0018 AR + 0.0564 AR + 0.3965 (5.8) −

2 κd = 0.001 AR + 0.036 AR + 0.4548 (5.9) − 34 Methodology of the Simulation Modelling

Finally, the 3D post-stall aerodynamic coeﬃcients are obtained using the correction factors as below.

CL = κl Cl (5.10)

CD = κd Cd (5.11)

CM = 0.25 CD (5.12) −

5.2.3 Aerodynamic data approximation script

Using both the XFLR5 software outputs and the Kirke Correlation method, a Python script is written to create the aerodynamic model of the PS model in KiteSim, shown in AppendixD. The XFLR5 analyses are used for the AoA range from -25 ◦ to 25◦ and Kirke Correlation is used from -180◦ to -30◦ and from 30◦ to 180◦. The AeroInterpolation function is written where a third degree interpolating spline is fitted to the data set of the aerodynamic coefficients of each surface using the InterpolatedUnivariateSpline class in the Interpolation sub-package of the SciPy library. The splines are further used to create aerodynamic coefficient matrices corresponding to the higher number of (N=1024) AoA values which are equally spaced between -180◦ to +180◦. Figure 5.5 shows the fitted splines of the aerodynamic coefficients of the wing, HT and VT. The obtained high resolution (N=1024) matrices for aerodynamic coefficients are used in the aerodata wing, aerodata ht and aerodata vt functions for the calculation of the CL, CD and CM as a function of AoA.

5.3 Rigid Wing Controller Implementation

A Logitech Extreme 3D Pro Joystick and a Futaba InterLink Elite Controller are added to the KiteSim for the manual control of the RW kite control surfaces which are ailerons, elevator and rudder. Either the joystick or the controller can be used for the RW kite simulations. The controllers give the control input as a value between -1.0 and +1.0 for each control surface with the common aircraft control surface sign conventions shown in Table 5.7. Each control surface is modelled to operate between the maximum and minimum deﬂection angles given in the input ﬁle, neglecting the actuator dynamics and delays.

Deﬂection Sense Primary Eﬀect Ailerons right-wing trailing edge down positive negative rolling moment Elevator trailing edge down positive negative pitching moment Rudder trailing edge left positive negative yawing moment

Table 5.7: Control Surface Sign Convention, from [34] 5.4 Atmospheric Model 35

Figure 5.5: Aerodynamic coeﬃcients as a function of AoA

5.4 Atmospheric Model

The simulation requires an atmospheric model to determine the wind speed at different kite and tether altitudes which is required for the calculation of the aerodynamic forces of the tether and the RW kite models. KiteSim is using the power law, the logarithmic law or a combination of both wind profiles for the simulation of the LEI kites [10]; these wind profiles are also used in the simulation of the RW kite.

Using the given reference altitude zref , reference wind speed vw,ref and the exponent α, the wind speed at the altitude can be calculated by the power law as shown in Equation 5.13. The logarithmic law given in Equation 5.14 requires the given reference altitude, reference wind speed and the surface roughness length z0 for the calculation. In KiteSim, the reference wind speed at the reference altitude is given at the altitude of zref =6 m for all of the wind proﬁle calculations. α is taken as 1/7 and z0 as 0.07 for power law and logarithmic law wind proﬁles respectively.

z α vw,exp = vw,ref (5.13) zref

log (z/z0) vw,log = vw,ref (5.14) log (zref /z0) 36 Methodology of the Simulation Modelling

The power and logarithmic law wind profiles can be further combined as shown in Equa- tion 5.15, in order to fit the wind profile to the measured wind speeds at three different altitudes from the tests [10]. The fit is achieved by altering the surface roughness z0 and K values, which yields for z0=0.0002 m and K=1.0 values. Accordingly, the exponent α value is chosen so that the wind speed calculated at the second altitude from power law and from logarithmic law will be equal. An example for the wind profiles obtained from the power and logarithmic laws are shown in Figure 5.6, along with the fitted wind profile which is obtained as a linear combination of the power and logarithmic law profiles.

vw = vw,log + K(vw,log vw,exp) (5.15) −

Figure 5.6: Power law, logarithmic law and fitted wind profiles along with three measured wind speeds at different altitudes, from [10]

The calculation of the air density at altitude is done with the formula below, where the 3 average sea-level air density ρ0 is assumed as 1.225 kg/m and Hρ value is 8.55 km.

z ρ = ρ0 exp − (5.16) Hρ

5.5 Tether Particle System Model

The tether model in KiteSim represents the single line tether that connects the Kite Control Unit of the LEI kite to the ground station [10]. The RW kite simulations in this thesis involve a single line tether conﬁguration that connects the cg of the RW kite with the ground station. Therefore, the current PS model for the tether is also used in the RW kite simulations. The PS model of the tether consists of discrete particles which are connected to each other by n number of spring-damper elements as shown as in Figure 5.7. The tpn particle represents the last particle of the tether which is connected to the cg of the RW kite; thus, the tpn particle of the tether and the m0 particle of the RW kite are merged into one 5.5 Tether Particle System Model 37

particle in the PS model. The particles are equally spaced on the tether; hence, the length of each spring-damper segment is equal. The mass of each segment is equally distributed to the particles at the ends of the segments. The tether length continuously changes in CWKP applications; the reel-out and reel-in of the tether results in the change of the tether length [10]. The tether model uses a constant number of particles throughout the simulation. Therefore, the masses of the particles and the lengths of the spring-damper segments discretizing the tether change during the simulation as shown in Eqns 5.17 to 5.19, with vt,ro as the tether reel-out speed, ti and lt,i the simulation time and tether th length at the beginning of the i time step and ls the segment length, moreover mtp being the mass of a tether particle and ρt the tether mass density.

lt = lt,i + vt,ro(t ti) (5.17) −

lt ls = (5.18) n

 2 ls ρt π(dt/2) for tp1 to tpn−1  mtp = (5.19) 2 ls ρt π(dt/2)  for tp0 and tpn 2

tpn (connected to RW cg location)

tpn−1

tpn−2

tp3

tp2 tp1 tp0 (connected to ground station)

Figure 5.7: Discrete tether particles (points tp0 to tpn) of the tether model

Corresponding to the change of the segment length, the spring constant and the damping coeﬃcient of each segment change with the below formula where ks0 and kd0 are the initial spring constant and damping coeﬃcient and ls0 is the initial segment length.

ls0 ks = ks0 (5.20) ls

ls0 kd = kd0 (5.21) ls 38 Methodology of the Simulation Modelling

In the PS model of the tether, each tether particle is exposed to the gravitational, spring- damper and aerodynamic drag forces. Gravitational forces are applied directly to the particles in the direction of the gravitational acceleration and calculated as below.

Fg = mtp g (5.22)

The tether particles are connected to each other by the spring-damper elements. As two particles are moving with respect to each other, the spring-damper segment connecting these two particles generates the spring-damper forces. These forces are applied to these two particles that are at the both ends of the segment. Figure 5.8 shows the spring-damper forces Fsd acting on two particles of a segment. These forces are calculated according to Hooke’s Law and damping force equation as follows.

s s Fsd = ks( s ls) + kd sv (5.23) k k − s · s k k k k where s = pi pi+1 (5.24) − and sv = vi vi+1 (5.25) −

vi+1 Fsd − tpi+1

vi tpi zW pi+1

sv Fsd yW pi

Figure 5.8: Representation of the position and velocity vectors along with the segment vector and segment relative velocity for a segment formed by tpi and tpi+1 particles

The spring constant ks that is used in the Eqn. 5.23 depends on the state of the segment: If the segment is on tension, the spring constant calculated in Eqn. 5.20 is directly used. 5.6 Rigid Wing Particle System Model 39

If the segment is on compression, the calculated spring constant is decreased in order to model the behavior of the ﬂexible tether line [10].

Spring Constant

tension ks

compression 0.1 ks

Table 5.8: Tether spring stiﬀness in tension and compression

The aerodynamic drag of the tether is a distributed load along the length of the tether. However, in the PS modelling approach, the forces can only be applied to the point masses. Therefore, the drag of the tether is modeled as discrete drag forces acting on the tether particles as follows: the drag of each segment is calculated and divided equally to each particles at both ends of the segment.

1 2 ρ va,s,⊥ s dt CD 2 k k k k t va,s,⊥ Fd = (5.26) 2 va,s,⊥ k k with vi + vi+1 vs = (5.27) 2 and

va,s = vw,s vs (5.28) −

In above equations, the center points of the segments are used as reference points; the air density ρ, segment velocity vs, wind velocity at the segment altitude vw,s and the apparent wind velocity of the segment va,s are obtained at these center points. The air density and the wind velocity at the segment altitude are obtained using the atmospheric model explained in Section 5.4. The drag of the tether is governed by the component of the apparent wind velocity that is perpendicular to the segment; thus, va,s,⊥ is used in k k the Eqn. 5.26.

5.6 Rigid Wing Particle System Model

The RW particles of the PS model are exposed to the gravitational, spring-damper and aerodynamic forces. This section presents the method used for the calculation of these forces.

The wind reference frame (FW) is used as the inertial reference frame in the KiteSim, for which the origin is at the ground station at the cable exit point. The xW axis points straight downwind, the yW axis points to the left looking downwind and the zW axis is upwards. All the vectors in KiteSim are represented in FW, as shown in the Figure 5.9. 40 Methodology of the Simulation Modelling

5.6.1 Calculation of the gravitational forces

The masses of the discrete particles m0 to m5 representing the RW kite are obtained as explained in Subsection 5.1.1. The gravitational forces acting on the particles in the direction of the gravitational acceleration are obtained as below.

Fg = mi g (5.29)

5.6.2 Calculation of the spring-damper forces

The particles of the PS model of the RW kite are connected to each other via the spring- damper elements which are formed as explained in Subsection 5.1.2. As in the tether model, a spring-damper element generates forces which are exerted to the particles at both ends of the segments. These forces are also calculated with Hooke’s Law and damping force equation, shown in Equation 5.30 where ls represents the initial length of the spring- damper element which connects two RW particles. The segment lengths of the RW kite are not changing significantly during the simulation. Therefore, unlike the tether model, the values of the spring constant ks and damping coefficient kd for the RW kite segments are initial values throughout the simulation. Moreover, the spring stiffness for the RW segments is the same during tension and compression.

vi+1

vi F sd mi+1 s mi − Fsd

pi+1 sv p zW i

Figure 5.9: Representation of the position and velocity vectors along with the segment vector and segment relative velocity for a segment formed by mi and mi+1 particles

s s Fsd = ks( s ls) + kd sv (5.30) k k − s · s k k k k with s = pi pi+1 (5.31) − and sv = vi vi+1 (5.32) − 5.6 Rigid Wing Particle System Model 41

5.6.3 Calculation of the aerodynamic forces

The six points (m0 to m5) of the PS model representing the RW kite are shown in Figure 5.10. The body-ﬁxed reference frame (FB) is used to deﬁne the kite reference frame. The origin of FB is the cg of the RW kite, which corresponds to the position of the m0 particle. The xB axis lies in the symmetry plane of the RW kite, positive points out the nose of the RW. The yB axis is perpendicular to the symmetry plane, positive points out the right wing. The zB axis is in the symmetry plane of the RW kite, perpendicular to the xB axis, positive is below the kite. The unit vectors of the FB axes are calculated as follows.

pm1 pm4 exB = − (5.33) pm pm k 1 − 4 k pm2 pm3 eyB = − (5.34) pm pm k 2 − 3 k

ez = ex ey (5.35) B B × B

Lm2

Dm2 m2

yB Lm3 m0 Lm D xB 4 m5 m5 m1 Dm3 Lm5 va zB Dm4 m3 m4

Figure 5.10: Representation of the RW particles with the lift and drag forces (L and D) acting on the particles m2, m3, m4 and m5; with apparent wind velocity va

The aerodynamic forces and moments acting on the PS model are applied to the particles associated with the right wing, left wing, HT and VT. Thus, the aerodynamic forces and moments are calculated at the location of the particles m2, m3, m4 and m5. As the particles do not have rotational DOF, the aerodynamic moments of the discrete aerodynamic surfaces cannot be directly applied to the associated particles. In order to represent the aerodynamic moments, a pair of induced forces is introduced for each aerodynamic surface that are parallel to each other and have the same magnitudes but the opposite directions. The discretization of the PS model is such that the particle m1 is the mirror symmetry of the particle m4 with respect to the yBzB plane; therefore, the aerodynamic moments generated at the aerodynamic surfaces are represented as induced forces applied to the particles m1 and m4. 42 Methodology of the Simulation Modelling

For the calculation of the aerodynamic forces and moments of the right and left wing surfaces, two wing axes systems (x2y2z2 and x3y3z3) are created at the right and left wing particle locations, in order to include the dihedral eﬀects of the wing in the calculations. These wing axes systems are created by rotating the body-ﬁxed axes system around the xB axis by the amount of the dihedral angle, the creation of these axes systems is shown in AppendixE. In order to calculate the angle of attack of each aerodynamic surface, the apparent wind velocities are projected to the x2z2, x3z3, xBzB and xByB planes, respectively for right wing, left wing, HT and VT.

