Assessment of Structural Loads in Wind Farms under Consideration of Wake Redirection Control A thesis accepted by the Faculty of Aerospace Engineering and Geodesy of the University of Stuttgart in partial fulfilment of the requirements for the degree of Doctor of Engineering Sciences (Dr.-Ing.) by Matthias Kretschmer born in Esslingen am Neckar, Germany Main referee: Prof. Dr. Po Wen Cheng Co-referee: Prof. Dr. Dominic von Terzi Date of defense: 17.02.2023 Institute of Aircraft Design University of Stuttgart 2024 Acknowledgements At the end of this journey, I would like to take a step back and express my deepest gratitude to a few people. First, I would like to thank Po Wen Cheng for your encouragement, freedom and advice. Thank you to Dominic von Terzi for providing the second opinion and especially for the detailed feedback and suggestions. I would like to thank my colleagues at the SWE for their support in all matters and for providing such a pleasant atmosphere. My sincerest gratitude goes to Matthias and Friedemann for kickstarting me in wind energy research as well as Birger, Umut, Kolja, Stefan, Ines, Maayen, Florian, Martin, Oliver, Alex, Ricardo, Holger, Tim, Sarah, Steffen, Frank, Fiona, Vasilis, Viola and all other colleagues. Further, I would like to thank all my students, in particular Nico, Julian, Omar, Frederik, German, Domenico, Philipp, Carsten, Felix. A special thanks goes to the people at NREL Jason, Kelsey, Emmanuel, Scott, Fabian, Jeremy, Liz, Caitlyn, Hannah with whom I had the pleasure of collaborating and sharing some enjoyable skiing trips. I want to extend my appreciation to the members of the RAVE consortium, who work diligently to provide the measurement data from the alpha ventus wind farm. I thank my family, Mama and Papa for always having my back. And finally, to the most important person in my life, Lena. Thank you for always being there, for your patience, encouragement and unlimited support. You and our wonderful children, Frida and Jakob, remind me of the really important things in life. Contents Abbreviations ix List of Symbols xi Abstract xiii Kurzfassung xv 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Background 5 2.1 Atmospheric boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Atmospheric stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Vertical wind speed profile . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Fatigue loading of wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Fatigue load representation and calculation . . . . . . . . . . . . . . . . . 13 2.2.2 Long-term fatigue evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Wake effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Velocity deficit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Wake meandering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Effects of yawed operation . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Wind farm control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Axial induction control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.2 Wake redirection control . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Simulation modelling 23 3.1 Atmospheric boundary layer modelling . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Mann uniform shear turbulence model . . . . . . . . . . . . . . . . . . . 23 3.1.2 Large eddy simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Wind farm flow modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Steady-state low-fidelity flow simulation . . . . . . . . . . . . . . . . . . 29 vi Contents 3.2.2 Coupled large eddy simulations . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Aeroelastic wind farm simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.1 FAST.Farm implementation . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.2 Aeroelastic simulation of a single wind turbine . . . . . . . . . . . . . . . 33 3.3.3 Wake dynamics: Implementation of the dynamic wake meandering model 34 3.3.4 Wake-added turbulence model . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.5 Spatial and temporal discretisation . . . . . . . . . . . . . . . . . . . . . 40 4 Simulation model calibration and validation 43 4.1 Calibration of wake-added turbulence model . . . . . . . . . . . . . . . . . . . . 43 4.1.1 Reference simulations: PALM-LES . . . . . . . . . . . . . . . . . . . . . 44 4.1.2 Wake-added turbulence model parametrisation . . . . . . . . . . . . . . . 46 4.1.3 Wake-added turbulence model discretisation . . . . . . . . . . . . . . . . 52 4.2 Aeroelastic turbine model validation for free-stream conditions . . . . . . . . . . 57 4.2.1 Alpha ventus measurement data base . . . . . . . . . . . . . . . . . . . . 57 4.2.2 Load case definition and simulation setup . . . . . . . . . . . . . . . . . . 59 4.2.3 Transfer of environmental conditions to simulations . . . . . . . . . . . . 61 4.2.4 Validation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Aeroelastic turbine model validation for single wake situations . . . . . . . . . . 65 4.3.1 Load case definition and simulation setup . . . . . . . . . . . . . . . . . . 67 4.3.2 Validation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Low-fidelity model calibration for wind farm optimisation . . . . . . . . . . . . . 78 4.4.1 Load case definition and simulation setup . . . . . . . . . . . . . . . . . . 78 4.4.2 Optimisation of FLORIS model parameters . . . . . . . . . . . . . . . . 81 5 Structural loads under consideration of wake redirection control 85 5.1 Wind turbine model and sensor specification . . . . . . . . . . . . . . . . . . . . 85 5.2 Definition of environmental conditions . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 Synthetic short-term atmospheric conditions . . . . . . . . . . . . . . . . 87 5.2.2 Measured short-term atmospheric conditions . . . . . . . . . . . . . . . . 87 5.2.3 Measured long-term atmospheric conditions . . . . . . . . . . . . . . . . 88 5.3 Structural loads of a free-stream turbine during yawed operation . . . . . . . . . 88 5.3.1 Load case definition and simulation setup . . . . . . . . . . . . . . . . . . 89 5.3.2 Structural load distributions at component level . . . . . . . . . . . . . . 90 5.3.3 Long-term evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Single wake characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4.1 Load case definition and simulation setup . . . . . . . . . . . . . . . . . . 96 5.4.2 Wake meandering characteristics . . . . . . . . . . . . . . . . . . . . . . 96 5.4.3 Wake profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.5 Structural loads of a waked turbine during yawed operation . . . . . . . . . . . . 105 5.5.1 Load case definition and simulation setup . . . . . . . . . . . . . . . . . . 105 5.5.2 Structural load distributions at component level . . . . . . . . . . . . . . 106 5.5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Contents vii 6 Optimisation of wind farm operation 113 6.1 Load case definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Optimisation of yaw angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2.1 Surrogate model for the structural loads . . . . . . . . . . . . . . . . . . 114 6.2.2 Cost functions for different objectives . . . . . . . . . . . . . . . . . . . . 116 6.3 Assessment of wind farm operation strategies . . . . . . . . . . . . . . . . . . . . 118 6.3.1 Long-term evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.3.2 Detailed evaluation of specific operational points . . . . . . . . . . . . . . 122 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7 Conclusions 131 7.1 Summary and main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1.1 Calibration and validation of the aeroelastic simulation model . . . . . . 132 7.1.2 Structural loads under consideration of wake redirection control . . . . . 133 7.1.3 Optimisation of wind farm operation . . . . . . . . . . . . . . . . . . . . 135 7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A Appendix 137 A.1 Additional results from the wake-added turbulence calibration . . . . . . . . . . 138 A.2 Additional results from the OpenFAST validation . . . . . . . . . . . . . . . . . 146 A.3 Additional results from the FLORIS calibration . . . . . . . . . . . . . . . . . . 147 A.4 Additional load distributions of a free stream turbine in yawed operation . . . . 149 A.4.1 Short-term evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A.4.2 Long-term evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.5 Additional load distributions of a waked turbine in yawed operation . . . . . . . 167 A.6 Additional results from the optimisation of operation strategies . . . . . . . . . . 182 A.7 Manual of the wake-added turbulence feature in FAST.Farm . . . . . . . . . . . 187 A.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A.7.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A.7.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Bibliography 191 Curriculum Vitae 205 Abbreviations 1P One-per-Revolution 3P Three-per-Revolution ABL Atmospheric Boundary Layer ACD Actuator Disc ACL Actuator Line AEP Annual Energy Production AS Atmospheric Stability AWAE Ambient Wind and Array Effects BEM Blade Element Momentum CFD Computational Fluid Dynamics DEL Damage Equivalent Load DLC Design Load Case DLL Dynamic Link Library DoE Design of Experiments DoF Degree of Freedom DWM Dynamic Wake Meandering ew edgewise fa fore-aft FAST Fatigue, Aerodynamics, Structures, and Turbulence FFoR Fixed Frame of Reference FFT Fast Fourier Transform FLORIS FLOw Redirection and Induction in Steady State fw flapwise IEC International Electrotechnical Commission IFB Institut für Flugzeugbau (Institute of Aircraft Design) IPC Individual Pitch Control JONSWAP Joint North Sea Wave Observation Project LCoE Levelised Cost of Energy LES Large Eddy Simulation LIDAR Light Detection and Ranging LSS Low-Speed Shaft MABL Marine Atmospheric Boundary Layer MBS Multibody Simulation MFoR Meandering Frame of Reference x Abbreviations NREL National Renewable Energy Laboratory PALM Parallelised Large-Eddy Simulation Model PDF Probability Density Function PSD Power Spectral Density RANS Reynolds-Averaged Navier-Stokes RAVE Research at alpha ventus SC Super Controller SCADA Supervisory Control and Data Acquisition SLSQP Sequential Least Squares Programming ss side-side SWE Stuttgart Wind Energy TKE Turbulent Kinetic Energy TSO Transmission Grid Operator WD Wake Dynamics List of Symbols Greek letters αk Kolmogorov’s spectral constant αshear Wind shear exponent of exponential power law approximation δ Lateral wake deflection from rotor-centre (m) ε Rate of viscous dissipation of specific turbulent energy (m2 s−3) Γ Anisotropy parameter of Mann’s spectral tensor κ Von Kármán constant, Poisson constant Λ Integral length scale (m) νT Eddy viscosity (m2 s−1) ϕ Wind direction (°) φ Angular coordinate (rad) σ Standard deviation θm Yaw misalignment angle (°) θw Wake skew angle (°) Θ Potential temperature (K) Θv Virtual potential temperature (K) ζ Dimensionless stability parameter Roman letters a Axial induction factor cT Thrust coefficient C Weibull scale parameter (m s−1) DW Wake deficit diameter (m) ∆S Load range from rainflow counting D Rotor diameter (m); damage fc Cut-off frequency (s−1) f Frequency (s−1) g Gravitational constant (m s−2) i Running variable j Running variable k Wavenumber (m−1) km1 Calibration factor of the wake-added turbulence model in FAST.Farm km2 Calibration factor of the wake-added turbulence model in FAST.Farm kmt Calibration factor of the wake-added turbulence model in FAST.Farm k Weibull shape parameter xii List of Symbols L Length scale (m) L∗ Obukhov length (m) LM Length scale of Mann’s spectral tensor (m) m Exponent for the calculation of equivalent load cycles N Reference number of load cycles p Pressure (Pa) r Spatial separation vector (m) r̃ Radial coordinate normalised by rotor radius R Rotor radius (m) r Radial coordinate (m); distance between points (m); mixing ratio of water vapor rL Mixing ratio of liquid water in the air t Time (s) T Time range (s); Temperature (K) TIWAT Turbulence intensity from wake-added turbulence TI Turbulence