UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
PROGRAMA DE MAESTRÍA Y DOCTORADO EN INGENIERÍA
INGENIERÍA CIVIL – HIDRÁULICA
ANCLAJE DE TURBINAS SUMERGIDAS Y FLOTANTES PARA OBTENCIÓN DE
ENERGÍA DE CORRIENTES MARINAS EN REGIONES DE FONDOS INCLINADOS
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN INGENIERÍA
PRESENTA:
MICHAEL RING
TUTOR PRINCIPAL:
DR. EDGAR GERARDO MENDOZA BALDWIN, INSTITUTO DE INGENIERÍA, UNAM
COMITÉ TUTOR:
DR. MOISÉS BEREZOWSKY VERDUZCO, INSTITUTO DE INGENIERÍA, UNAM
DR. RODOLFO SILVA CASARÍN, INSTITUTO DE INGENIERÍA, UNAM
DR. ÓSCAR POZOS ESTRADA, INSTITUTO DE INGENIERÍA, UNAM
DR. ISMAEL DE JESÚS MARIÑO TAPIA, ENES-MÉRIDA, UNAM
CIUDAD UNIVERSITARIA, CD. MX., 06 2023
 
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Tesis Digitales 
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JURADO ASIGNADO:
Presidente: Dr. Berezowsky Verduzco Moisés
Secretario: Dr. Silva Casarín Rodolfo
1er. Vocal: Dr. Mendoza Baldwin Edgar Gerardo
2do. Vocal: Dr. Pozos Estrada Óscar
3er. Vocal: Dr. Mariño Tapia Ismael De Jesús
Lugar o lugares donde se realizó la tesis: CIUDAD DE MÉXICO, MÉXICO
TUTOR DE TESIS
DR. EDGAR GERARDO MENDOZA BALDWIN
FIRMA

Acknowledgments
To my wonderful wife Paola, thank you for your unwavering support and patience during this time.
I could not have completed this work without the relentless support of my tutor Dr. Edgar Gerardo
Mendoza Balwin and the head of the department Dr. Rodolfo Silva Casarín, who always had time
and patience for questions, no matter how trivial they seemed to be, for the credit of trust, and for
introducing me to the topic.
The assistance provided by the personal of the laboratory during experiments was greatly appreci-
ated.
I am grateful to Julio Candela and Julio Sheinbaum for permission to use their empirical data,
and to the CEMIE-Océano, in particular to Dr. Ismael de Jesús Mariño Tapia, for gathering and
processing valuable data during the research expedition CEMIE-1 on board B.O. Justo Sierra.
I would also like to thank my family, especially my parents, who were always supportive and be-
lieved in me whichever path I took.
To my friends who were always there to remind me to have fun even in the most stressful times,
thank you.
I also appreciate very much the financial support provided by the Consejo Nacional de Ciencia y
Tecnología (CONACyT) and the CEMIE-Océano.

Contents
List of Figures xiii
List of Tables xvii
Nomenclature - Latin Letters xix
Nomenclature - Greek Letters xxvii
Abbreviations xxix
Abstract xxxi
Resumen xxxii
1 Introduction 1
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Structure of the Document . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Mooring Systems 5
2.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Line Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mooring Line Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Wire Rope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Synthetic Fiber Rope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Mooring Line Accessories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Acoustic Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Clump Weights and Buoyancy Aids . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Anchors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 Drag Embedment Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Pile Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
vii
viii CONTENTS
2.4.3 Micropile anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.4 Suction Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.5 Gravity Installed Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.6 Dead Weight Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.7 Screw Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Relevant Design Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Design Methodology 15
3.1 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Mooring Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Anchors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Design Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 Structural Design of Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.2 Design of the Mooring System . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.3 Fatigue Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.4 Limit States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Commercial Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 ProteusDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.2 Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Quasi-Static Mooring Analysis 25
4.1 Object Types for Modeling the Mooring System . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.2 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.3 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.4 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Linearized Equation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4.2 Material of Mooring Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.4 Experimental Series A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.5 Experimental Series B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5 Validation of Line Tensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Genetic Algorithm 47
5.1 Single Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.1 k-Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1.2 Recombination and Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
CONTENTS ix
5.2 Multi Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.1 Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Creating Next Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Site Characterization 57
6.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.1.1 Research Expedition CEMIE-1 . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.1.2 HYCOM Consortium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.1.3 CANEK Research Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.1.4 ERA 5 Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.1.5 HURDAT2 Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Estimation of Extreme Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2.1 Peaks over Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2.2 Extreme Annual Wind Speeds due to Hurricanes . . . . . . . . . . . . . . . . 62
6.2.3 Expected Wave Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2.4 Spacial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 Meteorological and Oceanographic Data . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3.1 Extreme Wind Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3.2 Significant Wave height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.3 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3.4 Bathymetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3.5 Storm Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.6 Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.7 Sea Level Rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.4 Geological and Geotechnical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7 System Design 81
7.1 General Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.1.1 Height-control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.1.2 Rotor Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.1.3 Buoyancy Providing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1.4 Elevation System for Maintenance . . . . . . . . . . . . . . . . . . . . . . . . 86
7.1.5 Mooring Line Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Design Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2.1 Turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2.2 Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.2.3 Mooring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2.4 Elevation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2.5 Internal Structure of Platform . . . . . . . . . . . . . . . . . . . . . . . . . . 94
x CONTENTS
7.2.6 Compartments and Layout of Platform . . . . . . . . . . . . . . . . . . . . . 95
7.2.7 Mooring Winch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3.1 Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3.2 Additional Environmental Conditions . . . . . . . . . . . . . . . . . . . . . . 99
7.3.3 Requirements of Limit States . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8 Optimization of Design Concept 103
8.1 Parameter Subject to Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.1.1 Principals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.1.2 Mooring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.1.3 Structural . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.1.4 Ballast Tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2.1 Preliminary Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2.2 Quasi Static Mooring Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2.3 Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.3 Automated Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.4 Manual Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.4.1 Three-Mooring Line Configuration . . . . . . . . . . . . . . . . . . . . . . . . 114
8.4.2 Four-Mooring Line Configuration . . . . . . . . . . . . . . . . . . . . . . . . . 122
9 Review of Final Design 129
9.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.1.1 Dynamic Analysis of Mooring System . . . . . . . . . . . . . . . . . . . . . . 129
9.1.2 Estimation of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.1.3 Structural Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.1.4 Design of Pile Anchor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.1.5 Fatigue Life of Mooring Lines and Components . . . . . . . . . . . . . . . . . 137
9.2 Dynamic Mooring Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.3 Natural Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.4 Fatigue Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.4.1 Low Frequency Tension Range . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.4.2 Probabilities of Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.4.3 Fatigue Life of System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.5 Anchor Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.6 Required Mooring Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10 Conclusions 153
10.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
CONTENTS xi
Bibliography 159
Appendices 169
A Quasi-Static Mooring Analysis 171
A.1 Example for Usage of OffshorePy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.1.1 Simple Mooring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.1.2 Multi-line Mooring System with Segmented Lines . . . . . . . . . . . . . . . . 173
A.1.3 Python Code to Generate OffshorePy model for Verification . . . . . . . . . . 175
A.2 Complementary Data for Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B Dimensions of structural members 181
C Publication in Frontiers in Marine Science 187

List of Figures
1.1 Location of the Cozumel Channel within the Mexican Caribbean. . . . . . . . . . . . 2
2.1 Typical profiles of mooring lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Common layouts of strands in wire ropes. . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 Flow chart of general design procedure. . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 Concept of single lline mooring case for verification of quasi-static mooring analysis
package OffshorePy. Not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Free body diagram of objects of single line mooring case. a: Free body diagram of
buoy; b: Free body diagram of mooring line; c: Free body diagram of anchor point.
Not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 General experimental setup for validation of numerical mooring analysis. . . . . . . . 33
4.4 Experimental setup for series A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 Current speeds measured during experiment series A. . . . . . . . . . . . . . . . . . 37
4.6 Comparison between numerical prediction of buoy position and experimental obser-
vation for experiment series A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.7 Deviation of z-position of the buoy for experiment series A . . . . . . . . . . . . . . 38
4.8 Experimental setup for series B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.9 Current speeds measured during experiment series B. . . . . . . . . . . . . . . . . . . 41
4.10 Comparison between numerical prediction of buoy position and experimental obser-
vation for experiment series B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.11 Comparison of z-position of the buoy between numerical prediction and experimental
observation for experiment series B. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.12 Comparison of line tensions between Dynamoor and OffshorePy. . . . . . . . . . . . 44
4.13 Water surface of the channel for different pump frequencies. . . . . . . . . . . . . . . 44
5.1 Process of optimization with a single objective genetic algorithm. . . . . . . . . . . . 48
5.2 Process of optimization with a multi objective genetic algorithm. . . . . . . . . . . . 53
5.3 Evaluation of genetic algorithm with Griewank-function. . . . . . . . . . . . . . . . . 55
xiii
xiv LIST OF FIGURES
6.1 Sections of measurement within the the Cozumel Channel carried out during research
expedition CEMIE-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Storms and hurricanes considered for statistical analysis. . . . . . . . . . . . . . . . . 64
6.3 Study on time window length for filtering peaks of data on wind at 10 m height.
a) Number of found peaks; b) Relative difference in number of found peaks to
subsequent time window length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4 Identified peaks, together with the limits between peaks and outliers (horizontal
lines) and occurrences of hurricanes (vertical lines). . . . . . . . . . . . . . . . . . . 66
6.5 Diagnostic plots for the GPD excess model fit. . . . . . . . . . . . . . . . . . . . . . 67
6.6 Study on time window length for filtering peaks of data on significant wave height. . 68
6.7 Identified peaks of significant wave height together with the limits between peaks
and outliers (horizontal lines) and close hurricanes (vertical lines). . . . . . . . . . . 69
6.8 Diagnostic plots for the GPD excess model fit to the significant wave height. a) p-p
plot; b) q-q plot; c) return level plot; d) density plot. . . . . . . . . . . . . . . . . . 70
6.9 Histogram of current as modeled with HYCOM. . . . . . . . . . . . . . . . . . . . . 71
6.10 Study on time window length for filtering peaks of data on current speed at surface.
(a) Number of found peaks (b) Relative difference in number of found peaks to
subsequent time window length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.11 Identified peaks of current speed at surface together with the limits between peaks
and outliers (horizontal lines) and close hurricanes (vertical lines). . . . . . . . . . . 73
6.12 Diagnostic plots for the GPD excess model fit to the current speed at surface. a)
p-p plot; b) q-q plot; c) return level plot; d) density plot. . . . . . . . . . . . . . . 74
6.13 Study on time window length for filtering peaks of data on current speed at a depth
of 50 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.14 Identified peaks of current speed at a depth of 50 m. . . . . . . . . . . . . . . . . . . 76
6.15 Diagnostic plots for the GPD excess model fit to the current speed at a depth of 50 m. 77
6.16 Position of the used transect in the Cozumel Channel and the position of data points
used for interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.17 Bathymetry of the transect in the Cozumel Channel. . . . . . . . . . . . . . . . . . . 79
7.1 Concepts for height control systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2 Example for a single hull system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 Example for a double hull system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4 Example for a triple hull system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.5 Exemplary cross-section of strut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.6 Mooring concepts used as basis for optimization. a) Three mooring line configura-
tion; b) Four mooring line configuration. . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.7 Procedure for reaching design depth after installation or service. . . . . . . . . . . . 94
7.8 Sketch of space frame of internal structure of the platform. . . . . . . . . . . . . . . 94
7.9 Layout of platform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
LIST OF FIGURES xv
8.1 Top view of the port side of the platform concept. . . . . . . . . . . . . . . . . . . . 104
8.2 Front view of the platform concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3 Concept of the mooring systems for optimization. . . . . . . . . . . . . . . . . . . . . 106
8.4 Possible configurations of space frame of hulls for the optimization algorithm. . . . . 107
8.5 Possible configurations of space frame of the strut for the optimization algorithm. . . 107
8.6 Plot of exemplary three mooring line configuration of the system. . . . . . . . . . . . 110
8.7 Plot of exemplary four mooring line configuration of the system. . . . . . . . . . . . 110
8.8 Performance of complete gene pool of mooring concepts. . . . . . . . . . . . . . . . . 112
8.9 Parametric study on G146I007 in three-mooring line configuration. . . . . . . . . . . 115
8.10 Parametric study on G146I007 in three-mooring line configuration (cont.). . . . . . . 116
8.11 Parametric study on G146I007 in three-mooring line configuration (cont.). . . . . . . 117
8.12 Parametric study on manually optimized G146I007 in three-mooring line configuration.118
8.13 Parametric study on manually optimized G146I007 in three-mooring line configura-
tion (cont.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.14 Parametric study on manually optimized G146I007 in three-mooring line configura-
tion (cont.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.15 Parametric study on G111I007 in four-mooring line configuration. . . . . . . . . . . . 123
8.16 Parametric study on G111I007 in four-mooring line configuration (cont.). . . . . . . 124
8.17 Parametric study on manually optimized G111I007 in four-mooring line configuration.125
8.18 Parametric study on manually optimized G111I007 in four-mooring line configuration
(cont.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.1 Mooring offset of platform in three mooring line configuration. . . . . . . . . . . . . 140
9.2 Mean rotation of platform in three mooring line configuration. . . . . . . . . . . . . 141
9.3 Maximum tension in mooring lines of platform. . . . . . . . . . . . . . . . . . . . . . 142
9.4 Results of free decay test to determine natural frequency of the system. . . . . . . . 144
A.1 Resulting mooring system of single line example. . . . . . . . . . . . . . . . . . . . . 173
A.2 Resulting mooring system of multi line example. . . . . . . . . . . . . . . . . . . . . 175

List of Tables
3.1 Selection guidelines for anchors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 Input data for calculations of single line mooring case. . . . . . . . . . . . . . . . . . 32
4.2 Relative error of numerical calculation with respect to analytical calculation of single
line mooring case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Measurements of for determining elasticity of mooring line. . . . . . . . . . . . . . . 34
4.4 Experiment matrix series A for validation of numerical mooring analysis software. . 36
4.5 Experiment matrix series B for validation of numerical mooring analysis software. . . 40
6.1 Return periods of extreme wind speeds due to hurricanes. . . . . . . . . . . . . . . . 64
6.2 Return levels of extreme wind speeds at height of 10 m. . . . . . . . . . . . . . . . . 66
6.3 Return levels of extreme significant wave height. . . . . . . . . . . . . . . . . . . . . 71
6.4 Return levels of extreme current speed at surface. . . . . . . . . . . . . . . . . . . . . 73
6.5 Return levels of extreme current speed at a depth of 50 m. . . . . . . . . . . . . . . . 76
7.1 Expected and estimated principal data of a reference 5 m-turbine. . . . . . . . . . . . 90
7.2 Complete load case matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Partial load safety factors for ultimate limit state (ULS) and a safety level of SL2. . 100
8.1 Diameter and thickness range of structural members. . . . . . . . . . . . . . . . . . . 108
8.2 Filtered design proposals and their performance for the three-mooring line configu-
ration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.3 Filtered design proposals and their performance for the four-mooring line configuration.114
8.4 Parameter changed during manual optimization of three-mooring line configuration . 121
8.5 Parameter changed during manual optimization of four-mooring line configuration . 127
9.1 Current amplitudes for low frequency fatigue life analysis. . . . . . . . . . . . . . . . 145
9.2 Tensions for fatigue life analysis of forward starboard mooring line. . . . . . . . . . . 145
9.3 Tensions for fatigue life analysis of forward port mooring line. . . . . . . . . . . . . . 145
9.4 Tensions for fatigue life analysis of aft center mooring line. . . . . . . . . . . . . . . . 146
9.5 Tensions for fatigue life analysis of aft starboard mooring line. . . . . . . . . . . . . . 146
xvii
xviii LIST OF TABLES
9.6 Tensions for fatigue life analysis of aft port mooring line. . . . . . . . . . . . . . . . . 146
9.7 Adjusted probabilities for load cases for fatigue life analysis. . . . . . . . . . . . . . . 147
9.8 Annual fatigue damage estimation of forward starboard mooring line. . . . . . . . . 147
9.9 Annual fatigue damage estimation of forward port mooring line. . . . . . . . . . . . 148
9.10 Annual fatigue damage estimation of aft center mooring line. . . . . . . . . . . . . . 148
9.11 Annual fatigue damage estimation of aft starboard mooring line. . . . . . . . . . . . 148
9.12 Annual fatigue damage estimation of aft port mooring line. . . . . . . . . . . . . . . 148
9.13 Loads for the design of a suitable pile anchor. . . . . . . . . . . . . . . . . . . . . . . 150
A.1 Statistical data on the current measured during the experiment series A. . . . . . . . 177
A.2 Signed relative error of experimental series A. . . . . . . . . . . . . . . . . . . . . . . 178
A.3 Statistical data on the current measured during the experiment series B. . . . . . . . 179
A.4 Signed relative error of experimental series B. . . . . . . . . . . . . . . . . . . . . . . 180
B.1 Diameter and thickness range of structural members. . . . . . . . . . . . . . . . . . . 181
B.1 Diameter and thickness range of structural members. . . . . . . . . . . . . . . . . . . 182
B.1 Diameter and thickness range of structural members. . . . . . . . . . . . . . . . . . . 183
B.1 Diameter and thickness range of structural members. . . . . . . . . . . . . . . . . . . 184
B.1 Diameter and thickness range of structural members. . . . . . . . . . . . . . . . . . . 185
Nomenclature - Latin Letters
Symbol Description
AS Swept area by turbine rotor
As Side surface area, outer surface area of the shaft
Ablst Cross-sectional area of a ballast tank
ablst Apothem of the cross-section of a ballast tank
Ac Cross sectional area
B̃ Net buoyancy force
CP Power coefficient
CQ Torque coefficient
CT Thrust coefficient
CD Drag coefficient
cMBL Coefficient for minimum breaking load
cshell Minimal clearance between outer shell and internal compo-
nents
cstruc Minimal clearance between structure and other components
d Diameter
dh Hamming distance
da Diameter of the anchor
dblst Outer diameter of rounded section of a ballast tank
dbt Outer diameter of a buoyancy tank
DF Expected annual fatigue damage of component
DF Design factor
DF,i Expected annual fatigue damage of component per environ-
mental state i
dH Outer diameter of the hulls
dI Outer diameter of longitudinal members in the hull’s space
frame (i.e., structural member type I)
dII Outer diameter of diagonal members in the hull’s space
frame (i.e., structural member type II)
di,j Euklidian distance between i and j
xix
xx Nomenclature - Latin Letters
Symbol Description
E Young’s modulus of the material used
wi Weight for parameter i
EA Axial stiffness per unit length
eabs Absolute error
Eir Initial Young’s modulus of rock
erel Relative error
Es Young’s modulus of steel
fi Fitness of an individual for parameter i
f Fitness or performance of an individual
f Frequency
f Generic function
Fd Design load
Fk Characteristic load
FA Anchor holding capacity
~FA Residual force on anchor
fA Factor for considering alteration of rock
FB Force due to buoyancy of the object
FD Drag force
fD Factor for considering rock quality designation
~FE External force on object
~FE,i External force i on object
fexp Value of experimental observations
f̄exp Mean value of experimental observations
f̄exp,0 Mean value of experimental observations for repetitions in
static equilibrium
~FF Residual force on fairlead
~FF,i Residual force on i-th fairlead
Fh Horizontal load
fi,lf Zero up-crossing frequency of the tension spectrum in en-
vironmental stat i
FMBL Minimum breaking load
~FML,i Residual force of mooring line i at one end
fn Natural frequency
fnum Value of numerical calculations
fnum,0 Value of numerical calculations for repetitions in static
equilibrium
~FR Residual force on the object
~FR,ML0 Residual force on mooring line at end 0
~FR,ML1 Residual force on mooring line at end 1
Nomenclature - Latin Letters xxi
Symbol Description
Fv Vertical load
FW Force due to weight of the object
g Gravitational acceleration
H Wave height
h Spacing between adjacent nodes of a uniform grid
h Outstand of shear key
h Water depth
I Second moment of area
i Variable for iteration
J Entry of the inertia matrix
j Variable for iteration
J1 Moment of inertia about its first principal axis
J2 Moment of inertia about its second principal axis
J3 Moment of inertia about its third principal axis
K Global stiffness matrix
k Intercept of T-N curve for fatigue life of mooring compo-
nents
k k-factor of genetic algorithm
kir Initial linear stiffness of the rock according to p-y curves
rrm Constant describing properties of rock
L Characteristic length
l Length
L′
BST Ratio between length of the ballast tanks and length of
buoyancy tanks
Le Embedded length of anchor
LBST Length of the ballast tanks
LBT Length of the buoyancy tanks
LF Fatigue life in years
LN Length of nose section
LOA Length over all
ls Stretched length of the mooring line
LS Length of the structure inside the hulls
LT Length of tail section
lus Unstretched length of the mooring line
m Slope of T-N curve for fatigue life of mooring components
m Mass
M0 Moment at groundline
mMLs Mass of all mooring lines
mP L Mass of platform
xxii Nomenclature - Latin Letters
Symbol Description
~MR Residual force on the body
n Shape parameter of nose section
n Number/count
n Sample size
nBLLST Number of ballast tanks
nk Number of positives/occurrences in sample
ngen Number of the generation
nH Horizontal coefficient of subgrade reaction
ni,lf Number of low frequency tension cycles per year for envi-
ronmental state i
no Number of the objectives for optimization
~P Vector of external forces of linearized equation system
P Mechanical power at the turbines shaft
~p Position in global reference frame
p Power factor for inverse distance weighted interpolation
~p0z z-position of end point 0 of the mooring line
~p1z z-position of end point 1 of the mooring line
~pA Position of anchor
~pF Position of fairlead
Pi Probability of occurrence of load case
~pRB Position of rigid body in global reference frame
pur Ultimate resistance of the rock
pvs Vertical pressure for sinking in rock
Q Torque
Qf Ultimate holding capacity due to friction of the shaft
Qha Ultimate axial holding capacity of the anchor
Qhl Lateral holding capacity of the anchor
Qsa Ultimate structural capacity of the anchor in axial direction
Qsl Ultimate structural capacity of the anchor in lateral direc-
tion
qt,r Tensile strength of rock
qu,g Unconfined compressive strength of grout
qu,r Compressive strength of rock
R Rotation matrix
R Ratio of tension range (double amplitude) to reference
breaking strength
~r Position in local reference frame
R Radius of rotor at blade’s tip
Rd Design resistance
Nomenclature - Latin Letters xxiii
Symbol Description
Rk Characteristic resistance
~rcob Position of center of buoyancy
~rcog Position of center of gravity
~rE,i Position of external force i
~rF,i Position of fairlead i
ri Inner radius
Rn Reynolds number
rn Node radius of the space frame of the hulls
rN Nose radius
ro Outer radius
rT Tail radius
S Final fitness score
s Lithoclase spacing of the rock
s Spacing or pitch height of shear keys
Sk Characteristic load effect
SASF Adjusted safety factor
Sbt Outer surface of a buoyancy tank
sc Scale of k-factor of genetic algorithm
SF Safety factor as required by the limit sate
SH Surface of a hull
Ri,lf Ratio of low frequency standard deviation tension range to
reference breaking strength
spe Height above groundline of the pad eye
st Stretch of k-factor of genetic algorithm
v∞ Undisturbed velocity
T Thrust
ta Wall thickness of the anchor
t′a Adjusted wall thickness of the anchor
Ti Time spent in environmental state i
d̃i Crowding distance
Tlf,h Tension in mooring line during low frequency cycle for fa-
tigue damage corresponding to higher current
Tlf,l Minimum tension in mooring line during low frequency cy-
cle for fatigue damage corresponding to lower current
tshell Thickness of outer shell
u Current speed
u Threshold for generalized Pareto distribution
~uF A Unit vector from fairlead to anchor
xxiv Nomenclature - Latin Letters
Symbol Description
uh Higher current for calculating tension range of low fre-
quency cycle for fatigue damage
ûk Amplitude of the current corresponding to component k of
the discrete Fourier transformation
ul Lower current for calculating tension range of low frequency
cycle for fatigue damage
uLC,i Current corresponding to the analyzed load case i
~uML01 Direction of mooring line from end point 0 to end point 1
~uML10 Direction of mooring line from end point 1 to end point 0
V Displaced volume
V̂ Coefficient of variation
v̂ Mean rate of occurrence
vbot Current speed at the bottom of the water column
Vbt Displaced volume of a buoyancy tank
VH Volume of a hull
VBLLST Displaced volume of a ballast tank
VMLs Displaced volume of all mooring lines
VP L Displaced volume of platform
~W Submerged weight
w′ Submerged weight per unit length
WML Submerged weight of the mooring line
WP Submerged weight of the pile anchor
~x Global position vector of linearized equation system
x Random variable
x X-position
x0 Desired minimum or maximum value of the i-th property,
whichever is further away from current value
X̄ Mean
x′
BST Relative x-position of the ballast tanks
x̂ Quantile
xBLLST Position of the ballast tanks in x-direction
xi Value of the i-th property
Xk Coefficients of the discrete Fourier Transform
xmax Maximum desired value of the i-th property
xmin Minimum desired value of the i-th property
xopt Desired optimum value of the i-th property
y Lateral deflection of pile
y0 Lateral deflection of pile at groundline
yg Gumbel reduced variate
Nomenclature - Latin Letters xxv
Symbol Description
yA Elastic limit deflection of the rock
yA,0 Elastic limit deflection of the rock at groundline
z Depth below surface