va,m ,x z = va,m (va,m ey ) ey (5.36) 2 2 2 2 − 2 · 2 2

va,m ,x z = va,m (va,m ey ) ey (5.37) 3 3 3 3 − 3 · 3 3

va,m ,x z = va,m (va,m ey ) ey (5.38) 4 B B 4 − 4 · B B

va,m ,x y = va,m (va,m ez ) ez (5.39) 5 B B 5 − 5 · B B

The apparent wind velocities are calculated for each particle by using the particle velocity and the wind velocity at the height of the RW kite as follows:

va,i = vw vi (5.40) − The AoA of each aerodynamic surface1 is calculated as follows:

va,m2,x2z2 αm2 = π arccos ex2 (5.41) − va,m ,x z · k 2 2 2 k va,m3,x3z3 αm3 = π arccos ex3 (5.42) − va,m ,x z · k 3 3 3 k va,m4,xBzB αm4 = π arccos exB (5.43) − va,m ,x z · k 4 B B k va,m5,xByB βm5 = π arccos exB (5.44) − va,m ,x y · k 5 B B k The RW aerodynamic model shown in Section 5.2 requires the AoAs of the aerodynamic surfaces to obtain the corresponding aerodynamic coefficients. However, the AoAs calculated by the Equations 5.41- 5.44 are the AoAs to which the particles are exposed. The incidence angles and the control surface deflections of the aerodynamic surfaces change the effective AoAs of the surfaces in the model. The incidence angle of the wing (iw)

1The angle shown by β in Figure 5.11 is denoted as ’sideslip angle’ in this thesis and it is used as ’angle of attack’ for the vertical surfaces. 5.6 Rigid Wing Particle System Model 43

xy plane y

m x va,x β va,y va,xy va,z α va,xz va

z xz plane

Figure 5.11: Apparent wind velocity and its components for the calculation of angle of attack α and sideslip angle β

and the HT (iHT) should be added in order to obtain the effective AoAs of the wing and HT particles. Moreover, the control surfaces (ailerons, elevator and rudder) are included by a simple method taken from [16], where the control surface deflections are accounted by a change in the AoA of the corresponding aerodynamic surface. The method replaces the aerodynamic surface by an equivalent flat plate, then calculates the effect of control surface deflection to the AoA of the aerodynamic surface as shown in Figure 5.12.

c ccs ccs − x αcs

αcs δcs ccs α ceﬀ va,xz

z Figure 5.12: Equivalent flat plate method used in the calculation of the control surface deflection effect on the AoA

p 2 2 ceﬀ = (c ccs) + (ccs) 2(c ccs)(ccs) cos(π δcs) (5.45) − − − − 44 Methodology of the Simulation Modelling

ccs sin δcs αcs = arcsin (5.46) ceﬀ

The eﬀective AoAs that are used to obtain the aerodynamic coeﬃcients are calculated as below for each of the aerodynamic surfaces:

αeﬀ,m2 = αm2 + αcs,m2 + iw (5.47)

αeﬀ,m3 = αm3 + αcs,m3 + iw (5.48)

αeﬀ,m4 = αm4 + αcs,m4 + iHT (5.49)

βeﬀ,m5 = βm5 + βcs,m5 (5.50)

Furthermore, the aerodynamic forces and moments are calculated as follows.

1 2 va,m2 ey2 Lm2 = ρ va,m2,x2z2 Sm2 CL(αeﬀ,m2 ) × (5.51) 2 va,m ey k 2 × 2 k

1 2 va,m3 ey3 Lm3 = ρ va,m3,x3z3 Sm3 CL(αeff,m3 ) × (5.52) 2 va,m ey k 3 × 3 k 1 v e L 2 a,m4 yB m4 = ρ va,m4,xB zB Sm4 CL(αeff,m4 ) × (5.53) 2 va,m ey k 4 × B k 1 v e L 2 a,m5 zB m5 = ρ va,m5,xB yB Sm5 CL(βeff,m5 ) × (5.54) 2 va,m ez k 5 × B k

1 2 va,m2 Dm2 = ρ va,m2 Sm2 CD(αeﬀ,m2 ) (5.55) 2 va,m k 2 k

1 2 va,m3 Dm3 = ρ va,m3 Sm3 CD(αeﬀ,m3 ) (5.56) 2 va,m k 3 k

1 2 va,m4 Dm4 = ρ va,m4 Sm4 CD(αeﬀ,m4 ) (5.57) 2 va,m k 4 k

1 2 va,m5 Dm5 = ρ va,m5 Sm5 CD(βeﬀ,m5 ) (5.58) 2 va,m k 5 k 1 2 Mm = ρ v Sm CM(α )c ¯m cos(Γm ) (5.59) 2 2 a,m2,x2z2 2 eﬀ,m2 2 2

1 2 Mm = ρ v Sm CM(α )c ¯m cos(Γm ) (5.60) 3 2 a,m3,x3z3 3 eﬀ,m3 3 3

1 2 Mm = ρ v Sm CM(α )c ¯m (5.61) 4 2 a,m4,xB zB 4 eﬀ,m4 4 5.6 Rigid Wing Particle System Model 45

1 2 Mm = ρ v Sm CM(β )c ¯m (5.62) 5 2 a,m5,xB yB 5 eff,m5 5 As mentioned previously the aerodynamic moment generated at an aerodynamic surface is represented by a pair of induced forces which is applied to the particles m1 and m4 as shown in Figure 5.13. The aerodynamic moments of the right wing, left wing and HT contribute as pitching moments to the RW in the body-fixed reference frame; therefore, the induced force pairs due to these aerodynamic moments apply to the particles m1 and m4 in the direction of zB axis. Similarly, the aerodynamic moment of the VT contributes as yawing moment; thus, the induced force pair applies in the direction of yB axis. The dihedral angle of the wings reduces the pitching moment contribution of the right and left wing aerodynamic moments, cosine terms in the Equations 5.59 and 5.60 establish this reduction. The yawing moment contribution coming from the dihedral angle of the right and left wings is neglected due to two facts: first, the design dihedral angles are not chosen large so the yawing moment contribution is small; second, the symmetric design of the RW ensures the counteraction of the yawing moment contribution from the right wing and left wing.

Mm2

Fi,m5 yB m m0 1 xB m5 F Mm3 Mm5 i,m2 Mm4 m4 va z Fi,m3 B m3 Fi,m5 Fi,m−2 Fi,m4 − Fi,m − 3 Fi,m − 4 Figure 5.13: The aerodynamic moments, shown with M, and aerodynamic moment induced forces, shown with Fi, explained on the PS model of the RW kite

Ultimately, the induced forces are calculated as shown in Equations 5.63-5.66, so that the moment caused by the induced forces is equal to the aerodynamic moment at the cg of the RW.

Mm2 Fi,m2 = ezB (5.63) − pm pm k 4 − 1 k

Mm3 Fi,m3 = ezB (5.64) − pm pm k 4 − 1 k

Mm4 Fi,m4 = ezB (5.65) − pm pm k 4 − 1 k 46 Methodology of the Simulation Modelling

Mm5 Fi,m5 = eyB (5.66) − pm pm k 4 − 1 k 5.7 Equation of Motion of the Particle System Model

The motion of the particle system is obtained according to Newtonian dynamics. The PS model of the tethered RW kite consists of the tether and RW particles. Each particle has a mass, a position, a velocity and a net force applying on it. The coupled ﬁrst-order ordinary diﬀerential equations of the PS model which are explained in Section 3.4 are written as shown below:

d p d p v = = (5.67) dt ˙p dt v A−1F(p, v, t) with A is the N-by-N diagonal matrix of the particle masses, F is the N-by-3 matrix of the net forces acting on the particles which are dependent on the N-by-3 matrices of the system states (position p and velocity v) of the particles and time t, where N is the number of particles in the PS model. These ﬁrst-order ordinary diﬀerential equations of the PS model can be written for N number of particles of the PS model as shown in Eqn. 5.68 below, where the forces F acting on each particle are calculated as shown in Section 5.5 and 5.6.

    p1,x p1,y p1,z v1,x v1,y v1,z  p2,x p2,y p2,z   v2,x v2,y v2,z       . . .   . . .   . . .   . . .      d pN,x pN,y pN,z  vN,x vN,y vN,z    =   (5.68) dt  v1,x v1,y v1,z   F1,x/m1 F1,y/m1 F1,z/m1       v v v   F /m F /m F /m   2,x 2,y 2,z   2,x 2 2,y 2 2,z 2   . . .   . . .   . . .   . . .  vN,x vN,y vN,z FN,x/mN FN,y/mN FN,z/mN

It should be noted that first particle of the PS model is the first tether particle, which is positioned at the origin of the wind reference frame and which is a stationary particle. Therefore, the force that is exerted to the first particle does not contribute to the motion of that particle. However, the force is used in the calculation of the tether acceleration by the winch model as explained in [10].

5.8 Formulation of the Implicit Problem

The ordinary diﬀerential equations obtained in the EOM of the PS model are solved using implicit numerical methods by formulating an implicit problem in the form as shown in Eqn. 5.69. The problem is formulated by the residual function R and the initial values of the time, states and state derivatives. The residual function R takes time t, states vector 5.9 Simulation Implementation 47

Y and state derivatives vector Y˙ as inputs. The residual function returns a zero matrix if the input values are consistent. Therefore, the initial values should be consistent as well in order not to experience problems with the implicit method.

R(t, Y, Y˙ ) = 0, Y(t0) = Y0, Y˙ (t0) = Y˙ 0 (5.69)

The state vector Y of the PS model is comprised of the positions and the velocities of the particles. Thus, the state derivatives vector Y˙ is formed by the position derivatives and the velocity derivatives of the particles.

Y = (p, v) (5.70)

Y˙ = (p˙ , v˙ ) (5.71)

As the set of ﬁrst-order ordinary diﬀerential equations of the PS model is given in Eqn. 5.68, the residual function R of the implicit problem for the PS model can be written as follows.

    p1,x p1,y p1,z v1,x v1,y v1,z  p2,x p2,y p2,z   v2,x v2,y v2,z       . . .   . . .   . . .   . . .      d pN,x pN,y pN,z  vN,x vN,y vN,z  R(t, Y, Y˙ ) =     (5.72) dt  v1,x v1,y v1,z  −  F1,x/m1 F1,y/m1 F1,z/m1       v v v   F /m F /m F /m   2,x 2,y 2,z   2,x 2 2,y 2 2,z 2   . . .   . . .   . . .   . . .  vN,x vN,y vN,z FN,x/mN FN,y/mN FN,z/mN

As mentioned before; due to the fact that the ﬁrst particle is a stationary particle, the residual of the ﬁrst particle is not calculated as shown in the Equation 5.72, but is directly given a zero value. The winch model used in the KiteSim includes the tether length and the tether reel-out velocity as the states of the implicit problem. Thus; the states, state derivatives and the residual matrix of the implicit problem is extended as explained in [10].

5.9 Simulation Implementation

5.9.1 Numeric solver

The implicit problem is solved by the Radau5DAE solver [13] of the Assimulo simulation package [3]. The solver is an implicit ﬁfth-order Runge-Kutta method with three stages and using variable step sizes. It is chosen considering that KiteSim uses the same solver for the simulation of the LEI kites; Fechner et al. state that Radau5DAE oﬀers the best performance [10]. The solver parameters used in the simulation are given in Table 5.9 48 Methodology of the Simulation Modelling

Solver Parameters Parameter Explanation Value inith initial step-size 0.003 thet Jacobian recalculation habit 0.09 fac1 step-size selection parameter 0.15 fac2 step-size selection parameter 10.0 quot1 step-size selection parameter 1.0 quot2 step-size selection parameter 1.5 atol (positions) absolute tolerances 1.8 10−2 × atol (velocities) absolute tolerances 3 10−4 × rtol relative tolerance 1 10−3 × Table 5.9: Radau5DAE solver parameters used in the simulation

5.9.2 Initial and simulation parameters

The simulation for the tethered RW kite is initiated with the initial parameters and the simulation parameters shown in the Table 5.10. The initialization is done in RW init v2.py and Settings.py scripts of the KiteSim, where the initial and simulation parameters required by other parts of the KiteSim are set. The particle discretization and the spring- damper elements formation are done in the the RW init v2.py script as explained in Subsections 5.1.1 and 5.1.2.

Initial Parameters Value Initial elevation angle 60 deg Initial total tether length 150 m Initial reel-out speed 0 m/s Initial azimuth angle 0 deg Initial kite velocity 0 m/s Simulation Parameters Value Number of tether segments, n 6 Simulation frequency, f 50 s−1 Simulation speed, relative to real time 1.0

Table 5.10: Initial and simulation parameters Chapter 6

Model Validation

In the development of computer simulation models, validation of the simulation model is required in order to check the accuracy of the model and to acquire an understanding on the model’s representation of the real system. The validation is usually done by comparing the simulation results with experimental results; in the case of this thesis, by comparing the simulation results with CWKP system flight tests. However, due to the lack of experimental testing for RW CWKP systems, the validation of the PS model is done by using the flight test results of a similar aircraft for which the experimental data is available. This chapter presents the validation method of the PS model and the related findings.