intensity TIamb,Rotor Spatial turbulence intensity around the rotor u Longitudinal wind speed (m s−1) ū Mean wind speed (m s−1) u′ Turbulent fluctuation component of the longitudinal velocity (m s−1) v′ Turbulent fluctuation component of the lateral velocity (m s−1) w′ Turbulent fluctuation component of the vertical velocity (m s−1) u∗ Friction velocity (m s−1) udef Velocity deficit u∞ Freestream wind speed (m s−1) v Lateral wind speed (m s−1) v Velocity vector (m s−1) VDiskAvg Rotor disk-averaged ambient wind speed normal to the disk (m s−1) Vr Radial velocity (m s−1) Vx Axial velocity (m s−1) w Vertical wind speed (m s−1); weighting factor w′T ′ Kinematic virtual heat flux (m K s−1) x Position vector (m) x Longitudinal coordinate (m); longitudinal displacement (m) y Lateral coordinate (m); lateral displacement (m) z0 Roughness length (m) z Vertical coordinate (m); vertical displacement (m) Abstract Wind farm control enables the operation of wind farms in a collective optimum that consid- ers all turbines instead of operating the individual wind turbines in their local optima. A collective optimum of a wind farm is dependent on the objective, which can be, for instance, increasing the energy yield, reducing the structural loading of the wind turbines or providing ancillary services to the electricity grid. The development of new wind farm control techniques requires knowledge in various disciplines that include wind turbine engineering (control design and implementation, structural and aerodynamic design, load assessment and validation), mul- tidisciplinary optimisation, wind resource modelling and atmospheric boundary layer modelling connected with the modelling of wakes. This thesis covers many of the aforementioned disciplines in order to investigate the wake redirection control concept as one option for wind farm control in detail. Wake redirection control is based on the concept of deflecting the wake behind a wind turbine through yaw- misaligned operation. The aim is to mitigate the wake effects on the downstream turbine by redirecting the wake. Hence, the downstream turbine can generate, for instance, more power that, ideally, compensates or exceeds the power losses at the upstream turbine caused by yaw- misalignment in below-rated conditions. For the numerical assessment, the aeroelastic simulation tool FAST.Farm is utilised. An adequate setup of the tool is developed to allow the investigation of various realistic operating conditions including different atmospheric stabilities. In addition, FAST.Farm is improved by implementing a model to include the wake-added small-scale turbulence. FAST.Farm is then calibrated against high-fidelity large eddy simulations and validated by using measurement data from the alpha ventus wind farm. Overall, good agreement between simulations and measurements is achieved for the structural loads in terms of statistical results and frequency response at the tower-base and blade-root. The inclusion of wake-added turbulence is crucial to avoid underestimation of the turbulence in the wake and consequently the loads. This is especially relevant in stable atmospheric conditions, where the ambient turbulence intensity is low and the meandering of the wake is weak. Without wake-added turbulence, the Damage Equivalent Loads (DELs) of the bending moment in the fore-aft direction are underestimated xiv Abstract by up to 87 % in full-wake conditions compared to the measurement data; the error reduces to 2 % when the wake-added turbulence module is activated in the simulations. An extensive investigation of the wake redirection control concept and its consequences on the structural loads is performed in a simulation study with the validated tool FAST.Farm. For a turbine in free-stream conditions, the fatigue loads at different turbine components are analysed, with changing atmospheric conditions and yaw misalignment angles. The largest effects of yaw misalignment on the load variations are found for stable atmospheric conditions with strong vertical wind shear and low turbulence intensity. In contrast, the influence of yaw misalignment on the fatigue loads becomes less important in unstable atmospheric stability with low vertical wind shear and high turbulence intensity. The investigation is extended for a turbine that is subjected to waked inflow conditions. Especially in partial wake situations, where the wake-centre lays between ±0.75 D (turbine diameter) measured from the rotor-centre of the waked turbine, the load distributions differ significantly from free-stream conditions. A directional dependency of the loads is found with respect to the lateral wake offset: The loads tend to be higher for negative lateral wake offsets compared to the loads from the same positive lateral wake offsets, because of higher load amplitudes over one rotor revolution. The gained knowledge is finally applied in the derivation of optimal operation strategies by using the wake redirection control approach for exemplary wind farm configurations and changing environmental conditions. An optimisation is performed using different objective functions including maximising the energy yield and reducing the structural loads at the wind turbines. The resulting optimal operation strategies are assessed by using aeroelastic simula- tions in FAST.Farm. The long-term evaluation suggests that the Annual Energy Production (AEP) of the considered turbine array setups can be increased compared to the baseline sce- nario without wind farm control, when the main objective is set to power maximisation while the fatigue loads at the turbines are not or equally weighted. Consequently, the strategies that focus on minimising the fatigue loads result into less AEP compared to the baseline strategy. The consideration of fatigue loads in the optimisation of operation strategies is realised with an efficient surrogate model. With this approach, strategies are derived that are able to reduce the fatigue loads at specific components significantly. Kurzfassung Im Gegensatz zur getrennten Betrachtung der einzelnen Windenergieanlagen (WEA) im Wind- park ermöglicht die Windparkregelung den Betrieb von Windparks in einem übergeordneten Optimalzustand, der alle WEA berücksichtigt. Dieser hängt von der gewünschten Zielvorgabe ab, die z.B. eine Erhöhung der Leistungsproduktion, eine Reduktion der Strukturlasten oder das Bereitstellen von Hilfsdienstleistungen für das elektrische Netz sein kann. Die Entwicklung von neuen Algorithmen zur Windparkregelung erfordert Kenntnisse in verschiedenen Bereichen, zu denen die Technik der WEA (Entwurf und Implementierung der Regelung, Entwurf der Struktur, Lastannahmen und deren Validierung), die Modellierung der Windressource sowie der atmosphärischen Grenzschicht in Verbindung mit Nachlaufeffekten gehören. In dieser Arbeit werden einige der genannten Bereiche abgedeckt, um das Windparkregelungs- konzept der Nachlaufablenkung detailliert zu untersuchen. Bei der Nachlaufablenkung wird der Nachlauf einer WEA durch gezielte Schräganströmung des Rotors abgelenkt, um Nachlaufef- fekte auf eine stromabwärts stehende Anlage zu reduzieren. Dadurch kann die stromabwärts stehende WEA beispielsweise eine höhere Leistung erzeugen, welche idealerweise die Verluste an der stromaufwärts stehenden Anlage, welche durch die Schräganströmung entstehen, kom- pensiert oder übersteigt. Für die numerische Beurteilung wird das aeroelastische Simulationsprogramm FAST.Farm eingesetzt. Für FAST.Farm werden geeignete Einstellungen erarbeitet, um unterschiedliche Betriebsbedingungen einschließlich verschiedener atmosphärischer Stabilitätsbedingungen re- alistisch abzubilden. Darüber hinaus wird zur Verbesserung von FAST.Farm ein Modell zur Berechnung der Effekte resultierend aus der nachlaufinduzierten kleinskaligen Turbulenz im- plementiert. Die empirischen Parameter in FAST.Farm werden mittels hochaufgelöster Large- Eddy Simulationen kalibriert. FAST.Farm wird mit Hilfe von Messdaten aus dem Windpark alpha ventus validiert. In der Auswertung der statistischen Strukturlastdaten sowie der Fre- quenzantwort der Struktur wird eine gute Übereinstimmung zwischen den Simulationen und Messdaten im Bereich des Turmfußes und der Blattwurzel festgestellt. Die Einbeziehung der nachlaufinduzierten Turbulenz ist dabei wichtig für die Berechnung der erhöhten Turbulenz im Nachlauf und somit auch für die Strukturlasten der WEA im Nachlauf. Dies ist besonders xvi Kurzfassung relevant in stabilen atmosphärischen Bedingungen, in welchen die Umgebungsturbulenz niedrig ist und das Mäandrieren des Nachlaufs nur schwach ausgeprägt ist. Ohne nachlaufinduzierte Turbulenz werden die schadensequivalenten Lasten des Turmfußbiegemoments im Vergleich zu den Messdaten um bis zu 87 % unterschätzt; der Fehler reduziert sich auf 2 % bei aktivierter nachlaufinduzierter Turbulenz. Mit dem validierten Simulationsprogramm FAST.Farm wird eine umfangreiche Simulations- studie durchgeführt, um die Auswirkungen des Regelungskonzepts der Nachlaufablenkung auf die Strukturlasten zu untersuchen. Zunächst werden die Ermüdungslasten an verschiedenen Komponenten einer WEA in freier Anströmung unter Berücksichtigung variierender Anströmbe- dingungen und Winkel der Schräganströmung (Gierversatz) analysiert. Die die größten Belas- tungsschwankungen resultierend aus der Schräganströmung treten in stabilen atmosphärischen Bedingungen mit starker vertikaler Windscherung und niedriger Turbulenzintensität auf. Im Gegensatz dazu ist der Einfluss der Schräganströmung auf die Ermüdungslasten in instabilen at- mosphärischen Bedingungen mit geringer Windscherung und hoher Turbulenzintensität weniger stark ausgeprägt. Im weiteren Verlauf wird die Untersuchung ausgedehnt für eine WEA, die im Nachlauf steht. Die Lastverteilungen bezüglich unterschiedlicher Gierversatzwinkel weichen dabei deutlich von den Lastverteilungen in freier Anströmung ab. Dies ist insbesondere der Fall in Nachlaufsituationen, in denen der Rotor partiell abgeschattet wird und das Nachlaufzentrum sich zwischen ±0.75 D (Rotordurchmesser) gemessen vom Rotorzentrum befindet. Zudem wird eine Richtungsabhängigkeit der Lasten in Bezug auf den seitlichen Nachlaufversatz festgestellt: Die Lasten sind tendenziell höher für negative seitliche Nachlaufversätze im Vergleich zu den Lasten aus denselben positiven seitlichen Nachlaufversätzen, weil die Lastamplituden über eine Rotorumdrehung höher sind. Die gewonnenen Erkenntnisse werden schließlich bei der Ableitung optimaler Betriebsstra- tegien unter Nutzung der Nachlaufablenkung für beispielhafte Windparkkonfigurationen und wechselnde Umweltbedingungen angewendet. Es wird eine Optimierung mit verschiedenen Ziel- funktionen durchgeführt, wie z.B. die Maximierung des Energieertrags oder die Reduzierung der strukturellen Belastung der WEA. Die daraus resultierenden optimalen Betriebsstrategien werden mit Hilfe von aeroelastischen Simulationen in FAST.Farm beurteilt. Die Auswertung der langfristigen Auswirkungen ergibt, dass der Energieertrag der betrachteten Turbinenanord- nungen im Vergleich zum Basisszenario ohne Windparkregelung erhöht werden kann, wenn die Zielfunktion die Maximierung des Energieertrags enthält; die Ermüdungslasten werden dabei entweder nicht berücksichtigt oder nur gleich gewichtet. Folglich senken die Strategien, die sich auf die Minimierung der Ermüdungslasten konzentrieren, den Energieertrag im Vergleich zum Basisszenario. Die Berücksichtigung der Ermüdungslasten bei der Optimierung der Betriebs- strategien über ein effizientes Ersatzmodell wird dabei erfolgreich demonstriert. Mit diesem Ansatz werden Strategien abgeleitet, mit denen die Ermüdungsbelastungen an bestimmten Komponenten signifikant reduziert werden können. 