Nomenclature - Greek Letters
Symbol Description
α Shape parameter of probability distributions
αr Strength reduction factor of the rock
α Angle between principal axis of inertia and reference axis
β Power law coefficient for current profile
β Scale parameter of probability distributions
β′ Adjusted scale parameter of probability distributions
χh Fineness ratio of hull
χRQD Rock quality designation in percent
ε Strain of the mooring line
εi Initial strain of the mooring line
γf Load factor
γM Material factor
Γ Gamma function
γF,F LS Single safety factor for fatigue limit state
λ Tip speed ratio
ν Cinematic viscosity
ω Angular velocity
ϕ Heel of floating body
φr Resistance factor
ψ Heading of floating body
ρB Average density of water
ρ Density
ρw Density of water
σ Scale parameter of generalized Pareto distribution
σcr Critical shear stress of the material in use
σlf Low frequency standard deviation tension
σwf Wave frequency standard deviation tension
σy Yield strength of the material in use
τB Return period
xxvii
xxviii Nomenclature - Greek Letters
Symbol Description
τf Maximum stress due to skin friction
θ Tail angle, that is half the included angle at the tail’s tip
θ Trim of floating body
ξ Shape parameter of generalized Pareto distribution
Abbreviations
ADCP acoustic doppler current profiler
ALS accidental limit state
AR acoustic release
ASD allowable stress design
ASF adjusted safety factor
AUV autonomous underwater vehicle
CI confidence interval
COB center of buoyancy
COG center of gravity
DFT discrete Fourier transform
DOF degree of freedom
ECMWF European Centre for Medium-Range Weather Fore-
casts
EVT extreme value theory
FLS fatigue limit state
GPD generalized Pareto distribution
HMPE high modulus polyethylene
HYCOM Hybrid Coordinate Ocean Model
IDW inverse distance weighted interpolation
IEC International Electrotechnical Commission
LRFD load resistance factor design
MBL minimum breaking load
MCEC marine current energy converter
MEC marine energy converter
MOGA multi objective genetic algorithm
NHC National Hurricane Center
NOAA National Oceanic and Atomspheric Administration
OCT ocean current turbine
POT peaks-over-threshold
PSFD partial safety factor design
xxix
xxx Abbreviations
RMS root mean square
RQD rock quality designation
SLS serviceability limit state
SOGA single objective genetic algorithm
TEC tidal energy converter
TSR tip speed ratio
ULS ultimate limit state
UNAM National Autonomous University of Mexico
WEC wave energy converter
Abstract
In the Mexican Caribbean, the Island Cozumel is separated from the mainland by a channel with
a width of approximately 18 km. This channel was found to have a relatively persistent current
with a heading in the northeastern direction. This current is part of the ocean-wide circulation and
converts later to the loop current in the Gulf of Mexico and the Gulf Stream after passing through
the stretch between Florida, USA and Cuba. It rarely reverses its direction and if it does, it does
so with very low magnitude. Due to its confinement between the island and mainland a very low
directional variation is reported. The rocky sea floor, steep slopes of approximately 10% degrees on
the channel’s side, and a wide and deep valley at its center with a depth of 400 m make this region
challenging for harnessing the energy contained in the ocean current.
In this work and as part of the requirements to obtain the degree of Doctor en Ingeniería, a
platform and mooring system was designed to fix a submerged, floating ocean current turbine at a
specific depth in the water column. To speed up the design process, a Python module for quasi-
static analysis of submerged offshore mooring systems with taut lines was created. It was shown
that especially the expected tensions in the mooring lines can be reliably predicted with this tool.
In conjunction with a Python module for evolutionary optimization, its performance was confirmed
with the 10th-order Griewank-function, the module for quasi-static analysis was used to generate
one mooring system design concept based on three anchors, and one concept based on four anchors.
Both design concepts were further optimized through a parametric design study, while the three-
anchor mooring concept was finally chosen for further evaluation. A dynamic mooring analysis
on all relevant load cases was performed with ProteusDS, to analyze involved stresses, tensions,
and deflection. Additionally, the model in ProteusDS was used to execute a free decay test for
estimating the system’s natural frequency. The fatigue life of the mooring lines was calculated
based on international design codes. Due to the absence of loads during operation originating from
higher frequency sources such as waves and wind, it was found that the fatigue life is not of concern
for submerged floating ocean current turbines. A reviewed methodology based on the literature
for designing pile anchors grouted to rock was presented and applied to two designs: one proposal
with and one without shear keys. The shear keys are designed to be a simple welding bead with
2.4 mm in height and helped to significantly reduce the required dimensions of the pile anchor from
115 mm in diameter to just 60 mm. For the design without shear keys, the limiting factor was the
friction between grout and steel, while the capacity of the pile with shear keys was limited by the
lateral structural capacity. The final design shows good agreement with the relevant design codes
and guidelines. At the moment, the design is based on the expected properties of an ocean current
turbine, thus it serves as a good starting point for adjusting the design to a turbine that is ready
for deployment. Even for different arrangements such as two vertical axis turbines side by side, the
efficient design methodology presented in this work is expected to produce a new design ready for
certification and fabrication within a few weeks.
xxxi
xxxii Abstract
Resumen
En el Caribe mexicano, la isla de Cozumel está separada del continente por un canal con un ancho de
aproximada 18 km. En este canal se encontró una corriente relativamente persistente con dirección
noreste. Esta corriente forma parte de la circulación de todo el océano y se convierte posteriormente
en la corriente de bucle del Golfo de México y en la Corriente del Golfo tras atravesar el tramo
entre Florida (EE. UU.) y Cuba. En los raros casos que se invierte su dirección lo hace con muy
baja magnitud y variación direccional debido a su confinamiento entre la isla y tierra firme. El
fondo marino rocoso, las fuertes pendientes de aproximadamente 10% en los lados del canal y un
amplio y profundo valle de 400 m en su centro hacen esta región un reto para el aprovechamiento
de la energía contenida en la corriente oceánica.
En este trabajo y como parte de los requisitos para obtener el grado de Doctor en Ingeniería, se
diseñó una plataforma y un sistema de amarre para fijar una turbina de corriente oceánica sumergida
y flotante a una profundidad específica en la columna de agua. Para acelerar el proceso de diseño,
se creó un módulo Python para el análisis cuasi-estático de sistemas de amarre sumergidos con
líneas tensas. Se demostró que, sobre todo, las tensiones esperadas en las líneas de amarre pueden
predecirse de forma fiable. Ese módulo se usó junto con un módulo de Python para la optimización
evolutiva, cuyo rendimiento fue confirmado a través de la función Griewank de décimo orden, para
generar un concepto de un sistema de amarre basado en tres anclas, y un concepto basado en
cuatro anclas. Ambos conceptos de diseño se optimizaron posteriormente mediante un estudio de
diseño paramétrico y finalmente se eligió el concepto de amarre con tres anclas para su subsecuente
evaluación. Con el programa ProteusDS se realizó un análisis dinámico del sistema de amarre en
todos los casos de carga relevantes, evaluando los esfuerzos, tensiones y deflexiones. Además, el
modelo en ProteusDS se empleó para ejecutar una prueba de decaimiento libre para estimar la
frecuencia natural del sistema completo. La resistencia a fatiga de las líneas de amarre se calculó
basándose en los códigos de diseño internacionales. Debido a la ausencia durante operación de
cargas originadas por fuentes de mayor frecuencia, como las olas y el viento, se comprobó que la
resistencia a fatiga no es preocupante para las turbinas de corrientes oceánicas sumergidas. Se
presentó una metodología revisada basada en la bibliografía para diseñar anclas de pilotes en roca y
se aplicó a una propuesta con y otra sin chavetas de cizallamiento. Las chavetas se diseñaron como
un simple cordón de soldadura de 2.4 mm de altura y ayudaron a reducir significativamente las
dimensiones requeridas del pilote de 115 mm de diámetro a solo 60 mm. Para el diseño sin chavetas,
el factor limitante era la fricción entre el concreto y el acero, mientras que la capacidad del pilote
con chavetas estaba limitada por la capacidad estructural lateral.
El diseño final muestra una buena concordancia con los códigos y directrices de diseño perti-
nentes. El diseño se basa en las propiedades esperadas de una turbina de corrientes oceánicas, por
lo que sirve como un buen punto de partida para ajustar el diseño a una turbina lista para su
despliegue. Incluso para disposiciones diferentes, como dos turbinas de eje vertical, una al lado de
la otra, se espera que la eficiente metodología de diseño presentada en este trabajo produzca un
nuevo diseño listo para su certificación y fabricación dentro de pocas semanas.
Chapter 1
Introduction
In recent decades, the production of electrical energy based on low-carbon energy sources such as
wind, and solar radiation has gained traction. Both sources have proven to be able to provide sig-
nificant amounts of electricity to the grid; however, they are not suitable for all regions around the
world due to several reasons such as sociological, geographical, or meteorological factors. This has
led to the endeavor to explore alternative renewable energy sources for the production of low-carbon
electricity. Various countries are actively funding research to develop new technology capable of
harnessing other sources. In Mexico, the centro de investigación en energía del océano (Spanish
for ocean energy research center) or short CEMIE-O – a consortium of Mexican universities and
Mexican industrial partners founded by the Mexican government – took the challenge to explore
ocean thermal energy conversion, wave energy conversion, energy conversion based on saline gradi-
ent, and ocean current conversion. All of those four energy sources are available within the Mexican
territory and can provide reliable energy of varying degree [50, 49].
A region with great potential of ocean currents is the Cozumel Channel, located offshore the
Caribbean coast of Mexico between the mainland and the island Cozumel (see figure 1.1). The
channel extends between the latitudes N20°15′ and N20°50′ and between the longitudes W86°40′
and W87°20′, oriented at roughly 40° azimuth [99]. A recent study was performed to validate the
energy potential [9]. A relatively steady, mono-directional current was found with a magnitude of
approximately 1 m s−1 [1] which makes it suitable to harness the energy of the ocean current. From
a technical point of view regarding energy extraction, the ocean current found at this site is similar
to tidal currents, with the main difference in cyclic bi-directional behavior and predictability of
tidal currents. Several confined tidal currents have been or are currently explored around the world
regarding their potential to produce low-carbon electric energy. Projects were deployed in different
key regions, such as in the Bay of Fundy, Canada [109, 71, 66, 107, 57], the Orkney Islands, UK
[2, 14], and the Kyoto stream, Japan [102]. In contrast to the ocean current found in the Cozumel
Channel, the tidal currents currently explored are much more energetic, presenting peak velocities
up to five times higher. Additionally, most of the used turbines are designed for cut-in velocities
for velocities that are barely encountered in the Mexican Caribbean, making it impossible to use
1
2 CHAPTER 1. INTRODUCTION
Figure 1.1: Location of the Cozumel Channel within the Mexican Caribbean. a) Location in
Northwestern Caribbean; b) Mexican Caribbean. Data on bathymetry, landmass, and political
borders are based on the data set provided by NaturalEarth.
existing technology [127]. At the University of Strathclyde, UK, a turbine specifically for low
energetic ocean currents such as they can be found in the Mexican Caribbean or the Philippines
is under active development [37], but it is in a very early design phase. Similarly, the Institute
of Engineering of the National Autonomous University of Mexico is designing a turbine capable of
harnessing low-energy ocean currents, which has demonstrated its capability in various laboratory
and real-world experiments.
To harness the power of the current, turbines must be placed at a specific height in the water
column. Various concepts have been used or proposed for tidal turbines, such as floating at the
surface [43] or mounted directly to the sea floor [126]. Other concepts involve placing turbines on
a platform that floats in the middle of the water column [79]. For instance, the PLAT-O platform
developed by Sustainable Marine Energy [58, 59] and the KAIRYU platform developed by the
IHI corporation [102, 52, 116] follow this concept. However, some recently developed projects are
inactive nowadays, such as the submerged, floating PLAT-O device which seems to be paused in
favor of a surface floating platform [111].
The PLAT-O and the KAIRYU prototypes are both submerged floating platforms but follow
different concepts. While both have two turbines mounted on a platform consisting of three cylin-
drical hulls, there are a few key differences. The PLAT-O prototype places the turbines between
the hulls and mounts them at the top of the platform to a structure that connects to the strut.
The turbines can be rotated around the transversal axis for easier access during maintenance. The
platform is fixed to the sea floor using several mooring lines and anchors. In contrast, the KAIRYU
prototype places the turbines directly inside the outer hulls on either side of the platform. The
turbines can adjust their pitch and rotate in different directions, the latter to counteract the reac-
tion moment generated. The platform is equipped with an altitude control device to maintain the
platform at the design depth. This is necessary because the platform is moored to a single anchor
which enables passive weathervaning of the complete system.
The mooring system for floating renewable energy converters poses a unique challenge. In
contrast to floating offshore oil and gas facilities, where the mooring system typically accounts for
1.1. OBJECTIVES 3
2% to 3% of the total budget, the mooring system for ocean current energy converters can consume
up to 30% [63, 121] of the budget. Additionally, the slope found in the Cozumel Channel imposes
limitations on the choice of the anchors suitable for installation [104].
1.1 Objectives
The objective of this work is to develop an optimized mooring system for a submerged turbine
that is designed to harness mono-directional ocean currents, as they can be found in the Mexican
Caribbean. The turbine should be fully submerged, except for maintenance, installation and dis-
mantling. Additionally, the platform should be free-floating, restrained only by the mooring lines
that are connected to the platform.
More specifically, the following objectives should be met:
• Elaborate recommendations for anchoring ocean current turbines (OCTs) for the bathymetric
profile found in the Cozumel Channel.
• Develop a methodology to efficiently evaluate early concepts for mooring systems, taking into
consideration sloped sea floors.
• Study the dynamics of a mooring system for sloped sea floors.
• Design a mooring system suitable for an OCT to harness the ocean currents in the Cozumel
Channel
1.2 Structure of the Document
The structure of the document and a brief summary of the chapters can be found in the following:
• Chapter 2: An overview of mooring systems with a focus on OCT is provided. The information
is derived from the offshore oil and gas sector and adapted for offshore renewable energy
devices. Various mooring configurations, materials, and anchors are discussed.
• Chapter 3: The design process and principles used in this work are described. This includes
the basic formulae used to evaluate the integrity of the structure, its fatigue life, and the load
capacity of the anchor. The required limit states according to the relevant design codes are
described and the commercial software used in this work is presented.
• Chapter 4: The Python package developed to analyze subsurface, offshore mooring systems
based on taut lines is described in detail. The numeric model is described, verified with simple
calculations, and validated with two experimental series and data available in the literature
on full-scale experiments with a spherical buoy.
• Chapter 5: The genetic algorithm which was used to optimize design proposals is explained.
This algorithm was evaluated with the Griewank-function as the benchmark.
4 CHAPTER 1. INTRODUCTION
• Chapter 6: The possible construction site in the Cozumel Channel is characterized. The data
sources and the methods for the extreme value analysis are described as well as the results
regarding extreme winds, waves, and currents. The currents were analyzed at the water
surface and a depth of 50 m, while the extreme wind speeds are based on hurricanes and data
provided by reanalysis. Information on the bathymetry is given and geotechnical information
available in the literature is reviewed.
• Chapter 7: The general choices on the system design, relevant load cases, and two potential
concepts are presented.
• Chapter 8: The optimization of the system design is performed for both concepts, and the
results are provided. The optimization is performed for both concepts, which were then
manually tweaked to find the optimum configuration for the given parameter.
• Chapter 9: The optimized design concept is evaluated for compliance with relevant design
codes. This includes dynamic mooring analysis, estimation of fatigue life, selection and design
of an appropriate anchor, and selection of the required mooring equipment.
The conclusions can be found in chapter 10.
Chapter 2
Mooring Systems
Moorings can be broadly categorized into two types: surface moorings and subsurface moorings.
The main difference between them is that while the subsurface mooring remains entirely underwater
during operation, the surface mooring has a visible part at or above the surface, typically in the form
of a buoy or a platform. For the surface mooring, the platform provides buoyancy and restoring
forces for heave, roll, and pitch, while the mooring lines provide restoring forces for surge, sway,
and yaw of the vessel [115].
Although certain applications require a surface mooring, a subsurface mooring is preferred as
it imposes less fatigue-related requirements on the system due to smaller environmental loads. In
addition, subsurface moorings benefit from being less visible, resulting in less problems with surface
vessels and vandalism [115].
A typical mooring system consist of three component groups: The flotation device or platform
at the upper end which in this case is connected to the marine current energy converter (MCEC),
the mooring line which transmits the loads from the platform to the anchor, and the anchor which
transmits the loads to the sea bed [115]. To achieve the desired safety level and dynamic behaviour,
mooring lines are often segmented, which is especially useful for inverse catenary moorings, where
positive and negative submerged weighed segments can be used to design the false bottom. The
lowest part of the mooring, which is connected to the anchor, is subject to high wear due to potential
contact with the sea bed, and is typically made up of steel chain. This means that wire rope or
fiber rope mooring lines will be segmented at least once [124].
2.1 Configuration
The mooring configuration typically refers the profile of the individual lines and the pattern in
which they are set up, including the number of lines. The length of the lines is derived from other
factors, among which is the line profile.
5
6 CHAPTER 2. MOORING SYSTEMS
a) b) c) d)
catenary
inverse
catenary taut vertical/tension
Figure 2.1: Typical profiles of mooring lines. a) Catenary profile; b) inverse caternay profile; c)
taut profile; d) vertical or tension leg profile. The positions of the anchors are marked with a cross.
2.1.1 Line Profile
For the mooring, a great variety of different mooring lines can be used to achieve the desired
behavior of the structure, yet, the line’s shapes can be traced back to a few basic profiles. Mooring
lines are classified as compliant or rigid [56], and the most common mooring systems are depicted in
figure 2.1. Catenary, inverse catenary, and taut moorings are used for compliant mooring systems,
while vertical moorings are categorized as rigid. Each of those types has its purpose and serves the
special needs of the individual site conditions. When it comes to mooring systems for MCEC, only
compliant systems are considered applicable, as noted by Weller et al. [121].
Catenary
Catenary moorings (see figure 2.1 a)) follow a simple design where the mooring line takes on a
catenary curve from the fairlead (the attachment point to the vessel) to the anchor. The section of
the line that connects to the anchor is intended to lie flat on the sea floor and should only be lifted
completely when the line is loaded to its breaking load [82]. Therefore the anchor is not loaded
vertically, which increases the variety of suitable anchors as the requirements are quite low [27].
These lines provide low horizontal stiffness to the system, while the vertical stiffness is provided
by the shape of the vessel. As the vessel moves horizontally, the line lifts partially from the ground,
resulting in more suspended line and restoring forces on the vessel [56]. Catenary lines are tolerant
to uncertainties in water depth and use low-cost materials like steel chains [27]. However, the
suspended section of the mooring line increases with greater water depths and larger mooring
offsets, leading to very heavy lines. The constant contact with the sea floor can cause damage to
both the sea floor and the mooring line.
In the offshore oil and gas industry, catenary moorings are used for depths of up to 500 m,
followed by a transitional depth where both catenary and taut profiles are used. For greater
depths, catenary profiles become too heavy, and taut moorings are mainly used. However, these
rules of thumb may not be applicable to MCEC, especially submerged MCEC, which have different
requirements that need to be reviewed.
2.1. CONFIGURATION 7
Inverse Catenary
Inverse catenary moorings (see figure 2.1 b)) follow an S-curve to provide the great compliance
to the vessel, which is achieved with clump weights and buoys connected at different arc lengths
to the line, or ropes of different weights [27]. This line profile is often used for buoys which are
required to ride the surface, such as wave buoys, and where the mooring offset is not a significant
concern. However, they may not be suitable for very large sea states and are difficult to operate
under low-tension conditions that can introduce shock loads. Strong currents can also reduce their
geometric compliance [27].
Taut
In contrast to a catenary mooring, a taut line (see figure 2.1 c)) is designed to avoid contact with
the sea floor [56]. These moorings are typically used for deep water moorings [27]. Vertical forces
are directly transferred to the anchor, while compliance is achieved by elasticity of the mooring line
material [56].
Taut mooring lines are typically made up of steel wire rope or synthetic rope such as nylon,
polyester, polypropylene, or polyolefin [27]. The mooring lines are design to be always loaded
with a minimum force, remaining in tension and reducing risk of snap loading [27]. They have
minimal impact on the sea floor since they do not make contact, but their design requires accurate
knowledge on the local bathymetry and are more challenging compared to catenary or inverse
catenary moorings.
Vertical Mooring
Vertical moorings or tension leg moorings (see figure 2.1 d)) are similar to taut moorings, but
they differ in that their mooring lines are considered very stiff due to their large dimensions [56].
Restoring forces result mainly from pretension and length of the mooring lines, with the pretension
typically around 20% the vessel’s displacement [56]. Therefore they are mostly suitable for large
vessels floating on the surface.
Vertical moorings are not well suited for subsurface moorings as horizontal displacement of the
vessel can only be counteracted by a geometric shift of the mooring line. This is not the case for
surface vessels since horizontal displacement introduces additional hydrostatic forces.
2.1.2 Pattern
For platforms used in the offshore oil and gas sector, moorings lines are often grouped together to
provide more space for risers and enhances dynamic behavior in case of mooring line damage [56].
The number of lines depends on various factors, especially if the condition of line loss has to be met
while still remain functioning. Achieving this with four or fewer lines is difficult, wherefore mooring
lines might be added and grouped. Typically they are named regarding the number of groups and
the number of mooring lines in each group [68]. For example, a vessel moored with five groups and
8 CHAPTER 2. MOORING SYSTEMS
two mooring lines in each group would be called a 5 x 2 pattern. A snapped line in a 5 x 2 mooring
pattern transfers nearly all its load to the other mooring line in its group, which could lead to the
consecutive failure of the remaining line. This can be avoided by increasing the lines per group
to the disadvantage of higher cost due to increased material and more installations. Therefore, in
general, the mooring lines are homogeneously spread around the platform, leading to a single angle
between different mooring lines. In this case, a 10 x 1 mooring patter would be preferred over a 5 x
3 mooring pattern, as the loads are more likely to be spread over more lines while still maintaining
the same number of material and installations.
2.2 Mooring Line Composition
The composition of a mooring line is an essential aspect that determines its behavior and per-
formance. A mooring line’s composition is determined by its nominal diameter and the material
used, which can be chain, wire rope, or synthetic rope, depending on the desired properties such as
stiffness, corrosion resistance, and abrasion resistance. Accessories such as weights, buoyant parts,
and connectors that might be fitted to the mooring lines are discussed in section 2.3.
2.2.1 Chain
Chains have been used in the offshore industry and naval architecture for many years and are
available in various shapes and grades. The grades of chains, which are made of steel, are defined
by DNV GL [31], which range from low to high strength: R3, R3S, R4, R4S, and R5. The grade
determines the chain’s minimum breaking load (MBL) for a given nominal diameter, which is the
diameter of the bar used to form the individual links.
The chain’s links can have so called studs, which form a connection of the opposing sides of the
same link. Those links are welded in place and prevent the chain from tangling when handling.
Stud link chains are mainly used for temporary moorings or moorings of ships, while studless chains
are preferred for permanent moorings due to better fracture toughness especially at the weld of the
studs.
Due to its greater resistance to abrasion and easier securing, chains are often used for the
segments of the mooring line that connect to the fairlead and the anchor, regardless of the material
used for the sections in between. Depending on the requirements, such as frequent change in length
or when ground contact can be precluded, it might be useful to refrain from the use of chain at one
or both ends of the line.
2.2.2 Wire Rope
In comparison to chains with same minimum breaking load (MBL), wire ropes are much lighter
with a higher elasticity. They are made by interweaving individual wires to form a strand, which
can be combined with more strands to form the rope or taken solely as a single strand rope, also
known as a spiral strand. The structure of wire ropes is an important consideration when selecting
2.2. MOORING LINE COMPOSITION 9
(a) Spiral strand (b) Six strand (c) Multi strand
Figure 2.2: Common layouts of strands in wire ropes.
a mooring line, as it affects the line’s behavior under load. Wire strands can be arranged as spiral
strand, six strand, eight strand, or multi-strand, following the definition of IEC 62600-10 [53] and as
shown in figure 2.2. Depending on the manufacturer, different variations of the multi-strand layout
are available, such as 7-wires, Seale, Filler and Warrington-Seale in the case of Bridon-Bekaert
Ropes Group [13]. This inner structure of wire rope defines its tendency to spin or twist under
load. Wire ropes with high tendency to spin or twist may form loops in unloaded conditions, which
can weaken the material when loaded again, and should be avoided if possible [115]. Six or eight
strand ropes, which have a pronounced tendency to spin or twist, are much more flexible and easier
to handle than spiral strand ropes and therefore typical for temporary moorings. In cases where
low rotation is required in the mooring system, typically multi-strand ropes are selected as they are
resistant to rotation due to their inner arrangement. Alternatively, the tendency to rotate under
load can be minimized by using different left and right handed layers or the ends of the mooring
line can be fitted with swivels.
Another differentiation between wire ropes is their rope lay, which is defined by the direction the
fibers are wound around the strands. The most common rope lays are Ordinary (sometimes also
called Regular) and Lang’s (or Langs). In Ordinary lays, the fibers are wound in opposite directions
as the strands are (sZ for right handed and Zs for left handed wires), where as for Lang’s lays, the
fibres are wound in the same direction (zZ for right handed and sS for left handed wires). Those
terms are also abbreviated with s and z, as the middle stroke of the letter indicates the direction
the strand or wire is laid: following from bottom to top, s goes from right to left and z from left to
right. Lang’s lay ropes have greater wear resistance than Ordinary lay ropes [13].
Fatigue bending resistance is an important factor to consider when designing a mooring line
that involves a significant number of sheaves or is frequently stored on a drum. The size of the
wire can increase the fatigue resistance as well as a smoothed surface of the outer strands [13]. A
smoothed surface additionally reduces the abrasive wear on the rope when being in contact with
itself or other parts of the system.
10 CHAPTER 2. MOORING SYSTEMS
Protection against Corrosion
To protect against corrosion, there are different techniques available for wire ropes. One common
method is to galvanize the wire rope. Another technique is to apply a plastic jacket or sheath made
up of Polyvinyl chloride, Polypropylene, or High-Density Polyethylene. The jacket might be kept
in yellow for increased visibility [115, 68]. Some wire ropes also include zinc wires and a special
grease in the strands to providing a sacrificial anode and prevent water ingress, further enhancing
their resistance to corrosion
2.2.3 Synthetic Fiber Rope
Although wire rope is less stiff than chains, using it in taut mooring systems can weaken or even
reverse this effect. In the 1990s, synthetic fiber rope became popular for deep waters, as it provides
the necessary compliance without the need for catenary moorings [68]. Synthetic rope also provides
good compliance as it improves the payload and the platforms offset due to its reduced weight
[124]. However, synthetic rope is less stiff than wire rope and may be prone to fish bite, which is a
common problem in synthetic materials at depths up to 2000 m [115]. Therefore it is recommended
only for line segments below 100 m, while the upper sections should be composed of chain or wire
rope [68].
The code IEC 62600-10 [53] allows for the use of Polyester, Aramid, or Dyneema. But also nylon
and high modulus polyethylene (HMPE) might be suitable, depending on the desired properties.
The synthetic yarn is typically interwoven to different strands, which are joined into a subrope.
The final rope is composed of several subropes, covered by a filler element and a jacket. The filler
element stops ingress of sediment while the jacket protects the rope against abrasion and helps
maintain the rope’s geometric shape. Within the subrope, the strands can be aligned parallel, and
can be braided or laid, each with its own purpose. While parallel and braided construction is torque
free, laid constructions will rotate under load. The improved contact between the strands in laid
subropes enhances its fatigue life [68].
Compared to wire ropes, synthetic fiber ropes provide a non-linear relation between tension and
strain, and exhibit construction stretch. Mooring lines made up of synthetic material will be greater
in length once installed than when they were manufactured. This plastic stretch further increases
after the first higher loading event on the line [68]. This is mainly due to the arrangement of the
yarns and strands within the rope as they tend to align themselves under load. Another factor can
be material creep, which increases the platform’s mean offset, and must be taken into account when
designing a mooring system based on synthetic ropes.
2.3 Mooring Line Accessories
A variety of accessories are available for mooring lines, including connectors, shackles, clump
weights, buoyancy aids, and acoustic releases.
2.4. ANCHORS 11
2.3.1 Acoustic Release
An acoustically commanded anchor release or simply acoustic release (AR) is primarily used to dis-
connect the mooring line from the anchor, especially for subsurface moorings, allowing for equipment
recovery. ARs can also be used for localizing the anchors exact position, which can be challenging
for subsurface moorings [115].
2.3.2 Clump Weights and Buoyancy Aids
Clump weights and buoyancy aids can be added to alter the characteristics of mooring lines. Both
can be used as concentrated or distributed loads. However, it is important to consider their potential
negative effects, such as ground contact, increased wear due to sharp angles at the support, and
increased drag [53].
Buoyancy aids are often made of hollow steel spheres, glass spheres with a protective plastic
layer, or synthetic foam spheres. The latter material have several advantages over steel or glass
floats as they can be molded into different shapes to reduce drag. Steel floats have a depth limitation
of around 1000 m due to excessive pressure [115]. The ability to create shapes with less drag makes
synthetic foam spheres an attractive option for high current environments.
2.4 Anchors
The offshore oil and gas sector and offshore renewable energy sector have developed various an-
chor types that have been proven in different projects. Experience has been gained for design,
installation, operation, and decommissioning, if required. The guidelines IEC 62600 [54, 53] were
developed by the International Electrotechnical Commission (IEC) to provide information specifi-
cally for marine energy converters (MECs) since the conditions for MECs are different from those
in the maritime oil and gas industry. Factors such as safety levels, environmental loads, and profit
margins differ significantly.
At most construction sites, sand or clay can be found although the region which this work
focuses on is known for rocky sea bed made up of lime stone. Typically, sediment has settled over
the years building up a sufficiently thick layer of sand or clay, allowing the anchor only to interact
with said material. In regimes with higher current, this layer of sediment may be much smaller
or even not existent, evoking the requirement of an anchor suitable for different kinds of rocks.
Many anchors are well proven for sand or clay and accepted design methods for rocky sea beds are
frequently non existent.
Some of the anchors suggested in IEC 62600-10 [53] are discussed below in more depth, keeping
in mind the high current and rocky sea bed at the site of interest.
12 CHAPTER 2. MOORING SYSTEMS
2.4.1 Drag Embedment Anchor
Drag embedment anchors are installed by anchor handling tugs which pull them horizontally after
placing them on the seabed. Due to their special shape, the anchor penetrates the sediment and
generates the desired holding capacity. However, their capacity is only valid for loads in direction
of installation with no or negligible vertical load component. IEC 62600-10 [53] differentiates
further between stockless anchors and high holding power anchors. High holding power anchors
have a much higher holding capacity but the downside is installation and recovery of such anchors.
They also require much higher precision in positioning before installation while the higher capacity
complicates their recovery.
2.4.2 Pile Anchor
Pile anchors are typically slender pipes which are widely used in the offshore oil and gas industry
and thus are well understood. The installation methods depend highly on the conditions found on
the construction site and range from hammering and vibrating into soil to drilling and grouting
into rock.
These anchors can withstand both, vertical and horizontal loads, but their design requires great
knowledge of the soil conditions in order to achieve the desired holding capacity and installation
depth, as stated in IEC 62600-10 [53]. Testing the design at full-scale is recommended when installed
in rock to gain detailed knowledge of their behavior and ultimate capacity [15].
2.4.3 Micropile anchor
Micropiles are widely used for onshore applications such as slope stabilization with their application
in offshore construction still being subject to investigation. A concept has been developed by
Meggitt et al. [74] with the focus on marine renewable energy. They suggest using a group of three
to four micropiles, drilled into the ground from a seabed drilling rig. For floating platforms they
suggest a plate which is fixed with several micropiles to the seabed, with the mooring line connected
to the plate instead to the actual anchor.
2.4.4 Suction Anchor
Suction anchors can vary heavily in their aspect ratio, from very slender, somewhat similar to pile
anchors, to short anchors with aspect ratios of 2:1. The installation to required depth is mainly
performed by generating a pressure differential between the inside of the anchor and the surround-
ing sea. This facilitates precise positioning on the seabed and relatively easy decommissioning.
However, they require soil which is easy to penetrate such as sand or clay and work best in soils
with low permeability.
2.5. RELEVANT DESIGN CODES 13
2.4.5 Gravity Installed Anchor
Gravity installed anchors rely on their own momentum to penetrate the seabed, generated by
dropping them from a sufficient height. The typical drop height of 100 m makes this anchor type
unsuitable for construction sites with low water depths. These anchors are effective at resisting
both horizontal and vertical loads, and work well in soft soils such as sand or clay.
2.4.6 Dead Weight Anchor
Dead weight or gravity anchors are not designed to penetrate the seabed but they are placed on top
of it. Prior treatment of the soil is commonly required to generate the designed holding capacity,
which is a combination of the anchors submerged weight and friction between it and the sea bed.
For increasing the holding capacity, shear keys fitted to the bottom of the anchor may be considered.
These keys penetrate the sea bed and mobilize shear within the soil or rock rather than friction on
its surface [88]. A simple form of these anchors is a dead weight made up of dense material such
as steel or concrete, or a box filled with heavy material after positioning. The box would be made
up of steel or concrete and filled with sand found in place or heavier material such as iron ore. For
small oceanographic moorings, metal pieces such as used train wheels can be used.
Modified versions of gravity anchors, known as grillage and berm anchors, use a grillage buried
under heavy material such as a rock-fill or an iron ore berm. For anchors with a predominant
loading direction, the grillage is buried on the back end of the berm, requiring the whole berm to
be moved in case of grillage failure [88]. Berm anchors are suitable for predominantly horizontal
loading conditions, which might not be the case for MCEC as it would require long and heavy
catenary mooring lines.
2.4.7 Screw Anchor
Screw anchors can withstand both horizontal and vertical loads, but are typically used only for
special applications and particular sediments. These anchors derive their holding capacity from the
friction generated with the seabed and from the reaction against the anchor’s structure.
2.5 Relevant Design Codes
The mooring system of MCEC should be analyzed in accordance with IEC 62600-10 [53] with similar
information available in DNVGL ST-0164 [33] and the general process of certification described in
DNVGL SE-0163 [30]. More general recommendations and requirements for mooring systems can
be found in API RP-2SK [5] and DNVGL OS E301 [32]. General information on foundations is given
in DNV CN-30.4 [20] and API RP-2GEO [6]. DNV RP-E303 [21] describes design and installation
of suction anchors in clay, whereas DNVGL OS-E302 [31] describes the requirements on offshore
mooring chains and accessories attached to those. Design of piles for marine structures in different
types of soils can be found in ROM 0.5-05 [86].
14 CHAPTER 2. MOORING SYSTEMS
Guidance on design and analysis of platforms for wave energy converters (WECs) and MCECs
is given in IEC 62600-2 [54], whereas information on the design of tidal energy converters is given
in DNV OSS-312 [22]. General recommendations on fixed offshore platforms can be found in API
RP-2A-WSD [4] or DNVGL OS-C101 [34]. Design regarding buckling of structures in general
and tubular members is provided by DNVGL CG-0128 [29] and DNV RP-C202 [25], respectively.
Information on environmental conditions in the southern Gulf of Mexico can be found in the code
PEMEX NRF-003:2000 [83] as it extends the API RP-2A-WSD by design values for fixed offshore oil
and gas platforms in this region. Although safety levels and dimensions might differ for platforms of
a different purpose, the return values are useful information during design phase. The recommended
practice for estimating environmental conditions and loads in general can be found in DNV RP-C205
[23], while the fatigue design of offshore steel structures in described in DNV RP-C203 [24].
Chapter 3
Design Methodology
3.1 Design Process
The design of a mooring system typically begins with determining the project specifications and
general system information, followed by defining the environmental conditions and the load case
matrix [68]. This process is illustrated in figure 3.1. The next step involves analyzing the platform
and the mooring system to ensure compliance with the national and international codes, as well as
the specified design criteria.
An appropriate anchor is selected based on the defined criteria such as the sea floor conditions,
mooring line profile, available installation equipment, and overall costs. Based on the loads de-
termined in the mooring analysis, the dimensions of the anchor can be defined, and the involved
stresses and deformations evaluated. Based on the results, the mooring components and required
equipment is selected.
3.1.1 Mooring Systems
The design of the mooring system itself starts by defining the specific configuration, including the
line profile, mooring pattern, and length of the lines. This involves defining the position of the
anchors, spread angle between the mooring lines, and their grouping, if applicable. The mooring
line composition, such as diameter and material, is selected and the loads are calculated based on
load cases specified in the load case matrix. The assumption is made that there is no distortion or
movement of the anchors. The design proposal is then evaluated based on relevant design codes,
and the principal data of the system are adjusted to optimize it with respect to involved loads and
the cost of material, installation, and maintenance, among other criteria.
Especially for submerged and floating platforms, their design significantly impacts the loads
experienced by the mooring lines. Therefore, the design of the platform and the mooring system
is performed simultaneously. The mooring system is analyzed with a quasi-static approach. The
result are then used as input for the structural analysis performed with Abaqus. After optimization,
15
16 CHAPTER 3. DESIGN METHODOLOGY
Load case matrix
Hydrodyn. analysis
Structural analysis
Evaluation
Principal data/dimensions
Structural capacity
Geotechnical capacity
Evaluation
Pincipal data/dimensions
Project specifications
General system information
Environmental Conditions
Mooring components Required equipment
Mooring system design
Project data
Final selections
Anchor design
Figure 3.1: Flow chart of general design procedure.
3.2. DESIGN PRINCIPLES 17
a hydrodynamic mooring analysis is conducted with the software ProteusDS (see section 3.3.1).
Subsequently, a structural model of the platform is created in Abaqus, taking into account the
loads estimated by ProteusDS.
3.1.2 Anchors
Many different anchor types suitable for permanent moorings have been developed in the past
decades due to the vast activity in the offshore oil and gas sector. A selection of anchor types is
discussed in section 2.4, each with its unique advantages and disadvantages that must be evaluated
based on the project’s specific requirements. Bathymetry and geological and geotechnical conditions
should also be taken into account when selecting a specific type. Skop [104] provides some selection
guidelines for anchors, dividing them into four groups: Drag embedment anchors, gravity anchors,
pile anchors, and direct embedment anchors. For regions with an inclined sea floor, pile anchors
or direct embedment anchors are a viable option. Both work well for a broad range of sea bed
material, except for hard boulders and in case of pile anchors for mud or soft clay. Gravity anchors
are suitable for all types of sea bed material, but not recommended for sloped sea floors such as
those found around Cozumel Island (see section 6.3.4). The same is true for drag embedment
anchors, which are not designed to withstand vertical loads and do not work on rocky sea beds.
Detailed information is given in table 3.1.
Regardless of the anchor type, their design typically involves assessing the structural and
geotechnical (or holding) capacity separately. These are further divided into horizontal and vertical
capacity (or axial and lateral capacity for cylindrical anchors), which are all evaluated individu-
ally due to the common assumption of uncoupled behavior between vertical and horizontal load
capacity of the soil (i.e uncoupled interaction between soil and structure in horizontal and vertical
direction) [88]. Drag embedment and direct embedment anchors are often proprietary designs that
are selected based on their load capacity. Gravity and pile anchors are typically custom-made to
the specific requirements of the project. After defining their principal data, their capacity can be
evaluated on many design guidelines available in literature and international design codes. A design
procedure for pile anchors in weak rock is described in section 9.1.4.
3.2 Design Principles
The design of the position keeping system of an ocean current turbine consists of two different parts:
designing the platform according to IEC 62600-2 [54], and designing the mooring system including
the mooring lines and the anchor according to IEC 62600-10 [53].
3.2.1 Structural Design of Platform
The structural design of the platform for a MCEC can be done using various methods, as per the
guidelines IEC 62600-2 [54]. These include the load resistance factor design (LRFD) method, the
partial safety factor design (PSFD) method, the allowable stress design (ASD) method, or direct
18 CHAPTER 3. DESIGN METHODOLOGY
Table 3.1: Selection guidelines for anchors, after Skop [104].
Parameter D
ra
g
em
b
ed
m
en
t
G
ra
vi
ty
P
ile
D
ir
ec
t
em
b
ed
m
en
t
Sea bed material:
Sand or gravel • • • •
Mud or soft clay • • ◦ •
Stiff clay • • • •
Very stiff clay or glacial till ◦ • • •
Soft rock or coral − • • •
Hard, monolithic rock − • ◦ ◦
Boulders − • − −
Sea bed topography:
< 10° slope • • • •
> 10° slope − − • •
Loading direction:
Unidirectional • • • •
Omnidirectional − • • ◦
Large vertical component − • • •
Horizontal load range:
< 45 kN • • ◦ •
45 kN − 450 kN • ◦ • ◦
> 450 kN − − • −
• functions well
◦ normally not a good choice
− does not function
3.2. DESIGN PRINCIPLES 19
evaluation of limit states regarding the reliability. All these methods use the same limit states to
define acceptable ultimate, accidental, service, and fatigue conditions. Following the international
standards such as IEC 62600-2 [54] or DNV GL OS C101 [34], the LRFD method is widely used
and accepted by the industry and will be used in this work. The design may also be assisted by
testing, while additionally a probability based design method may be used [34].
Load Resistance Factor Design Method The basic concept of the LRFD method demands
the design resistance Rd of material and structure to exceed the design loads (Fd), as defined in
IEC 62600-2 [54] and DNV GL OS C101 [34]:
Fd ≤ Rd (3.1)
The design load is the sum of the characteristic load effects Ski and the corresponding load factor
γf i by
Fd =
n∑
i=1
(
γf iSki
)
. (3.2)
The design resistance is calculated by the characteristic resistance (Rk) and a resistance factor
(φr) as
Rd = φrRk , (3.3)
with the resistance factor defined by
φr =
L
γM
, (3.4)
where L denotes the characteristic length of the structural member such as stiffener span and γM
the material factor. According to IEC 62600-2 [54], the characteristic length can be set to 1. DNV
GL OS C101 [34] provides more detailed information on the material factors and specifies slightly
more conservative values than IEC 62600-2 [54]. However, this standard is developed with a wider
scope for offshore structures made up of steel in general, while IEC 62600-2 [54] focuses specifically
on MEC.
3.2.2 Design of the Mooring System
The design of the mooring system is described in IEC 62600-10 [53] and follows a much simpler
approach by comparing the achieved safety factor to the adjusted safety factor (ASF). For both
required limit states, the ULS and the accidental limit state (ALS), a safety factor (SF ) is defined
based on the type of analysis conducted. Quasi-static calculations generally require a higher safety
factor than dynamic mooring analyses. The safety factors for holding capacity of different anchor
types such as anchor piles and gravity anchors are provided for intact conditions and redundancy
checks. This safety factor is multiplied by the design factor (DF ) to obtain the ASF (SASF ) as
follows
SASF = SFDF . (3.5)
20 CHAPTER 3. DESIGN METHODOLOGY
According to IEC 62600-10 [53], the mooring lines must then satisfy the following condition
SASF <
FMBL
Fd
, (3.6)
where FMBL is the MBL in corroded state and Fd the design tension. The condition which has to
be satisfied by the anchors follows a similar definition and is defined by
SASF <
FA
Fd
, (3.7)
where FA is the holding capacity of the anchor and Fd the design tension at the anchor.
3.2.3 Fatigue Life
Different approaches to estimate the fatigue life of offshore structures and mooring systems can be
found in common design codes such as API RP-2SK [5], DNVGL OS-E301 [32], or ISO 19902 [55].
Most widely used is Minor’s rule for fatigue assessment. In this work, estimation of the fatigue life
(LF ) of all components is based on Minor’s rule which is defined by
LF =
1
γF,F LSDF
, (3.8)
where γF,F LS is the safety factor for the fatigue limit state and with the expected annual fatigue
damage of the analyzed component (DF ) given by
DF =
∑
DF,i , (3.9)
where DF,i is the annual fatigue damage per environmental state.
3.2.4 Limit States
Modern design philosophies are base on limit states, which define the frame of acceptable range
of loads and displacements for the installed device and its equipment, such as mooring lines and
anchors. Exceeding these limit states may cause failure as the established design requirements are
no longer satisfied. In the case of safety factors for evaluating the holding capacity of anchors, IEC
62600-10 [53] distinguishes between intact conditions and conditions for redundancy checks, but
does not strictly differentiate between different load cases.
Ultimate Limit State
The load combinations of the ULS define the loads a structure should be able to ultimately resist.
For floating MCEC, the ULS includes loss of platform stability, yielding of critical components
(e.g., the platform structure, mooring lines, and anchors), buckling of structural members, loss of
station-keeping, and sinking of the structure, as defined by IEC 62600-2 [54] and IEC 62600-10 [53].
3.3. COMMERCIAL SOFTWARE 21
Permanent loads, variable functional loads, environmental loads, and deformation loads are
combined with particular load factors into two different combinations. The load factors are specified
in table 7.3 as defined by in DNV OS-J101 [26]. In accordance with the standard, the load factor
for combined environmental loads was reduced to 1.15 as the structure is unmanned.
The required safety factors vary depending on the design category for the specific load case and
the component that is reviewed. For the platform, partial load safety factors can be found in IEC
62600-2 [54, p. 29], while the necessary safety and design factors for the mooring lines and anchors
are available in IEC 62600-10 [53, p. 39f.].
Accidental Limit State
The ALS consideres accidental loads that may occur during installation and operation of the struc-
ture. In case of a floating MCEC, this includes the loss of a mooring line, flooding of compartments
and assessment of stresses in the components and whether the platform retains its ability to raise
to the surface. For loads on the platform, IEC 62600-2 [54] suggests a partial load safety factor of
1.0, while the safety factors for the mooring lines and anchors are given in IEC 62600-10 [53, p.
39f.].
Fatigue Limit State
Under the fatigue limit state (FLS) the cumulative damage caused by repetitive loads is taken
into consideration. IEC 62600-10 [53] recommends using Minor’s rule for evaluating the systems
fatigue life. According to IEC 62600-2 [54], welded joints and stress concentrations should be
assessed for fatigue following ISO 19902 [55]. General information on the fatigue design of offshore
structures based on S-N curves can be found in DNV RP-C203 [24]. In case of the MCEC, exciting
frequencies may arise from from various sources such as vortex induced vibrations on mooring lines,
the rotational frequency of rotor, interaction of tip vertices with structure, waves and wind (during
maintenance), fluctuations in current, and the number of maintenance cycles.
Serviceability Limit State
In case of MCEC, the serviceability limit state (SLS) is used for evaluating the accelerations,
displacements, and distortions affecting the performance of the system [54]. In case of the platform
design in this work, the displacements from neutral position in all 6 degree of freedom (DOF) are
reviewed.
3.3 Commercial Software
The commercial software used in this work is limited to ProteusDS for the dynamic mooring analysis
and to Abaqus for the structural analysis.
22 CHAPTER 3. DESIGN METHODOLOGY
3.3.1 ProteusDS
ProteusDS is developed by DSA Ocean and is used generally for marine dynamic analysis. The
effect of the environment such as wind, waves, currents, and the sea floor on floating and submerged
objects can be assessed in the time domain. The software models mooring lines with cubic finite-
elements that are able to interact with the air, water, the sea floor, and other objects in the model.
Environment
Several profiles are available for modeling environmental effects such as current, wind, and waves.
The temporal onset of the environmental effects can be set for each effect individually or for all
effects at the same time. This parameter is called ramp time and is mainly used to give the model
time to settle before environmental effects take place and this way possibly avoid unwanted spikes
in loads.
Current The current can be represented with a uniform profile, linear shear current, a power-law
current profile, and a custom profile. For all currents, heading and nominal speed can be controlled.
The uniform profile applies the current speed to the complete water column, while the linear shear
current linearly decreases the current speed from its nominal speed at the water surface to zero at
the sea bed. The power-law is defined by
u(z) = u
(
h− z
h
)β
, (3.10)
where h is the water depth, u(z) the current speed at depth below surface z, β the power law
coefficient, and u the nominal current speed [36]. The power law can either be applied to the
hole depth or to just a lower fraction of the water column which is defined by the parameter
CurrentPowerLawPercentDepth. In the latter case, the upper section is covered with a uniform
current.
Waves Waves can be modeled based on Airy-wave theory, Stokes second order waves, Stokes
fifth order waves, a Pierson-Moskowitz wave spectrum, and a JONSWAP wave spectrum [35]. The
definition of the Stokes wave model of fifth order can be found in [39].
Wind Constant wind, the Ochi-Shin spectrum and the Froya/NPD spectrum are available as
wind types, while the profile can be modeled with a uniform profile, a logarithmic profile, and a
power law-profile, among others. For more information, see the manual of ProteusDS [36].
Mooring Lines
In ProteusDS, mooring lines are modeled with 1-dimensional finite-elements with cubic shape func-
tions. The mooring lines, referred to as Cable in ProteusDS, can be segmented into different
materials. Each cable is discretized into several lumped masses along its arc length with two Frenet
3.3. COMMERCIAL SOFTWARE 23
frames. One in axial direction and one fixed to the cable’s cross section to determine its elastic tor-
sion. Lumping the mass of the elements to its adjacent nodes neglects the curvature inertia in favor
of computational efficiency. The hydrodynamic drag is calculated at the nodes of each element and
linearly interpolated along the element. It is based on the normal and tangential drag coefficient of
the line’s material and hydrodynamic load coefficients, that consider the angle between the normal
and tangential current components. The added mass of each element is calculated based on the
hydrodynamic mass coefficient in the normal direction while the tangential component is neglected.
It is then added to the physical mass to resolve acceleration. For more detailed information refer
to the manual of the software [35].
Added Mass
ProteusDS is capable of calculating the added mass of all wetted surfaces; however, it does not
account for the added mass and their added mass inertia of surfaces with their normal vector
perpendicular to the flow. Additionally, the software is unable to calculate frequency dependent
added mass, which has greater effect on large, surface piercing objects.
3.3.2 Abaqus
The structural analysis of the platform was performed using the finite-element feature of the widely
accepted commercial software suite Abaqus. To model space frames, Abaqus offers a range of
different beam and pipe element, such as 2-node linear beams (B31), 3-node quadratic beams
(B32), 2-node cubic beams (B33), and 3-node quadratic pipes (PIPE32), among others. These
beam elements are mostly based on the Timoshenko beam theory, although several elements are
also available in Euler-Bernoulli formulation. As the use of Abaqus is prevalent in the industry,
a detailed description of the software and its features is not provided in this work. For more
information, the reader is encouraged to consult the user manual of the software [19].