Figure 6.1: Modiﬁed Schweizer SGS 1-36 sailplane used by NASA

49 50 Model Validation

6.1 Validation Aircraft

The aircraft used for the validation of the PS model is Schweizer SGS 1-36 Sprite sailplane, Figure 6.1. A modified version of this sailplane is used for deep stall research in NASA and the flight tests data are published in [19] and [33]. The general data and the mass properties of the Schweizer SGS 1-36 are obtained from [6], [19] and [33]; these are shown in Appendix F.2. There is some minor discrepancy in the data obtained from different sources. Therefore, the data used in the simulation is chosen by considering all of the sources and in case of discrepancy by using educated guesses. Also, in order to comply with the limitations of the PS model structure, the T-tail configuration of the SGS 1-36 sailplane is changed to conventional tail. The input file that is created for the simulation of the SGS 1-36 sailplane is given at Appendix F.1. Moreover, XFLR5 software models of the wing, horizontal tail and vertical tail for the SGS 1-36 are given in Appendix F.3.

6.2 Validation Methodology

6.2.1 Simulation of non-tethered ﬂight in KiteSim

Schweizer SGS 1-36 Sailplane test results are for the free flight results; therefore, there is not any tether attached to the sailplane during the experiments. In order to be able to generate the free flight simulation cases in the KiteSim framework, the tether diameter is −6 decreased to dt=1.0 10 meters. This tether diameter results in having very low tether × particle masses, tether spring constant and tether damping coefficient for the PS model of the tether. Therefore, the forces acting on the tether PS model become very small, that results in having very small tether forces applying from tether PS model to the RW PS model. The maximum tether force that is obtained during the validation simulations is 0.042 N. Thus, the tether can be assumed negligible and the free flight simulation cases can be obtained for the RW PS model. The validation cases are conducted with an initial total tether length of 500 meters.

6.2.2 Fuselage drag estimation

The fuselage of the RW kites is neglected in the development of the PS model. This limitation of the model can be applied for unmanned RW kites with thin fuselages; however, SGS 1-36 sailplane is a manned sailplane with a large fuselage. Hence, the fuselage affects the aerodynamic performance of the sailplane. In order to include the effect of fuselage in the simulation cases, the fuselage drag is determined using the method from [24], with the assumption that the flight is low-subsonic and the flow over the fuselage is turbulent. The fuselage drag Df is exerted to the m0 particle of the RW PS model, and is calculated as shown in Equation 6.1. The wetted area of the fuselage Swet,f , the turbulent flat-plate skin friction drag coefficient Cf,t and the fuselage form factor FF are calculated as given in Equations 6.2 to 6.6 using the Raymer’s method from [24].

1 2 va,m0 Df = ρ va,m0 Swet,f Cf,t FF (6.1) 2 va,m k 0 k 6.3 Validation Results and Discussions 51

Atop + A S = 3.4 side (6.2) wet,f 2 0.455 Cf,t = 2.58 (6.3) (log Ref ) 60 f FF = 1 + + (6.4) f 3 400 where va,m l Re = 0 f (6.5) f ν and l f = f (6.6) df

Table 6.1 shows the values of the parameters that are used for the fuselage drag calculation and the obtained wetted area, turbulent flat-plate skin friction coefficient and the form factor values. Cf,t is a function of Reynolds number, so it depends on the flight conditions via the apparent wind speed and the kinematic viscosity. However, in the validation case a constant Cf,t is used, which is calculated for an apparent wind speed of 30 m/s and kinematic viscosity of 1.5 10−5 m2/s. × Parameters Description Value

lf Fuselage length 6.278 m

df Fuselage maximum diameter 0.991 m 2 Atop Fuselage top projected area 2.628 m 2 Aside Fuselage side projected area 3.342 m 6 Ref Fuselage Reynolds number 12.5 10 × FF Form Factor 1.252

Cf,t Turbulent ﬂat-plate skin friction coeﬃcient 0.0029 2 Swet,f Wetted area 10.149 m

Table 6.1: SGS 1-36 Sailplane fuselage drag calculation parameters

6.3 Validation Results and Discussions

Sim presents lift-to-drag ratio versus airspeed curve and sink rate versus horizontal velocity curve that is obtained from the flight tests of SGS 1-36 [33]. Several simulation cases are conducted with the SGS 1-36 model in order to compare the simulation results with the data presented from the flight tests. In these simulation cases, the simulation model is manually piloted until it reaches a satisfactory gliding flight. The corresponding L/D ratio, airspeed, sink rate and horizontal velocity of the PS model are obtained for each simulation case. These are collected over the time period of the gliding flight, then 52 Model Validation

the mean values are calculated. Figure 6.2 and 6.3 shows the plots for L/D versus airspeed and sink rate versus horizontal velocity. The dashed lines show the flight test data obtained from [33] and the diamonds are the KiteSim simulation results. The dash-dot line in Figure 6.2 represent the adjusted baseline SGS 1-36 sailplane data which are obtained from other flight tests by Johnson [14] prior to the modifications done by NASA. Sim relates the lower aerodynamic performance of the modified SGS 1-36 sailplane to the extensive modifications done to the original SGS 1-36 sailplane [33]. Figure 6.2 shows that the simulation results are overestimating the L/D ratio by approximately 10% for low airspeed and as the airspeed increases, overestimation increases up to approximately 32% compared to the NASA flight tests. When compared to the baseline data, the simulation results show a matching L/D ratio for low airspeed; and there is an overestimation which increases up to approximately 19% for high airspeed. Figure 6.3 shows that the maximum difference of the sink rate between the simulation results and the NASA flight test is an approximately 36% overestimation which occurs at low airspeed. It is also seen that at high airspeed, the sink rate obtained from the simulation shows an underestimation up to approximately 28%. The L/D ratio and the sink rate curves are coherent. This is expected, because the L/D ratio is the aerodynamic efficiency factor, and higher L/D ratio results in higher glide ratio in steady conditions. In other words, for the same horizontal velocity, higher L/D ratio provide a glide with lower sink rate. An important discussion about the L/D versus airspeed curve is the fuselage drag estimation. Although a fuselage drag estimation is implemented, the turbulent flat-plate skin friction coefficient Cf,t is calculated using Eqn. 6.3 for an apparent wind speed of 30 m/s and then Cf,t is assumed to be constant throughout the simulation. In fact, the turbulent flat-plate skin friction coefficient is a function of Reynolds number and it is decreasing with the increasing Reynolds number. Therefore, the choice of using a constant Cf,t which is obtained for 30 m/s apparent wind speed results in overestimating the fuselage drag for higher airspeed and underestimating for lower airspeed. Hence, it can be said that the simulation overestimates the L/D, especially at the higher airspeeds. The overestimation of the L/D can be related to the drag estimation by several aspects. First, it is given that the XFLR5 software methods tend to underestimate the drag [1]. Therefore, the drag of the aerodynamic surfaces of the model could have been underestimated. Second, the fuselage drag estimation method could give an underestimation for the fuselage drag. Third, the interference drag and the miscellaneous drag are not included in the simulation model, this could also give an underestimated drag. Last, the modifications on the SGS 1-36 sailplane that are mentioned by Mahdavi [19] employ a pushrod system, and a modified cockpit, canopy and nose cone which introduce additional drag. The additional drag coming from these modified parts are not included; hence, this leads for an underestimation of the drag. Additionally Sim presents time history plots for directional, lateral and longitudinal maneuvers that are performed in the flight tests [33]. These are performed by giving rudder, aileron and elevator inputs for the directional, lateral and longitudinal maneuvers respectively. The control surface inputs used in the flight tests are given by Sim, the same control inputs are generated in the KiteSim in order to obtain the time history plots in the simulation. In the simulation cases, the simulation model of the SGS 1-36 is manually piloted until it reaches a satisfactory gliding flight, then the control inputs are applied to 6.3 Validation Results and Discussions 53

Figure 6.2: L/D versus airspeed plot obtained from the ﬂight tests and the KiteSim

Figure 6.3: Sink rate versus horizontal velocity plot obtained from the flight tests and the KiteSim the model and the resultant time history for each maneuver is compared with the time history of the flight tests. The directional and lateral time histories are obtained consecutively in the flight tests [33]. Hence, they are also obtained consecutively in the KiteSim, by first giving the rudder control input following with the aileron control input in the simulation. The longitudinal time history is obtained by giving the elevator control input in the simulation. Time history of the directional and lateral maneuvers are shown in Figures 6.4 and 6.5. The aileron control input has a high directional coupling; whereas, the rudder control input has a slight lateral coupling. Time history of the longitudinal maneuver is shown in Figure 6.6. The elevator control input does not have a coupling in other axes.

The rudder control input δr results in the yaw body rate r and the yaw body rate derivative r˙ responses as given in Figure 6.4. The KiteSim simulation results are matching with the ﬂight test results in terms of the magnitude of the response for the yaw body rate and the yaw body rate derivative. However, simulation results have an oscillatory response 54 Model Validation

where the flight test result have a damped response. This is anticipated to be related to the fact that fuselage directional effects are not included. Generally, the fuselage has a destabilizing yaw moment [21]; hence, the directional response is more oscillatory in the absence of the fuselage. The coupling of the rudder input into the lateral axis can be seen in Figure 6.5; the coupling is satisfactory in terms of magnitude of the roll body rate and the roll body rate derivative, however the simulation results have an oscillatory behaviour. This oscillatory behaviour results in the difference in the roll angle response in comparison with the flight test.

The aileron control input δa results in the roll body rate p, the roll body rate derivative p˙ and the roll angle Φ responses as given in Figure 6.5. KiteSim results have lower magnitude of the roll body rate in comparison to the ﬂight test results. This is expected due to the fact that the PS model of RW kite has full span ailerons which is a limitation of the model as given in Section 5.1. Although the SGS 1-36 sailplane and the PS model have the same aileron area, the rolling moment generated in the SGS 1-36 sailplane is higher for a same aileron control input. Because the ailerons in the SGS 1-36 sailplane are located towards the tips of the span, hence have a longer moment arm compared to the PS model. Roll body rate derivative is similarly lower in magnitude in the simulation results. Furthermore, the roll angle Φ response shows that the roll angle change due the aileron input is also lower in the simulation, due to the same reason. The coupling of the aileron input into the directional axis can be clearly detected in Figure 6.4, the full span ailerons limitation is thought to be the reason for obtaining lower aileron coupling into the directional axis in the simulation.

The elevator control input δe results in the angle of attack α, pitch body rate q, the pitch body rate derivativeq ˙ and the pitch angle Θ responses as shown in Figure 6.6. The pitch body rate and the pitch body rate derivative responses obtained from the simulations are satisfactory in comparison with the flight tests. However the angle of attack and the pitch angle responses have an offset. This offset is anticipated to be due to the difference in the trim of the glider for the simulations and the flight tests. Although the longitudinal maneuver is performed at a starting α ∼= 6.1 degrees at the flight test, the gliding flight of the simulation model is reached at a starting α ∼= -0.5 degrees at the simulations. In the absence of experimental data of a CWKP system with a RW kite, this validation method is found sufficient for checking the accuracy of the simulation model. As a result of the validation simulations, the developed PS model is decided to be adequately accurate for representing a sailplane which is similar to a RW kite. 6.3 Validation Results and Discussions 55

Figure 6.4: Rudder input and directional time history

Figure 6.5: Aileron input and lateral time history 56 Model Validation

Figure 6.6: Elevator input and longitudinal time history Chapter 7

Simulation Results and Discussions

This chapter presents a variety of numerical simulations that are done using the developed RW PS model in the KiteSim framework. The input data for the simulated RW kite is coming from AP2 Powerplane of Ampyx Power, with the mass and geometry properties as shown in AppendixA andB. Winch model and winch controller parameters that are used in the simulation cases are shown in AppendixG.

7.1 Mass Properties Results

The rigidity of the PS model depends on the selection of the spring constant and the damping coefficient of the spring-damper elements as mentioned in Subsection 5.1.2. As suggested earlier, examination of the mass properties gives an understanding about the rigidity of the model throughout the simulation. The maximum speed of AP2 PowerPlane is given as 50 m/s in the flight envelope by Ampyx Power. Therefore, the rigidity of the PS model is examined for the kite beyond the maximum speed. Two different cases are simulated in order to check the mass properties of the PS model of AP2 PowerPlane.

7.1.1 Case 1: Tethered, non-maneuvering kite at very high atmospheric wind speed

For case 1; a tethered, non-maneuvering kite which is exposed to a high atmospheric wind speed is simulated. The 3D trajectory of the simulation can be seen in Figure 7.1, with the projections of the ﬂight path on xWyW, xWzW and yWzW planes. A reference wind speed of 35 m/s at the height of 6 m is set in the simulation. Although high wind speeds as such are not in the design wind speed range for the kite operations, this wind speed is set in order to get a high apparent wind speed on the PS model. The apparent wind speed plot can be seen in Figure 7.2; where the model is exposed to apparent wind speeds higher than 50 m/s during the simulation. Figure 7.3 shows the tether reel-out speed. The mass properties of the PS model throughout the simulation is given in Figure 7.4.

57 58 Simulation Results and Discussions

250

z 200

[m] 150

100

0 200 30 20 150 10 0 100 [m] yW [m] 10 50 xW − 20 − 30 0 − Figure 7.1: Flight trajectory; case 1: White and black circles represent start and end position, blue circles represent the position at every 5 s. Yellow triangle shows the ground station.