1 Introduction 1.1 Motivation Wind energy is one of the key technologies to transform the energy system from fossil energy sources to renewable energy sources. In the European Union, a significant share of the electricity generation with wind energy is planned to be achieved by the deployment of offshore wind farms; the current goal is set to an installed capacity of 60 GW in 2030 [1] (compared to 12 GW in 2021). One example of offshore wind farms already in operation is the German Bight, where ≈ 1300 installed wind turbines with a capacity of ≈ 7 GW in total have been in operation at the end of 2020 [2]. This comes along with a dense spacing of the turbines, which has the advantage of sharing the infrastructure (e.g. grid connection, operation and maintenance). However, the associated wake losses are larger compared to a sparser spacing of the turbines. The optimisation of the operation of wind farms aims to reduce the Levelised Cost of Energy (LCoE) given the boundaries of a fixed turbine layout. Different objectives can be formulated and include, for instance, the maximisation of power, a life time management that considers the structural loading of all wind turbines and a better integration into the electricity grid. In order to achieve these objectives, an integrated wind farm control strategy is necessary, which optimises the operation of all wind turbines in a wind farm collectively rather than finding local optima for each turbine individually. The research on wind farm control strategies connected with system integration into the future electricity grid is part of the “Grand challenges in the science of wind energy” formulated in [3]. The development of wind farm control techniques requires knowledge in various disciplines that include [4]: wind turbine engineering (control design and implementation, structural de- 2 1 Introduction sign, load assessment and validation), wind resource modelling, atmospheric boundary layer modelling connected with the modelling of wakes. For the acceptance and so-called bankability (which is the willingness of established financial institutions to finance a project at a reasonable interest rate [4] ), a substantial factor is the demonstration of the capability to simulate the implementation and impact of wind farm control on the energy and loading of the turbines with robust modelling techniques. A deeper understanding of the structural loads occurring in conjunction with wind farm control is also identified as a major research gap in [5] and the main motivation of this thesis. 1.2 Research objectives The work performed within this thesis contributes to the research area of wind farm control. In particular, the wake redirection control strategy is assessed with respect to the associated struc- tural loads using an integrated, aeroelastic simulation environment. Therefore, the following objectives are defined: • Validating the aeroelastic simulation environment in terms of the prediction of power and structural loads in wake conditions against measurement data from operating turbines. • Increasing the accuracy of the load calculations of waked turbines by the implementation of a module to model the wake-added small-scale turbulence and verification against high-fidelity simulations. • Analysing the effects from yawed operation on the structural loads of free-stream and waked turbines under consideration of different environmental conditions. • Developing and evaluating optimised wind farm operation strategies with the goals of power maximisation as well as fatigue load management. 1.3 Related Work The numerical simulation of wind farm effects has greatly advanced in the past years. Wind farm simulations are primarily considered to be a problem of fluid mechanics, because of the in- teractions of the wind turbines with the Atmospheric Boundary Layer (ABL). The involvement of a wide range of turbulent scales (small scales, in the order of millimetres, around the blades to large scales, in the order of kilometres, for the entire wind farm) makes it a complex problem that should be tackled with models of different fidelity [6]. High fidelity models are applied to gain deeper understanding in the flow physics [7, 8] and the turbine’s response [9], but they are only applicable to a few selected situations due to their high computational costs. In contrast, low fidelity models require significantly lower computational resources, have a lower accuracy 1.3 Related Work 3 and are typically used only for steady-state calculations in many cases (e.g. AEP calculations) [10]. A trade-off between accuracy and computational costs is the aim for the development of midfidelity models. Here, the Dynamic Wake Meandering (DWM) model presented first in [11] is a prominent representative, which has been gradually improved by different groups [12, 13, 14] in the past years and is also accepted by the IEC 61400-1 standard [15]. It enables the aeroelastic simulation of wind farms within a reasonable time-frame and is the basis for the newly developed simulation software FAST.Farm [12] used in this thesis. In the past, the optimisation of the operation of wind farms was predominantly motivated by the maximisation of the total power output. Different approaches were investigated, under which the wake redirection control strategy was identified as most promising [16]. Therefore, this strategy received most attention by the research community. Significant contributions are made in [17] by developing yaw-based optimisation strategies as well as control-oriented low- fidelity flow models. Wake steering is further improved by using of high-fidelity simulations [18], wind tunnel experiments [19] and field test campaigns [20]. The potential benefits of the application of Light Detection and Ranging (LIDAR) devices for tracking the wake position are explored in [21]. In recent years, the objectives of structural load mitigation and electricity grid integration [22] gained more popularity and dedicated wind farm control strategies were designed [23]. In the state of the art in the design of new wind farm control strategies, the structural loading of the turbines is typically either neglected [24] or it is considered in a simplified way. In the latter case, the structural loads are usually determined using a database which is created beforehand based on the results of aeroelastic single turbine simulations for a variety of environmental conditions [25, 26]; the loads are then interpolated with respect to a calculated wind farm state that is derived by the use of a control-oriented low-fidelity flow model. The numerical test of the developed control strategies is performed with high-fidelity simulations for a few load cases in some studies [18]. However, the detailed numerical testing is often omitted because of the high computational costs. In more recent studies, the testing of the new control concepts is performed with midfidelity models, which enable the analysis of the turbine’s aeroelastic response for many load cases [27, 28]. This is especially considered to be crucial for the future certification of new wind farm control strategies [22]. 4 1 Introduction 1.4 Outline After the introduction (Chapter 1), this thesis continues with the description of the necessary theoretical background information in Chapter 2. It includes explanations related to the ABL, in which wind turbines and farms are operated. Furthermore, the load mechanisms leading to fatigue on wind turbines are discussed and the interactions between the wind turbines and the ABL flow as well as the concepts to control the flow are explained. In Chapter 3, the methods are introduced which are applied to simulate the mutual effects between the ABL and the wind turbines including their structural loading. In Chapter 4, the aeroelastic simulation environment used in this thesis is calibrated and validated using high-fidelity simulations as well as measurement data; in particular, newly added functionalities to the simulation tool are addressed. Furthermore, the simulation tool is calibrated to be suitable for the application in the optimisation of wind farm operation strategies considered. Chapter 5 is dedicated to the investigation of the effects connected with wake-redirection control. The associated wake characteristics are examined and the structural loads of a wind turbine subjected to yawed operation in free-stream and waked conditions are assessed. The knowledge and results from the previous chapters are eventually applied in the optimi- sation of wind farm operation strategies in Chapter 6. The wake-redirection control concept is employed to optimise the wind farm operation according to different objectives that include the maximisation of the wind farm power output as well as the reduction of the fatigue loads. The results from the optimisation are evaluated in terms of short-term and long-term effects by the usage of aeroelastic simulations. A summary of the thesis is given in Chapter 7; the main conclusions are drawn and an outlook for further research is provided. 2 Background The theory and concepts applied in this thesis are introduced and related literature reviewed. In Section 2.1, the structure and characteristics of the ABL are presented in order to provide information on the regime, in which wind turbines are operated. In Section 2.2, the load mechanisms leading to fatigue on wind turbines are defined and the methods for the calculation of fatigue loads are explained. The influence of the wind turbines on the ABL flow is the topic of Section 2.3, where the wake effects caused by normal operation and yawed operation are discussed. An introduction to different wind farm control approaches is given in Section 2.4 and their potential is reviewed. 2.1 Atmospheric boundary layer The ABL defines the lower part of the troposphere that is affected by the earth’s surface [29]. In the ABL, the wind speed is decreased compared to the geostrophic wind leading to a vertical wind profile. The height of the ABL varies from 0.1 to 3 km and is dependent on atmospheric conditions, which include thermal stratification as well as low/high pressure conditions [29, 30, 31]. Shear stresses, created by wind shear and thermal conditions, induce turbulence in the ABL. In an idealised way, the structure of the ABL layer can be divided into three layers in the vertical direction [32]. The lowest layer is laminar and its vertical dimension is a few millimeters; for wind energy applications this layer is not relevant. On top of the laminar layer is the surface layer (also called Prandtl layer) which covers 10 % of the ABL on average [31]. It is often described as the constant flux layer because all vertical energy and momentum fluxes 6 2 Background vary by less than 10 % of their magnitude [29, 31]. Compared to the other layers, the strongest vertical wind speed gradients occur in the surface layer, reaching approximately 70-80 % of the geostrophic wind at its top [33]. The mixed layer (also called Ekman layer) takes the largest portion of the ABL with approximately 90 %. Turbulent fluxes decrease with increasing height and shear stresses tend to be zero at the top of the mixed layer [33]. The Coriolis force induces a variation of the wind direction with height. In contrast to flow over a land surface, the atmospheric boundary layer over sea surface (Marine Atmospheric Boundary Layer (MABL)) shows little variations over a diurnal cycle. Due to the large heat capacity of water, energy from the sun is absorbed to a high extent without increasing the temperature of the sea surface much [29]. The MABL has a distinct seasonal cycle. 