Chapter 4
Quasi-Static Mooring Analysis
Several commercial software is available for detailed mooring analysis, such as ProteusDS (see
section 3.3.1) or OrcaFlex, with the focus on dynamic simulation of the object in question and its
connected mooring lines.
While these software packages are accurate, ProteusDS was found to to slightly overestimate
the loads in the mooring lines [12, 58, 128]. Their main disadvantage, however, is the required
and significant computational time. In early design stages, and especially when working with
evolutionary algorithms, time efficient computation is crucial. During the first design steps, a
quasi-static mooring analysis is performed using linear-elastic straight springs with zero compressive
stiffness to replace the mooring lines.
To address this need for time-efficient computation, the Institute of Engineering of the National
Autonomous University of Mexico has developed a Python module called OffshorePy. This tool is
specifically designed for submerged systems with taut lines that can be approximated by straight
lines. By using a quasi-static approach, OffshorePy provides an efficient and accurate method for
analyzing the mooring system during the initial design stages.
4.1 Object Types for Modeling the Mooring System
The mooring system is modeled with 3 different base types of objects: Rigid bodies, points and
(taut) lines. The points themselves are divided into three sub types of points, which are fairleads,
anchor points, and floats.
4.1.1 Rigid Body
Rigid bodies in OffshorePy are different from the other objects in that they posses 6 DOF, and have
a designated center of gravity (COG) and center of buoyancy (COB), at which their weight and
buoyancy forces are applied. External forces, such as thrust and moment of turbines or drag forces
of the body, can be added as external forces along with their point of application. The definition
25
26 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
of external forces is given in the global reference frame while their point of application is specified
relative to the local reference frame, i.e., with respect to the position and orientation of the rigid
body. As a result, the force vector remains unchanged upon translation or rotation of the rigid
body, while the point of application moves along with body. Rigid bodies interact with the mooring
system solely through Fairleads, which must be added to the rigid body. The residual force (~FR)
on the RigidBody is calculated by
~FR =