75 70 65

[m/s] 60 a

v 55 50 0 5 10 15 20 25 30 2016-07-23 13-42-59 Time [s]

Figure 7.2: Apparent wind speed versus time; case 1

12 10 8

[m/s] 6

ro 4 , t

v 2 0 0 5 10 15 20 25 30 2016-07-23 13-42-59 Time [s]

Figure 7.3: Reel-out speed versus time; case 1 7.1 Mass Properties Results 59

8 10− 6 3 × − 0.4 8 2 − 0.3 10 1 − 0.2 location [m] 12 location [mm] 0 location [mm] − cg cg cg y z x 0.1 14 1 − 0 5 10 15 20 25 30 − 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2016-07-23 13-42-59 2016-07-23 13-42-59 2016-07-23 13-42-59

(a) xcg vs time(s) (b) ycg vs time(s) (c) zcg vs time(s)

+3.772 101 +2.1722 101 +5.9298 101 × × × 0.022 0.006 0.008 0.020 0.007 ] ] ] 0.005 2 2 0.018 2 0.006 0.016 0.004 0.005 [kg m [kg m [kg m 0.004 0.014 0.003 zz xx yy I I 0.012 I 0.003 0.002 0.010 0.002 0.008 0.001 0.001 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30 2016-07-23 13-42-59 2016-07-23 13-42-59 2016-07-23 13-42-59

(d) Ixx vs time(s) (e) Iyy vs time(s) (f) Izz vs time(s)

10 8 4 × − 0.003 0.470 2 0.469 ] ] ] 2 0.002 2 2 0.468 0

[g m 0.467 [kg m 0.001 [kg m

xy 2 xz yz I

I 0.466 − I 0.000 0.465 4 − 0.464 6 0 5 10 15 20 25 30 0 5 10 15 20 25 30 − 0 5 10 15 20 25 30 2016-07-23 13-42-59 2016-07-23 13-42-59 2016-07-23 13-42-59

(g) Ixy vs time(s) (h) Ixz vs time(s) (i) Iyz vs time(s)

Figure 7.4: PS model mass properties versus time; case 1 60 Simulation Results and Discussions

Examining the mass properties results in Figure 7.4 (a), (c), (d), (e), (f) and (h); it is seen that there are oscillations on the mass properties which is damping out as the simulation continues. These oscillations are thought to be related to the oscillation of the apparent wind speed that can be seen in Figure 7.2, as the aerodynamic forces and moments are proportional to the square of the apparent wind speed. The oscillations at the beginning of the simulation are due to the initial condition of the simulation, where the kite and the tether are initiated as stationary at their initial location. The changes in the moments and products of inertia of the PS model during the simulation are less than 0.05%, except for the Ixz which has a 1.5% change. Moreover, the changes in the xcg and zcg are up to 0.4 mm and -7.5 mm respectively. It is important to realize that the kite is exposed to very high apparent wind speeds which are beyond the ﬂight envelope of the kite during the simulation case 1. Considering these results, it can be evaluated that the mass properties of the PS model are maintained accurately throughout the simulation where a high atmospheric wind speed exist.

7.1.2 Case 2: Tethered, maneuvering kite at high atmospheric wind speed

For case 2; a tethered, maneuvering kite which is flying a FoE trajectory for a reel-out phase of a pumping cycle operation is simulated. The 3D trajectory of the simulation case can be seen in Figure 7.5, with the flight paths projections on the xWyW and xWzW planes. The reference wind speed at the height of 6 m is set to 15 m/s in the simulation; this wind speed is in fact beyond the design wind speed and it is set in order to simulate very high apparent wind speeds on the kite during the FoE patterns. The apparent wind speed plot is shown in Figure 7.6. It can be seen that high peaks in the apparent wind speed are obtained during the simulation; where the model exceeds the airspeed of 50 m/s which is the maximum speed specified in the flight envelope. Tether reel-out speed is shown in Figure 7.7. The mass properties of the PS model throughout the simulation is given in Figure 7.8. The plots at Figure 7.8 show that the changes in the moments and products of inertia of the PS model during the simulation are less than 0.03%, except the Ixz, which has a 1.1% change. The changes in the xcg and zcg are up to 0.33 mm and -5.5 mm respectively. Also the trend of the change in the mass properties is again similar to the apparent wind speed seen at Figure 7.6, for which it is known that the aerodynamic loading is highly dependent on the apparent wind velocity as the aerodynamic forces and moments 2 are a function of va. The results show that during a crosswind FoE flight with a high atmospheric wind speed, where the model experiences fluctuating aerodynamic loading, the PS model represents the mass properties accurately throughout the simulation. 7.1 Mass Properties Results 61

300

250

W 200

[m] 150 100 50 0 350 300 100 250 50 200 y 150 W [m] 0 100 [m] 50 50 xW − 0

Figure 7.5: Flight trajectory; case 2: White and black circles represent start and end position, blue circles represent the position at every 5 s. Yellow triangle shows the ground station.

55 50 45

[m/s] 40

a 35 v 30 25 0 10 20 30 40 50 2016-07-23 14-05-15 Time [s]

Figure 7.6: Apparent wind speed versus time; case 2

14 12 10 8 [m/s] 6

ro 4 , t 2 v 0 0 10 20 30 40 50 2016-07-23 14-05-15 Time [s]

Figure 7.7: Reel-out speed versus time; case 2 62 Simulation Results and Discussions

5 − 0.35 0.015 6 − 0.30 7 0.010 − 0.25 8 0.005 − 0.20 9 0.000 − 0.15 10 − location [mm] location [mm] 0.005 location [mm] 11

cg 0.10 cg − cg − z y x 0.010 12 0.05 − − 0.015 13 0 10 20 30 40 50 − 0 10 20 30 40 50 − 0 10 20 30 40 50 2016-07-23 14-05-15 2016-07-23 14-05-15 2016-07-23 14-05-15

(a) xcg vs time (b) ycg vs time (c) zcg vs time

+3.772 101 +2.1722 101 +5.9298 101 × 0.005 × 0.007 × 0.018 0.006 0.004 ] ] 0.016 ] 2 2

2 0.005 0.014 0.003 0.004 [kg m [kg m 0.012 [kg m 0.003 zz xx yy I

I 0.002 0.010 I 0.002 0.008 0.001 0.001 0.000 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 2016-07-23 14-05-15 2016-07-23 14-05-15 2016-07-23 14-05-15

(d) Ixx vs time (e) Iyy vs time (f) Izz vs time

0.15 0.470 0.010 0.10 ] ] 2 0.469 ] 2 2 0.005 0.05 0.468 [g m [g m [kg m

yz 0.000 xy xz I I 0.00 0.467 I

0.05 0.466 0.005 − − 0.465 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 2016-07-23 14-05-15 2016-07-23 14-05-15 2016-07-23 14-05-15

(g) Ixy vs time (h) Ixz vs time (i) Iyz vs time

Figure 7.8: PS model mass properties versus time; case 2 7.2 Reel-Out Phase Results 63

7.2 Reel-Out Phase Results

The reel-out (traction) phase is the power generation phase for a CWKP system operating in a pumping cycle mode. During the reel-out phase, the kite is flying in a crosswind pattern while reeling the tether out of the drum of the ground station. The reel-out phase ends when the kite reaches a predefined tether length and/or altitude. Two simulation cases are conducted by manually flying FoE trajectories in order to examine the reel-out phase of the PS model of AP2 PowerPlane with two different atmospheric wind speeds.

7.2.1 Case 3: Tethered, reel-out, ﬁgure-of-eight trajectory ﬂight at strong wind

For case 3; a tethered, reeling-out phase is simulated by flying the PS model in FoE crosswind maneuvers, where the model is flown up at the corners of the FoE trajectory. A reference wind speed of 8.9 m/s at the height of 6 m is set in the simulation. The 3D trajectory of the simulation case can be seen in Figure 7.9, with the projections of the flight path on xWyW and xWzW planes. In the initial 10 seconds of the simulation, the transition of the model from the initial condition to the reel-out phase occurs. The reel-out phase continues until 100 seconds.

200

z 150

[m] 100

0 500 150 400 100 300 50 y 0 200 [m] W [m] 50 100 xW − 100 − 0 Figure 7.9: Flight trajectory; case 3: White and black circles represent start and end position, blue circles represent the position at every 5 s. Yellow triangle shows the ground station.

Time history of the PS model is plotted in Figure 7.10. During the reel-out, the apparent wind speed changes significantly between 23 m/s and 49 m/s. As the model is flying upwards at the corners of the FoE trajectory, it is slowing down. Similarly, the model is accelerating flying from one corner to the other. The x, y, z position of the model given in the FW shows that the kite is moving downwind and gradually gaining altitude during the 64 Simulation Results and Discussions

reel-out phase. The angle of attack αkite varies between -2.8 to 11.5 degrees, the sideslip angle βkite varies between -14.9 to 13.6 degrees. The roll-pitch-yaw body rates p, q, r are given along with the Tait-Bryan yaw-pitch-roll angles Ψ, Θ, Φ for the kite orientation which are obtained between FW and FB. The control surface deflections are shown. The turns of the kite are mostly done using the coordinated turns with the ailerons and the rudder. Also a nose-up elevator command is given during FoE maneuvers. The power output of the simulation is plotted in Figure 7.11 along with the other parameters that are related with the power output. The tether is reeled out approximately 310 meters. Tether reel-out speed has peaks and troughs during the reel-out phase, which are matching with the tether force. The tether force is varying so much during the operation, that is caused by the force parameters of the winch and winch controller. As the mechanical power is a function of the tether reel-out speed and the tether force, it also has the peaks and troughs during the reel-out. A maximum mechanical power peak of 44.5 kW is reached where the mean mechanical power output is 9.8 kW during the reel-out phase. This leads to obtain a maximum power harvesting factor ζ peak of 11.5 and a 2.3 mean power harvesting factor. Kite lift and drag coefficients CL,kite, CD,kite and the kite lift-to-drag ratio (L/D)kite are shown. Also the equivalent tether drag coefficient CD,t,eq and the lift-to-drag ratio of the kite and tether combined (L/D)kite+tether are plotted. Equivalent tether drag coefficient is obtained as in Equation 7.1, taken from [9]. The (L/D)kite+tether shows that the tether drag has an important factor on the overall lift-to- drag ratio. The (L/D)kite has a mean value of 23 where the (L/D)kite+tether has a value of 8 at the beginning of the reel-out and decreases drastically as the reel-out continues.

lt dt CD,t,eq = 0.32 CDt (7.1) Swing 7.2 Reel-Out Phase Results 65

50 40

[m/s] 30 a v 20 0 20 40 60 80 100 2016-07-23 14-56-32 500 200 400 150 100 [m] 300 [m] 50 W 200 W 0 x y 50 100 −100 − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32 250 100 ]

o 80 200 [ 60 [m] 150

W 40 z 100 elevation 20 β 50 0 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

80 15 ]

o 60 ]

[ 10 o 40 [ 5 20 kite 0 α azimuth 0 φ 20 5 − − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

15 100

] 10 o 50 [

5 /s] 0 o 0 kite p [ β 5 −10 50 −15 − − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32 100 200 50 ] 100 o /s] [ 0 o

Φ 0 q [ 100 50 − − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

80 100 60

] 40 50 o /s] [ 20 o 0

Θ 0 20 r [ 50 −40 −100 − − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

40 200 30

] 20 ]

100 o o [

[ 10

0 a 0 δ Ψ 10 100 −20 − −30 0 20 40 60 80 100 − 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

40 40 30 30 20 ] 20 ] o o [ 10 [ 10 r e 0 0 δ δ 10 10 −20 −20 −30 − − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32 Time [s] Time [s]

Figure 7.10: Time history; case 3 66 Simulation Results and Discussions

500 6000 450 5000 400 4000 350 [m]

3000 t 300 l 250

Force [N] 2000 1000 200 0 150 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

10 50 40 5 30 [m/s] 20 ro , t

v 0 10 Power [kW] 0 5 − 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

14 12 10 1.5 8 kite

, 1.0 ζ

6 L

4 C 2 0.5 0 0.0 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

0.12 35 0.10 30 25 0.08 kite ) kite , 20

D 0.06

C 15 L/D 0.04 ( 10 0.02 5 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

0.30 10 0.25 8 eq

, 6 t , 0.20 kite+tether D )

C 4 0.15 L/D

( 2 0.10 0 0 20 40 60 80 100 0 20 40 60 80 100 2016-07-23 14-56-32 2016-07-23 14-56-32

Figure 7.11: Simulation results for the power related parameters; case 3 7.2 Reel-Out Phase Results 67

7.2.2 Case 4: Tethered, reel-out, ﬁgure-of-eight trajectory ﬂight at moderate wind

For case 4; a tethered, reeling-out phase is simulated by flying the model in FoE crosswind maneuvers and flying up turns at the corners. A reference wind speed of 6.92 m/s at the height of 6 m is set in the simulation. The 3D trajectory of the simulation case can be seen in Figure 7.12, with the projections of the flight path on xWyW and xWzW planes. The transition of the kite model from the initial condition to the reel-out phase occurs in the first 10 seconds of the simulation. The reel-out phase continues until approximately 68 seconds.

120 100

z W 80

[m] 60 40 20 0 300 100 250 200 y 50 150 W [m] 0 100 [m] xW 50 50 − 0 Figure 7.12: Flight trajectory; case 4: White and black circles represent start and end position, blue circles represent the position at every 5 s. Yellow triangle shows the ground station.