2.1.1 Turbulence Turbulence in fluid flows describes the chaotic and random behaviour of a flow that is super- posed over the mean flow [34]. Turbulent flow is unsteady, three-dimensional and rotational. In contrast to laminar flow, momentum exchange perpendicular to the mean flow direction occurs. These characteristics make turbulent flow a multi-scale problem, which is illustrated by the en- ergy spectrum in Figure 2.1. It shows the kinetic energy distribution over the wavenumber k; the wavenumber describes the oscillations per length unit L and is defined as k = 2π L . (2.1) The region of small wavenumbers (the so-called integral range), equivalent to large length scales, contains the bulk of the turbulent energy and is where energy production takes place. In ABL flows, turbulent energy is produced by wind shear and buoyancy forces. The maximum of the spectrum indicates the typical size of turbulent eddies; the integral length-scale Λ can be approximated by k ∼ 1/Λ [30]. Towards higher wave numbers, the energy cascade follows. In the so-called inertial subrange, energy is neither produced nor dissipated, but larger eddies are decomposed to smaller eddies. The energy distribution should be proportional to E(k) ∼ k−5/3 [30] for isotropic decaying turbulence. Eventually, in the dissipation range the kinetic energy of the smallest eddies is dissipated into heat. Under the assumption of Taylor’s frozen turbulence, wavenumber and frequency are directly proportional. For the mathematical description of a turbulent flow quantity, the Reynolds decomposition is commonly used. For example, the wind speed u can be split into its mean wind speed component ū and its turbulent fluctuation component u′: u = ū+ u′ (2.2) 2.1 Atmospheric boundary layer 7 ln(k) E( k) energy production inertial subrange energy dissipation energy dissipation k-5/3 Figure 2.1: Schematic illustration of an energy spectrum of a turbulent flow in the ABL. Adapted from [30]. 8 2 Background The mean value of the turbulent fluctuation component over a time T equals zero: u′ ≡ lim T→∞ 1 T T∫ 0 u′ dt = 0 (2.3) A common measure in wind energy for the statistical description of turbulence is the Tur- bulence Intensity (TI) defined as the standard deviation of the wind speed σu divided by the mean wind speed ū: TI = σu ū . (2.4) 2.1.2 Atmospheric stability The ABL can be in different states that are dependent on the thermal stratification. In general, three basic types are distinguished [32]. An unstable ABL appears when cool air flows over warm surfaces. This leads to an upward directed heat flux and air rises due to buoyancy forces. In higher layers, pressure is reduced leading to an expansion of the risen air with combined adiabatic cooling. If no thermal equilibrium with the surrounding air is achieved, the air will keep rising. In a stable ABL, warm air flows over cool surfaces. A downward directed heat flux is observed and vertical movement by air is damped. Compared to unstable ABL, the vertical dimension of the ABL and vertical momentum exchange are less. The neutral ABL is characterised by a thermal equilibrium of risen air particles with the surrounding air due to adiabatic cooling. In order to classify the state of the atmosphere a commonly used parameter in micro me- teorology is the Obukhov length L∗ [31]. It relates contributions from shear and buoyancy generated turbulence to the turbulent kinetic energy in the surface layer by defining: L∗ = − u3 ∗ κ g Θv w′T ′ . (2.5) where κ is the von Kármán constant and the friction velocity u∗ represents the contribution from shear generated turbulence (see Equation 2.12). The buoyancy production term w′T ′g/Θv is formed by the gravitational acceleration constant g, the kinematic virtual heat flux of the surface layer w′T ′ and a reference virtual potential temperature Θv: Θv = Θ(1 + 0.61r − rL). (2.6) In Equation 2.6, r is the mixing ratio of water vapor and rL is the mixing ratio of liquid water 2.1 Atmospheric boundary layer 9 Table 2.1: General classification of atmospheric stability Atmospheric stability kinematic virtual heat flux L∗ ζ Unstable w′T ′ > 0 < 0 < 0 Neutral w′T ′ = 0 ∞ 0 Stable w′T ′ < 0 > 0 > 0 Table 2.2: Specific classification of atmospheric stability in this thesis adopted from [35] Atmospheric stability class L∗ interval [m] Very unstable −100 ≤ L∗ ≤ −50 Unstable −200 ≤ L∗ ≤ −100 Near unstable −500 ≤ L∗ ≤ −200 Neutral |L∗| ≥ 500 Near stable 200 ≤ L∗ ≤ 500 Stable 50 ≤ L∗ ≤ 200 Very stable 10 ≤ L∗ ≤ 50 in the air; Θ is the actual potential temperature that is calculated by: Θ = T (p0/p)κ, (2.7) with the temperature T , the pressure p, the standard pressure p0 and the Poisson constant κ. L∗ defined in Equation 2.5 can be interpreted as a characteristic height for thermal stratifi- cation where the production of turbulent energy from shear equals the production of turbulent energy from thermal processes [33]. When L∗ is related to the height over the ground z, the dimensionless stability parameter ζ can be formed: ζ = z L∗ . (2.8) The direction of the kinematic virtual heat flux determines the sign of L∗ and the state of the ABL correspondingly (see Table 2.1). By defining appropriate intervals for the Obukhov length L∗ different states of the ABL can be classified. In this thesis, the classification from [35] is adopted and shown in Table 2.2. In particular, the three atmospheric stability class definitions of unstable, neutral and stable from Table 2.2 are used for most of the presented analyses to limit complexity. Besides the Obukhov length, atmospheric stability can be determined by using the Richardson 10 2 Background number [31]. There exist different definitions of the Richardson number, from which the bulk Richardson number is used in this thesis. It relates the vertical potential temperature gradient ∆Θ to the wind speed gradient ∆u by defining: RiB = − g Θ0 ∆Θ∆z (∆u)2 , (2.9) where Θ0 is the potential temperature at sea surface and ∆z is the height between the tem- perature measurements. Over the ocean, the bulk Richardson number can be used to derive ζ with the empirical relationship provided by [36]: ζ =  10RiB (1−5RiB) , ζ > 0 10RiB, ζ ≤ 0. (2.10) Atmospheric stability is correlated with wind speed and the vertical wind profile [32]. Unsta- ble conditions occur often at lower wind speeds and induce a more uniform vertical wind profile because of strong turbulent mixing. Stable conditions happen more frequently at higher wind speeds and come along with a highly sheared wind profile. At very high wind speeds, neutral conditions are observed mostly because shear generated turbulence dominates over buoyancy generated turbulence. 2.1.3 Vertical wind speed profile Atmospheric boundary layer flows develop a vertical wind shear profile due to the surface friction implying zero wind speed at the surface. The shape of the profile depends on the mechanical surface characteristics such as the shape and roughness. In addition, thermodynamic properties of the surface (e.g. heat emissivity, heat capacity, albedo) have an influence on the energy and momentum exchange between the surface and the atmosphere [32]. Together with the thermal stratification of the ABL, they all determine the shape of the vertical wind shear profile. For the description of the vertical wind shear profile, the logarithmic law and power law are the most commonly used approaches in wind energy. They are strictly valid in the surface layer only [32], where the strongest wind gradients occur. The logarithmic wind profile for a neutral ABL can be derived from physical considerations based on Prandtl’s mixing length hypothesis [32]. Its mathematical description is given in Equation 2.11: u(z) = u∗ κ ln z z0 , (2.11) where the van Kármán constant is defined as κ = 0.4 and z0 is the roughness length that is dependent on the surface. The friction velocity u∗ is a measure of the vertical momentum exchange and can be simplified to Equation 2.12 when the main wind direction is the same as 2.1 Atmospheric boundary layer 11 the longitudinal wind speed component u: u2 ∗ = −u′w′, (2.12) where u′ and w′ are the turbulent fluctuations of the longitudinal and vertical wind speed component. For non-neutral atmospheric conditions, a stability dependent correction function Ψ can be introduced that is added to Equation 2.11 as follows: u(z) = u∗ κ ln z z0 + Ψ(ζ). (2.13) Ψ is dependent on the stability parameter ζ (Equation 2.8). Different formulations of Ψ exist in the literature from which the following is chosen as explained in [37] and [38]: Ψ = −aζ, when stable 2 ln 1+χ 2 + ln 1+χ2 2 − 2 arctanχ+ π 2 , when unstable, (2.14) where a = 5 and χ = (1− bζ)1/4 with b = 15. Exemplarily, vertical wind shear profiles for different atmospheric stabilities and roughness lengths are shown in Figure 2.2. An increase of roughness length causes stronger wind shear because of higher surface friction. This implies additionally a higher friction velocity u∗ leading to the same reference wind speed at the reference height zref , thus the vertical momentum exchange is higher. The changes of the wind speed profile in non-neutral conditions compared to neutral conditions become visible: The thermal stratification in unstable conditions amplifies the vertical momentum exchange, which reflects in less vertical wind shear and a higher friction velocity for the same surface roughness. In contrast, stable atmospheric conditions damp the vertical momentum exchange; this is reflected in a lower friction velocity and leads to stronger vertical wind shear. The logarithmic wind profile can be approximated with the empirical power law, which can be written as: u(z) = uref ( z zref )αshear , (2.15) where the reference wind speed uref at the reference height zref and the exponent αshear are used to define the vertical wind speed profile. The quality of its approximation of the logarithmic profile is discussed in more detail in [32]; it shows that the power law provides good approximations of the wind speed profile in the surface layer for very smooth terrain (e.g. offshore). Therefore, it is applied for the description of the vertical wind speed profile in this thesis. 12 2 Background 0.4 0.6 0.8 1 1.2 u/uref [-] 0 50 zref 100 150 200 z [m ] u =0.28 shear=0.09 u =0.47 shear=0.16 0.4 0.6 0.8 1 1.2 u/uref [-] u =0.31 shear=0.06 u =0.52 shear=0.13 0.4 0.6 0.8 1 1.2 u/uref [-] u =0.24 shear=0.22 u =0.40 shear=0.36 Figure 2.2: Vertical wind shear profiles normalised by the reference wind speed uref = 8 m s−1 at zref = 90 m. Left panel: neutral Atmospheric Stability (AS); middle panel: unstable AS; right panel: stable AS. Colour code: blue: z0 = 0.001 m; green: z0 = 0.1 m. Solid lines represent the logarithmic law, dashed lines are derived from the power law. 2.2 Fatigue loading of wind turbines Fatigue plays a major role in the design and operation of wind turbines. The design lifetime of wind turbines is usually between 20-30 years; a continuous operation leads to approximately 108-109 rotor revolutions per lifetime [39]. Different load effects can be distinguished that lead to fatigue of wind turbine components [40]: • Periodic loads: The regular recurrence or variation of excitation forces due to the rotation of blades leads to periodic loads. The blades are subject to gravitational loads that induce a cyclic loading mainly in the edgewise direction. Imbalances in the rotor, e.g. due to different blade masses or blade pitch offsets, cause additional periodic loads in the support structure. Further periodic aerodynamic excitation comes from tower blockage effects as well as external inflow conditions such as wind shear and yawed inflow. • Random loads: The turbulence in the incoming wind is the dominating source for ran- dom loads. The turbulent structures vary strongly and are, for instance, dependent on atmospheric stability and surface/terrain conditions. Another source of random loads are stochastic waves which mainly affect the support structure and tower of offshore wind turbines. 2.2 Fatigue loading of wind turbines 13 • Transient loads: Characterised by temporarily limited extension are transient events. These events are triggered for example by start-up and shut-down events as well as sudden wind direction or wind speed changes (wind gusts); in many cases high peak loads occur during transient events. In this thesis, transient loads are out of scope; only periodic and random loads are taken into account because the considered wind farm control strategies are applied in normal oper- ating conditions. Furthermore, fatigue loading is examined from a system perspective. This means that the term loading is referred to internal forces and moments acting on the turbine’s structure. More detailed considerations of fatigue loading such as stress concentration, crack propagation or fracture mechanics are not covered. In the following, the calculation of fatigue loads performed throughout this thesis and corresponding fatigue life evaluation is discussed. 