0
0
1

V ρwg −


0
0
1

mg +
∑
~FE,i +
∑
~FF,i (4.1)
and the residual moment ( ~MR) by
~MR =


0
0
1

V ρwg × ~rcob −


0
0
1

mg × ~rcog +
∑
~FE,i × ~rE,i +
∑
~FF,i × ~rF,i , (4.2)
where V is the body’s displaced volume, m its mass, ρw the density of water, g the gravitational
acceleration, ~FE,i the external force i, ~FF,i the residual force on the i-th fairlead, ~rcob the position of
the COG, ~rcob the position of the COB, ~rE,i the point of application of the i-th external force, and
~rF,i the position of the i-th fairlead. All positions are assumed to be in the local reference frame of
the rigid body and the operator × denotes the cross product. To transform coordinates from the
local reference frame of the rigid body into the global reference frame of the mooring system, the
rotation matrix is used, which is defined by
R =


cos θ cosψ sinϕ sin θ cosψ − cosϕ sinψ cosϕ sin θ cosψ + sinϕ sinψ
cos θ sinψ sinϕ sin θ sinψ + cosϕ cosψ sinψ sin θ cosϕ− sinϕ cosψ
− sin θ sinϕ cos θ cosϕ cos θ

 , (4.3)
where ϕ is the heel of the platform, θ is the trim of the platform, and ψ is the heading of the
platform. The final transformation is than performed by the dot product as follows:
~p = R~r + ~pRB , (4.4)
where ~p is the position in the global reference frame (i.e., the mooring system), ~r the position in
the local reference frame (i.e., the rigid body), and ~pRB the position of the rigid body in the global
reference frame.
4.1.2 Points
Points have no expansion in space and do not account for their mass and volume. However,
external forces can be added to all type of points. Floats can move freely in all three DOFs and
4.1. OBJECT TYPES FOR MODELING THE MOORING SYSTEM 27
the submerged weight can be specified. Fairleads are also able to move in three direction but
their movement is coupled to a rigid body, meaning that their position does not contribute to the
global DOF-vector. Their position is defined in the local reference frame of the coupled rigid body
rather then in terms of the global coordinate system. Anchor points (AnchorPoint) are fixed in
space and do not move. While technically external forces can be added, they have no effect on the
mooring system and are simply used as reference point for mooring lines. Since points have three
DOFs, only the residual force (~FR) needs to be calculated, which is defined by
~FR = ~FE + ~W +
∑
~FML,i , (4.5)
where ~FE is an external force on the point, ~W the submerged weight of the point (if applicable),
and ~FML,i the residual force of the i-th mooring line at the end it is connected to the point.
4.1.3 Lines
Mooring lines are modeled as linear-elastic springs and require two connection points. They follow
a straight line between those points. If the distance between those two points is shorter than the
unstretched length of the line, the stiffness of the mooring line is set to zero. The pretension of the
mooring line can be adjusted using the initial strain (εi) of the line, that is defined by
εi =
ls
lus
− 1 , (4.6)
where ls is the stretched length of the line (i.e., the distance between the adjacent points) and lus
the unstretched length of the line. The submerged weight of the line is carried by the adjacent point
with greater z coordinate or is equally shared between both if their height is equal. The residual
force on the mooring line at end point 0 (~FR,ML0) is given by
~FR,ML0 = ~uML01 (ls − lus)EA+



[
0 0 1
]T
lusw
′ for ~p0z > ~p1z
[
0 0 1/2
]T
lusw
′ for ~p0z = ~p1z
0 for ~p0z < ~p1z
(4.7)
and at end point 1 (~FR,ML1) by
~FR,ML1 = ~uML10 (ls − lus)EA+



[
0 0 1
]T
lusw
′ for ~p1z > ~p0z
[
0 0 1/2
]T
lusw
′ for ~p1z = ~p0z
0 for ~p1z < ~p0z
, (4.8)
where ~uML01 is the unit vector from end point 0 of the mooring line to end point 1, ~uML10 the unit
vector in opposite direction, EA the mooring line’s axial stiffness per unit length, w′ ist submerged
weight per unit length, ~p1z the z-position of end point 0, and ~p0z the z-position of end point 1.
28 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
4.1.4 Material
Objects are provided to store the material properties for line segments. These properties include
the nominal diameter, its axial stiffness per unit length, the MBL, the submerged weight per unit
length, and the normal and tangential drag coefficient. The drag coefficients and MBL are currently
not used by the software but help the user to keep track of those values. In case specific material
properties are not available, they can be estimated by the following formulae as they are published
by Ma et al. [68]. The MBL (FMBL) in N is calculated by
FMBL =



cMBL d
2
(
44 × 10−6 − 8 × 10−11d
)
for chain,
9 × 10−7d2 for wire rope,
2.5 × 10−7d2 for polyester rope
, (4.9)
where d is the nominal diameter given in m, and cMBL is the coefficient for considering the chain’s
strength. The coefficient cMBL is 22.6 for chains of grade R3, 24.9 for chains of grade R3S, 27.4 for
grade R4, 30.4 for grade R4S, and 32.0 for grade R5. The submerged weight per unit length (w′)
in N m−1 is given by
w′ =



187 × 10−9d2 for stud chain,
171 × 10−9d2 for studless chain,
43 × 10−9d2 for wire rope,
1.7 × 10−9d2 for polyester rope,
, (4.10)
where the nominal diameter (d) is given in m. The axial stiffness per unit length (EA) in N is
defined by
EA =



101 × 10−3d2 for stud chain,
85.4 × 10−3d2 for studless chain,
88.7 × 10−3d2 for wire rope,
1.1 × 10−3d2 for polyester rope,
, (4.11)
where the nominal diameter (d) is given in m.
4.2 Linearized Equation System
The global stiffness matrix (K) is created using the central difference scheme with a default pertur-
bation of 1 × 10−5. The unit for rotation of the rigid body is radians while distances are unitless.
The vector of external forces (~P ) is composed of the resulting forces on all floats and rigid bodies.
Fairleads, anchor points, and lines are ignored in terms of the DOFs of the system but are con-
sidered regarding the forces acting on them. Lines exert a force on the adjacent points whereas
fairleads transmit the forces acting on them to the coupled rigid body via a force and a moment
4.3. VERIFICATION 29
with respect to the origin of the rigid body. Anchor points only serve the purpose as reference
point for the stretched length and direction of a line, and to evaluate the forces at this point. The
equation system can be represented by
K~x = ~P , (4.12)
where ~x is the change in position, and which is solved using the function numpy.linalg.lstsq
which is part of the numpy-Package for Python 3.9 and was used in version 1.22.4. In case the
matrix is of full rank, the numerically exact solution for ~x is returned, otherwise this function
minimizes ||~P − K~x|| and returns the minimum value of ||~x|| in case of various solutions, where
||~a|| denotes the Euclidean 2-norm of a vector (see Harris et al. [48]).
4.3 Verification
The software has been verified with a simple mooring system which can be calculated by hand. A
spherical subsurface buoy is moored to the sea floor with a single mooring line while the current
speed (i.e. load on the buoy) is varied between 0.0 m s−1 and 2.0 m s−1 in 0.5 m s−1 increments. In
the second case the buoy is held in place with four mooring lines. The software was validated with
model scale experiments and full scale experiments published in literature, its documentation can
be found below in section 4.4.
External Forces The drag force (FD) of the body can be calculated by
FD = 1/2CDρu
2A , (4.13)
where CD is the specific drag coefficient, u the speed and A the reference area of the body, in this
case the projected area parallel to the direction of the current. A formula for estimating the drag
coefficient of a sphere as a function of the Reynolds number is defined by Morrison [76] as
CD =
24
Rn
+
2.6Rn
5.0
1 +
(
Rn
5.0
)1.52 +
0.411
(
Rn
263000
)−7.94
1 +
(
Rn
263000
)−8.0 +
0.25 Rn
106
1 + Rn
106
(4.14)
with the Reynolds number (Rn) defined by
Rn =
uL
ν
, (4.15)
and where u denotes the relative speed between (undisturbed) fluid and body, L the characteristic
length of the body, and ν the cinematic viscosity of the fluid. The buoyant force (FB) on the sphere
is calculated by
FB = ρwgV , (4.16)
30 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
where V is the displaced volume of water, ρw the density of water, and g the gravitational ac-
celeration. Since the sphere is completely submerged, its displaced volume can be calculated by
V = π
3 d
3, where d is the diameter of sphere. The weight of the buoy (FW ) can be calculated by
simply substituting the density of the water in equation (4.16) by the buoys average density (ρB):
FW = ρBgV (4.17)
Considering the direction of the forces, the vector of external force (~FE) on the buoy can be
assembled to
~FE =


1
0
0

FD +


0
0
1

FB −


0
0
1

FW . (4.18)
The current is assumed to be parallel to the channel’s floor, thus has only a horizontal component.
The buoyant force acts in positive z-direction while the weight acts in the opposite direction.
Although the cases are completely 2-dimensional, 3-dimensional vectors are used since it simplifies
the calculation of the moment around the y-axis using the cross product.
Mooring Line The mooring line is modeled as linear elastic spring, connecting both end points,
the anchor point and the fairlead, with a straight line. The strain (ε) of the mooring line is defined
as
ε =
‖~pF − ~pA‖
lus
− 1 , (4.19)
where ~pF is the position vector of the fairlead, ~pA the position vector of the anchor, and lus is
the unstretched length of the mooring line. The submerged weight of the mooring line (WML) is
assumed to be completely carried by the higher end point (i.e., the fairlead) and is estimated by
WML = w′lus, where w′ is the weight per unit length. The unit vector in direction of the mooring
line from fairlead to anchor (~uF A) is defined by
~uF A =
~pA − ~pF
‖~pA − ~pF ‖ . (4.20)
The force on the anchor point (~FA) can be calculated by
~FA = −~uF A
EA
lus
(ls − lus) , (4.21)
where EA denotes the mooring lines stiffness per unit length and ls the stretched length of the
mooring line (i.e., the distance between anchor point and fairlead). The force on the fairlead (~FF )
is given by
~FF = ~uF A
EA
lus
(ls − lus) −