Time history of the PS model is plotted in Figure 7.13. During the reel-out, the apparent wind speed changes between 22 m/s and 42 m/s. The kite is progressing downwind and ﬂying between 40 m and 115 m altitude. The angle of attack αkite varies between -3 to 10.4 degrees, the sideslip angle βkite varies between -11.3 to 13.7 degrees. The control surface deﬂections are shown. The turns of the kite are mostly done using the coordinated turns with ailerons and rudder. A nose-up elevator command is kept during reel-out. The power output of the simulation is plotted in Figure 7.14 along with the other parameters. The tether is reeled out approximately 170 meters. Tether reel-out speed, tether force and mechanical power shows peaks and troughs during the reel-out phase. A maximum mechanical power peak of 28.3 kW is reached where the mean mechanical power output is 7.6 kW during the reel-out phase. This leads to obtain a maximum power harvesting factor ζ peak of 17.7 and a 4.5 mean power harvesting factor. The (L/D)kite has a mean value of 20 where the (L/D)kite+tether has a value of 8 at the beginning of the reel-out and decreases as the reel-out continues. 68 Simulation Results and Discussions

45 40 35 30 [m/s] 25 a

v 20 15 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 350 300 150 250 100 [m] 200 [m] 50 W W x 150 y 0 100 50 − 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

140 70 ] o 120 [ 60 50 [m] 100 40 W 80 30 z elevation

60 β 20 10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

40 ] 15 o ] [

20 o [ 10 0 5 kite α

azimuth 20 0 φ −40 5 − − 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46 100 15

] 50

o 10 [

5 /s] o 0

kite 0 p [ β 5 50 − − 10 100 − 0 10 20 30 40 50 60 70 − 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46 100 200 80 60 ] 100 o /s] 40 [ 0 o 20 Φ

q [ 0 100 20 − −40 0 10 20 30 40 50 60 70 − 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

60 100 40

] 20 50 o /s] [ 0 o 0

Θ 20 −40 r [ 50 −60 − − 100 0 10 20 30 40 50 60 70 − 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

200 30 20 ] ]

100 o 10 o [ [ 0 0 a δ Ψ 10 100 −20 − −30 0 10 20 30 40 50 60 70 − 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

30 30 20 20 ] ] o 10 o 10 [ 0 [ 0 r e δ δ 10 10 −20 −20 −30 −30 − 0 10 20 30 40 50 60 70 − 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46 Time [s] Time [s]

Figure 7.13: Time history; case 4 7.2 Reel-Out Phase Results 69

350 4000 300 3000 250 [m] t

2000 l 200 Force [N] 1000 150 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

8 30 6 4 20

[m/s] 2

ro 10 ,

t 0 v

2 Power [kW] 0 −4 − 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

2.0 20 15 1.5 kite

10 , ζ 1.0 L

5 C 0.5 0 0.0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

0.16 35 0.14 30 0.12 25 kite

0.10 ) kite

, 20

D 0.08

C 15 L/D

0.06 ( 10 0.04 5 0.02 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

10 0.20 0.18 8 eq , 0.16 6 t , kite+tether D 0.14 )

C 4 0.12 L/D 0.10 ( 2 0.08 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 2016-07-23 15-33-46 2016-07-23 15-33-46

Figure 7.14: Simulation results for the power related parameters; case 4 70 Simulation Results and Discussions

7.2.3 Comparison of reel-out cases results and theoretical analysis

Schmehl et al. present a theoretical analysis of pumping kite power systems in [29]. They derive the tether force Ft formula which is given in Equation 7.2. Moreover, the traction power P is given as in Equation 7.3. They assume a massless kite in quasi-steady motion, a uniform and constant wind speed vw in the direction of xW, a straight inelastic tether and constant aerodynamic coeﬃcients. Detailed derivation procedure can be found in the reference. Relevant ﬁgures are included in AppendixH.

" 2# 1 2 L 2 Ft = ρ v Swing CR 1 + (sin θele cos φazi f) (7.2) 2 w D − and P = Ft vt,ro (7.3) where, q 2 2 CR = CL + CD (7.4)

◦ θele = 90 βele (7.5) −

vt,ro f = (7.6) vw

For the comparison of the reel-out cases and the theoretical analysis; theoretical tether force Ft,theo and traction power Ptheo are calculated using the averaged values of the variables on the right-hand sides of the Equations 7.2 and 7.3. These theoretical results are compared with the averaged tether force and power of the simulation cases. The averaged data are obtained from the simulation cases for a small time interval during which the PS model has an approximately horizontal flight state with a course angle χ ≈ 90◦ or 270◦. Definition of the course angle χ is shown in Figure H.1. It should be noted that for CD in Equation 7.4, the sum of kite drag coefficient CD,kite and equivalent tether drag coefficient CD,t,eq is used. Similarly, for (L/D) ratio in Equation 7.2,(L/D)kite+tether is used. Detailed averaged data obtained from the simulation cases 3 and 4 can be seen at Table H.1. There is a significant difference between the simulation and theoretical values for the tether force and traction power. Table H.2 shows that the tether force and traction power of the simulation cases are drastically lower then the theoretical calculations. The average percentage difference (δ¯%) for case 3 is -69%, for case 4 it is -78%. In fact, such a difference is anticipated. Because the theoretical calculations assume a massless kite; whereas the simulation cases include the mass. Further interpretation of the results is inessential; however, it is worth to mention that an iterative solution analysis is also presented in [29] and it combines the gravity effects to the quasi-steady analysis. This comparison could be highly interesting and promising results could be obtained. The iterative solution analysis is not performed in this thesis as it remains beyond the scope of this thesis. 7.3 Plausibility Checking Results 71

7.3 Plausibility Checking Results

Plausibility checking is a method to investigate a mathematical model by examining the coherence of certain phenomena that occur in the model, in order to see if these phenomena are in accordance with physical logic and experience [28]. In this respect, two simulation cases are conducted for plausibility checking of the PS model of AP2 PowerPlane performing a glide flight and a stall maneuver by manual piloting. During these plausibility checking cases, in order to be able to generate a non-tethered flight, the tether diameter is decreased to 1.0 10−6 meters. By doing so, very low tether × forces are obtained that apply to the cg of the PS model. The maximum tether force that is obtained during the gliding case is 0.145 N and during the stalling case is 0.004 N. Those tether forces applying to the cg are considered satisfactory for representing a non-tethered flight. For the simulation cases, a reference wind speed of 6.92 m/s at the height of 6 m is set.

7.3.1 Case 5: Non-tethered, gliding ﬂight

In the gliding flight case, PS model is manually piloted to have a gliding flight using elevator inputs. 3D trajectory of the simulation case can be seen in Figure 7.15. Due to the manual piloting, having a steady gliding flight is a difficult task. Therefore, the piloting is done using careful elevator commands and small unsteadiness during the gliding flight is tolerated. A satisfactory gliding flight is obtained between the simulation time of 55 and 85 seconds. Elevator deflection angle δe, apparent wind speed va, angle of attack αkite, pitch angle Θ, pitch body rate q, resultant aerodynamic force Lkite, weight Wkite, glide ratio and L/D ratio of the RW kite during the gliding phase are plotted in Figure 7.16.

120 100

z W 80

[m] 60 40 20 0 200 150 100 50 0 y 0 100 W [m] 50 300 200− − 100 400 −[m] − 150 500− −xW − 200 600− − − Figure 7.15: Flight trajectory; case 5: White and black circles represent start and end position, blue circles represent the position at every 5 s. 72 Simulation Results and Discussions

3.0 2.5 ] o [ 2.0 e δ 1.5 1.0 55 60 65 70 75 80 85 2016-08-27 15-29-48 18.0 2.0 −

17.9 ] 1.9 o 17.8 [ −1.8

[m/s] −

17.7 kite 1.7 a α v 17.6 −1.6 17.5 −1.5 55 60 65 70 75 80 85 − 55 60 65 70 75 80 85 2016-08-27 15-29-48 2016-08-27 15-29-48 4.0 0.4 3.8 0.2 ] o

3.6 /s] [ o 0.0 3.4 Θ

q [ 0.2 3.2 −0.4 3.0 − 55 60 65 70 75 80 85 55 60 65 70 75 80 85 2016-08-27 15-29-48 2016-08-27 15-29-48 330 36 Rkite [N] Kite glide ratio 328 Wkite [N] 34 (L/D)kite

326 32

324 30

322 28

320 26 55 60 65 70 75 80 85 55 60 65 70 75 80 85 2016-08-27 15-29-48 2016-08-27 15-29-48 Time [s] Time [s]

Figure 7.16: Time history for gliding phase; case 5

During a steady gliding flight, where the resultant aerodynamic force is equal to the weight, and in the absence of an atmospheric wind speed, the glide ratio of a kite should be equal to the L/D ratio of the kite. Glide ratio is the ratio of the horizontal distance traveled to the vertical drop in altitude or the ratio of the horizontal speed to the sink rate. However, considering the atmospheric wind speed that is set in the simulation, the glide ratio is calculated as the ratio of the apparent horizontal speed to the apparent sink rate. As explained earlier, the simulation case is not a steady case. This can be seen in Figure 7.16; there are small variations in the plots except for the weight of the kite, which is constant. The mean value of the glide ratio is 29.59 and the mean value of the L/D is 30.82 during the gliding phase, there is an approximately 4% relative difference between the mean values of glide and L/D ratios. The gliding flight simulation case shows that the behaviour of the PS model in a gliding flight is accurate and in accordance with the experience. 7.3 Plausibility Checking Results 73

7.3.2 Case 6: Non-tethered, stalling maneuver

For the stalling maneuver, PS model is manually piloted by elevator; a gradual pitch-up command is given in order to increase the angle of attach and have a stall. Once the stall is reached and the nose of the model starts to pitch down, the elevator input is terminated and no more elevator input is given in order to let the model dive and recover from the stall. 3D trajectory of the simulation case can be seen in Figure 7.17.

120 100

z W 80

[m] 60 40 20 0 200 150 100 50 70 y 0 50 60 W [m] 50 40 − 100 20 30 [m] − 150 10 xW − 200 0 − Figure 7.17: Flight trajectory; case 6: White and black circles represent start and end position, blue circles represent the position at every 5 s.

Figure 7.18 shows the time history of the PS model during the stall maneuver. It is seen that the model is stalling around 17.5 degrees of kite angle of attack (αkite) which corresponds to 21.5 degrees of effective wing angle of attack (αeff,m2 ) as a result of the given 4 degrees of wing incidence angle, AppendixA. The stall angle of the kite in the simulation corresponds with the aerodynamic coefficients of the AP2 PowerPlane that are obtained previously and shown in Figure 5.4. The stall speed is approximately 8.5 m/s. The stall behaviour of the PS model of the kite is found reasonable and in accord with the physical logic. 74 Simulation Results and Discussions

] 0 o [ 10 e δ −20 −30 − 6 8 10 12 14 16 18 2016-08-31 18-01-09 25 20

] 15

20 o [ 10 15 [m/s]

kite 5 a α v 10 0 5 5 6 8 10 12 14 16 18 − 6 8 10 12 14 16 18 2016-08-31 18-01-09 2016-08-31 18-01-09 30 30 20 20

] 10 10 o /s] [ 0 o 0

Θ 10 10 −20 q [ −20 −30 −30 − 6 8 10 12 14 16 18 − 6 8 10 12 14 16 18 2016-08-31 18-01-09 2016-08-31 18-01-09 25 8 ] ]

o 20 o 6 [ [

2 15 4 4 ,m 10 ,m 2 eﬀ eﬀ 0 α 5 α 2 0 − 6 8 10 12 14 16 18 6 8 10 12 14 16 18 2016-08-31 18-01-09 2016-08-31 18-01-09 1.8 0.20 1.6 1.4 0.15 kite

kite 1.2 , ,

L 0.10 1.0 D C 0.8 C 0.6 0.05 0.4 6 8 10 12 14 16 18 6 8 10 12 14 16 18 2016-08-31 18-01-09 2016-08-31 18-01-09 Time [s] Time [s]

Figure 7.18: Time history for stall maneuver; case 6 7.4 Discussions 75

7.4 Discussions

This section contains the discussion on the results that are given in the previous sections. The discussion is presented in several subsections, where diﬀerent aspects of the results are discussed.