2.2.1 Fatigue load representation and calculation Fatigue loads can be represented with different level of detail: A complete fatigue load infor- mation of a component is stored in the load time history. However, the processing of large time series is time consuming and time histories are not directly comparable, thus fatigue load information is often condensed associated with a loss of information. Load cycles are obtained from the full load time series by using cycle counting algorithms; widely used rainflow-counting algorithms determine the number of load cycles with different load amplitudes. In this work, the algorithm from [41] is used. By assuming linear damage accumulation, the total damage of a component can be estimated by the combination of all partial damages or, alternatively, a damage equivalent load can be calculated for easy comparison of load histories. Linear damage accumulation A widely used approach in fatigue damage calculation, e.g. [42], is the assumption of linear damage accumulation often referred as Miner’s rule [39]. It states that the total damage D can be calculated by linearly adding the damage caused by the load cycles ni of a specific amplitude specified by index i: D = ∑ i=1 ni Ncrit,i . (2.16) The critical load cycle count Ncrit,i is obtained from the material’s S/N-curve and gives the maximum allowed cycles for a specific stress range before failure. A failure of the structure occurs when D ≥ 1. 14 2 Background Damage equivalent load The idea of the damage equivalent load is motivated by the shape of a S/N-curve that is a straight line with the slope 1/m when plotted on a double logarithmic scale; the constant m is material specific. The equivalent load range Seq,Nref,m with the reference load cycle count Nref causes, in theory, the same damage as the damage induced by the load history with a combination of different stress ranges and cycles: ∆Seq,Nref,m = m √ ∆Sm N Nref (2.17) Using the linear damage accumulation law, a damage equivalent load range with the reference cycle number Nref can be obtained by combining different load ranges ∆Si with corresponding Ni load cycles from a load distribution: ∆Seq,Nref,m = m √∑n i=1 ∆Smi Ni Nref (2.18) This damage equivalent load range is often referred to as Damage Equivalent Load (DEL) and reduces a complicated load distribution into a single value. Compared to a full load time series or load spectrum, DEL is certainly a simplification. However, it is very useful when comparing different load cycle distributions. Hereby, the exponent m plays an important role because it weights the individual load ranges exponentially. By varying m, an interpretation of the contribution of individual load cycle ranges is possible. DELs calculated with large exponents are sensitive to a very few large load cycles. On the contrary, DELs derived with small exponents are mainly influenced by a large number of small load amplitudes. In this thesis, the short-term DEL ∆Seq,Nref,m is computed as 1-Hz DEL, which is equivalent to Nref = 600 load cycles for a 10-min time series. Hereby, the indication of the index Nref is omitted. 2.2.2 Long-term fatigue evaluation A wind turbine experiences a large variety of operational conditions in its lifetime. The typical fatigue life prediction of a wind turbine component can be summarised by the following steps [39]: 1. A short-term load distribution is derived by calculating the elastic response of the wind turbine for a specific environmental condition (e.g. wind speed, wind turbulence). 2. The contribution of the short-term load distributions is added up according to the long- term site-specific conditions and weighted with the probability of occurrence of those 2.3 Wake effects 15 specified conditions. 3. Additional load contributions from transient events (e.g. start-up shut-down) are added to the fatigue life. 4. Partial safety factors from certification guidelines are applied. 5. An appropriate damage model for the considered component is chosen and the fatigue lifetime is calculated. In this work, a complete fatigue life assessment for specific components is not performed, thus only steps one and two are of relevance. For the calculation of a long-term DEL, the short- term DELs derived for specific environmental conditions are weighted with their probability of occurrence wi. The long-term DEL is determined as: ∆Seq,m,long = m √√√√ n∑ i=1 wi ∆Smeq,Nref,m,i , (2.19) where the short-term DELs are derived with the same reference cycle number and exponent. 2.3 Wake effects A wind turbine extracts kinetic energy from ABL flows and converts it to electrical energy. The flow region behind the turbine, that is affected by it, is commonly referred to as wake. The main wake effects, due to the interaction of the turbine with the wind, are a decreased flow velocity (wake deficit) and an increased TI compared to the incoming wind. The increased turbulence in the wake includes a often called “apparent turbulence” [43], which originates from the meandering of the wake leading to wind speed fluctuations at a fixed spatial position (see also Figure 2.3). In addition, a wake-added turbulence, which is of rather small scale nature, contributes to the total turbulence in the wake of a turbine. Besides the introduced wake effects, the effects of a yawed turbine operation on the wake are discussed in more detail in the following. 16 2 Background Figure 2.3: Instantaneous flow field of the longitudinal wind speed component u in a horizontal plane at hub height from a large eddy simulation. The wake of a single wind turbine is visualised including the wake meandering. The turbine rotor location is indicated by the white line. 2.3.1 Velocity deficit The wake region is characterised by a decreased flow velocity compared to free-stream conditions due to the extraction of kinetic energy by the wind turbine rotor. The so-called wake shear layer develops at the boundary of the wake flow and the undisturbed surrounding air; turbulent mixing is enhanced and momentum from the undisturbed flow is transferred into the wake region leading to the recovery of the velocity deficit. The downstream evolution of the wake can be separated into different regions that are as- sociated with the velocity deficit [45] (see Figure 2.4). The near-wake region starts directly behind the rotor and extends to approximately 1-2 rotor diameters (D) downstream. Following one-dimensional stream-tube theory [46], the static pressure, increased before passing the ro- tor, drops behind the rotor and recovers to the ambient pressure; a commonly used definition marks the point of full pressure recovery as the end of the near-wake region [45]. In addition, the velocity decreases further until the full deficit is formed at the point of complete pressure recovery and the wake expands correspondingly due to mass conservation. The tip and root vortices typically break down within the first 2 D [47]; in low turbulent atmospheric conditions, they can exist longer downstream [47, 48]. In the intermediate (transition) wake region, the wake shear layer expands and reaches the wake centre axis after 2-5 D. Turbulent mixing transfers momentum from the surrounding flow into the wake leading to the recovery of the velocity deficit. The maximum TI is observed 2.3 Wake effects 17 Figure 2.4: Illustration of the different wake regions and their characteristics. Taken from [44] in accordance with the Creative Commons Attribution 4.0 License without alterations. between 2-6 D depending on atmospheric conditions [49]. At the point of maximum TI, the turbulence in the wake is considered to be in quasi-equilibrium with the mean flow [50]. The velocity deficit and turbulence profile in the far-wake region have a nearly Gaussian- shape caused by turbulent mixing. The properties of the far-wake are mainly functions of the rotor radius, rotor thrust, wind speed and ambient TI. Other turbine properties such as the blade profile, hub and nacelle geometry as well as the tower have no significant effect on the far-wake. The recovery rate of the velocity deficit depends on the magnitude of the momentum en- trainment from the surrounding flow into the wake zone, thus it is largely dependent on the turbulence characteristics in the atmospheric flow. The turbulence characteristics are different for changing atmospheric stability. In particular, the momentum fluxes in the vertical direc- tion (expressed with u′w′) and in the lateral direction (expressed with u′v′) are relevant to the momentum entrainment [51]. Higher momentum fluxes imply higher momentum exchange in the corresponding direction which eventually leads to a faster wake recovery. Typically, the momentum fluxes in unstable atmospheric conditions are higher compared to neutral AS; they are the lowest in stable conditions due to the negative surface buoyancy fluxes, which damp the turbulence [51]. 18 2 Background 2.3.2 Wake meandering The wake profile is not fixed at a centred position downstream but oscillates randomly in the horizontal and vertical directions. A widely accepted explanation is given by [11] postulating that the velocity deficit follows the turbulent eddies of the ambient flow that are larger than two rotor diameter. In full-scale experiments, the phenomenon of wake meandering was identified by utilising a LIDAR device; it was measured and correlated with the incoming large scale turbulence in [52] and [53]. In wind tunnel experiments [54], wake meandering was only observed when turbulent eddies with dimensions of much larger than a rotor diameter exist in the incoming flow confirming the aforementioned postulation by [11]. The magnitude of wake meandering is dependent on atmospheric conditions. In experimental work by [54], it was found that increasing ambient TI leads to larger amplitudes of the wake meandering. This was confirmed by numerical analyses in [49]. In addition, both investigations show that the horizontal wake meandering is larger than the vertical meandering which is attributed to the larger turbulent kinetic energy in the lateral direction. They also report in agreement with [55], that the amplitude of the wake meandering becomes larger with increasing downstream distances. Besides the TI, the turbulent energy distribution over length scales in the flow (see Figure 2.1) is equally important for the wake meandering. The large turbulent eddies (>2 D)influence the wake meandering, while small eddies affect the velocity deficit evolution through turbulent diffusion. The turbulent length scale correlates with the atmospheric stability; in an unstable ABL, turbulent structures with large length scales are present, whereas in stable conditions the length scales are typically smaller (see Table 3.1 for an example of length scales in flat and homogeneous terrain). The influence of atmospheric stability on the wake meandering was studied in [56] using numerical simulations by explicitly keeping the TI constant. They observed a much stronger wake meandering during unstable atmospheric conditions compared to stable conditions emphasising the importance of including atmospheric stability as a parameter in wind farm analyses. 2.3.3 Effects of yawed operation The operation of a wind turbine under yawed conditions (i.e. the rotor is not aligned with the incoming wind direction) has not only implications on the turbine itself but also on the downstream wake. The mean centre position of the wake that originates from a yawed turbine is deflected compared to the wake of an aligned turbine. This is illustrated in Figure 2.5, where the wake is deflected in the negative direction caused by a positive (anti-clockwise) yaw- misalignment angle θm. In addition, the shape of the velocity deficit in the far-wake differs from a Gaussian-shape, especially in the vertical direction. 