0
0
1

w′lus . (4.22)
4.3. VERIFICATION 31
Anchor point
Water surface
Sea floor
Current Buoy
1
 m
x
z
8
0
 m
39
.5
 m
Mooring line
Figure 4.1: Concept of single lline mooring case for verification of quasi-static mooring analysis
package OffshorePy. Not to scale.
a) b) c)
Buoy
FF
FB
FW
FD
Mooring line
FF
FA
FAAnchor point
Figure 4.2: Free body diagram of objects of single line mooring case. a: Free body diagram of
buoy; b: Free body diagram of mooring line; c: Free body diagram of anchor point. Not to scale.
If no data are given, the stiffness (EA) in Pa m2 and the submerged weight per unit length (w′) in
N m−1 of the wires can be estimated with equations (4.10) and (4.11), following Ma et al. [68].
Single Line Mooring
The buoy is placed initially at half the local water depth of 80 m. The anchor point is at the center
line and is assumed to remain fixed to the sea floor. The buoy has a diameter of 1 m and the
mooring line is connected directly to its outer shell. The principal arrangement is illustrated in
figure 4.1. For demonstration purposes, the sketch is not to scale. Although the current is depicted
as uniform over the whole height of the water column, only the buoy experiences drag forces, not
the mooring line. The corresponding free body diagrams for the anchor point, mooring line, and
the buoy can be seen in figure 4.2. The buoyancy force FB , the weight FW , the drag force due to
the current FD and the force of the mooring line at fairlead ~FF are acting on the buoy. The former
three are acting on the COG and COB, respectively, which are at the same position in this case.
The force of the mooring line is acting at the position of the fairlead ~pF . The input data for the
32 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Table 4.1: Input data for calculations of single line mooring case.
Water depth (h) 80 m
Water density (ρw) 1025 kg m−3
Water viscosity (ν) 1.0023 × 10−6 m2 s−1
Diameter of buoy (dB) 1 m
Average density of buoy (ρB) 200 kg m−3
Unstretched length of line (lus) 39.5 m
Diameter of line (dML) 0.02 m
Stiffness of line (EA) 3.548 × 107 Pa m2
Submerged weight per unit length (w′) 1.72 N m−1
Table 4.2: Relative error of numerical calculation with respect to analytical calculation of single
line mooring case.
Current speed 0.0 m s−1 0.5 m s−1 1.0 m s−1 1.5 m s−1 2.0 m s−1
x position of buoy 0.00* 6.70 × 10−8 2.81 × 10−9 2.49 × 10−11 0.00
z position of buoy 3.38 × 10−15 4.56 × 10−13 6.02 × 10−13 3.80 × 10−14 0.00
y rotation of buoy 0.00* 2.66 × 10−10 3.26 × 10−11 1.35 × 10−14 0.00
Stretched length 3.42 × 10−15 0.00 0.00 0.00 0.00
Strain 3.26 × 10−11 3.14 × 10−13 2.49 × 10−13 1.38 × 10−12 1.15 × 10−12
Tension 2.83 × 10−11 3.65 × 10−15 6.02 × 10−13 5.22 × 10−13 4.96 × 10−13
* Absolute error
calculations are shown in table 4.1; the code used to create the OffshorePy-model can be found in
section A.1.3.
4.4 Experimental Validation
For validation, a spherical buoy was moored in an open circulation tank with four mooring lines.
Two experimental series were conducted due to findings in previous experiments, which were similar
in terms of setup. The first series involved a vertical section of the mooring line as shown in figure 4.3,
but without the load cell. Due to excessive deflection of the vertical section, a second series was
conducted without the vertical section and with the lines being connected directly to the channel’s
floor instead. Due to the elasticity of the mooring lines, the buoy was closer to the water surface
in equilibrium than in initial condition. A current meter was positioned at the center line of the
channel 1.2 m upstream of the buoy. On one side of the channel a high speed camera was set up
together with two to three LED light sources to capture the scene. The circulation tank is equipped
with three pumps, their frequency was varied through out the experimental series.
4.4. EXPERIMENTAL VALIDATION 33
Current
0.66 m
0
.5
0
 m
1.19 m
LED
LED
Camera
Reference plane
Vectrino
M
o
o
ri
n
g
 l
in
e
s
Water surface
Buoy
z y
x
Figure 4.3: General experimental setup for validation of numerical mooring analysis.
The buoy had a diameter of 0.08 m and was mainly 3D printed with glycol-modified polyethylene
terephthalate (PETG). It was covered with white paint to properly seal the surface and generate a
smooth finish. Depending on the experimental series, the eye bolts or some precisely defined points
on the buoy’s hull were painted black to be visible against a white background and the buoy’s
surface.
4.4.1 Equipment
The current was measured with a Nortek Vectrino which is able to measure the current speed at a
point just below the probes in all three directions via the Doppler effect. The sampling rate was
set to 80 Hz with a nominal velocity range of ±1.0 m s−1. The transmit length was set to 1.8 mm,
the sampling vol. to 7.0 mm and the Power level to HIGH+.
The position of the buoy was recorded with the high speed camera HiSpec 2G Mono by Fastec
Imaging at 80 frames per second and a shutter time of 800 µs. The lens used was the NAVI-
TAR 50mm F0.95 TV with the aperture set to F8. The images were processed with ImageJ/Fiji
(Schindelin et al. [95]) and the plugin Trackmate (Tinevez et al. [114] and Ershov et al. [38]).
34 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Table 4.3: Measurements of for determining elasticity of mooring line.
Force Pixels Deformation
0.000 N 448 0.1677 m
1.956 N 551 0.2062 m
3.912 N 654 0.2448 m
The resulting trajectories of the fairleads as well as the current data obtained via the Vectrino
were analyzed with Python 3 and the modules NumPy in version 1.22.4 [48], and Pandas in version
1.4.3 [112, 72].
4.4.2 Material of Mooring Line
Monofilament fishing line was used as mooring line material with a nominal diameter of 0.3 mm
and that was later determined to 0.32 mm with a digital caliper. The weight per unit length
was measured as 9.862 500 × 10−5 kg m−1 with the analytical balance Ohaus Pioneer PX224. The
fishing line is sold in Mexico under the brand name Mustad and has a nominal strength of 4.6 kg.
Elasticity The elasticity of the line was measured with 2 calibration masses of 200 g each. The
deformation of the line with known length was captured with a digital camera and later analyzed
with a raster image manipulation software. With the known length of the mount, the relation
between pixels and length was determined and a deformation could be assigned to each load. The
measurements can be found in table 4.3. The function stats.linregress of the Python module
SciPy in version 1.8.0 [120] was used to calculate the linear regression for the three points. The
stiffness of the samples was calculated to 5.0741 × 101 N m−1 with a corresponding r-value of 1.0000.
The stiffness per unit length (EA) was then estimated to 1.138 634 × 102 Pa m2.
4.4.3 Methodology
The absolute error (eabs) between the mean of the experimental observations (f̄exp) and the nu-
merical calculation (fnum) is defined by
eabs = fnum − fnum,0 − f̄exp + f̄exp,0 (4.23)
and the relative error (erel) is defined by
erel =
fnum − fnum,0
f̄exp − f̄exp,0
− 1 , (4.24)
where fnum,0 and f̄exp,0 are the mean position of the first three repetitions (e.g., A01F1_A,
A02F1_B and A03F1_C). In case of the experimental observations, the mean position for the
corresponding experiment is taken, which is indicated by the bar on top of the symbol.
4.4. EXPERIMENTAL VALIDATION 35
Figure 4.4: Experimental setup for series A.
4.4.4 Experimental Series A
The experimental setup is shown in figure 4.3. In figure 4.4, a photo of the setup before the exper-
iments started can be seen, clearly displaying the black painted eye bolts, that serve as fairleads,
as well as two of the four anchor points at the lower right side of the image. After passing through
roller and swivel of the anchor point, a vertical section of a mooring line is visible at the right side of
the image. The black marks on the background below the the buoy are reference points with known
height for determining the buoys vertical position. Initially, the buoy was placed at the channels
center line, 0.2 m above the reference plane. The distance between the virtual anchor point and the
fairleads was 0.68 m. The frequency of the pumps was changed between 0 Hz and 60 Hz in 10 Hz
increments. With three repetitions, 21 experiments were conducted in total. The complete matrix
can be found in table 4.4. The mooring lines were attached to metallic, black eye bolts. To account
for the vertical section, which cannot be modeled with OffshorePy, the apparent stiffness per unit
length (EA′) was calculated to 39.3096 Pa m2.
36 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Table 4.4: Experiment matrix series A for validation of numerical mooring analysis software.
Experiment FP Experiment FP Experiment FP
Hz Hz Hz
A01F1_A 0 A08F3_B 20 A15F5_C 40
A02F1_B 0 A09F3_C 20 A16F6_A 50
A03F1_C 0 A10F4_A 30 A17F6_B 50
A04F2_A 10 A11F4_B 30 A18F6_C 50
A05F2_B 10 A12F4_C 30 A19F7_A 60
A06F2_C 10 A13F5_A 40 A20F7_B 60
A07F3_A 20 A14F5_B 40 A21F7_C 60
Current Speed
The histogram of the current speeds measured for each experiment with current are shown in
figure 4.5. Unfortunately, the measurements of the vectrino were very poor for the first three
experiments (A01F1_A, A02F1_B and A03F1_C), resulting in the detection of a non-existent
current. For the simulations, these measurements were disregarded and the current was assumed
to be zero. For all other experiments, the mean current speed of the respective repetition was used.
For the experiments A13F5_A and following, the distribution presents a much greater standard
deviation. While the standard deviation was below 0.020 m s−1 for all repetitions before A13F5_A,
it ranges from 0.024 m s−1 to 0.36 m s−1 for the experiments A13F5_A to A21F7_C (see table A.1).
Position of Buoy
The position and the rotation of the buoy was calculated from the position of the fairleads. The
numerical model predicted the x-position of the buoy very accurately (see figure 4.6 a)) up until a
current speed of 0.42 m s−1. However, the corresponding absolute error in figure 4.6 b) indicates that
the simulation slightly overestimates the x-position until this threshold, followed by a significant
underestimation. The relative error was highest for the second lowest current speeds which can
be attributed to the low absolute values. For the two following current speeds, the relative error
is at maximum 0.151 (see table A.2). Similarly, the rotation around the y-axis (see figure 4.6
d)) was accurately predicted by the numerical model for low current speeds, but showed a clear
overestimation of rotation for higher current speeds. This discrepancy was less pronounced than
for the x-position. The error ranged between 0.25 and 0.35 for the current speeds of 0.21 m s−1
and 0.32 m s−1. However, the z-position of the buoy shows a huge error (see figure 4.7). While the
simulation predicts nearly no change in the z-position, a clear decrease of 0.005 m was observed
during the experiments.
4.4. EXPERIMENTAL VALIDATION 37
Figure 4.5: Current speeds measured during experiment series A. The 2.5th and the 97.5th quantile
(shaded area) as well as the mean current speed (dashed line) are shown for each experiment. a)
First repetitions; b) Second repetitions; c) Third repetitions.
38 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Figure 4.6: Comparison between numerical prediction of buoy position and experimental observa-
tion for experiment series A: a)-c) Deviation in x-direction, and corresponding absolute and relative
error; d)-f) Rotation around y-axis, and corresponding absolute and relative error;
Figure 4.7: Deviation of z-position of the buoy as predicted by numerical prediction and observed
in experiments for experiment series A
4.4. EXPERIMENTAL VALIDATION 39
Figure 4.8: Experimental setup for series B, taken during experiments with the pumps at 35 Hz.
4.4.5 Experimental Series B
The experimental series B featured a different mooring configuration than series A, with the mooring
lines directly attached to the channel’s bottom via the anchor. However, the general setup was
similar to that shown in figure 4.3. In figure 4.8, the four mooring lines are visible as well as
one anchor point at the lower right corner (white box), slightly obscured by a cable. Despite the
image being taken at the maximum pump frequency (i.e., maximum current speed), the mooring
lines remained straight between the anchor point and the fairleads. Since the mooring lines were
directly glued to the holes inside the buoy, dedicated fairleads similar to experimental series A were
unnecessary. Instead, two black dots were painted on the surface of the buoy at a distance of 0.03 m
arc length to aid in determining the buoy’s position and orientation. The same reference points
used in experimental series A to calculate the buoy’s vertical position were also used in series B,
visible as the black dot on the background below the buoy. The scene was illuminated with 5 light
emitting diodes, their reflection is visible in the image. Similar to series A, the buoy was placed
at half the water depth of 0.5 m at the channel’s center line, resulting in a line length of 0.6937 m.
The frequency of the pumps was changed between 0 Hz and 30 Hz in 5 Hz increments. With three
repetitions, 21 experiments were conducted in total, leading to the experiment matrix shown in
40 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Table 4.5: Experiment matrix series B for validation of numerical mooring analysis software.
Experiment FP Experiment FP Experiment FP
Hz Hz Hz
B01F1_A 0 B08F3_B 10 B15F5_C 20
B02F1_B 0 B09F3_C 10 B16F6_A 25
B03F1_C 0 B10F4_A 15 B17F6_B 25
B04F2_A 5 B11F4_B 15 B18F6_C 25
B05F2_B 5 B12F4_C 15 B19F7_A 30
B06F2_C 5 B13F5_A 20 B20F7_B 30
B07F3_A 10 B14F5_B 20 B21F7_C 30
table 4.5. The roller at the anchor points and the subsequent vertical sections of the mooring lines
were omitted by gluing the lines to the anchor points.
Current Speed
The measured current speeds during this series are presented in figure 4.9. The mean of the cur-
rent changes significantly between repetitions especially for experiments B04F2_A, B05F2_B and
B06F2_C. The correlation as reported by the Vectrino (see table A.3) seems to have great influ-
ence on the standard deviation of the measured current. For some experiments, the measured 95%
confidence interval (shadows in figure 4.9) does overlap for adjacent experiments (e.g. B17F6_B
and B20F7_C, see figure 4.9 c)). Noticeable is also the steady increase in standard deviation with
increasing pump frequency (i.e., current speed), with the exception of the experiments without
current and experiments B05F2_B and B06F2_C, which showed a much higher standard deviation
than expected. Similar to the experimental series A, the measurements for the experiments without
current seem to be inaccurate (B01F1_A, B02F1_B and B03F1_C), as currents were detected in
the range of roughly −0.1 m s−1 to 0.1 m s−1. For the simulations, these measurements were ignored
and the current was set to zero while for all other experiments, the mean current speed was chosen.
Position of Buoy
The position of the buoy was determined by analyzing the relative position of the dots painted on the
buoy’s surface. To minimize errors due to camera tilt, misalignment of dots, or minimal differences
in line lengths, the position of the buoy was compared to its initial position. The deviation in x
direction and the rotation around the y-axis, together with their respective absolute and relative
error, can be seen in figure 4.10, and the exact values of the signed relative error can be found
in table A.4. The numerical model underestimates the deviation in x-direction by approximately
0.5 for the repetitions of current speeds ≥ 0.1 m s−1 (07F3_A and higher). For the first repetition
4.4. EXPERIMENTAL VALIDATION 41
Figure 4.9: Current speeds measured during experiment series B. The 95% confidence interval is
indicated by the shadows and the mean current speed by the dashed line, while the colors correspond
to the experiment. a) First repetitions; b) Second repetitions; c) Third repetitions.
42 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Figure 4.10: Comparison between numerical prediction of buoy position and experimental observa-
tion for experiment series B: a)-c) Deviation in x-direction, and corresponding absolute and relative
error; d)-f) Rotation around y-axis, and corresponding absolute and relative error;
of each experiment the error is increased. For the first three repetitions, the error is one as the
numerical model naturally does not predict a deviation from its initial position. The repetitions for
the second experiment (04F2_A through 06F2_C) present a great variation of errors. However, for
the first six repetitions, the 95%-confidence interval crosses zero, wherefore no qualified statement
can be made for low current speeds.
For higher current speeds, the numerical model shows a much better fit on the rotation around
the y-axis. However, a similar pattern as for the x-deflection can be found, where the first repetition
of the experiments shows a significantly higher error. For example for the last experiment, the first
repetition shows a relative error of -0.4, while the relative error of the second and third repetition
were reported to be -0.04 and -0.06 respectively. For the first four experiments (u ≤ 0.2 m s−1) a
very large 95%-confidence interval was reported (see figure 4.10 c)). However, the for the majority
of the repetitions, the absolute error was found close to 0, while the 95%-confidence intervals are
significantly larger (see figure 4.10 c)), suggesting that the errors reported for lower currents might
be the result of the experimental setup.
The deviation in z-direction is shown separately in figure 4.11, as the predictions of the numerical
model are far off of the observed deviations. With increasing current speed, the buoy moved
upwards, reaching the maximum position at around 0.2 m s−1 before it begins to slowly drop again.
In contrast, the numerical model predicts a deflection of 3 µm towards the bottom of the channel.
4.5. VALIDATION OF LINE TENSIONS 43
Figure 4.11: Comparison of z-position of the buoy between numerical prediction and experimental
observation for experiment series B.
4.5 Validation of Line Tensions
Accurately predicting tensions in mooring lines is crucial for designing effective mooring systems.
Since during the experimental validation, the tensions were not measured, the python module is
compared to data found in literature. Specifically, data from full-scale experiments conducted by
Berteaux et al. [11], who moored a spherical subsurface buoy to the sea floor at a depth of 27.5 m
using wire rope with a nominal diameter of 5/16 in. The diameter of the buoy was 48 m, while its
buoyancy was varied between 400 lb, 800 lb, and 1050 lb. The mass of the buoy was adjusted to
match the buoyancy for a perfectly spherical buoy with the given diameter. The initial depth was
set to 5 m below the surface. To compare the prediction by OffshorePy to the reported tensions,
the properties of the wire rope were calculated based on equations (4.10) and (4.11), provided by
Ma et al. [68]. The results of the calculations can be seen in figure 4.12, with the relative error
defined by
erel =
fnum
fexp
− 1 , (4.25)
where fnum refers to the tension as predicted by OffshorePy and fexp to the value from literature.
It should be noted, that the values reported in [11] for the experiment (here called Dynamoor) are
the mean tensions. The tension ranges were reported to 573 lb, 780 lb, and 786 lb, respectively, fully
embracing the predictions by OffshorePy and SSMOOR. As can be seen in figure 4.12, OffshorePy
overestimates the tensions with respect to SSMOOR, and underestimates the tensions with respect
to the measurements. The mean relative error with respect to SSMOOR was calculated to 0.040
0.04 and the mean error with respect to the measurements (i.e., Dynamoor) are -0.073.
4.6 Discussion
The impact of current speed on the quality of the current measurements is evident from the results
presented in figures 4.5 and 4.9. As the current speed increases, particularly for the experiments
A13F5_A and following with pump frequencies exceeding 30 Hz, the current quality deteriorates.
44 CHAPTER 4. QUASI-STATIC MOORING ANALYSIS
Figure 4.12: Comparison of line tensions between Dynamoor and OffshorePy.
Figure 4.13: Water surface of the channel for different pump frequencies. a) 20 Hz; b) 40 Hz; c)
60 Hz. The surface appears to be inclined due to the perspective of the photos.
This behavior may be attributed yo the quality of the current in the channel as shown in figure 4.13.
The surface appears to be completely flat at 20 Hz, but as the frequency rises, small ripples appear
at 40 Hz, and eventually a noticeable disturbance is observed at 60 Hz. Since the current is the
only environmental influence on the water, this indicates a decrease in the alignment of the stream
lines. Previous studies have reported a significant vertical profile of the current flow in this channel,
where the vertical position of the peak velocity varies with the frequency of the pumps. However,
since the current speed was only measured at a single point upstream of the buoy at the same
depth during the experiments, the vertical profile of the current during the experiments remains
unknown.
Regarding the experimental series A, it is important to note that the numerical model does not
include the fairleads (i.e., eye bolts of the buoy) and their effect on the buoy. The fairleads are
positioned at the bottom of the buoy and their drag force generates a opposing moment to the
4.6. DISCUSSION 45
moment of the drag force of the buoy itself. This effectively reduces the rotation around the y-axis,
which could explain the overestimation of the pitch angle as seen in figure 4.6 c). Additionally,
the reduced quality of the current could be the source of the error for currents above 0.42 m s−1.
During the experiments, a significant deflection of the vertical section of the mooring lines was
noted, effectively shortening the diagonal part of the mooring line. This resulted in lowering the
position of the buoy, as it was observed during the experiments (see figure 4.6 b)). However,
this effect cannot be represented by the mooring software as such mooring configuration is rather
untypical.
For the above mentioned reasons, a second experiment series, series B, was performed; to address
the limitations of the first series and to eliminate the fairleads and the vertical sections of the
mooring lines. In the series B the buoy was positioned slightly higher than in the experimental
series A. Noticeable is the unexpected behavior of the buoy to first raise its position with increasing
velocity and to drop again after reaching its maximum height at roughly 0.22 m s−1. A similar
behavior was observed in preliminary experiments at this height, suggesting that it may be linked
to the vertical position of the buoy above the channel’s floor. The influence of this effect on the
x-position of the buoy is unclear, but the fact that the relative error during series A was much lower
for the x-position suggests a possible link. Nonetheless, the agreement regarding the trim of the
buoy is much better. The numerical model showed a high relative error for some repetitions and
good agreement for others, indicating the need for further investigation.
Both experimental series showed high standard deviations for position and rotation of the buoy,
especially at lower current speeds. It is possible that some of the reported deviations could be
attributed to the camera sensor noise since the observed deflections are at the order of magnitude
of a pixel. For future experiments a higher elasticity of the mooring system is recommended to
reduce the error due to spacial resolution of the sensor. However, during preliminary experiments
with a single mooring line the buoy never reached quasi-static equilibrium and oscillated with
an amplitude of several diameters perpendicular to the current. As shown in the experimental
series B, the estimation of the tilt of the buoy was very good, while the horizontal position was
underestimated by approximately 0.5. It remains unclear, weather this is an error of the numerical
model or due to the experimental setup. Nonetheless, for the purpose of the software developed
in this work, the angle is of much greater importance than the mooring offset. Additionally, the
error of the tensions in comparison to the software SSMOOR and measurements of the Dynamoor
experiments was very, with a relative error of 0.04 and -0.073, respectively. This suggests that the
tensions in subsurface mooring lines can be predicted reliably with OffshorePy.

Chapter 5
Genetic Algorithm
For the concept design of the position keeping system a multi objective genetic algorithm is used
as they are described by Zhou et al. [125]. In this chapter the genetic algorithm is explained while
its application can be found in chapter 8. This approach has been successfully applied by several
authors in the past. For example Xu et al. [123] proposed a design methodology based on the non-
dominated sorting algorithm for very large floating structures. Xia et al. [122] optimized a single
point mooring system based on static equilibrium model with a multi-objective genetic algorithm.
Shafieefar and Rezvani [101] proposed a design procedure to optimize parameters such as heading
and mooring pattern of floating structures with the help of a genetic algorithm. Another work has
been presented by Pillai et al. [84], who optimized the mooring system for a floating offshore wind
turbine. Therefore, the problem is generalized and simplified, only described by a few parameters,
which are registered and joined in the genome of the system. During the optimization process, the
algorithm modifies the parameters and evaluates the obtained position keeping system until the
termination criterion is met. A brief description of the overall principle is given in figure 5.1. The
process beings with the initialization, where the first generation is randomly created, obtaining a
wide spread gene pool. Each set of parameters (i.e. genome) represents one variation of the position
keeping system (i.e. individual). In the next step, all variations (i.e. the whole generation) are
evaluated and sorted according to their individual fitness. The fitness can be evaluated with respect
to a single objective or multiple objectives. Single objective genetic algorithms (SOGAs) optimize
the problem for a single objective which can include multiple objectives via a fitness function, that
reduces the fitness to a single score, and is discussed in more detail in section 5.1. Multi objective
genetic algorithms (MOGAs) typically aim to find the Pareto-Optimized Front, which is not bound
to a specific weighting between the different objectives, as described in section 5.2. The individuals
are sorted and only a few individuals are selected to serve as breeders for the next generation. For
the selection, many different algorithms can be selected, where tournament-selection and roulette-
wheel-selection are among the most used for genetic algorithms. These breeders are randomly
recombined to create new individuals and mutation is applied to maintain a wide spread gene pool.
If elitism is used, the next generation is mixed with the previous generation and cut down to the size
47
48 CHAPTER 5. GENETIC ALGORITHM
Recombination
Mutation
Selection
Evaluation
Initialization
First generation
Sorted generation
Breeders
New generation
Next generation
Figure 5.1: Process of optimization with a single objective genetic algorithm.
5.1. SINGLE OBJECTIVE OPTIMIZATION 49
of the original generation size by discarding the worst performing individuals. This helps maintain
a good set of information within the gene pool until better individuals are found which outperform
their breeders.
5.1 Single Objective Optimization
The fitness function aims to evaluate the performance of an individual in a given situation by
considering multiple properties, which are assigned different weights based on their relative im-
portance. Each evaluated property has a minimum, maximum and desired optimum value and an
associated weight. To calculate the fitness score for a given individual, the fitness function applies a
mathematical formula that takes into account the values of each property, their associated weights,
and the chosen function type (quadratic or sinusoidal). This process results in a single value that
represents the overall quality of the individual’s performance in the evaluated situation. The used
genetic algorithm provides either a quadratic function or a sinusoidal function.
The quadratic function is defined by
fi =



0 for xi < xmin(
−1
(x0−xopt)2 (xi − xopt)
2 + 1
)
for xmin ≤ xi ≤ xmax
0 for xi > xmax
, (5.1)
where xi denotes the value of the property, x0 the min or max x-value, whichever is further away
from the optimum value, xopt the optimum value and xmin and xmax the minimum and maximum
value, respectively. The function has its apex at the desired optimum and a simple zero at the
minimum or maximum value, whichever is further away from the optimum value.
The sinusoidal function is given by
fi =