7.4.1 Discussion on the developed particle system model

There are several aspects that are to be discussed about the developed PS modelling approach. These include not only the limitations and the assumptions that are made during the formation of the model as given in Sections 5.1 and 5.2; but also the choices made during the implementation of the simulation. Having a conventional configuration for the RW kite is one of these limitations. This limitation is set considering most of the CWKP systems using a RW kite include a conventional design which looks like an unmanned sailplane. Hence, the PS model structure is created for a conventional RW kite configuration. The major finding on this aspect is that, for a conventional RW configuration, the kite can be modelled as a PS. Each of the aerodynamic surfaces of the RW kite can be represented as a point mass which is located at the mean aerodynamic center of the corresponding surface. The lift and the drag forces of that aerodynamic surface are directly applied to the point mass. Moreover, it is found that the aerodynamic moments of the surfaces can be added to the PS model, by having two particles which are placed as the mirror symmetry of each other with respect to the yBzB plane of the RW kite. Introducing a force couple that are applied to these two point masses generates an equivalent moment having the same magnitude of the aerodynamic moment about the cg of the RW. Lastly, it is found useful that the PS model to have a point mass at the location where the tether is connecting to the RW kite. In a PS model, the forces can be only exerted to the point masses; thus, having a point mass at the tether connection point is favorable in order to avoid complex parametric relations for representing the tether forces. Considering these findings, although the current PS model structure does not allow unconventional RW kite configurations, it is suggested that different RW configurations can be modelled as a PS by creating the structure of the PS model accordingly. The fuselage is neglected in the current PS model. This can be considered as an acceptable assumption for the small scaled RW kites, where the fuselage is basically a very thin rod. However, this assumption results in having an underestimation of the RW drag. Moreover, the fuselage effects on the pitching and yawing characteristics are automatically excluded due to the assumption. For the future studies, it is highly suggested to not neglect the fuselage. A simple fuselage drag estimation is already employed for the validation simulation cases as explained in 6.2 and the results are satisfactory. Several limitations and assumptions are done in the view of the aerodynamic model that is chosen to be used for the RW kite. Implementing a detailed aerodynamic model is beyond the scope of this thesis, therefore the aerodynamic model is chosen considering the easiness to implement. For this respect, the aerodynamic model is chosen to be pre- calculated functions that take the angle of attacks of the aerodynamic surfaces as inputs and give the corresponding 3D aerodynamic coefficients. Moreover, Reynolds number 76 Simulation Results and Discussions

effects and sideslip angles of the aerodynamic surfaces are neglected in the calculation of the aerodynamic coefficients. This is acceptable, as long as the model does not experience high sideslip angles and is not designed for a large airspeed range. Thereby, an average Reynolds number can be used in the construction of the aerodynamic model. The limitation of having full span control surfaces could be considered as a valid limitation for the elevator and rudder. However, full span ailerons configuration is not a common ailerons design for airplanes or RW kites. Usually, the ailerons are located towards the ends of the wing span. The limitation of the full span control surfaces is made for the ease in the calculation of the aerodynamic forces when there is a control surface deflection. By having the full span control surfaces and using the equivalent flat plate method, the effects of the control surface deflections are included simply as a change in the angle of attack of the aerodynamic surfaces. Lastly, the interaction effects are also neglected in the current aerodynamic model. There- fore, wing-tail interactions are not included. This is in fact a critical assumption, because downwash and upwash effects do play an important role for the tail and wing respectively. The usual outcome of the downwash and upwash effects is to increase the local effective angle of attack of the wing and to decrease the local effective angle of attack of the horizontal tail. This is indeed a shortcoming of the current modelling method. However, a simple empirical method can be added which could be satisfactory.

7.4.2 Discussion on the controlling and piloting

The actuator dynamics for the control surfaces are neglected; as a result, the control inputs given from the manual controller are not smoothed and show discontinuities depending on the piloting technique. These non-smooth control deflections can be seen in Figure 7.10. A smoothing function should be implemented which ensures less discontinuous control deflections. Moreover, during the simulation cases, it has been observed that; abrupt control inputs make the simulation usually slower. Another essential point is that all of the simulation cases are manually piloted; hence, the piloting technique plays a significant role in each case. In other words, different pilots could obtain different simulation results, depending on their piloting technique. Therefore, the implementation of an automatic pilot is highly recommended for the future optimization studies, so that a reference flight case can be explicitly defined.

7.4.3 Discussion on the mass properties

PS modelling is not an idealization to represent a perfectly rigid body. Under a loading of external forces, deformations occur which also aﬀect the rigidity of the model. The results show that the PS model structure representing a RW kite preserves the mass properties of the model within a satisfactory margin as shown in the Figure 7.4 and 7.8. Hence, it is proven that the PS modelling approach is a successful representation of a rigid body, and small deformations of the model can be neglected. The reference wind speeds are intentionally set high for both of the mass properties simulation cases. As a result of this, high apparent wind speeds are obtained which result in high aerodynamic loadings on the PS model. As a matter of fact; during simulation 7.4 Discussions 77

case 1, an instantaneous lift force which is 18.2 times of the kite weight is seen. Similarly, during simulation case 2, an instantaneous lift force which is 14.3 times of the kite weight is seen. In AppendixI, the plots of the lift force for case 1 and case 2 are given. As mentioned previously in Subsection 5.1.2, the selection of the spring constant and the damping coefficient of the spring-damper elements of the PS model has utmost importance 7 in the rigidity of the model. They are selected by trial and error method as ks = 5 10 × N/m and kd = 100 Ns/m. For the spring constant, lower values such as in the order of magnitude of 6 cannot maintain the rigidity and the model deforms immediately under the loading. Higher values such as in the order of magnitude of 8 give better rigidity; however, the computation time increases which affects the real-time capabilities of the simulation. For the damping coefficient, the values up to the order of magnitude of 5 give satisfactory behaviour in representation of the rigidity of the model. Starting with the values in the order of magnitude of 6, the simulation solver crashes. The results shows that the selected values for the spring-damper coefficients are satisfactory for the PS model of AP2 PowerPlane, in fact also for the PS model of the Schweizer SGS 1-36 sailplane at the validation simulations. However, the selection of the coefficients is expected to be highly dependant on the RW kite and the rigidity of the model should be double-checked in prior of the simulation of different RW kite designs. The spring-damper coefficients for all of the segments representing the RW PS model are taken the same in this study. An interesting further study can be made on the selection of different spring-damper coefficients for each segment. A more interesting future work could be to investigate the possibility of tuning the spring-damper coefficients so that some aeroelastic phenomena could be modelled.

7.4.4 Discussion on the power generation

The simulation cases 3 and 4 constitute the reel-out phases results in this study, in order to examine the capability of the KiteSim framework to simulate the power generation phase of the developed PS model. For this respect, FoE trajectory reel-out simulation flights are conducted with two different reference wind speed 6.92 m/s and 8.9 m/s, which are set at the reference height of 6 meters. The reel-out simulation cases show that the developed PS model for the RW kites is satisfactorily integrated to the KiteSim framework. Moreover, successful FoE and circular trajectory reel-out simulation cases are flown by manual piloting in the KiteSim. These two simulation cases are not sufficient to conduct a detailed analysis of the power production capabilities of a RW kite model. In fact, the analysis of the full cycle flight and the detailed examination of the power production phases are beyond the scope of this thesis. However, several comparison and discussion can be made using these reel-out simulation cases. All of the reel-out cases have fluctuations for the tether force, the tether reel-out speed and consequently the mechanical power during the reel-out. Moreover, the peak values of the fluctuations are very high in comparison to the mean values. Such behaviour is undesirable in CWKP systems, because the systems are designed considering the peak conditions. Hence, if there is such a high difference between the mean and the peak values, then the system will be designed considering the peak values and it will not be efficient 78 Simulation Results and Discussions

and optimized. These ﬂuctuations and the high peaks are anticipated to be related to the winch and winch controller parameters that are shown in AppendixG. These parameters are needed to be properly selected and tuned for a desirable power generation behaviour. It is not possible to compare the simulation results with experiments due to the lack of experimental data. However, the mean mechanical power values that are obtained can be considered as realistic values; in the view of the information given by Cherubini about the fact that Ampyx Power demonstrated an average power production of 6 kW with the peaks of 15 kW at a test campaign in 2012, he also adds that previous tests showed the peaks of 30 kW [5]; where the prototypes with a wingspan of 5.5 meters are used. It should be also noted that AP2 PowerPlane is a prototype which is not optimized for power production (R. Ruiterkamp, personal communication, August 31, 2016) and optimization would give notable diﬀerence in performance parameters. It could be helpful to design a small scale rigid wing prototype that can operate with the current ground station of the TU Delft Kite Power research group. Therefore, experimental validation studies can be conducted for the developed PS model. Chapter 8

Conclusions and Recommendations

In view of this thesis project, the following conclusions can be made. Particle system modelling approach has been successfully used to model tethered rigid wing kites. KiteSim framework has reached the capability to simulate kite power systems employing a rigid wing kite via the developed particle system model. Considering the current status of the particle system model; KiteSim can be used to test rigid wing designs, to check the flight characteristics of rigid wings and to train rigid wing pilots using the manual controller. For the further applications such as performance analyses or trajectory optimizations; an automatic control system must be developed. A conventional rigid wing configuration with a single tether attachment point which is located at the center of gravity of the kite can be modelled as a particle system by means of six point masses and 13 spring-damper elements. The wings and tails of a rigid wing can be modelled as point masses located at the mean aerodynamic centers of the corresponding surfaces. Mass of the kite can be distributed to six point masses by formulating an optimization problem, so that the mass properties of the particle system model accurately match with the mass properties of the real rigid wing. Aerodynamic lift and drag forces can be exerted to the particles of the model representing the aerodynamic surfaces. On the other hand, aerodynamic moments can be represented by introducing force couples acting on two point masses; creating an equivalent moment about the kite center of gravity. Lifting line theory method from XFLR5 software can be used along with Kirke’s correlation method for the construction of a full angle of attack envelope aerodynamic model which is a must for dynamic simulations, especially in the presence of a highly manoeu- vrable rigid wing. The effect of kite’s control surface deflections can be approximated by a change in the effective angle of attack of the corresponding aerodynamic surface, by applying the equivalent flat plate method.

79 80 Conclusions and Recommendations

The equation of motion of a particle system model can be written as a set of first-order ordinary differential equations. From which, an implicit problem can be formulated to solve the motion of the particle system model. The implicit problem can be simulated by using an implicit fifth-order Runge-Kutta method. The selection of a spring constant of 5 107 N/m and a damping coefficient of 100 Ns/m for × the spring-damper elements of particle system model proved to be accurate in representing the mass properties of the model within a very small margin during simulations. Most sensitive mass property of the model is the zcg location. Mass properties of the particle system are correlated with the apparent wind speed. Effect of the magnitude and the frequency of the apparent wind speed variations can be observed in the mass properties of the model. Simulating Ampyx Power’s AP2 prototype in the KiteSim for power generation, instantaneous mechanical power peaks of 45 kW and mean mechanical power outputs of 10 kW are obtained in presence of strong atmospheric wind speeds. Instantaneous peaks of 30 kW and mean outputs of 8 kW are obtained in presence of moderate atmospheric wind speeds.

The following recommendations can be made for possible future research. When modelling a new rigid wing design, it is useful to conduct a simulation case for checking the mass properties response of the model to be certain on the rigidity of the model. A smoothing function should be added for control surface deflections, the current implementation results in high discontinuities. As a result, high non-linearities can occur depending on the piloting technique. An automatic aerodynamic model creation can be added to KiteSim. The current application requires to manually create the aerodynamic model which is not convenient. Strip line method could be a suitable method. A good update of the current aerodynamic model could be to obtain the aerodynamic coefficients of the surfaces not only for a range of angle of attack but also for a range of control surface deflections. Implementing a drag estimation calculation for the fuselage is a must, especially if the rigid wing has got a large fuselage. Reference [21] presents a promising method that can easily be implemented. References

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Input File for the Initialization Script of AP2 PowerPlane

#This file contains the input data for the #Rigid Wind initialization script.

#Kite Mass(in kg), Ixx(kgmˆ2), Iyy(kgmˆ2), Izz(kgmˆ2), #Ixy(kgmˆ2), Ixz(kgmˆ2), Iyz(kgmˆ2) respectively.

33.01#kite mass in[kg] 37.75#Ixx in[ kgmˆ2] 21.73#Iyy in[ kgmˆ2] 59.24#Izz in[ kgmˆ2] 0 . 0#Ixy in[ kgmˆ2] 0.47#Ixz in[ kgmˆ2] 0 . 0#Iyz in[ kgmˆ2]

#Wing Geometry

5 . 5#wing span in[m] 0.667#wing taper ratioc t i p w/c ro ot w 0.666#wing root chord in[m] 0 .#wing dihedral angle in[deg] 1.15617604970478#wing leading edge sweep angle in[deg] 4 .#wing incidence angle[deg]

#Horizontal Tail Geometry

1.35#ht span in[m] 1 .#ht taper ratioc t i p h t/c r o o t h t 0.29#ht root chord in[m] 0 .#ht dihedral angle in[deg] 0 .#ht leading edge sweep angle in[deg] 0 .#ht incidence angle[deg]

#Vertical Tail Geometry

85 86 Input File for the Initialization Script of AP2 PowerPlane

0.65#vt height in[m] 0 . 7#vt taper ratioc t i p h t/c r o o t h t 0 . 4#vt root chord in[m] 0 .#vt dihedral angle in[deg] 2.6425452932398#vt leading edge sweep angle in[deg]

#Location of CG, Wing, HT, VT #x is measured in Fuselage Station(FS) #y is measured in Buttock Line(BL) #z is measured in Water Line(WL) #FS origin at nose, positive to back #BL origin at centerline, positive to right #WL origin at cg level, positive to upwards

1.696#1.5#x cg in[m] 0 .#y cg in[m] 0 .#z c g in[m]

1.487#x w l e in[m] 0 .#y w l e in[m] 0 .#z w l e in[m]

3.532#3.497#x h t l e in[m] 0 .#y h t l e in[m] 0 .#z h t l e in[m]

3.499#3.497#x v t l e in[m] 0 .#y v t l e in[m] 0 .#z v t l e in[m]

#Control surface dimensions 8 . 2#chord of the ailerons[% chord of the wing] 2 4 .#chord of the elevator[% chord of the HT] 3 4 .#chord of the rudder[% chord of the VT]