2.3 Wake effects 19 x y Ambient wind flow Rotor-centre line θm δ Wake-centre θ𝑤 Figure 2.5: Illustration of wake deflection caused by yawed operation. The turbine rotates anti- clockwise with the positive yaw-misalignment angle θm. The wake-centre is deflected by the distance δ(x) and the skew angle θw(x). Motivated by new wind farm control methods (see Section 2.4.2), research on the far-wake originating from a yawed turbine has gained more attraction recently. By utilising high-fidelity numerical simulations as well as wind tunnel and full-scale experiments, the following depen- dencies of the wake centre deflection were found in various studies: • Turbine’s yaw angle: The wake centre deflection increases with higher yaw-misalignment angles as shown in the numerical study by [57] and in the full-scale experiment by [58]. • Yaw direction: The yaw direction has an influence on the magnitude of wake centre deflection which is connected with the wake rotation caused by the rotation of the rotor as seen in [57] and explained in [59]: Assuming the same yaw angle in the positive and negative direction, the wake centre is deflected more for positive yaw angles compared to negative yaw angles (assuming clockwise rotation of the rotor). • Turbine’s thrust coefficient: Increasing the rotor thrust leads to increased wake centre deflections as reported by [60]. • Downstream distance: In numerical investigations [57] and fullscale tests [61], it is ob- served that larger wake deflection occurs at farther downstream distances. • Ambient TI: With low incoming TI the wake deflection effect is found to be stronger as shown in [59]. • Atmospheric stability: In stable atmospheric conditions, larger wake deflection is observed compared to neutral and unstable conditions [62, 63]. 20 2 Background For larger yaw-misalignmets (> 20◦), the shape of the velocity deficit in the far-wake cannot be considered Gaussian anymore, but it looks more like a kidney due to the presence of a counter-rotating vortex pair [64]. This is observed in Large Eddy Simulation (LES) by [65] [66] and confirmed in wind tunnel experiments by [59]. 2.4 Wind farm control The basic concept of wind farm control is to operate individual wind turbines inside a wind farm in a coordinated way in order to achieve predefined objectives. This is motivated by the wake effects and by the desire to reduce their impact. In general, there are two methods of wind farm control, the axial induction control (see Section 2.4.1) and the wake redirection control (see Section 2.4.2). The application of wind farm control can reduce the LCoE; to be more precise, the objectives of wind farm control can be split into three categories: • Power maximisation: The first goal is power maximisation as it is directly connected with higher revenue. • Load reduction/balancing: Wake effects impose higher structural fatigue on downstream turbines. Wind farm control can reduce the impact of the wakes and to balance structural load distributions between the individual wind turbines inside a wind farm evenly. • Provision of grid services: Wind farms are connected to the electrical grid. With wind farm control, the wind farm can provide services for stabilising the grid. 2.4.1 Axial induction control Static axial induction control aims to mitigate wake effects on downstream turbines by reducing the thrust and thus, decreasing the aerodynamic efficiency of the upstream turbine compared to normal operation. The tip-speed ratio of the upstream turbine is altered by changing the generator torque and/or pitching the blades [67]. In the past, literature reported very limited potential of the method in increasing a wind farm’s total power output; power gains of a few percent or even power loss were observed in high fidelity simulations [68], wind tunnel experiments [69] and fullscale experiments [70]. In more recent work [71, 72], dynamic axial induction control was explored as a variant of Individual Pitch Control (IPC) by introducing modulated cyclic pitching of the blades leading to faster recovery of the wake deficit. First investigations show better results in maximising the total power output of a wind farm albeit coming at the cost of increased fatigue. Axial induction control offers higher potential for structural load alleviation within a wind farm [73]. It can be applied to balance the fatigue consumption of the individual turbines inside 2.4 Wind farm control 21 a wind farm [26]. This is often combined with active power control where, for instance, a given power output of the wind farm is prescribed by the Transmission Grid Operator (TSO) [23]. 2.4.2 Wake redirection control Wake redirection control intends to reduce wake effects on the downstream turbines by redi- recting the wake. Different approaches can be utilised to introduce a deflection of the wake centre position: • Yaw-based redirection control exploits the wake characteristics that evolve during yawed operation of a wind turbine (see Section 2.3.3). Hereby, the turbine’s yaw motor is used as an actuator to intentionally misalign the rotor with respect to the incoming wind, thus redirecting the wake centre position downstream [60]. • The rotor plane can be tilted to deflect the wake centre position up- or downwards [74]. The concept is based on the same principles as yaw-based wake redirection control, but would require a redesign of current state-of-the-art wind turbines. • IPC can be modified in order to induce a moment on the wake flow causing a redirection of the wake centre [57]. An evaluation of the concept in [57] revealed limited potential of this method when purely applied for wake redirection. However, in combination with the control concept of modulated cyclic pitching of the blades mentioned in Section 2.4.1 wake redirection can be triggered as a secondary effect that can potentially make the former strategy even more effective. The most common approach is the yaw-based wake redirection control, hence it is applied in this thesis and discussed in more detail in the following. It is mainly used to maximise the wind farm’s total power output [75]. In high fidelity simulations, power gains of up to 15 % could be achieved for a three turbine setup in certain conditions compared to a non-controlled scenario [18]. This is in line with the results from wind tunnel experiments [76]. In recent field test campaigns, varying results for power maximisation were obtained. For example in [77], in certain conditions, an increase in power of up to 15 % for a three turbine setup and up to 35 % for a two turbine setup were achieved. However, in other scenarios no improvement or even power loss was observed. This was also seen in other fullscale studies, e.g. [78, 79, 80], demonstrating a large uncertainty range in the current implementations. In previous studies, structural load reduction/balancing was rarely a primary objective for yaw-based wake redirection control, e.g. [81, 82], but it was investigated in conjunction with power maximisation or active power control as secondary objective. In general, the space of optimal yaw-angle configurations in a wind farm for power maximisation is reduced, when structural loads are taken into account [83]. It is possible to maximise the collective power 22 2 Background output while reducing the loading due to partial wake overlap [84]. However, possible fatigue load reductions are also dependent on the component and on the implemented cost function, when both power and loads are optimised [85]. Yaw-based wake redirection control has inherently slow actuation times, thus it is only ap- plicable for the provision of certain grid services. It can be applied to provide tertiary control reserves, which for instance require a complete activation within 15 minutes in Germany. First simulation results indicate a general functionality of the concept but also show large uncertain- ties that must be further analysed [27]. Overall, taking the findings from Section 2.3.3 into account, the wake redirection control strategy for wind farms is most effective for the maximisation of the wind farm’s power output under the following circumstances: An increase of the overall power production is only possible in below-rated wind conditions, which corresponds to high thrust conditions. Additionally, the power maximisation at the downstream turbines is potentially higher when the turbines are located slightly misaligned to each other. Furthermore, the strategy is more effective in conditions of low ambient turbulence and for positive yaw-misalignment angles due to the effects from wake rotation. 3 Simulation modelling The modelling approaches are introduced and described. In Section 3.1, the methods for sim- ulating flows in the ABL including turbulence are explained. The modelling of the interaction effects between the ABL and the wind turbine aerodynamics is explained in Section 3.2. The description of the applied aeroelastic wind farm simulation approach is in Section 3.3; in this Section, the modelling approach applied in this thesis is presented. 3.1 Atmospheric boundary layer modelling For the aeroelastic simulation of wind farms, an adequate representation of the ABL and its flow structures is required as input. The aim is to reproduce important characteristics of the ABL such as the stability, turbulence distribution and wind profile. In this thesis, the modelling of ABL flows is performed with a spectral tensor method and LES; they are explained in the following sections. 3.1.1 Mann uniform shear turbulence model The Mann spectral model [86] is one of the recommended models by the IEC 61400-1 Ed. 4 standard [15] to reproduce the turbulence in the ABL. It is a semi-empirical model that is strictly valid for homogeneous turbulence in the neutral atmospheric surface layer. In general, for a homogeneous turbulent velocity field, the correlation of two points in space 24 3 Simulation modelling is only a function of the separation vector r; the covariance tensor Rij can be defined as [87]: Rij(r) = 〈u′i(x) u′j(x + r)〉, (3.1) where the indices {i,j}=1,2,3 denote the velocity components and 〈·〉 takes the ensemble average. The covariance tensor for the single point (r = 0) statistics is: Rij(r = 0) = 〈u ′u′〉 〈u′v′〉 〈u′w′〉 〈v′u′〉 〈v′v′〉 〈v′w′〉 〈w′u′〉 〈w′v′〉 〈w′w′〉  =  σ 2 u σuv σuw σvu σ2 v σvw σwu σwv σ2 w  , (3.2) where the matrix elements represent the variances and covariances of the three-dimensional velocity field. The Fourier transform of the covariance tensor Rij defines the spectral velocity tensor and can be written as: Φij(k) = 1 (2π)3 +∞∫∫∫ −∞ Rij(r) exp (−ik · r) dr, (3.3) with the wave number k = (k1, k2, k3). The spectral velocity tensor contains information about the second order statistics of the three velocity components. The Mann model derives the spectral velocity tensor by using the isotropic von Kármán turbulent energy spectrum (Equation 3.4) as initial condition. It depends on the Kolmogorov spectral constant αk, the rate of viscous dissipation of specific turbulent energy ε and a length scale L. E(k) = αkε 2 3L 5 3 (Lk)4 (1 + (Lk)2) 17 6 (3.4) The initially isotropic spectral tensor is then transformed to an anisotropic spectral tensor using rapid distortion theory and uniform velocity shear. In combination with the assumption on the turbulent eddy’s life-time, the anisotropic spectral velocity tensor reaches a stationary state. The complete derivation of the spectral velocity tensor and the full set of equations are presented in [86]. In total, three parameters are needed to determine the spectral velocity tensor with Mann’s approach: • αε2/3: The product of Kolmogorov’s constant αk and the dissipation rate ε raised to the power of two-thirds. • LM : A length scale that defines the size of the most energy-containing turbulent eddies. • Γ: The non-dimensional anisotropy parameter Γ is related to the eddy lifetime and is 3.1 Atmospheric boundary layer modelling 25 used to describe the degree of anisotropy of the turbulence. When Γ = 0, turbulence is isotropic. With Mann’s spectral velocity tensor, the coherence of a turbulent wind field is described in three dimensions. In order to compare the model with measurements that are usually carried out at one point, the cross-spectra χij between two points in a y-z plane separated by a distance dy and dz can be derived. Hereby, the spectral tensor is numerically integrated over the wave numbers in the directions k2 and k3: χij(k1, dy, dz) = +∞∫∫ −∞ Φij(k, αkε2/3, LM ,Γ) exp (ik2dy + ik3dz) dk2 dk3. (3.5) By setting the distance to zero (dy=0, dz=0) the one-point spectra Fi(k1) = χij(k1, 0, 0) can be calculated. The spectral velocity tensor in combination with the discrete Fourier series can be applied to generate a turbulent wind field; the process is explained in detail in [88]. It is numerically implemented in the turbulence generator [89] which is used in this thesis to generate three- dimensional turbulent wind fields. Model parametrisation In this thesis, three-dimensional wind fields are generated with the Mann model. The model parameters LM and Γ change with the atmospheric stability. They are based on the results from [35], where the Mann model is fitted to measurements from a met mast; it is shown, that the Mann model predicts well the measured spectra from different atmospheric stabilities with an appropriate parameter selection, although the model is derived for neutral conditions. The values of the parameters LM and Γ are summarised in Table 3.1. For the wind field generation, the parameter αε2/3 is modified according to the target TI, thus it depends on the wind speed as well as LM and Γ. The parameters are taken at the height z = 90 m that equals approximately the hub height position of the turbines used in this thesis. The resulting one-point spectra are shown in Figure 3.1. Hereby, the values for αε2/3 are chosen to match the variance of the longitudinal wind component u for each AS: σ2 u,unstable = σ2 u,neutral = σ2 u,stable. It can be seen that an increased length scale LM results in a shift of the spectra towards lower wave numbers. Additionally, increasing the anisotropy parameter Γ leads to lower variances in the lateral σ2 v and vertical σ2 w wind component, hence lower turbulent kinetic energy. 