0 for xi < xmin
1/2
(
cos
(
xi−xopt
xmin−xoptπ
)
+ 1
)
for xmin ≤ xi < xopt
1/2
(
cos
(
xi−xopt
xmax−xoptπ
)
+ 1
)
for xopt ≤ xi ≤ xopt
0 for xi > xmax
, (5.2)
where the variables are reserved for the same value as in equation (5.1). The function is bell shaped
with its maximum at xopt and zero at xmin and xmax, while the shape in between is sinusoidal.
Outside the acceptable range of the property, the fitness is set to zero, while the ordinate of
the quadratic function of its apex is the weight (wi) of the specific property. The values of the
property’s fitness (fi) are added up and then normalized by the total weight in order to limit the
maximum fitness to 100:
f =
100∑
wi
∑
wifi , (5.3)
where wi is the weight associated to the property.
50 CHAPTER 5. GENETIC ALGORITHM
5.1.1 k-Factor
The k-factor (k) is an additional factor which is added to the fitness of an individual to obtain its
overall score. It is used to promote a wide spread gene pool particularly in the first generations
by the use of the scale factor (sc), the stretch factor (st), and the hamming distance (dh) between
the individual and the whole generation, that is how different the individual is from its generation
based one the entries in its genome. Thus, the score S is calculated by
S = f + kdh , (5.4)
with
k = sc
st + 1
ngen
, (5.5)
where ngen denotes the number of generation.
5.1.2 Recombination and Mutation
The recombination process involves selecting the best breeder as a pattern, and then randomly
selecting a value for each parameter from the pool of available breeders.
To introduce variability and avoid convergence to local optima, a mutation step is applied to
each parameter with a probability determined by the globally specified mutation rate. Depending
on the parameter type, the mutation strength is taken into account or not. For list types, a random
item is selected from the list, which may be the same or different. For integers and floating point
numbers, a random value between 0 and the mutation strength times the original value is added
or subtracted from the current value. If the new value falls outside the acceptable range for the
parameter, the procedure is up to 100 times repeated until a valid value is found or the value
remains unchanged.
5.2 Multi Objective Optimization
In multi-objective optimization, it is often beneficial to avoid aggregating the different objectives
into a single score, as is typically done in single-objective optimization. Instead, a Pareto-Optimized
front can be searched, which consists of individuals that represent the best possible trade-off between
different objectives. The Pareto front is defined as the set of individuals whose improvement in one
objective comes at the cost of a degradation in another objective. This means that any individual on
the Pareto front represents the best possible trade-off between the different objectives, and further
optimization of one objective would necessarily lead to a deterioration in performance with respect
to the other objectives.
5.2. MULTI OBJECTIVE OPTIMIZATION 51
5.2.1 Sorting
In comparison to the single objective evaluation, no absolute value can be given to determine
whether an individual is superior to another or not. MOGAs usually sort the individuals by their
rank and use the crowding distance to order individuals of equal rank.
Rank A rank-based fitness assignment gives each individual a rank based on the number of
individuals it is dominated by. An individual is considered to be dominated (i.e. it is inferior) if
there exists another individual that is at least as good as it in all objectives, and strictly better
in at least one objective. If all objectives are fulfilled equally good by two compared individuals,
neither of both individuals is dominated by the other (i.e. non-inferiority) [41].
There are different ways to define the rank of an individual, depending on the specific algorithm
used. One common definition is based on the work of Fourman [42], who defines the rank of an
individual as the number of individuals which are superior elevated by one. For example, individuals
of the Pareto optimal set have rank 1, as they are dominated by no other individual. Individuals of
rank 2 would only be dominated by one other individual and so on. Another definition for the rank,
proposed by Golberg [45], defines the rank as the number of individuals of different ranks, which
dominated the individual in question. For example, an individual dominated by two individuals of
rank 1 would be of rank 3 following Fourman and of rank 2 following Golberg.
Crowding Distance The crowding distance of an individual is a measure of its diversity with
respect to the other individuals in the population for a particular objective. Specifically, it is cal-
culated as the difference in performance for that objective (j) between the two adjacent individuals
(f i+1,j and f i−1,j) when sorted regarding their performance for the objective and normalized by
the range of performance values between for that objective in the population. The total crowding
distance of an individual i is the simple sum of the crowding distance for each objective. Those
individuals which only have one neighbor, which are the best and the worst performing individuals
regarding the analyzed objective, have an infinite crowding distance.
d̃i =
no∑
j=1
f i+1,j − f i−1,j
fmax,j − fmin,j
, (5.6)
where no is the total number of objectives, fmax,j the fitness of the best performing individual, and
fmin,j the fitness of the worst performing individual in the same population.
5.2.2 Creating Next Generation
The breeder for the recombination are selected through Round Robin tournaments, where individ-
uals are selected randomly from the complete gene pool. The best individual from each tournament
is selected regarding rank and crowding distance, and chosen as breeder.
All individuals of the last generation are then copied to the new generation. An equal number
of new individuals is then generated through recombination of the breeders and mutation of the
52 CHAPTER 5. GENETIC ALGORITHM
newly generated individuals. For recombination, each parameter is copied from a randomly selected
breeder. This process is performed for each parameter and each new individual independently,
increasing the probability of a unique combination. Afterwards, a random number between 0 and 1
is drawn for each parameter of the newly generated individuals. If this number is smaller or equal
to the specified mutation rate, the value of the parameter is mutated. In case the parameter is a
list of values, a random value of the list is drawn. If the parameter is an integer or a floating point
number, a second number between 0 and 1 is randomly drawn, multiplied by the mutation strength
and the current value of this parameter. The sign of the value is also set randomly before being
added to the current value. If the new value falls outside the permitted range for the parameter,
this procedure is repeated up to 100 times until mutation of this value is abandoned.
After mutation, the newly generated individuals are evaluated by an external function that must
be provided during initialization of the gene pool. Since the current generation contains double the
desired number of individuals per generation, they are sorted regarding rank and crowding distance
and the inferior half is discarded.
5.3 Implementation
The pattern of the genome is defined in an XML file, which is given as input to the GenePool
Python class responsible for managing genetic optimization. The GenePool class parses the XML
file and uses the information to generate the initial population of individuals and to determine the
parameters for the genetic optimization algorithm. The XML document begins with information
on the XML version and encoding, followed by the root element genome, as can be seen in the
following example:
1 <?xml version ="1.0" encoding ="UTF -8"?>
2 <genome v="0.1">
3 </ genome >
Within the genome element, the parameters subject to change are defined by providing the name
of the parameter, its type, and the range. The name is used to reference to the specific parameter
within Python. The type of the parameter can be integer, indicated by int, a floating point number,
indicated by float, and a comma separated list, indicated by list. The range attribute defines
the possible range of values, the parameter may reproduce. The min and max value are separated
by a dash "-" and the direction of the square bracket indicates if the boundary is part of the range
(enclosing bracket) or not (faced away from boundary), as can be seen in the following listing:
1 <?xml version ="1.0" encoding ="UTF -8"?>
2 <genome v="0.1">
3 <parameter name="line numbers " type="int" range ="[1 -5]"/>
4 <parameter name="line diameter " type=" float " range =" ]0.005 -0.2[ "/>
5 <parameter name="line material " type="list" range ="dyneema , steel "/>
6 </ genome >
5.3. IMPLEMENTATION 53
Selection
of breeder
Discarding
inferior half
Recombination
Mutation
Joining sets of
individuals
Evaluation
Sorting
Current
generation
Breeder
New
individuals
Mutated
individuals
Evaluated
individuals
Preliminary
generation
Preliminary
generation
Empty
gene pool
Next
generation
Final
generation
Initialization
Finished?
Figure 5.2: Process of optimization with a multi objective genetic algorithm.
54 CHAPTER 5. GENETIC ALGORITHM
The parameter "line numbers" is an integer and ranges from one to five, including the boundary,
whereas the parameter "line diameter" is a floating point number ranging from 0.005 to 0.2, not
including the boundaries. The line material may be one of "dyneema" or "steel".
Sometimes one or more parameter define one object which can be used multiple times in different
appearances. To obtain this behavior, the corresponding parameter are child elements of a block
element.
1 <block name=" hulls " prefix ="hull" range ="[1 -3]">
2 <parameter name=" diameter " type=" float " range ="]0.1 -1]"/>
3 <parameter name=" length " type=" float " range ="]0.1 -5["/>
4 </ block >
The number of objects can be accessed via the name, in this case "hulls", which always is of type
integer. The prefix is used to get access to the objects, follow by an underscore, the serial number
of the object, a dot, and finally the parameter’s name. For example, to obtain the diameter of the
second hull, the string "hull_1.diameter" has to be used. The range and type attributes of both, the
block element and the parameter elements of the object, follow the same rules as described above.
5.3.1 Performance
The performance of the algorithm was evaluated with the Griewank-function of 10-th order as
suggested by Digalakis and Margaritis [28]. The function (f) is given by
f = 1 +
10∑
i=1
x2
i
4000
−
10∏
i=1
cos
xi
i
, (5.7)
where xi is the i-th parameter. As can be observed, the function attains its absolute minimum at
0 when each parameter is also 0. The optimization process involved 150 generations, each with 50
individuals, a mutation rate of 0.6, a mutation strength of 1.0, and a randomly generated initial
generation. Each parameter was restricted to 6 decimal digits. The maximum parameter value per
generation is shown in figure 5.3 a) while figure 5.3 b) illustrates the function value. Both maximum
parameter and the function values approach zero rapidly. However, the absolute minimum is not
reached until the 102nd generation.
5.3. IMPLEMENTATION 55
Figure 5.3: Evaluation of genetic algorithm with Griewank-function. a) Maximum parameter; b)
Value of Griewank-function. The vertical line marks the 102nd generation at which the absolute
minimum was reached.

Chapter 6
Site Characterization
For the design process, comprehensive data on the construction site are essential to reliably pre-
dict required dimensions of the mooring system, select suitable anchors, and estimates the loads
occurring during operation and maintenance. These data should cover meteorological and oceano-
graphic conditions, water depth, geotechnical conditions, seismicity, biology, and human activities
[33]. However, since this work concentrates on the design of the mooring system and the platform
for mounting the turbine, not all topics will be addressed. Previous research has already been
conducted on topics such as selecting a suitable location and estimating power potential [1]. This
chapter concentrates on the extreme values to be expected regarding wind, waves and currents.
The data sources used are described in section 6.1, while the methodology applied for estimating
extreme return levels can be found in section 6.2. The values obtained from the extreme value anal-
ysis are presented in section 6.3. Despite the lack of data specifically for the potential construction
site, a brief review of the geological and geotechnical characteristics of the region can be found in
section 6.4.
6.1 Data Sources
For the extreme value analysis, data on the currents have been taken from the Hybrid Coordinate
Ocean Model (HYCOM) Consortium, data regarding wind and waves from the ERA 5 database,
and hurricane data from the HURDAT2 database. Data from the research expedition CEMIE-1
was used for the bathymetric analysis. In the following, all sources are explained in detail.
6.1.1 Research Expedition CEMIE-1
The research expedition CEMIE-1 on board the research vessel B.O. Justo Sierra took place from
the 25th April 2019 and the 5th may 2019. The primary objective of this expedition was to gather
information around Cozumel on the soil composition, the currents, the bathymetry and various
biological parameters. The following systems were used during the campaign for the parameter
57
58 CHAPTER 6. SITE CHARACTERIZATION
with importance for this work:
• Sub-bottom profiler Kongsberg TOPAS PS 120 used for the soil composition
• Multibeam echo sounder Kongsberg ES 302 used for bathymetry and soil composition of
uppermost layer
• Acoustic doppler current profiler (ADCP) by Teledyne-RDI used for current profile
In this work, the most important data was obtained during three transects crossing the Cozumel
Channel, which were carried out roughly from the 200 m isobath on the island’s coast to the 200 m
isobath on the continental coast. As depicted in figure 6.1, three transects were carried out, starting
at the channel’s north and finishing at the channel’s south end. The transects are arranged in an
alternating pattern, perpendicular to the prevailing current and therefore in southeastern (transect
one and three) and northwestern (transect two) direction. During the transects, current, soil com-
position, and bathymetry were measured. However, during the connection passages, only current
and bathymetry measurements were carried out, as determining soil composition was not possible
in due to the pronounced slopes in these areas.
6.1.2 HYCOM Consortium
The HYCOM consortium develops and evaluates a hybrid isopycnal-sigma-pressure (generalized)
coordinate ocean model. For the Gulf of Mexico including the Cozumel Channel, they provide
information on the northward and eastward water velocity, salinity, temperature, surface elevation
and temperature, among others. The consortium provides the data based on reanalysis in different
projects, each with specific spacial and temporal coverage as well as resolution. For this work, the
data of the model HYCOM + NCODA GOMNu0.04 experiment 50.1 were used, which are are
publicly available in https://www.hycom.org/data/gomu0pt04/expt-50pt1. The model provides
data for the Gulf of Mexico and parts of the Mexican Caribbean, covering 1st January 1993 to 31st
December 2012. The temporal resolution is three hours and the spatial resolution is 0.04°, while
the data are reported for 40 discrete depths.
6.1.3 CANEK Research Project
Measurements have been taken of the current in the Cozumel Channel by the CANEK project [16].
The Centro de Investigación Científica y Educación Superior de Ensenada coordinated the CANEK
research project since its foundation in 1996, also named Estudio de la circulación y el intercambio
a través del Canal de Yucatán (Study of circulation and exchange through the Yucatan Canal). A
stationary, long-range acoustic Doppler current profiler at N20°32.218′ W87°2.738′ was anchored
at a depth of approximately 400 m to obtain the data. The measurements were recorded every half
hour, from 9th April 2009 to 14th May 2011 at 16 vertical cells. The cell closest to the surface was
located at a depth of 49 m.
6.1. DATA SOURCES 59
Figure 6.1: Sections of measurement within the the Cozumel Channel carried out during research
expedition CEMIE-1, based on nautical chart S.M. 922 by Secretaria de Marina [99]. A solid red
line indicates transects, while a dotted red line indicates the trajectory for connecting different
transects.
60 CHAPTER 6. SITE CHARACTERIZATION
6.1.4 ERA 5 Database
The ERA 5 database is based on reanalysis and provided by the European Centre for Medium-
Range Weather Forecasts (ECMWF) [51]. At time of analysis, the data covered the complete earth
from 1978 up to date and were later updated to include data from 1950 and later. The values
are reported on an hourly basis and a grid with a spacial resolution of 0.25° for atmospheric and
land-surface parameter (such as the wind) and 0.5° for (ocean) wave parameter.
6.1.5 HURDAT2 Database
The HURDAT2 database is a valuable resource for the analysis of tropical cyclones. The database
is compiled by the National Hurricane Center (NHC) after the cyclone has passed, as real-time
data may not be available for all aspects of the storm. The database is constantly updated and
reevaluated with modern methodologies, particularly for older records. At time of analysis, the
database contained data on all recorded tropical cyclones from 1851 up to date [64], including
information such as the central pressure, position, intensity (i.e. wind speeds) and the storms’
spacial extend. The data are recorded every six hours with interpolation performed for records
earlier than 1956, for which raw data are only available on a 12-hour basis. Furthermore, newer
records may contain asynoptic points to mark important events (e.g. landfall). It is important to
note, that the database is not complete and does not contain all tropical cyclones occurred since
1851. The reporting stations and ships that provide data were limited in number in the early
years, and satellite-based imagery was not available. Therefore, some events may be missing in the
database or are not fully documented.
Studies have been performed to improve understanding of the uncertainty associated with hur-
ricane detection in the years before 1965. These studies have suggested that the apparent increase
in hurricane frequency in the HURDAT2 database may be partly due to increased detection capa-
bility [119]. The failure to detect some hurricanes, particularly those of category 4 and 5, can result
in overestimation of the return period of hurricanes and underestimating the correct design wind
speeds.
6.2 Estimation of Extreme Values
For estimating extreme values corresponding to a specific return period, the application of extreme
value theory (EVT) is necessary. EVT can follow two different procedures: Counting extreme values
in a certain time span (i.e. block), known as Block Model, or counting the extreme values above
a certain threshold, known as Threshold Model or peaks-over-threshold (POT) [92, 46]. The block
model is widely used for analyzing metocean data, where annual maxima, minima or averages are
computed by selecting a block size of one year. However, it neglects the typical temporal spread
of extreme events as only one value is picked, which can lead to biased results. For variables with
temporal spread, the POT-model is a better choice, which establishes a threshold and considers all
values above it [92]. The selection of a suitable threshold is crucial for the accuracy of the results.
6.2. ESTIMATION OF EXTREME VALUES 61
Regardless of the chosen model, the obtained extreme values need to be fitted by a suitable
distribution function. Distributions for normalized maxima can be of different types. Mallor et al.
[69] suggest three different types:
Type I Gumbel distribution
Type II Fréchet distribution
Type III Weibull distribution
However, the choice of distribution should be based on the best fit of the data. The selected
distribution is then used to estimate the value corresponding to a specific return period.
Besides those two widely adopted methods of the EVT, specific methods have been developed
using the whole sample for predicting extreme values rather than just a selection of the sample.
Various methods for evaluating extreme values are available in the relevant literature. For example,
for extreme wind speeds due to hurricanes a specific methodology is presented below while for
extreme wave heights a methodology is available in [17].
6.2.1 Peaks over Threshold
A general methodology for an extreme value analysis is the POT approach, which involves fitting an
extreme model to all peaks above a certain threshold. Typically, a generalized Pareto distribution
(GPD) is chosen as excess model. However, the selection of a suitable threshold has a significant
influence on the results. In this methodology, an automated threshold selection method is used
which was originally published by Thompson et al. [113]. This is to reduce the influence and
possible bias of the person conducting the analysis. This methodology was modified and extended
to suit the specific requirements of this work and is described in detail in Ring et al. [90]. The GPD
used is defined following Liang et al. [67] by
F (x) =