#Control surface deflections 3 0 .#absolute value of the max and min deflection angle of aileron[deg] 3 0 .#absolute value of the max and min deflection angle of elevator[deg] 3 0 .#absolute value of the max and min deflection angle or rudder[deg]

#End of the file Appendix B

XFLR5 Models for Wing, HT and VT of AP2 PowerPlane

Figure B.1: AP2 PowerPlane Wing

87 88 XFLR5 Models for Wing, HT and VT of AP2 PowerPlane

Figure B.2: AP2 PowerPlane HT

Figure B.3: AP2 PowerPlane VT Appendix C

Correction Factor Plots for Post-Stall Aerodynamic Coeﬃcients

89 90 Correction Factor Plots for Post-Stall Aerodynamic Coeﬃcients Appendix D

Approximate AeroData RW Script

# ∗ coding: utf8 ∗ ””” This module provides the functions aerodata wing, aerodata h t and aerodata v t f o r the fast calculation of the aerodynamic coefficients as function of the angle of attack. ””” # pylint: disable=C0326, E1101

from Timer import Timer from scipy.interpolate import InterpolatedUnivariateSpline from numba import autojit import numpy as np import matplotlib.pyplot as plt

from os.path import expanduser PATH = expanduser(”˜”) + ’/00PythonSoftware/KiteSim ’

”””Input Aerodynamic Data from XFLR5””” INPUT WING = np. loadtxt(PATH+”/input/ap002 wing v05 modified.txt”) INPUT HT = np.loadtxt(PATH+”/input/ap002 ht v05 modified.txt”) INPUT VT = np.loadtxt(PATH+”/input/ap002 vt v05 modified.txt”) #INPUTWING= np.loadtxt(PATH+”/input/RW i n p u t f i l e s/sgs WING v02.txt”) #INPUT HT= np.loadtxt(PATH+”/input/RW i n p u t f i l e s/sgs HT v02.txt”) #INPUT VT= np.loadtxt(PATH+”/input/RW i n p u t f i l e s/sgs VT v02.txt”)

”””Post Stall Correlations””” BJC = 1# Beans and Jakubowski Correlation KC = 2# Kirke Correlation

#Choosea Post Stall Correlation by hard coding. #If chosen KC, camber of the airfoils are needed to be hard coded. CORRELATION = KC CAMBER WING = 4 .# The% camber of the wing airfoil (4% for FDD) CAMBER HT = 0 .# The% camber of the ht airfoil (0 for NACA0012) CAMBER VT = 0 .# The% camber of the vt airfoil (0 for NACA0012) #Hard code the AR of the surfaces for finite wing corrections. AR WING = 9 . 9 1# Approx AR wing AR HT = 4 . 6 6# Approx AR HT AR VT = 1 . 9 1# Approx AR VT

”””Interpolation Function””” d e f AeroInterpolation(INPUT, CAMBER, AR): ALPHA = np. zeros(INPUT.shape [0])

91 92 Approximate AeroData RW Script

CL = np. zeros(INPUT.shape[0]) CD = np. zeros(INPUT.shape[0]) CM = np. zeros(INPUT.shape [0])

f o ri in range (0,INPUT.shape[0]): ALPHA[ i ] = INPUT[ i , 0 ] CL[ i ] = INPUT[ i , 2 ] CD[ i ] = INPUT[ i , 5 ] CM[ i ] = INPUT[ i , 8 ]

#POST STALL AERODYNAMIC CORRELATIONS ALPHA PS POSAOA = np.arange(ALPHA[ 1]+5 , 181) ALPHA PS NEGAOA = np.arange( 180 , ALPHA[0] 5)

i f CORRELATION == BJC : CL PS POSAOA = 2∗( np . s i n (ALPHA PS POSAOA∗np.pi/180))∗∗2∗ \ ( np . cos (ALPHA PS POSAOA∗np.pi/180)) CD PS POSAOA = 2∗( np . s i n (ALPHA PS POSAOA∗np.pi/180))∗∗3 CM PS POSAOA = 0 . 2 5 ∗ CD PS POSAOA

CL PS NEGAOA = 1 ∗ 2 ∗ ( np . s i n (ALPHA PS NEGAOA∗np.pi/180))∗∗2∗ \ ( np . cos (ALPHA PS NEGAOA∗np.pi/180)) CD PS NEGAOA = 1 ∗ 2 ∗ ( np . s i n (ALPHA PS NEGAOA∗np.pi/180))∗∗3 CM PS NEGAOA = 0.25∗CD PS NEGAOA

e l i f CORRELATION == KC : CL PS POSAOA = (1 + 0 . 0 5 ∗ CAMBER) ∗ \ ( np . s i n (2∗ALPHA PS POSAOA∗np.pi/180)) CD PS POSAOA = (0.9 + 0.025 ∗ CAMBER) ∗ \ ( 1 . 5 ∗ ( np . s i n (ALPHA PS POSAOA∗np.pi/180))∗∗3 + \ 0 . 5 ∗ ( np . s i n (ALPHA PS POSAOA∗np.pi/180)) + \ 0 . 0 5 ∗ CAMBER) CM PS POSAOA = 0 . 2 5 ∗ CD PS POSAOA

CL PS NEGAOA = 1 ∗ (1 + 0 . 0 5 ∗ 1 ∗CAMBER) ∗ \ ( np . s i n (2∗ np . abs (ALPHA PS NEGAOA)∗ np.pi/180)) CD PS NEGAOA = (0.9 + 0.025 ∗ 1 ∗CAMBER) ∗ \ ( 1 . 5 ∗ (np. sin (np. abs(ALPHA PS NEGAOA)∗ np.pi/180))∗∗3 \ + 0 . 5 ∗ (np. sin (np. abs(ALPHA PS NEGAOA)∗ np.pi/180)) \ + 0 . 0 5 ∗ 1 ∗CAMBER) CM PS NEGAOA = 0.25∗CD PS NEGAOA

CL CORR FACT = 0.0018 ∗ AR∗∗2 + 0.0564 ∗ AR + 0.3965 CD CORR FACT = 0 . 0 0 1 ∗ AR∗∗2 + 0.036 ∗ AR + 0.4548

CL PS POSAOA = CL CORR FACT ∗ CL PS POSAOA CD PS POSAOA = CD CORR FACT ∗ CD PS POSAOA CM PS POSAOA = CD CORR FACT ∗ CM PS POSAOA CL PS NEGAOA = CL CORR FACT ∗ CL PS NEGAOA CD PS NEGAOA = CD CORR FACT ∗ CD PS NEGAOA CM PS NEGAOA = CD CORR FACT ∗ CM PS NEGAOA

ALPHA=np . concatenate ( (ALPHA PS NEGAOA, ( np . append (ALPHA, ALPHA PS POSAOA) ) ) ) CL = np.concatenate((CL PS NEGAOA, (np.append(CL, CL PS POSAOA ) ) ) ) CD = np.concatenate((CD PS NEGAOA, (np.append(CD, CD PS POSAOA ) ) ) ) CM = np.concatenate((CM PS NEGAOA, ( np . append (CM, CM PS POSAOA) ) ) )

c a l c cl = InterpolatedUnivariateSpline(ALPHA, CL, k=3) c a l c cd = InterpolatedUnivariateSpline(ALPHA, CD, k=3) calc cm = InterpolatedUnivariateSpline(ALPHA, CM, k=3)

N = 1024 INTERP ALPHA = np. linspace(ALPHA[0] , ALPHA[ 1] , N, endpoint=True) INTERP CLS = c a l c c l (INTERP ALPHA) INTERP CDS = c a l c c d (INTERP ALPHA) INTERP CMS = calc cm (INTERP ALPHA) SIZE = l e n (INTERP ALPHA) 93

r e t u r n ALPHA, CL, CD, CM, \ c a l c c l , c a l c c d , calc cm , \ N, INTERP ALPHA, INTERP CLS , INTERP CDS, INTERP CMS, SIZE

#”””Wing Aerodynamic Interpolation””” ALPHA WING, CL WING, CD WING, CM WING, \ c a l c c l w i n g , c a l c c d w i n g , calc cm wing , \ N WING, INTERP ALPHA WING, INTERP CLS WING, INTERP CDS WING, INTERP CMS WING, \ SIZE WING = AeroInterpolation(INPUT WING, CAMBER WING, AR WING)

#”””HT Aerodynamic Interpolation””” ALPHA HT, CL HT, CD HT, CM HT, \ c a l c c l h t , c a l c c d h t , calc cm ht , \ N HT, INTERP ALPHA HT, INTERP CLS HT , INTERP CDS HT, INTERP CMS HT, \ SIZE HT = AeroInterpolation(INPUT HT, CAMBER HT, AR HT)

#”””VT Aerodynamic Interpolation””” ALPHA VT, CL VT, CD VT, CM VT, \ c a l c c l v t , c a l c c d v t , calc cm vt , \ N VT, INTERP ALPHA VT, INTERP CLS VT , INTERP CDS VT, INTERP CMS VT, \ SIZE VT = AeroInterpolation(INPUT VT, CAMBER VT, AR VT)

@autojit(nopython=True) d e f aerodata wing(angle): ””” Approximate the aerodynamic coefficient through linear interpolation i n pre calculated array. alpha: angle of attack in degrees. ””” assert (angle <= INTERP ALPHA WING [ 1 ] ) , ”AOA o f wing i s beyond the range ” assert (angle >= INTERP ALPHA WING [ 0 ] ) , ”AOA o f wing i s below the range ” s i z e = l e n (INTERP ALPHA WING) s l o p e c l = 0 . 0 s l o p e c d = 0 . 0 slope cm = 0 . 0 s t e p = INTERP ALPHA WING [ 1 ] INTERP ALPHA WING [ 0 ] idx = int((angle INTERP ALPHA WING[0]) / step) 1 i f idx < 0 : idx = 0 while idx < s i z e 1 : i f angle >= INTERP ALPHA WING[ idx ] and angle <= INTERP ALPHA WING[ idx+1]: s l o p e c l = (INTERP CLS WING [ idx +1] INTERP CLS WING[idx]) / step s l o p e c d = (INTERP CDS WING [ idx +1] INTERP CDS WING[idx]) / step slope cm = (INTERP CMS WING[ idx +1] INTERP CMS WING[idx]) / step break idx += 1

r e t u r n INTERP CLS WING[idx] + slope c l ∗ ( angle INTERP ALPHA WING[ idx ]) , \ INTERP CDS WING[idx] + slope c d ∗ ( angle INTERP ALPHA WING[ idx ]) , \ INTERP CMS WING[idx] + slope cm ∗ ( angle INTERP ALPHA WING[ idx ] )

@autojit(nopython=True) d e f aerodata h t ( angle ) : ””” Approximate the aerodynamic coefficient through linear interpolation i n pre calculated array. alpha: angle of attack in degrees. ””” assert (angle <= INTERP ALPHA HT [ 1 ] ) , ”AOA o f ht i s beyond the range ” assert (angle >= INTERP ALPHA HT [ 0 ] ) , ”AOA o f ht i s below the range ” s i z e = l e n (INTERP ALPHA HT) s l o p e c l = 0 . 0 s l o p e c d = 0 . 0 slope cm = 0 . 0 s t e p = INTERP ALPHA HT [ 1 ] INTERP ALPHA HT [ 0 ] idx = int((angle INTERP ALPHA HT[0]) / step) 1 94 Approximate AeroData RW Script

i f idx < 0 : idx = 0 while idx < s i z e 1 : i f angle >= INTERP ALPHA HT[ idx ] and angle <= INTERP ALPHA HT[ idx +1]: s l o p e c l = (INTERP CLS HT [ idx +1] INTERP CLS HT[idx]) / step s l o p e c d = (INTERP CDS HT [ idx +1] INTERP CDS HT[idx]) / step slope cm = (INTERP CMS HT [ idx +1] INTERP CMS HT[idx]) / step break idx += 1

r e t u r n INTERP CLS HT[idx] + slope c l ∗ ( angle INTERP ALPHA HT[ idx ] ) , \ INTERP CDS HT[idx] + slope c d ∗ ( angle INTERP ALPHA HT[ idx ] ) , \ INTERP CMS HT[idx] + slope cm ∗ ( angle INTERP ALPHA HT[ idx ] )

@autojit(nopython=True) d e f aerodata v t ( angle ) : ””” Approximate the aerodynamic coefficient through linear interpolation i n pre calculated array. alpha: angle of attack in degrees. ””” assert (angle <= INTERP ALPHA VT [ 1 ] ) , ”AOA o f vt i s beyond the range ” assert (angle >= INTERP ALPHA VT [ 0 ] ) , ”AOA o f vt i s below the range ” s i z e = l e n (INTERP ALPHA VT) s l o p e c l = 0 . 0 s l o p e c d = 0 . 0 slope cm = 0 . 0 s t e p = INTERP ALPHA VT [ 1 ] INTERP ALPHA VT [ 0 ] idx = int((angle INTERP ALPHA VT[0]) / step) 1 i f idx < 0 : idx = 0 while idx < s i z e 1 : i f angle >= INTERP ALPHA VT[ idx ] and angle <= INTERP ALPHA VT[ idx +1]: s l o p e c l = (INTERP CLS VT [ idx +1] INTERP CLS VT[idx]) / step s l o p e c d = (INTERP CDS VT [ idx +1] INTERP CDS VT[idx]) / step slope cm = (INTERP CMS VT [ idx +1] INTERP CMS VT[idx]) / step break idx += 1

r e t u r n INTERP CLS VT[idx] + slope c l ∗ ( angle INTERP ALPHA VT[ idx ] ) , \ INTERP CDS VT[idx] + slope c d ∗ ( angle INTERP ALPHA VT[ idx ] ) , \ INTERP CMS VT[idx] + slope cm ∗ ( angle INTERP ALPHA VT[ idx ] )

# call the functions once for pre compilation aerodata wing ( 1 0 . 0 ) a e r o d a t a h t ( 1 0 . 0 ) a e r o d a t a v t ( 1 0 . 0 ) Appendix E

Wing Axes System Creation for Wing Dihedral Eﬀects

y2 m2 x2 yB y2 z2 m0 x2 xB y3 m3 zB x3 yB xB z3

y3 x3

y2 yB y3

z2 zB z3 rotated by Γ rotated by Γ around xB− around xB

95 96 Wing Axes System Creation for Wing Dihedral Eﬀects Appendix F

Schweizer SGS 1-36 Sailplane

F.1 Input ﬁle for SGS 1-36 Sailplane

#This file contains the input data for the #Rigid Wind initialization script.