26 3 Simulation modelling Table 3.1: Values of the Mann model parameters used within this thesis Atmospheric stability Length scale LM [m] Anisostropy parameter Γ [-] IEC 61400-1 Ed.4 - 33.6 (for heights ≥ 60 m) 3.9 Høvsøre parameter set at z = 90 m from Figure 4 in [35] unstable 69.2 2.09 neutral 33.1 2.57 stable 11.6 2.79 10-4 10-2 100 k1 (m-1) -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 k 1 F ij unstable AS 10-4 10-2 100 k1 (m-1) neutral AS 10-4 10-2 100 k1 (m-1) stable AS k1 F11 k1 F22 k1 F33 k1 Re(F13) Figure 3.1: One-point spectra Fij for different atmospheric stabilities (AS) calculated with the pa- rameters given in Table 3.1. The parameter αε2/3 is chosen to match the variance of the longitudinal wind component u for all AS: σ2 u,unstable = σ2 u,neutral = σ2 u,stable. 3.2 Wind farm flow modelling 27 3.1.2 Large eddy simulations Large eddy simulations (LES) solve the Navier-Stokes equations in order to simulate turbulent flows. Hereby, the turbulent energy spectrum (see also Figure 2.1) is split in a way that only the large turbulent eddies in a flow are resolved directly, whereas eddies that are smaller than a defined cut-off wave number are modelled with a subgrid-scale model [34]. LES are computa- tionally expensive and require typically high-performance clusters. In this thesis, results from LES performed with the open-source code Parallelised Large-Eddy Simulation Model (PALM) are used as reference for model calibration purposes. The PALM code solves the non-hydrostatic, filtered, and incompressible Boussinesq approx- imation of the Navier-Stokes equations by using central differences on a structured grid. It is particularly suited for the simulation of atmospheric and oceanic flows. More information can be found in [90]. 3.2 Wind farm flow modelling Inside a wind farm, the wakes from individual wind turbines interact with the ABL flow which leads to new inflow conditions at downstream located turbines. With the modelling of wind farm effects, it is aimed to understand the physics and to predict the response of individual wind turbines as well as the wind farm as a whole to various environmental conditions and corre- sponding states. The gained knowledge is applied for the optimisation of turbine design, micro siting and farm operation, thus the uncertainties and the associated technical and economic risks are reduced. The modelling of wind farm flows is complex because of the wide range of turbulent scales that influence the flow dynamics. Therefore, different model complexities are developed: High- fidelity models resolve the underlying physics accurately but require high computational re- sources. Models of lower fidelity introduce simplifications to the underlying physics and/or reduce the temporal and spatial resolution. A comprehensive review on wind farm flow mod- elling is given in [64]. In the following, a brief overview of different model fidelities is provided and the classification of the models used in this thesis is performed; they are then explained in the subsequent sections in more detail. • High fidelity models: They solve the governing Navier-Stokes equations typically with a LES approach (see also Section 3.1.2) or by a hybrid simulation combining LES with the Reynolds-Averaged Navier-Stokes (RANS) method [91]. They resolve the physics of the large turbulent structures directly and are capable of simulating ABL flows with different thermal stratification. There are different methods for the representation of the rotor. The most detailed ap- proach is to model the blade’s geometry directly as solid boundaries in the CFD domain. 28 3 Simulation modelling This requires very fine meshes and normally hybrid LES-RANS or unsteady RANS ap- proaches are applied, where the turbulence in the blade’s boundary layer is modelled. The main purpose is to understand the flow physics around the rotor, in the near-wake and how the far-wake is affected by the near-wake flow. Investigations using a fully re- solved rotor are typically performed for single turbine cases [92, 93] or small wind farm configurations [91]. The Actuator Line (ACL) method replaces the physical representation of the blades by body forces that capture the influence of the blades on the flow. They are calculated by using tabulated blade profile data and the local inflow conditions. Compared to the fully resolved rotor, computational resources are significantly reduced because the blade’s boundary layer is not resolved. Nevertheless, the flow physics in the wake are captured accurately [94], which makes it a reasonable simplification to investigate the wake physics. Examples of analyses of the wake physics are performed for single wind turbines [51, 95] and multiple turbines [96, 97]. The Actuator Disc (ACD) concept is a simplification of the ACL method and introduces an azimuthal averaging of the body forces. This enables a coarser time step, thus a speed-up in simulation time. The calculation of the body forces with the ACD can be, on the one hand, based on tabulated blade profile data. On the other hand, the calculation can be based on the integral thrust force of the rotor, which requires only the turbine’s cT-curve. The flow behind an ACD does not include all the details in the near-wake (e.g. vortex structure) but this is acceptable for the far-wake. Thus, it is typically applied in simulations of large wind turbine clusters [98, 99, 100, 101] or even to study intra-farm effects [102]. • Medium fidelity models: Three different models are classified as medium fidelity models which are physics based but also introduce simplifications to flow features to reduce computational times. For example, models based on the RANS-equations are steady-state and calculate a time averaged solution of the wind farm flow. This implies challenges to the turbulence modelling which can be performed in different levels of detail. Examples of RANS-simulations are provided in [103] and [104]. Another type of medium fidelity models is the free vortex wake model which is based on potential-flow theory [105]. It is capable of simulating the rotor induction and near-wake aerodynamics including the vortex structures but lacks accuracy in the far-wake due to the assumption of inviscid flow. The third type is the DWM model, which is mainly used in this thesis and explained in detail in Section 3.3.3. It introduces a split in the turbulent scales and treats small- scale turbulence based on a reduced form of the Navier-Stokes equations and large-scale turbulence based on the passive tracer analogy suggested by [11]. It requires the use 3.2 Wind farm flow modelling 29 of three-dimensional turbulent wind fields and calculates the wind farm response in the time domain. This makes it an ideal model for aeroelastic wind farm calculations while keeping the computational time in the order of minutes (for 5-10 turbines on a normal desktop computer). • Low fidelity models: They are typically based on some form of reduced order physics to achieve fast computational run times in the order of (milli-)seconds (for large wind turbine clusters on a normal desktop computer). They calculate typically a two-dimensional steady state solution of the flow field and involve empirical calibration constants that need to be changed for different operational conditions. Examples are the Jensen wake model [106] and the Gaussian wake model [107]. The latter is applied in this thesis and discussed in Section 3.2.1. An example of a dynamic reduced-order model is given in [108]. 3.2.1 Steady-state low-fidelity flow simulation A low-fidelity wind farm flow simulation model is usually required in the context of wind farm design or optimisation of operational strategies to simulate many parameter variations. In this thesis, the FLOw Redirection and Induction in Steady State (FLORIS) wind plant optimisation tool [109] in the version 2.4.0 developed by the National Renewable Energy Laboratory (NREL) is used. It includes a variety of models to calculate the velocity deficit behind a wind turbine as well as the wake-centre deflection caused by yaw-misaligned operation. In this thesis, the Gaussian velocity deficit and wake-centre deflection model is used. Its full description is given in [110]. The model is based on the research in [107, 51, 111, 59], but it includes secondary effects from wake steering. Secondary steering means the deflection of the wake of a downstream, aligned turbine caused by the counter-rotating vortices developed behind an upstream turbine under yawed operation. The model is derived for the far-wake region and uses calibrated boundary conditions from the near-wake region. According to [59], it should be applied for yaw-misalignment angles that are smaller than |θm| < 30°. The authors in [59] reason this limitation with the usage of the Gaussian distribution for the velocity profile that is not valid for high yaw angles due to the kidney shape of the velocity profile in the z-direction (see Section 2.3.3). The Gaussian velocity deficit and wake-centre deflection models are introduced in the follow- ing. Several calibration parameters are used within these models; their tuning is performed in Section 4.4. 30 3 Simulation modelling Gaussian velocity deficit model The velocity deficit udef (x, y, z) = u(x, y, z)/u∞ in the far wake region behind a wind turbine is approximated with a Gaussian function, u(x, y, z)/u∞ = 1− C exp (y − δ)2 2σ2 y exp −(z − zh)2 2σ2 z , (3.6) with the velocity deficit at the wake center C: C = 1− √ 1− (σy0σz0)cT σyσz . (3.7) Here, δ is the lateral wake deflection from the rotor centre, zh is the hub height of the turbine, cT is the turbine’s thrust coefficient, σy and σz define the wake width in the y- and z-directions respectively. The subscript “0” attributes the initial values at the start of the far-wake; for the calculation of these values which depend on the ambient TI and cT, the reader is referred to [59]. The Cartesian coordinate system (x,y,z) has its origin in the rotor centre of the wake-emitting turbine. The wake width in the y- and z-direction (σy and σz) becomes wider with increasing down- stream distance due to the turbulent mixing with the free-stream flow. It is calculated by using the equations σy D = ky x− x0 D + σy0 D , (3.8) σz D = kz x− x0 D + σz0 D , (3.9) where D is the rotor diameter and ky and kz are parameters that define the expansion rate of the wake. In the current implementation of FLORIS, ky equals kz leading to the same expansion rate in the lateral and vertical directions: ky = kz = ka TI + kb, (3.10) where ka and kb are calibration constants. The calculation of the resulting wind speed profile from multiple wakes is based on the sum of squares method presented in [112]. 