1 −
(
1 + ξ x−u
σ
)
−1/ξ
for ξ 6= 0
1 − exp
(
− x−u
σ
)
for ξ = 0
, with x ≥ u , (6.1)
where x represents the random variable, u the threshold, ξ the shape parameter, and σ the scale
parameter. Due to the temporal resolution of the time series, the data cannot be considered as truly
random and the data is filtered with a moving time window. The window length is determined by a
sensitivity study. To identify and eliminate outliers in the univariate dataset, the quartile method
suggested by Laurikkala et al. [65] is applied. For identification of a suitable threshold, potential
thresholds are equally spaced between the 25th and 98th percentile. For each of the potential
thresholds, a reparameterized scale parameter and its difference to this value of the next higher
threshold is determined. A normal distribution with zero mean is fitted through the determined
differences, eliminating one difference after another, starting with the difference corresponding to
the lowest potential threshold. As soon as the p-value corresponding to the normal distribution
is above the significance level of 5%, this threshold is selected as suitable for the extreme value
62 CHAPTER 6. SITE CHARACTERIZATION
analysis. More details and information regarding the estimation of the extreme values and the
corresponding confidence intervals can be found in Ring et al. [90, chapter 2.3], which can also be
found in appendix C.
6.2.2 Extreme Annual Wind Speeds due to Hurricanes
Extreme wind speeds result from low pressure centers such as storms, hurricanes, or typhoons,
depending on the geographical region. The estimation of the wind speed due to hurricanes with a
certain return period can be challenging as the number of hurricanes is typically too low for this
type of analysis. However, taking certain assumptions for distributions makes it possible to reduce
this to a relatively simple procedure which was suggested by Salvadori et al. [92]. The data required
is the number of hurricanes (nk, i.e. occurrences in the sample) within a certain time span of years
(n, i.e. sample size). Furthermore, the mean wind speed (X̄) and the coefficient of variation (V̂ )
need to be calculated for the sample. The mean rate of occurrence (v̂) is then calculated by
v̂ =
nk
n
. (6.2)
Assuming Pareto distributed wind speeds, the shape parameter (α) is defined by
α = 1 +
√
1 +
1
V̂ 2
(6.3)
and the scale parameter (β) by
β =
X̄ (α− 1)
α
. (6.4)
The annual maximum wind speed is modeled with a Fréchet distribution, its parameters can be
obtained from the Pareto distribution. The shape parameter (α) remains the same, while the scale
parameter (β) should be adjusted according to
β′ = βv̂
1/α . (6.5)
The Gumbel reduced variate (yg) is defined as
yg = − ln
[
− ln
(
1 − τB
−1
)]
, (6.6)
where τB is the return period for which the wind speed is calculated. The quantile (x̂) of the
maximum annual wind speed corresponding to the return period (τB) is calculated by
x̂ = β′ exp
(yg
α
)
. (6.7)
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 63
6.2.3 Expected Wave Frequency
Extreme waves are calculated regarding their wave height with the methodology described above.
According to relevant literature and design codes such as DNV OS-J101 [26] and IEC 62600-2 [54],
the wave period corresponding to the wave height is in the range of
11.1
√
H
g
≤ T ≤ 14.3
√
H
g
, (6.8)
where H is the wave height and g the gravitational acceleration. Depending on the purpose of the
calculation, the conservative wave period is chosen.
6.2.4 Spacial interpolation
Several methodologies are available for geo-spacial interpolation [75]. One commonly used method is
the inverse distance weighted interpolation, which involves calculating the values at the surrounding
nodes and weighing them according to their inverse distance and a specific power factor. Typically,
this method considers 10 to 30 surrounding nodes and uses a power factor of 2 [75]. The depth (hj)
at the new point j is calculated by
hj =
∑
hi (di,j)
−p
∑
(di,j)
−p , (6.9)
where di,j denotes the Euclidean distance between the point j and node i, p the power factor, and
hi is the water depth at node i.
6.3 Meteorological and Oceanographic Data
The results of the extreme value analysis for the wind speed, significant wave height, and ocean
currents are presented below. Furthermore, the bathymetry derived from the expedition data is
described and values for storm surge, tides, and sea level rise are reported based on available
literature.
6.3.1 Extreme Wind Speed
The extreme wind speeds are estimated based on the hurricanes found in this region and based on
data provided by the HURDAT2 database. Additionally, the wind speeds at 10 m height, as they
are reported by the ERA5 reanalysis model, are analyzed.
Hurricanes
Due to the geographical location, Cozumel Island and the Cozumel Channel are hit by hurricanes
with significant severity. To analyze the hurricanes which have passed near Cozumel, the HURDAT2
64 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.2: Storms and hurricanes considered for statistical analysis.
Table 6.1: Return periods of extreme wind speeds due to hurricanes.
Return period, in years 2 5 10 15 20 25 50 75 100
Wind speed, in m s−1 22.63 32.53 41.37 47.38 52.09 56.04 70.20 80.02 87.79
database elaborated by the National Oceanic and Atomspheric Administration (NOAA) was used
[64]. Hurricanes and tropical storms which passed the location within 2° with wind speeds ≥ 34 kt
were considered in the analysis. The results of this analysis are shown in figure 6.2, where the
horizontal lines delimit the threshold for the different categories of tropical depressions. The data
reveals that in the early years, there were very few hurricanes, particularly those with higher wind
speeds. As mentioned in section 6.1, this could potentially be attributed to the methods used for
data acquisition during this period, which differs significantly from the methods employed nowadays.
From 1851 to 2018, 1893 hurricanes were registered in the HURDAT2 database in the Atlantic basin,
152 of which passed close to Cozumel. The mean wind speed during passage was 30.511 m s−1 with
a standard deviation of 16.290 m s−1. The rate of occurrence was estimated to 0.910 and shape
and scale parameter to 3.123 and 20.742, respectively, resulting in an adjusted scale parameter of
20.127. More information on the parameters can be found in section 6.2.2. The obtained extreme
wind speeds due to hurricanes for selected return periods are presented in table 6.1.
Wind at a Height of 10 meters
The numerical data are provided by the ERA5 reanalysis model were developed by the ECMWF.
For the following analysis the node at N20.418° W87.114° was selected as it is the closest node to
a potential site [1] for which both, wind and wave data exist.
The results of the study regarding the time window length can be seen in figure 6.3. The
relative difference in figure 6.3 b) dips for the first time below the 5% mark at a time window
length of 9 days, which is equivalent to 216 (numerical) observations. All identified peaks are
shown in figure 6.4. Vertical lines indicate the occurrence of a hurricane within the distance of 2°
with the color intensity related to the maximum wind speed. The horizontal dashed lines indicate
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 65
Figure 6.3: Study on time window length for filtering peaks of data on wind at 10 m height. a)
Number of found peaks; b) Relative difference in number of found peaks to subsequent time window
length.
the upper and lower limits for what is considered a legitimate peak and what is considered an
outlier. However, the majority of possible outliers occur during hurricanes for which in this analysis
outliers were not removed from the data set. The peaks appear to be fairly equally distributed in
the domain, indicating that the time window of 9 days is sufficient to capture extreme events. A
total of 1506 peaks were used for the study.
The threshold of 10.1379 m s−1 was found to be the most suitable, with a corresponding p-value
of 0.0778 and 133 peaks exceeding this threshold. The shape parameter resulted to 0.4418 with
the 95% confidence interval ranging from 0.138 to 0.745, indicating that the corresponding GPD
excess model is unbounded [18]. The scale parameter was calculated to be 0.6659. The diagnostic
plots presented in figure 6.5 show some deviations between the excess model and empirical data,
particularly in the p-p plot (figure 6.5 a)) and the q-q plot (figure 6.5 b)). The p-p plot shows
that the excess model overestimates lower values and underestimates higher values. The q-q plot
indicates that the most rare peaks are underestimated by the excess model. However, the return-
level plot (figure 6.5 c)) shows that 98.5% of the peaks lie within the estimated 95%-confidence
interval (CI) of the return level, suggesting that the excess model is a good fit to the data. For
some specific return periods the corresponding extreme wind speeds are presented in table 6.2.
66 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.4: Identified peaks, together with the limits between peaks and outliers (horizontal lines)
and occurrences of hurricanes (vertical lines).
Table 6.2: Return levels of extreme wind speeds at height of 10 m.
R
et
ur
n
p
er
io
d,
in
ye
ar
s
E
xt
re
m
e
w
in
d
sp
ee
d,
in
m
s−
1
95
%
C
I,
in
m
s−
1
0.50 10.497 10.170 . . . 10.824
1.00 11.166 10.404 . . . 11.927
2.00 12.074 10.880 . . . 13.268
3.00 12.749 11.321 . . . 14.178
4.00 13.308 11.723 . . . 14.892
5.00 13.792 12.093 . . . 15.492
10.00 15.642 13.569 . . . 17.715
15.00 17.017 14.618 . . . 19.417
20.00 18.154 15.389 . . . 20.918
25.00 19.141 15.972 . . . 22.309
30.00 20.022 16.421 . . . 23.623
40.00 21.566 17.055 . . . 26.077
50.00 22.906 17.464 . . . 28.349
75.00 25.707 17.963 . . . 33.451
100.00 28.022 18.076 . . . 37.967
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 67
Figure 6.5: Diagnostic plots for the GPD excess model fit. a) p-p plot; b) q-q plot; c) return level
plot; d) density plot.
68 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.6: Study on time window length for filtering peaks of data on significant wave height.
a) Number of found peaks; b) Relative difference in number of found peaks to subsequent time
window length.
6.3.2 Significant Wave height
The data provided by the ERA5 reanalysis model, developed by the ECMWF describe the hourly
significant wave height of combined wind sea and swell. For the analysis, the same node as for the
analysis of the wind (N20.418° W87.114°) was selected as it is the closest node to a possible site
identified by Alcérreca-Huerta et al. [1].
The results of the study regarding the time window length are shown in figure 6.6. The figure
reveals that the relative difference in figure 6.6 b) dips for the first time below the 5% mark at a time
window length of 7 days, corresponding to 168 (numerical) observations. The resulting peaks are
shown with black dots in figure 6.7. The upper and lower limits for what is considered a legitimate
peak and what is considered an outlier are represented by horizontal dashed gray lines. The thin
vertical lines indicate hurricanes which passed the node within a radius of 2°, where the color
intensity represents the maximum wave height was during the passage of the hurricane. Similarly
to the analysis of the wind at a height of 10 m, many of the peaks which would be considered as
outliers correlate with the occurrence of hurricanes, wherefore no peak was filtered out for this
study. All 1760 peaks are well distributed, suggesting that the time window length of 7 days is
suitable for this analysis.
Following the methodology described in section 6.2, the most suitable threshold was estimated
to 1.5189 m with a corresponding p-value of 0.0576 and 179 peaks above the threshold. The shape
parameter resulted to 0.3817 with the 95% confidence interval ranging from 0.065 to 0.698, resulting
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 69
Figure 6.7: Identified peaks of significant wave height together with the limits between peaks and
outliers (horizontal lines) and close hurricanes (vertical lines).
in an unbound excess model [18]. The scale parameter was calculated to be 0.2023. The p-p plot
shown in figure 6.8 a) has a z-shaped deviation. This results in overestimating low values and
underestimating higher values, particularly for the most extreme values the q-q plot in figure 6.8
b) supports these findings. With 98.3% the vast majority of the peaks are withing the estimated
95%-CI of the return level (see figure 6.8 c)). Despite the differences, the excess model is a good
fit for the numerical data. For selected return periods the corresponding extreme significant wave
heights are presented in table 6.3.
6.3.3 Currents
The data on the current are taken from the HYCOM and evaluated at N20.40° and W87.12°,
the closest node to the data on wind and wave as provided by the ERA5-reanalysis model of the
ECMWF. The current was examined at the surface and at a depth of 50 m. Characteristic values for
the currents at the sea bed are provided by Flores [40], who suggests vbot = 0.3 m s−1 . . . 0.4 m s−1.
The overall prevailing current direction is NNE with a share of 94%, while they stay 96% in the
direction between N and NE as stated by Flores [40]. For the specific node analyzed, the HYCOM
model predicts the surface current mostly in eastern direction as can be seen in figure 6.9 a). The
current at a depth of 50 m (see figure 6.9 b)) is also closely confined, but appears to be dependent
on the current speed. For lower current speeds, the current has a heading of east-northeast, while
for higher currents, it has a higher eastward component.
Currents at Surface
The study on the time window length for peak identification shows the same behavior as wind and
waves (see figure 6.10). The relative difference constantly decreases as shown in figure 6.10 (b)
and falls for the first time below the 5% mark at a time window length of 8 days, equivalent to
64 (numerical) observations. The resulting peaks are shown with black dots in figure 6.11 while
70 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.8: Diagnostic plots for the GPD excess model fit to the significant wave height. a) p-p
plot; b) q-q plot; c) return level plot; d) density plot.
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 71
Table 6.3: Return levels of extreme significant wave height.
R
et
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n
p
er
io
d,
in
ye
ar
s
E
xt
re
m
e
si
gn
ifi
ca
nt
w
av
e
he
ig
ht
,
in
m
95
%
-C
I,
in
m
0.25 1.537 1.507 . . . 1.567
0.50 1.703 1.624 . . . 1.782
1.00 1.919 1.797 . . . 2.041
2.00 2.201 2.045 . . . 2.357
3.00 2.404 2.209 . . . 2.599
4.00 2.568 2.321 . . . 2.814
5.00 2.708 2.402 . . . 3.014
10.00 3.229 2.602 . . . 3.856
15.00 3.604 2.669 . . . 4.539
20.00 3.907 2.687 . . . 5.128
25.00 4.167 2.680 . . . 5.653
30.00 4.396 2.658 . . . 6.133
40.00 4.791 2.590 . . . 6.992
50.00 5.129 2.503 . . . 7.755
75.00 5.822 2.253 . . . 9.391
100.00 6.383 1.988 . . . 10.778
Figure 6.9: Histogram of current as modeled with HYCOM. a) Current at the water surface; b)
Current at a depth of 50 m. Both values for the node at N20.40° and W87.12°.
72 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.10: Study on time window length for filtering peaks of data on current speed at surface.
(a) Number of found peaks (b) Relative difference in number of found peaks to subsequent time
window length.
the red dots indicate outliers which have been filtered out. The upper and lower limits for what is
considered a legitimate peak and what is considered an outlier are represented by horizontal dashed
gray lines. The thin vertical lines indicate hurricanes which passed the node within a radius of 2°.
The more intense their color is, the greater their maximum wind speed was during their passage. In
contrast to wind and wave, only a few outliers correlate with the occurrence of hurricanes for which
the outliers were filtered out at this node. In total 736 peaks were found with 9 outliers above the
upper threshold and 3 outliers below the lower threshold. The distribution of the peaks appears to
be random for which the time window length of 8 days is considered as adequate.
The best threshold was estimated to 1.1519 m s−1 with a p-value of 0.0520, resulting in 379 peaks
over threshold. The corresponding shape parameter is −0.4044 with its 95 %-CI between −0.519
and −0.290. Due to the negative CI of the shape parameter it is safe to assume that a bound excess
model is the best fit [18]. The scale parameter was calculated to 0.2080. Following the p-p plot
in figure 6.12 a), the excess model resembles the numerical observations quite good, with a slight
overestimation of the values at the lower end, also visible in the density plot (see figure 6.12 d)). At
the upper end, the model deviates from the empirical data, as can be seen in figure 6.12 b) and c).
Nevertheless all peaks are within the 95 %-CI. For some specific return periods the corresponding
extreme significant wave heights are presented in table 6.4.
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 73
Figure 6.11: Identified peaks of current speed at surface together with the limits between peaks
and outliers (horizontal lines) and close hurricanes (vertical lines).
Table 6.4: Return levels of extreme current speed at surface.
R
et
ur
n
p
er
io
d,
in
ye
ar
s
E
xt
re
m
e
si
gn
ifi
ca
nt
w
av
e
he
ig
ht
,
in
m
95
%
-C
I,
in
m
0.25 1.392 1.269 . . . 1.515
0.50 1.459 1.310 . . . 1.608
1.00 1.510 1.345 . . . 1.674
2.00 1.548 1.375 . . . 1.721
3.00 1.566 1.389 . . . 1.742
4.00 1.577 1.399 . . . 1.755
5.00 1.585 1.406 . . . 1.763
10.00 1.604 1.424 . . . 1.785
15.00 1.614 1.433 . . . 1.794
20.00 1.620 1.439 . . . 1.800
25.00 1.624 1.444 . . . 1.804
30.00 1.627 1.447 . . . 1.806
40.00 1.631 1.451 . . . 1.810
50.00 1.634 1.455 . . . 1.813
75.00 1.639 1.460 . . . 1.817
100.00 1.642 1.464 . . . 1.820
74 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.12: Diagnostic plots for the GPD excess model fit to the current speed at surface. a) p-p
plot; b) q-q plot; c) return level plot; d) density plot.
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 75
Figure 6.13: Study on time window length for filtering peaks of data on current speed at a depth
of 50 m. a) Number of found peaks; b) Relative difference in number of found peaks to subsequent
time window length.
Currents at a Depth of 50 meters
For the currents at a depth of 50 m, the study on the time window length is depicted in figure 6.13.
As expected, the optimum window length is 8 days. The relative difference between a time window
length at the subsequent one (see figure 6.13 b)) decreases with minor fluctuations. As illustrated
in figure 6.14, only 3 outliers were found at the upper end and one at the lower end. Given the fact,
that only one of those three outliers coincides with the occurrence of a hurricane, all outliers were
filtered out. Hurricanes which passed at a distance of no more than 2° are marked with vertical
lines with the color intensity representing the hurricane’s maximum wind speed during the passage.
At the depth of 50 m, 694 peaks were found which seem to be randomly distributed in the domain.
A suitable threshold was calculated to 0.9823 m s−1 with a p-value of 0.0527, resulting in 250
peaks. The shape parameter and its 95 %-CI was estimated to −0.3730 and −0.460 to −0.286,
respectively, indicating a bound excess model similarly to surface currents. The corresponding
scale parameter is 0.1542. The diagnostic plots in figure 6.15 show a good fit, although higher
values are slightly overestimated by the excess model (see figure 6.15 b) and c)). All peaks were
found within the estimated 95 percent-CI of the return level. For some specific return periods the
corresponding extreme currents speeds are presented in table 6.5.
76 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.14: Identified peaks of current speed at a depth of 50 m together with the limits between
peaks and outliers (horizontal lines), close hurricanes (vertical lines) and outliers (red dots).
Table 6.5: Return levels of extreme current speed at a depth of 50 m.
R
et
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n
p
er
io
d,
in
ye
ar
s
E
xt
re
m
e
cu
rr
en
t
sp
ee
d,
in
m
s−
1
95
%
-C
I,
in
m
0.25 1.125 1.039 . . . 1.212
0.50 1.187 1.069 . . . 1.305
1.00 1.235 1.095 . . . 1.374
2.00 1.271 1.117 . . . 1.426
3.00 1.289 1.128 . . . 1.449
4.00 1.300 1.136 . . . 1.464
5.00 1.307 1.141 . . . 1.474
10.00 1.327 1.156 . . . 1.499
15.00 1.337 1.163 . . . 1.511
20.00 1.343 1.168 . . . 1.518
25.00 1.347 1.172 . . . 1.523
30.00 1.350 1.174 . . . 1.527
40.00 1.355 1.178 . . . 1.532
50.00 1.358 1.181 . . . 1.535
75.00 1.364 1.186 . . . 1.541
100.00 1.367 1.189 . . . 1.544
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 77
Figure 6.15: Diagnostic plots for the GPD excess model fit to the current speed at a depth of 50 m.
a) p-p plot; b) q-q plot; c) return level plot; d) density plot.
78 CHAPTER 6. SITE CHARACTERIZATION
Figure 6.16: Position of the used transect in the Cozumel Channel and the position of data points
used for interpolation.
6.3.4 Bathymetry
The position of the transects is shown in figure 6.16. It starts on the mainland coast close to
Plaza del Carmen and ends near Punta Norte on Cozumel Island. The gray dots indicate the data
points which have been used for interpolation. They result from measurements taken during the
expedition CEMIE-O 1 in 2019, from the nautical chart 922-004 published by the Mexican Navy,
and NaturalEarth in case of the coast line. After interpolation with the inverse distance weighted
interpolation (IDW) method, the depth has been filtered with a Savitzky–Golay filter in order to
eliminate strong fluctuations. The results are shown in figure 6.17. The Cozumel Channel shows
a pronounced valley with a relatively flat plain at approximately 450 m of depth. The valley is
delimited by steep slopes on either side, with an inclination ranging between 5 and 10 degrees and
with peaks over 20 degrees. Although the wide valley in the center of the channel provides space for
a lot of turbines, the slopes are interesting for installation as they significantly reduce the length of
the mooring lines and the required length of subsea cables. During early phases of harnessing the
Cozumel Current, it is expected that turbines are installed at the slopes of the Channel in much
6.3. METEOROLOGICAL AND OCEANOGRAPHIC DATA 79
Figure 6.17: Bathymetry of the transect in the Cozumel Channel with the black, dashed lines
indicating the positions with depths of 50 m. a) Water depth,; b) Density distribution of water
depth; c) Angle of the slope; d) Density distribution of absolute angle of the slope.
shallower waters and closer to the coast.
6.3.5 Storm Surge
No exact data are available on the storm surge, however, observations during the passage of Hur-
ricane Gilbert in 1988 suggest that the storm surge may vary between 3 m and 6 m for the coast
of Quintana Roo, including the Cozumel Channel [10]. Thus, the maximum storm surge can be
considered with 6 m due to a hurricane with return period of 125 years.
6.3.6 Tides
The tidal range is generally consistent throughout the Caribbean. On the Mexican coast, a mixed
mainly semidiurnal tide is encountered with an astronomical tidal range of approximately 20 cm
to 30 cm. [61, 62]. This prediction is based on harmonic amplitudes and epochs for six tidal
constituents. Comprehensive data is available from measurements by the Servicio Meteorológico
Nacional (Mexican National Weather Service) at a site near Puerto Morelos (N20°52′6′′, W86°52′)
and Cozumel (N20°31′37.2′′, W86°56′24.6′′) [105].
6.3.7 Sea Level Rise
The study conducted by Ruiz-Ramírez et al. [91] reported a sea level rise of 2.8 mm y−1 for the
entire coast in the Mexican Caribbean between 1993 and 2015. However, more regional estimates
were obtained for two specific locations during the same period. The sea level rise in Puerto Morelos
80 CHAPTER 6. SITE CHARACTERIZATION
was found to be 0.6 mm y−1, while Isla Cozumel experienced a sea level rise of 1.0 mm y−1.
6.4 Geological and Geotechnical Data
The site of interest has not been subject to comprehensive geological or geotechnical surveys, but
previous studies have explored the geology of Cozumel, the Mexican Caribbean, and the Gulf of
Mexico.
The upper most layer or the layer immediately below the ground surface across the entire Island
of Cozumel can be characterized as Caliche II crust, according to the findings of Spaw [106]. Since
slopes generally transport sediment from higher to lower regions in the oceans [98], the upper most
layer in the pronounced slope at the eastward side of the Cozumel Channel can be expected to
have similar sediments to these on the coast of Cozumel Island. Although marine sediments are
similar to land based sediments, there are a few difference such as the pore water presenting a
higher saturation of dissolved gases. According to Schwab et al. [98], the sediments found in the
Gulf of Mexico and the western Caribbean is mainly Carbonate Ooze with skeleton remains greater
than 30%, while close to Cozumel Island terrigenous sediment can be expected. An extensive
study on the modern sediments in the Gulf of Mexico has been performed by Balsam and Beeson
[7] analyzing 186 top core and grab samples, a lot of which were located on the shelf north of the
Yucatán Peninsula and the Campeche Bank which is in the wake of the Yucatan Straight. According
to the findings, the sediment on the shelves near Yucatán is mainly composed of calcareous sands
while the sediment on the slopes can be mainly classified as carbonate mud (micrite). Inspection
of the reefs around the southern part of Cozumel Island, showed the sea floor being covered in
calcareous sediment, coarse sand, and coral debris, which eventually will be transported down the
insular slope [77]. Gischler and Lomando [44] studied reefs south of Cozumel Island, which are
composed of grainstone, wackestone and packstone. Similar to the reefs in the northern region of
the mainland of Belize, Tertiary and Quaternary limestones are present. The foundation of the
reefs in this region was reported to be limestone, according to Purdy [87], Mazzullo et al. [70], and
Shinn et al. [103]. For the Elmer Reef in the north of Belize, limestone was found 1.6 m below the
reef’s top, with coral framestone, partially lithified gravelly sands, and unconsolidated sediments
on top of the limestone [70]. Measurements of samples of limestone taken at the main land close to
Cozumel have been conducted by Santiago Sierra [94], reporting density, unconfined compressive
strength, and Young’s modulus. A work based on measurements to relate tensile strength to the
compressive strength of limestone has been done by Nazir et al. [80] and is defined by
qt,r =
( qu,r
9.25
)1/0.947
(6.10)
.
The limestone in the region of the Yucatan Channel has a density of 2110.2 kg m−3, a uniaxial
unconfined compressive strength of 27.73 MPa, and a minimum Young’s modulus of 4.3185 GPa
[94]. The coefficient of subgrade reaction can be assumed to be 80 MNm−3 [8].
Chapter 7
System Design
The design of the position keeping system for a MCEC is performed mainly via an automated
optimization process based on the genetic algorithm described in chapter 5. To ensure efficacy, it is
important to start with a sound design concept. The choices behind the design concepts used and
the considerations made during this process are discussed in this chapter.
7.1 General Design Considerations
For submerged mooring systems, design of platform and mooring system are strongly coupled. A
decision made for one will inevitably impact the other’s optimum. To systematically evaluate the
options, the design process is separated into the following sub-problems:
• Height-control system
• Rotor alignment
• Buoyancy providing system
• Elevation system for maintenance purpose
• Mooring line configuration
Those problems will be addressed separately in the following although some solutions will overlap
an consequently solve more than one design problem. Other solutions depend on the choices taken
for another problem, as e.g. the elevation system depends on the mooring configuration.
7.1.1 Height-control system
The height-control system mainly controls the vertical position of the platform within the water
column. Maintaining a constant depth of the system despite environmental changes is important
for ensuring productivity and safety of the system. In the upper section of the water column, wind
81
82 CHAPTER 7. SYSTEM DESIGN
a) Spread mooring b) Buoyancy excess
c) Passive-dynamic lift d) Active flight control
Figure 7.1: Concepts for height control systems.
and wave induced currents might have a significant impact on the conditions. Accurate knowledge
of the design depth and its variability over the lifespan is also necessary for designing pressurized
compartments and estimating the fatigue life of various components. In figure 7.1 various systems
to control the platform’s height are shown, which are discussed below.
Multiple Mooring Lines One straightforward and mechanically simple approach for maintain-
ing a constant platform height above seabed, or depth below surface, is a mooring system with
multiple lines as depicted in figure 7.1 a). In this configuration, a slight excess of buoyancy causes
the platform to rise towards the surface, while a minimum of two mooring lines prevent the sys-
tem from floating upwards. The system’s buoyancy is enough to maintain all mooring lines under
tension, even in the most severe load cases, ensuring the platform maintains a constant height
regardless of the load conditions.
Excess of Buoyancy Another simple approach is a system that utilizes a single mooring line with
an excess of buoyancy, illustrated figure 7.1 b). In this system, horizontal and vertical movements
are coupled, as the platform’s possible positions are roughly situated on the surface of a sphere
surrounding the anchor point. Therefore, the tension in the mooring line controls both horizontal
and vertical movements. To significantly reduce the mooring offset and maintain the platform’s
position, even in the severe conditions, a substantial excess in buoyancy is required.
Passive Lift Control An elegant, yet slightly more complex solution involves using integrated
foils in the platform’s structure with a single mooring line, as shown in figure 7.1 c). With a slight
7.1. GENERAL DESIGN CONSIDERATIONS 83