#Kite Mass(in kg), Ixx(kgmˆ2), Iyy(kgmˆ2), Izz(kgmˆ2), #Ixy(kgmˆ2), Ixz(kgmˆ2), Iyz(kgmˆ2) respectively.

396.4397#kite mass in[kg] 1374.7994#Ixx in[ kgmˆ2] 869.0793#Iyy in[ kgmˆ2] 2214.0507#Izz in[ kgmˆ2] 0 . 0#Ixy in[ kgmˆ2] 66.9774#Ixz in[ kgmˆ2] 0 . 0#Iyz in[ kgmˆ2]

#Wing Geometry

14.0726#wing span in[m] 0.45#wing taper ratioc t i p w/c ro ot w 1.2802#wing root chord in[m] 4 .#wing dihedral angle in[deg] 1.432#wing leading edge sweep angle in[deg] 0 . 0#wing incidence angle[deg]

#Horizontal Tail Geometry

2.414#ht span in[m] 0.742#ht taper ratioc t i p h t/c r o o t h t 0.5669#ht root chord in[m] 0 .#ht dihedral angle in[deg] 5 .#ht leading edge sweep angle in[deg] 1 . 9#ht incidence angle[deg]

#Vertical Tail Geometry

97 98 Schweizer SGS 1-36 Sailplane

1.3198#vt height in[m] 0.69#vt taper ratioc t i p h t/c r o o t h t 0 . 8#vt root chord in[m] 0 .#vt dihedral angle in[deg] 2 7 .#vt leading edge sweep angle in[deg]

1.99864#x cg in[m] 0 .#y cg in[m] 0 .#z c g in[m]

1.59059#x w l e in[m] 0 .#y w l e in[m] 0 .#z w l e in[m]

5.45859#x h t l e in[m] 0 .#y h t l e in[m] 0 .#z h t l e in[m]

5.14990#x v t l e in[m] 0 .#y v t l e in[m] 0 .#z v t l e in[m]

#Control surface dimensions 7 . 7#chord of the ailerons[% chord of the wing] 3 4 .#chord of the elevator[% chord of the HT] 4 5 .#chord of the rudder[% chord of the VT]

#Control surface deflections 3 2 .#absolute value of the max and min deflection angle of aileron[deg] 2 0 .#absolute value of the max and min deflection angle of elevator[deg] 3 0 .#absolute value of the max and min deflection angle or rudder[deg]

#End of the file F.2 General Data for SGS 1-36 Sailplane 99

F.2 General Data for SGS 1-36 Sailplane

Figure F.1: SGS 1-36 general data, from [33]

Figure F.2: SGS 1-36 mass properties, from [33] 100 Schweizer SGS 1-36 Sailplane

Figure F.3: SGS 1-36 general data, from [19]

Figure F.4: SGS 1-36 general data (continued), from [19]

Figure F.5: SGS 1-36 mass properties, from [19] F.2 General Data for SGS 1-36 Sailplane 101

Figure F.6: SGS 1-36 three-view drawing , from [33] 102 Schweizer SGS 1-36 Sailplane

Figure F.7: SGS 1-36 side drawing , from [6] F.3 XFLR5 models for wing, HT and VT of SGS 1-36 Sailplane 103

F.3 XFLR5 models for wing, HT and VT of SGS 1-36 Sailplane

Figure F.8: Schweizer SGS 1-36 Wing

Figure F.9: Schweizer SGS 1-36 HT

Figure F.10: Schweizer SGS 1-36 VT 104 Schweizer SGS 1-36 Sailplane Appendix G

Winch Model and Winch Controller Parameters

G.1 Winch Model Parameters

{ ”CLASS”: ”WinchModel” , ”name”: ”winch 8000 RW”, ” f max ” : 8000 , ” d tether”: 1.0e 6, ” f min ” : 600 , ” f d e l t a high”: 1000, ” f d e l t a l o w ” : 200 , ” f high”: 2500, ” f l o w ” : 600 , ”p nom”: 32000, ”p max”: 48000, ” g e a r b o x ratio”: 6.2, ” drum radius”: 0.1615, ”inertia”: 0.328, ” c f ” : 0 . 7 9 9 , ” t a u s ” : 3 . 1 8 , ”inductivity”: 2.97, ”speed nom” : 4 . 0 9 , ”u nom” : 231 }

105 106 Winch Model and Winch Controller Parameters

G.2 Winch Controller Parameters

{ ”CLASS”: ”WinchControl” , ”name”: ”winch c o n t r o l 8 0 0 0 5 . 2 WC3 RW” , ”controller”: ”WinchController3”, ” f l o w ” : { ”P”: 0.000144, ”I”: 0.0075, ”K b” : 1 . 0 , ”K t” : 8 . 0 , ” t b l e n d ” : 3 . 0 } , ” f h i g h ” : { ”P”: 0.00231, ” I ” : 0 . 0 0 1 , ”D” : 0 . 0 , ”N” : 1 5 . 0 , ”K b” : 1 . 0 , ”K t” : 10.0 } , ”operational”: { ”prediction”: false , ” f ro max ” : 2500 , ” f r i m a x ” : 1625 , ” v ro min ” : 0 . 5 , ” v r o r e l ” : 0 . 2 5 , ” v ri max ” : 5 . 0 , ” v r i r e l ” : 1 . 5 , ” beta 0 ” : 25.0 } } Appendix H

Additional Figures and Data for Quasi-Steady Analysis

Figure H.1: Representation of kite velocities, with azimuth angle φazi, polar angle θele and course angle χ shown; from [29]

107 108 Additional Figures and Data for Quasi-Steady Analysis

Figure H.2: Representation of aerodynamic forces and velocities, with the azimuth angle φazi and polar angle θele shown; from [29]

Figure H.3: Representation of aerodynamic forces, gravity force and velocities, with the azimuth angle φazi and polar angle θele shown; from [29] 109 theo 952 [W] 8328 6949 9411 9460 6222 5807 4192 1132 4986 5656 7199 7616 4061 5359 2282 4187 2911 P 19002 14743 12525 11386 theo , [N] t 4601 7917 4316 7263 5228 5345 4233 3723 2653 2903 3216 4489 3525 6200 5839 5413 5684 4275 4060 3803 3913 3550 F sim 277 134 808 420 636 [W] P 2648 6043 2045 4008 2325 2752 1628 1966 1957 2166 1517 3726 3306 1560 1648 1335 1010 sim , t [N] 703 503 853 698 945 778 1464 2512 1268 1975 1291 1557 1107 1259 1235 1398 1205 1846 1693 1173 1228 1010 F ro , t v 1.81 2.40 1.61 2.03 1.80 1.77 1.47 1.56 1.58 0.39 1.55 1.26 0.27 2.02 1.95 1.33 1.34 0.95 1.32 0.60 1.07 0.82 [m/s] ] azi ◦ [ φ -9.47 -5.66 -8.34 -5.23 -7.92 13.10 16.50 19.60 17.76 16.83 10.83 16.49 21.09 13.82 14.52 -24.03 -11.90 -16.43 -26.34 -17.27 -22.71 -17.49 ] ele ◦ [ β 50.86 33.78 43.17 30.32 35.86 27.43 29.85 35.76 35.78 25.19 24.47 14.52 52.20 32.64 27.43 25.42 23.67 24.28 26.09 20.78 18.92 18.39 - 7.95 7.40 6.16 6.48 5.64 5.35 4.65 4.59 3.67 3.29 3.44 3.76 7.58 7.79 7.32 6.60 6.34 5.77 5.43 5.00 5.01 4.75 L/D R - C 1.08 1.14 0.90 1.13 1.01 1.07 0.99 1.14 1.02 0.94 1.04 1.28 1.00 1.34 1.34 1.19 1.21 1.11 1.11 1.05 1.12 1.10 w v 13.54 13.10 13.71 13.32 13.83 13.55 13.95 14.51 14.86 14.30 14.37 13.45 10.59 10.14 10.06 10.11 10.14 10.28 10.49 10.22 10.22 10.26 [m/s] ] 3 ρ 1.209 1.212 1.207 1.211 1.206 1.209 1.205 1.199 1.194 1.202 1.201 1.209 1.209 1.212 1.213 1.213 1.213 1.211 1.209 1.212 1.212 1.212 [kg/m z [m] 91.48 88.59 84.00 86.13 87.08 95.36 93.27 90.63 92.80 111.99 124.18 101.21 130.54 114.68 138.99 184.41 219.38 165.59 169.69 111.00 115.82 110.50 ] ◦ χ [ 88.03 88.51 91.71 89.09 90.71 92.01 90.87 91.70 89.22 89.88 89.74 89.98 270.46 270.05 270.86 271.09 270.21 271.56 272.86 270.99 271.94 271.31 2 t [s] 13.6 18.5 22.2 27.3 31.8 38.0 44.2 57.6 74.7 82.9 89.9 99.5 11.8 17.5 22.4 27.7 32.5 38.3 44.3 49.7 55.2 61.3 Averaged values of thepower, system for parameters simulation during cases the 3 and approximate horizontal 4. ﬂight Row state, numbers along 1 with to theoretical 12 traction are case force 3 and and row number 13 to 22 are case 4 comparisons. 1 t [s] 13.4 18.4 22.1 27.2 31.6 37.9 44.0 57.4 74.5 82.8 89.7 99.4 11.7 17.4 22.3 27.6 32.3 38.2 44.1 49.6 55.0 61.1 Table H.1: No.1 No.2 No.3 No.4 No.5 No.6 No.7 No.8 No.9 No.10 No.11 No.12 No.13 No.14 No.15 No.16 No.17 No.18 No.19 No.20 No.21 No.22 110 Additional Figures and Data for Quasi-Steady Analysis

Ft,sim Ft,theo Psim Ptheo δ%,F = − 100 δ%,P = − 100 Ft,theo × Ptheo ×

Ft,sim [N] Ft,theo [N] δ%,F Psim [W] Ptheo [W] δ%,P No.1 1464 4601 -68.2 2648 8328 -68.2 No.2 2512 7917 -68.3 6043 19002 -68.2 No.3 1268 4316 -70.6 2045 6949 -70.6 No.4 1975 7263 -72.8 4008 14743 -72.8 No.5 1291 5228 -75.3 2325 9411 -75.3 No.6 1557 5345 -70.9 2752 9460 -70.9 No.7 1107 4233 -73.9 1628 6222 -73.8 No.8 1259 3723 -66.2 1966 5807 -66.1 No.9 1235 2653 -53.4 1957 4192 -53.3 No.10 703 2903 -75.8 277 1132 -75.5 No.11 1398 3216 -56.5 2166 4986 -56.5 No.12 1205 4489 -73.1 1517 5656 -73.2 No.13 503 3525 -85.7 134 952 -85.9 No.14 1846 6200 -70.2 3726 12525 -70.3 No.15 1693 5839 -71.0 3306 11386 -71.0 No.16 1173 5413 -78.3 1560 7199 -78.3 No.17 1228 5684 -78.4 1648 7616 -78.4 No.18 853 4275 -80.1 808 4061 -80.1 No.19 1010 4060 -75.1 1335 5359 -75.1 No.20 698 3803 -81.6 420 2282 -81.6 No.21 945 3913 -75.8 1010 4187 -75.9 No.22 778 3550 -78.1 636 2911 -78.1

Table H.2: Tether force and traction power comparison of the simulation cases 3 and 4 with the theoretical analysis. Row numbers 1 to 12 are case 3 and row number 13 to 22 are case 4 comparisons. Appendix I

Additional Simulation Results

6000 Lkite [N]

Kite weight, Wkite [N] 5000

4000 [N] kite

W 3000 [N],

kite 2000 L

1000

0 0 5 10 15 20 25 30 2016-07-23 13-42-59 Time [s]

Figure I.1: Kite lift and kite weight during the simulation case 1

Lkite [N] 5000 Kite weight, Wkite [N]

4000 [N]

kite 3000 W [N], 2000 kite L

1000

0 0 10 20 30 40 50 2016-07-23 14-05-15 Time [s]

Figure I.2: Kite lift and kite weight during the simulation case 2

111 112 Additional Simulation Results