3.2 Wind farm flow modelling 31 Gaussian wake-centre deflection model The lateral position of the wake centre δ with respect to the rotor centre behind a turbine operating with a yaw-misalignment can be calculated with the equation derived in [59]: δ = δ0 + θmE0 5.2 √ σy0σz0 kykzcT ln (1.6 +√cT ) ( 1.6 √ σyσz σy0σz0 −√cT ) (1.6−√cT ) ( 1.6 √ σyσz σy0σz0 +√cT ) , (3.11) where θm is the turbine’s yaw-misalignment angle (see Figure 2.5) and E0 = C2 0 − 3e1/12C0 + 3e1/3 with C0 indicating the initial far-wake velocity deficit at the wake centre. The initial wake-centre deflection at the start of the far-wake δ0 that is developed in the near-wake is calculated with δ0 = x0 tan θw,rotor, (3.12) where x0 is the length of the near-wake and θw,rotor is the initial wake deflection angle at the rotor due to yaw-misalignment, which is approximated by θw,rotor ≈ 0.3 θm cos θm ( 1− √ 1− cT cos θm ) . (3.13) 3.2.2 Coupled large eddy simulations The PALM code (see Section 3.1.2) is capable of performing wind farm simulations by including the wind turbines with actuator discs or actuator lines. The aerodynamic forces exerted by the wind turbines are applied as momentum sources in the discretised Navier-Stokes equations by projecting them onto the underlying grid. The results from the PALM simulations used in this thesis are based on an actuator sector model that is coupled to the aeroelastic simulation code Fatigue, Aerodynamics, Structures, and Turbulence (FAST) (see Section 3.3.2). At the turbine’s location, the flow field velocities are transferred from PALM to FAST which calculates the resulting blade forces. The aerodynamic forces are then distributed on the actuator disc in PALM, which is divided into radial and azimuthal segments to consider spatial variations. The azimuthal segmentation of the rotor allows the use of different time steps depending on the azimuthal span of the segments, thus it increases the simulation efficiency compared to a rotor representation by actuator lines. A more detailed description of the coupling method is given in [113]. The simulations results are produced with PALM in the revision 3401 and FAST in the version FASTv8. The aerodynamic force calculation in FAST is conducted with the module Aerodyn in the version 14. 32 3 Simulation modelling 3.3 Aeroelastic wind farm simulation For the full numerical assessment of wind turbines located in wind farms, an aeroelastic sim- ulation model is required that accounts for wake interaction effects on turbine performance and structural loading. In this thesis, aeroelastic wind farm simulations are performed with the multiphysics tool FAST.Farm developed by NREL. FAST.Farm adds functionalities for the calculation of wake effects to the existing simulation tool OpenFAST, which solves the aero-hydro-servo-elasto dynamics of individual turbines. An overview of FAST.Farm as well as explanations of important features and model extensions are presented in the following sections. A comprehensive description of FAST.Farm can be found in [12] and in the user’s guide and theory manual [114]. 3.3.1 FAST.Farm implementation The general structure of FAST.Farm is modular and the different calculation steps of a wind farm simulation are performed in dedicated submodules. A governing driver code interconnects the submodules, enables information exchange between the submodules and drives the time- domain solution forward. In particular, the following submodules exist: • OpenFAST: Each turbine inside a FAST.Farm simulation is modelled with OpenFAST. It solves the aero-hydro-servo-elastic dynamics for an individual turbine and is explained in more detail in Section 3.3.2. • Wake Dynamics (WD): For each rotor, the WD module calculates the wake effects in- cluding wake advection, deflection and meandering. The implementation is based on the DWM model and is described in Section 3.3.3. • Ambient Wind and Array Effects (AWAE): This module processes ambient wind and wake interactions across the wind farm domain. It includes a wake-merging submodel, which identifies zones of wake overlap and calculates the resulting wake deficit of multiple wakes. Wind fields can be used from a synthetic turbulence generator (e.g. TurbSim or the Mann model) or from a high-fidelity LES. • Super Controller (SC): The SC module allows the connection to an external controller library that calculates wind farm wide setpoints for the individual turbines. The module enables the communication of such a wind farm controller with each turbine in OpenFAST. All presented results within this thesis were produced with the governing version 2.4.0 of the OpenFAST framework. The specific version of FAST.Farm, which includes the implementation of a wake-added turbulence model (see Section 3.3.4), is made available under [115]. 3.3 Aeroelastic wind farm simulation 33 3.3.2 Aeroelastic simulation of a single wind turbine The dynamic response of a wind turbine can be calculated by means of a coupled Multibody Simulation (MBS) model. The MBS methodology is applied in the simulation software Open- FAST [116], which is used for the load calculations of single wind turbines in this thesis as well as for the load calculations of individual turbines inside a FAST.Farm simulation. Multibody simulation In general, a MBS is used to calculate the kinematics and dynamics of multiple bodies that experience large displacements with respect to each other. The bodies can be rigid or flexible and are connected with joints. External forces can be applied on the MBS model to represent excitation from, for instance, aerodynamic forces. In this work, the blades and the tower are modelled as flexible bodies. The flexibility is de- scribed using a linear modal representation. The structural properties are specified by stiffness and mass distributions along the span of the members. In addition, for each flexible mode, mode shape functions are defined as polynomials, which are generated beforehand by a finite-element code. The flexibility of the blades is defined with two flapwise and one edgewise bending-mode as Degrees of Freedom (DoFs). For the tower, two bending-modes are used each in the fore-aft and side-side directions. In OpenFAST, external forces from aerodynamics, hydrodynamics and controller dynamics are computed in dedicated submodels and applied to the equations of motion. Rotor aerodynamics Rotor aerodynamics are modelled using the Blade Element Momentum (BEM) theory, which is commonly accepted as state-of-the-art. It calculates on the one hand the local forces at the blade elements distributed over the radius based on the local inflow conditions and airfoil characteristics (e.g. lift- and drag-polars). On the other hand, the induced velocities at the rotor are derived by using an one-dimensional stream tube, satisfying momentum conservation. The relation of the local blade forces and the global momentum conservation is solved iteratively until an equilibrium is reached. The BEM is implemented in the Aerodyn v15 code that is a submodule of OpenFAST. For the aerodynamic calculation of wind turbine rotors, several correction models are usually applied to overcome the limitations that are present in the original BEM theory. For example, they include the correction for root and tip loss effects, the inclusion of wake dynamics and the correction for the turbulent wake state. For wake redirection control, the correction models for yawed inflow and dynamic stall are considered as especially relevant. They are discussed here in more detail: 34 3 Simulation modelling • Yawed inflow: In yawed operation, the wake behind the rotor cannot be assumed to be axisymmetric (see Section 2.3.3), thus the induction distribution over the rotor is inhomogeneous. In this case, pure BEM theory is not valid and a correction model becomes necessary to calculate the induction distribution over the rotor. It should capture decrease of the induced velocity towards the leading edge of the inclined rotor and the increase towards the trailing edge of the rotor. For the aerodynamic calculation in this work, the correction model for yawed inflow is based on Glauert, which states ayaw = a ( 1 +K(χ) r R sin(ψ) ) , (3.14) where ayaw is the corrected local induction factor, a is the induction from axisymmetric momentum balancing, K is a model dependent empirical factor that is normally depen- dent on the wake-skew angle χ and ψ is the azimuthal rotor position. The complete model is described in [117] and defines the factorK(χ) based on the Pitt and Peters model. In addition, it takes into account the effects on both axial and tangential induction. • Dynamic stall: The non-axisymmetric induction distribution over the rotor during yawed operation leads to cyclic variations of the local blade-airfoil aerodynamics (similar be- haviour is observed from wind shear and rotor tilt). The changing local inflow angle can lead to a separation of the vortices that causes a hysteresis of the lift coefficient and is referred to as dynamic stall. In order to account for the dynamic stall effect, the Beddoes-Leishman model is used with extensions from [118] and [119]. The details of the model are quite complicated, hence the reader is referred to [120] who describe the model and its implementation in the Aerodyn v15 code. 3.3.3 Wake dynamics: Implementation of the dynamic wake meandering model The calculation of wake aerodynamics in FAST.Farm is based on the DWM model, which was originally introduced in [11]. It essentially splits the turbulent scales present in ABL flows into two parts: large turbulent eddies (greater than two rotor diameters) affect the wake meandering (see Section 2.3.2) and small turbulent eddies (less than two rotor diameters) have an influence on the wake deficit evolution. The core parts of the original DWM model include the calculation of the wake deficit, the wake meandering and the wake-added turbulence. In FAST.Farm, additional models were developed to consider the wake deflection due to yawed operation and to correct the wake-deficit calculation in the near-wake. The aforementioned models are described in the following. The wake-added turbulence model is implemented into 3.3 Aeroelastic wind farm simulation 35 FAST.Farm within this work and is explained in Section 3.3.4. The DWM model is a semi-empirical model which is based on physics but also includes simplifications of the governing equations that require the use of calibration parameters. The calibration of the parameters used in FAST.Farm is performed against LES and is explained in more detail in [121] as well as in [114]. The calibrated default values are not changed in this thesis unless explicitly specified. Wake meandering The DWMmodel treats the wake meandering (see also Section 2.3.2) in a way that the wake acts as a passive tracer following the large turbulent structures in the ABL flow. In order to model wake meandering, axial, transversal and vertical wake transport velocities are derived. These velocities essentially define the movement of the wake planes in the Fixed Frame of Reference (FFoR) and depend on the local flow features that are varying with changing downstream distances. In FAST.Farm, they are achieved by calculating the weighted spatial average of the wind velocity within the circle having the diameter of CMeander DW around the wake centre on a wake plane; DW is the wake diameter and CMeander is a calibration constant which is set to 2 in the original definition of the DWM model [11] following from the characteristic dimension for wake meandering. The wake transport velocity vector is: vplane = N∑ n=1 wn vwind,n N∑ n=1 wn , (3.15) where vplane is the vector giving the wake transport velocities for a wake plane; N is the total number of considered points within the circle; vwind is the vector defining the wind speed components at point n. Different implementations for the spatial weighting factor wn are available in FAST.Farm, wn = 1 provides a simple spatial average. Wake d