buoyancy excess, the platform can maintain its position during low or no loads. During high-load
phases, the position can be regulated using the dynamic lift of the foil-shaped members of the
platform. Under loading, the surfaces contributing to the dynamic lift increase the tension in the
mooring line due to the increased flow. Furthermore, a reduction in the pitch of the platform (nose
up) generates more lift which further reduce the mooring offset. It is crucial to ensure that the
pitch angle is decreased to prevent downforce, which could push the platform towards the sea floor.
A significant change in the pitch angle could also lead to performance loss in the turbine.
Active Lift Control/Flight-Control System Alternatively to the passive-dynamic lift, an
active flight control system could be used as depicted in figure 7.1 d). This approach requires a
nearly neutrally buoyant platform and a mechanism for adjusting the platform’s static buoyancy
or dynamic lift. Since the platform is nearly neutrally buoyant, the flight control system only
needs to counteract environmental forces. If the leading mooring line is sufficiently long, surfacing
the platform for maintenance would be a simple task. However, the active control requires more
moving parts, which reduces the overall system reliability. Additionally, the mooring radius is
highly dependent on the length of the leading mooring line. Like all systems with a single mooring
line, the lack of redundancy in the mooring line and anchor poses a risk and requires greater safety
factors.
7.1.2 Rotor Alignment
As this system is designed for a horizontal-axis turbine, it should be installed with its rotational axis
parallel to the current, resulting in the current passing through the rotor area perpendicularly. In
the industry, two configurations are commonly used for horizontal turbines. The first configuration
places the rotor upwind of the rotor’s body and is more prevalent, particularly in the wind industry.
The second configuration can sometimes be observed on pod drives of large ships, which require
high maneuverability, or in concept studies for newer wind turbines.
Upstream Upstream turbines position the rotor ahead of the turbine’s body and are the most
common among wind and ocean current turbines. This configuration has the advantage of providing
undisturbed flow to the rotor. However, the platform, or at least parts of it, is subject to cyclic
loading due to the rotor wake. In wind turbines, this phenomenon is often referred to as rotor
shedding, as the frequency of loading on each member depends on rotational frequency, number
of blades, and the number of elements in the turbine’s wake. Blades have a relatively low flexural
stiffness, which leads to high deflections under great load. Depending on the distances between the
platform and the rotor plane, the blade’s tips might get close to the platform, further increasing
the cyclic loads and the risk of damage due to tip vortex cavitation.
Downstream Especially the problem with reduced distance due to flex of the blades can be
avoided by placing the rotor plane downstream. For wind and marine currents, downstream turbines
are rather rare, yet are very common for ship propulsion systems. Just as the platform has to operate
84 CHAPTER 7. SYSTEM DESIGN
a) Front view b) Side view
Figure 7.2: Possible arrangement for a single hull system with hull above turbine. a) Front view;
b) Side view.
in the rotors wake on upstream turbines, downstream turbines place the rotor in the wake of the
turbine housing and its struts. Depending on the design of the platform, this results in a highly
unsteady inflow, potentially increasing vibration and noise, cyclic loading, and the possibility for
cavitation of the turbine blades. The produced noise or vibration is a result of the cyclic loading
due to circumferential speed gradients in the current field. Its frequency depends directly on the
number of struts, the number of blades and the rotor’s rotational frequency and can hardly be
modified during operation. Special care has to be taken on the structural blade design and the
control mechanism to avoid resonance or to block certain rotational frequencies by the control
system of the turbine if it results impossible to avoid resonance.
7.1.3 Buoyancy Providing System
Since the turbine itself is not neutrally buoyant, the platform must provide the required buoyancy.
This is typically accomplished through hydrodynamically shaped floaters or hulls that are inte-
grated into the platform’s structure. Various formulae have been developed to create streamlined
underwater bodies, such as torpedoes or autonomous underwater vehicle (AUV)s, which will be dis-
cussed later. However, one mayor question is the number and geometrical arrangement of the hulls
within the platform’s structure. While the number of hulls can principally be infinite, more than
three hulls might not significantly improve the platform’s properties, reducing the possibilities to
one, two, or three hulls. Furthermore, they should be arranged symmetrically (with respect to the
x-z plane) with their main axis parallel to the current’s flow to reduce drag. These constraints limit
the number of options to only a few, which are discussed below. Nonetheless, several parameters,
such as the hull’s relative position in forward (x), sideways (y), and vertical (z) directions can still
be modified, to suit the project’s specific requirements.
Single Hull To satisfy the symmetry, the single hull should be placed either above or below the
turbine, symmetrically placed in the x-z plane of the rotor as depicted in figure 7.2. A mayor
7.1. GENERAL DESIGN CONSIDERATIONS 85
a) Front view b) Side view
Figure 7.3: Possible arrangement for a double hull system with hull above turbine. a) Front view;
b) Side view.
problem with this arrangement is that it does not provide a restoring moment through the mooring
line, causing the platform to tilt to either the starboard or port side. As a result, the orientation
of the platform becomes less well-defined, making difficult to predict loads on the entire system.
Double Hull A double hull system is a better choice in terms of providing a restoring moment
via the mooring lines. In this arrangement, the hulls are placed at an equal height and an equal
distance from the x-z plane, as shown in figure 7.3. Compared to single hull systems, the double
hull system allows for greater freedom in positioning the hulls, making it easier to avoid the rotor’s
swept area. Additionally, maintenance at the surface is made easier as the platform’s roll is reduced
and the turbine can be placed closer to the surface, improving accessibility for service in water.
Placing the hulls at center line as variation of the double hull system offers no significant benefits
over double hull systems as described above, but carries the drawbacks of both the single and the
double hull systems.
Triple Hull A combination of the single hull systems with the hull at center line and the double
hull systems with the hull each at the turbine’s sides could potentially improve some characteristics
of the platform. In such a system, the center hull does not necessarily need to be of the same size as
the ones on the sides and could house some of the mechanical and/or electronic parts that do not fit
into the turbine’s housing. In the system shown in figure 7.4, two main hulls and a third auxiliary
hull are used. Alternatively, the third hull could be increased in size to house the turbine as well
as a buoyancy tank. However, the advantage of increased redundancy is offset by the disadvantage
of a greater complexity due to more tanks and hulls. The benefits of increased redundancy might
be rather small in comparison to a well-designed two hull system. Moreover, this arrangement does
not provide any significant improvement regarding the platform’s mooring offset and orientation.
86 CHAPTER 7. SYSTEM DESIGN
a) Front view b) Side view
Figure 7.4: Possible arrangement for a triple hull system with hull above turbine. a) Front view;
b) Side view.
7.1.4 Elevation System for Maintenance
Not least because of the permanent currents, in water survey and maintenance while the platform
is submerged is challenging. The best and safest option is to elevate the platform to the surface
and conduct the necessary service either on the surface or by lifting the platform out of the water.
Elevating the platform to the surface is relatively easy in principle as the platform requires either
a slight excess in buoyancy or an active flight system. However, lowering the platform to its design
depth without heavy equipment poses a significant challenge, especially for smaller systems like
marine renewable energy platforms. Unlike large systems from oil and gas, off-the-shelf tensioning
systems such as chain jacks, wire winches, and in-line tensioners are typically unsuitable for smaller
systems. Nevertheless, the functional principle of these larger systems can be applied to such smaller
systems, which is discussed below.
Winch One or more small-scale winches can be integrated into the platform’s structure to facil-
itate maintenance. Since the turbine requires connectivity to the grid even when the platform is
not operating, supply of electricity is not a problem. During operation, one or more mooring lines
would be required to be stored on the winch, with the excess in length of the order of magnitude
of the design depth of the system. For maintenance, the break of the winch would be released and
due to the platform’s buoyancy, it would rise to the water surface. To lower the platform back to
its design depth, the mooring line would be hauled in, and the brake finally engaged. Depending
on the size of the buoyancy tanks, significant force might be required to haul the mooring line,
for which ballast tanks might be necessary. These tanks would be flooded after finishing service
at water surface to reduce the platform’s buoyancy. Once the final position is reached, the ballast
tanks would be evacuated, for example, by pressurized gas tanks or a temporary gas line to the
surface.
To reduce system complexity while maintaining redundancy, a single winch could be considered
if two or more mooring lines have to be lengthened for maintenance. However, this would require
positioning the winch at the center line of the platform to reduce the heel. Furthermore, the mooring
7.1. GENERAL DESIGN CONSIDERATIONS 87
lines need to be fed and guided through the structure, as the fairleads and the attachment point of
the winch would not be close.
In-Line Tensioner In-line tensioners can be particularly useful for systems with a single mooring
line, as they allow for external mechanical power to be used for shortening the mooring line. After
maintenance, a service vessel would pull the free end of the mooring line to tension it and then lower
the platform to its planned position. Before raising the platform again for the next maintenance
cycle, the mooring lines must be picket up by the service vessel or designed to be accessible from
the water surface at all times. However, in-line tensioners may not be feasible for systems with
multiple mooring lines or when the required tools are not available during maintenance cycles. In
particular, achieving a specific heel and trim angle can be a challenge for multiple mooring lines.
7.1.5 Mooring Line Configuration
The mooring line configuration can vary from a single line to four mooring lines. In this section,
the advantages and disadvantages of a configuration with one, two, three, and four mooring lines
are discussed.
Single Mooring Line Configuration In a single mooring line configuration, the platform con-
nects to a single anchor, with the mooring line bridled at its top end and connected to the platform
at various points. If designed correctly, the platform is able to swivel around the anchor pointand
align itself always with the direction of the current. This configuration requires less installation
complexity and material, but lacks redundancy in the system, which could increase the material of
the single anchor. Depending on the exact circumstances and requirements, this might not neces-
sarily result in a reduced cost and thus needs a closer look whether this concept might be the better
choice for a project.
Two Mooring Line Configuration The implementation of a two mooring line configuration for
the system requires one mooring line upstream and one mooring line downstream of the platform,
each connected to their respective anchor. Bridled sections at the top end of both mooring lines
control the heel and heading of the platform. That way, each mooring line connects to both forward
and both aft ends of the platform, respectively.
In comparison to the mooring configuration with a single mooring line, the mooring offset can
be reduced without the need for great forces in the mooring lines. Especially in farms with various
turbines, the mooring offset has to be either very narrowly defined or the distance between the
devices has to be increased significantly to avoid contact or entanglement of the mooring line.
Additionally, the trim of the platform is more closely defined. Nevertheless, the platform looses
its ability to follow the current’s direction on its own. This would either require a more advanced
tracking system or accept the loss in performance due to the current not coming directly from the
front.
88 CHAPTER 7. SYSTEM DESIGN
Three Mooring Line Configuration The third mooring configuration uses three mooring lines
with two anchors upstream of the platform and one anchor located on the center line aft of the
platform. The forward mooring lines connect the anchor directly to the corresponding fairleads,
while the aft mooring lines connects to both aft fairleads via a bridled section at its top end.
The two forward mooring lines lead to better behavior of the platform under currents coming
from either side of the platform and not straight from the front. Furthermore, they help significantly
control the heel of the platform. In comparison to the two mooring line configuration, an additional
anchor is needed. As turbines are typically installed in a farm, sharing anchors might be an option,
depending on the conditions of the sea floor and the layout of the farm.
Four Mooring Line Configuration To furthermore control the platform’s position, an addi-
tional mooring line can be added at the aft end of the platform. Each fairlead located at the four
ends of the hulls connects to an anchor placed radially outwards, with two anchors upstream of the
platform and two anchors downstream of the platform.
Similar to the three mooring line configuration, sharing anchors might be possible, if the condi-
tions of the farm allow for it. The additional anchor and mooring line increase the redundancy of
the system as in case of a line loss, three of four lines absorb the forces, while for the three mooring
line configuration, the forces are distributed between the two remaining mooring lines.
7.2 Design Concept
The description of the design concept for the complete mooring system can be found below. This
concept was used as basis for the subsequent optimization process, described in chapter 8. Its final
review to ensure compliance with the important design codes is documented in chapter 9.
7.2.1 Turbine
In recent years, various concepts have been developed for MCEC that can be classified based on the
alignment of the main rotary axis with respect to the current. The rotor of many recently developed
devices share the same basic design with wind turbines as both types extract kinetic energy from a
fluid. The majority of these devices have their main rotational axis in the horizontal plane, which
is parallel to the current. However, there have been efforts to design cross-flow turbines with an
axis of rotation perpendicular to the current, resulting in a vertical axis of rotation that eliminates
the need to track the current’s direction. Despite this, current designs continue to primarily use a
horizontal axis of rotation.
Available Designs
Different MCECs suitable for this zone and the projected conditions are available, however all of
them show a lack of detailed information publicly available. One such device is being developed by
the Mexican company Potencia Industrial.
7.2. DESIGN CONCEPT 89
A small MCEC has been developed by Schottel HYDRO with a diameter of 5 m, while a bigger
version with 6.3 m in diameter seems to be in development. Sustainable Marine Energy provides two
platform options to moor the turbine to the sea floor: the PLAT-I for surface mooring or PLAT-
O for full submersion of the device. However, the latter seems to be abandoned as no updates
have been published in recent years. The turbine is optimized for higher energy currents typically
found in confined tidal currents or rivers, and adoptions might be necessary for low energetic ocean
currents.
The University of Strathclyde in Glasgow, Scotland is developing a turbine optimized for slow
ocean currents with speeds up to 1.5 m, s−1 found in regions like the Philippines or the Cozumel
Channel in Mexico. However, no final design is available as the turbine is still in the early stages
of technological readiness.
Finally, at the Institute of Engineering of the National Autonomous University of Mexico
(UNAM), Mexico, two prototypes for low energetic ocean currents are under development. The
first is a cross-flow turbine, and an early prototype at scale has already undergone testing in a
circulation tank, towing tank, and natural conditions. The second development is a parallel flow
turbine, similar to the prototype developed by the University of Strathclyde, but in a much earlier
stage of development as tank test have not been carried out yet.
Estimation of Turbine Parameter
Due to the lack of verified data of a turbine suitable for the Cozumel Channel, the platform will be
developed based on a conservative estimate for a parallel flow turbine.
Key Coefficients A key parameter used for characterizing a parallel flow turbine is the power
coefficient (CP ) that is defined by
CP =
2P
ρASv3
∞
, (7.1)
where P denotes the power at the shaft, ρ the water’s density, AS the swept area of the rotor, and
v∞ is the undisturbed velocity of the current at hub height [93]. The power coefficient is a function
of the tip speed ratio
λ =
ωR
v∞
, (7.2)
where ω is the angular velocity of the rotor and R its tip radius. The torque coefficient (CQ) is
directly related to the power coefficient and tip speed ratio by their ratio (CQ = CPλ
−1) [93], thus
can be calculated by
CQ =
2Q
ρASv2
∞
R
, (7.3)
where Q is the torque at the shaft. The thrust coefficient is typically calculated by
CT =
2T
ρASv2
∞
, (7.4)
90 CHAPTER 7. SYSTEM DESIGN
Table 7.1: Expected and estimated principal data of a reference 5 m-turbine.
Description Symbol Unit Value
Rotor diameter D m 5
Total length L m 2.0
Hub diameter d m 0.3
Mass m kg 1200
Displaced volume V m3 0.15
Maximum power coefficient CP − 0.45
Maximum thrust coefficient CT − 0.8
Maximum torque coefficient CQ − 0.075
Design TSR λ − 6.5
where T is the thrust generated by the turbine [93].
Mass and Volume The Schottel Tidal Generator is a tidal turbine developed with a diameter of
5 m Schottel Hydro [96, 97]. The turbine supposedly operates at high velocities as its cut-in velocity
is 0.7 m s−1 and its cut-out velocity is 4.6 m s−1. As a turbine in the Cozumel Channel will work at
much lower current speeds, the rotational speeds are equally lower, possibly requiring a bigger, thus
heavier generator. The Schottel Tidal Generator has a mass of approximately 800 kg to 1000 kg.
Therefore, the reference mass of the turbine for the platform design is estimated conservatively to
1200 kg, while the reference volume is assumed to be 0.15 m3.
Power and Thrust Considering that the use case of the aforementioned turbine is similar, it’s
coefficients are expected to be quite similar. During field trials, the power coefficient of the turbine
SIT250 was found to be CP = 0.44 at a tip speed ratio (TSR) of λ ≈ 5.0 and the thrust coefficient
was found to be CT = 0.78 at λ ≈ 5.5 [108]. As conservative estimation, the power coefficient
is taken as CP = 0.45 and the thrust coefficient as CT = 0.8, both at λ = 6.5, as suggested by
Encarnacion et al. [37]. The torque coefficient was then estimated to CQ = 0.070. The results are
summarized in table 7.1.
Power Control Mechanism For sake of this work, no pitch control is considered. Nevertheless,
it is assumed that the frequency of the generator is adjusted to maintain the turbine always in
its optimal working conditions as described in table 7.1. This might be achieved via a frequency
converter.
7.2.2 Platform
As the turbine itself is not neutrally buoyant it has to be mounted on a platform which provides the
necessary buoyancy for maintaining its position. Single hull, double hull, and triple hull systems
7.2. DESIGN CONCEPT 91
have been considered and are discussed in detail in section 7.1.3. Single hull systems, such as the
one shown in figure 7.2, stand out for their simplicity, as they have much less parts and consequently
in much less joints. Independently on the technology used for joining one or more parts, each joint
is prone to failure due to different effects, including fatigue, corrosion and stress concentration.
However, dual hull systems can reach further upstream without significantly increasing the pitching
moment due to the hulls drag. They also offer the possibility to connect various mooring lines or a
bridled mooring line to compensate the reaction moment due to the generators torque or moments
by current not coming directly from front. For a system with three hulls, the advantages as stated
in section 7.1.3 do not offset its disadvantages in comparison to a two-hull system. As the volume of
a body increases with the third power of its longitude and the surface only with the second power,
fewer but bigger hulls tend to have a lower drag then more but smaller hulls.
Shape of Hull
The purpose of the hull is providing buoyancy while simultaneously minimizing their drag. Myring
[78] developed the so called Myring-profiles for the nose- and tail-section, which provide the possible
shape of low-drag bodies for fully submerged, cylindrical bodies. The nose shape is defined by the
position dependent radius (rN ) as
rN =
1
2
dH
(
1 −
(
x
LN
− 1
)2
)1/n
, (7.5)
with its derivation defined by
drN
dx
=
1
2
dH
2
(
x
LN
− 1
)(
2LN x−x2
LN
2
)1/n−1
nLN
(7.6)
where x denotes the distance from the nose’s tip, dH the maximum diameter of the hull, LN the
length of the nose and n the shape parameter. The tail’s position dependent radius (rT ) is given
by
rT =
1
2
dH −
(
3dH
2LT
2
− tan θ
LT
)
x2 +
(
dH
LT
3
− tan θ
LT
2
)
x3 , (7.7)
with its derivation given by
drT
dx
= −2
(
3dH
2LT
2
− tan θ
LT
)
x+ 3
(
dH
LT
3
− tan θ
LT
2
)
x2 , (7.8)
where x is the distance from the front end of the tail (counting towards the tail’s aft end), LT the
length of the tail, dH the maximum diameter of the hull, and θ is half the included angle at the
tail’s tip.
Uppal et al. [117] did some research on the optimal parameter for the nose and tail section for
an autonomous underwater vehicle with a total length of 1.4 m and in the range of 0.6 m s−1 to
1.4 m s−1. A fineness ratio of 0.1 was used, with a relative nose length of 5/28LOA and a relative tail
92 CHAPTER 7. SYSTEM DESIGN
z
Space frame
Shell
ForwardAft
x
Figure 7.5: Exemplary cross-section of strut.
length 9/28LOA, where LOA is the length over all of the hull. The optimal nose shape parameter
was found to be 3, while a tail angle (θ) of 15° gave the lowest volumetric drag coefficient of
CdV = 0.0382, CdV = 0.0415 and CdV = 0.0476 for speeds of v = 1.4 m s−1, v = 1.0 m s−1 and
v = 0.6 m s−1, respectively.
The equations for volume (VH) and surface (SH) of the hull can be derived by
VH = π
∫ LN
0
r2
N dx+
π
4
dH
2 (LOA − LN − LT ) + π
∫ LT
0
r2
T dx (7.9)
and
SH = 2π
∫ LN
0
rN
√
1 +
(
drN
dx
)2
dx+ πdH (LOA − LN − LT ) + 2π
∫ LT
0
rT
√
1 +
(
drT
dx
)2
dx .
(7.10)
Strut Design
The strut connects both hulls and provides the mounting point of the turbine, transmitting the
forces exerted on the turbine to the hulls and in turn to the mooring lines. It primarily comprises
a steel space frame placed between both hulls. The shell of the strut follows the shape of the space
frame, with the cross-section shaped as depicted in figure 7.5.
7.2.3 Mooring System
As discussed in chapter 2, the offshore industry currently employs various types of systems . Cate-
nary systems are simple and easy to install, but they become less efficient on deeper installations
due to their weight, and their mooring system’s footprint is significantly larger compared to other
systems. Additionally, catenary systems do not provide vertical restoring forces required for sub-
surface moorings. Tension leg platforms, on the other hand, require a significant excess in buoyancy
and are not suitable for fully submerged platforms. For this reason, a taut mooring system will be
used for the platform in this work. In total, two different system configurations will be compared,
7.2. DESIGN CONCEPT 93
Platform
Forward
b)a)
Aft
Bridled section
Current direction
Anchor
Mooring line
Figure 7.6: Mooring concepts used as basis for optimization. a) Three mooring line configuration;
b) Four mooring line configuration.
one with three mooring lines and one with four mooring lines.
Both configurations (see figure 7.6) consist of four fairleads, one at each of the hull’s ends,
which connect to the mooring lines. At the forward ends of the structure, the fairleads are directly
connected to the anchor via a mooring line. In the three mooring line configuration, the aft fairleads
connect to a bridled section that joins into a single aft mooring line at the centerline of the platform.
In contrast, the four mooring line configuration has two aft mooring lines, each directly connecting
the fairlead to the corresponding anchor.
All mooring lines and bridled section are composed of steel wire with the same diameter. Al-
though the mooring lines in a taut system should never make contact to the sea floor, for the last
two meters of each mooring line connecting to an anchor, studless chain is used as precautionary
measure.
7.2.4 Elevation System
To raise the platform to the water surface and lower it back to design depth, a combined system
of ballast tanks and winches for the aft mooring lines is chosen. The mooring lines are attached to
the platform via four fairleads. Since three are sufficient to define the platform’s position in space,
two mooring lines have to be adjusted in length. The complete process of reaching design depth
is depicted in figure 7.7. In the first stage the ballast tanks are flooded to reduce the platforms
overall buoyancy and decrease the required power of the winches (see figure 7.7 b)). Afterwards, the
94 CHAPTER 7. SYSTEM DESIGN
a) b) c) d) e)
Figure 7.7: Procedure for reaching design depth after installation or service. a) Platform floating
on surface for maintenance; b) Ballast tanks flooded to reduce buoyancy; c) Hauling in aft mooring
lines with winch; d) Evacuating ballast tanks with pressurized gas when design depth reached; e)
Turbine in operation.
b)a)
Figure 7.8: Sketch of space frame of internal structure of the platform. a) View from behind, with
internal structure of port hull visible; b) Top view with section of port hull visible.
winches lower the platform by reducing the length of the aft mooring lines (figure 7.7 c)) until the
design depth is reached. The ballast tanks are evacuated by the help of small tanks with pressurized
gas, carried within the structure. These can be either exchanged during service or recharged if the
service vessel can be equipped with the necessary devices. This method helps reducing the power
requirements on the winches. They can be smaller, lighter, and especially cheaper than the winches
which would be required for lowering the platform without reducing its buoyancy. Furthermore,
the ballast tanks do not need to be design to withstand excessive pressure, as they are supposed to
be at ambient pressure at all times.
7.2.5 Internal Structure of Platform
In case an outer shell is needed for a structure or a vessel, it is typically also designed to considerably
contribute to the overall strength of the object. However, it is also possible to separate this specific
task from the outer shell, what requires additional structural members to transmit forces and
moments. Several approaches exist to design such structures, but due to their efficient use of
material for a given strength, space frames or space trusses are widely used.
The cross section of the space frame within the hulls is of quadratic shape with the corners
7.2. DESIGN CONCEPT 95
Outer shell
Structure
Buoyancy tank
Ballast tank
Winch
CL
CL
PortStarboard
Buoyancy tank Ballast tanks
Control unit
Control unit
a)
b)
Outer shellStructure
Turbine
Turbine
Figure 7.9: Layout of platform. The space frame and the outer shell are depicted in black, The
buoyancy tanks in blue, the ballast tanks in orange, the position of the winch is marked in green,
the placement of the control units in red and the position of the turbine in gray. a) Top view; b)
Front view.
aligned vertically and horizontally. This way, the space frame of the strut can connect to nodes of
the hull’s space frame. A sketch of the principal arrangement of the members is shown in figure 7.8.
The structure of the space frame is similar to the Warren truss configuration.
7.2.6 Compartments and Layout of Platform
The general layout of the platform is shown in figure 7.9. To counteract the submerged weight of
the structure, two buoyancy tanks are fitted within the structure of the hulls. This design feature
provides protection against external forces during unexpected events, such as collisions with other
vessels or objects and the platform is more likely to keep its ability to surface without the need for
special equipment. The ballast tanks are placed between the outer shell and the structure. The
number of the ballast tanks is subject to optimization. The upper outer face of the space frames
is left blank for control equipment to be placed. This location was chosen as it provides the best
accessibility during service. Equipment for controlling emergency functions may be placed on the
strut close to the turbine, as this position is best protected against external influence.
96 CHAPTER 7. SYSTEM DESIGN
Buoyancy Tanks
To maximize the volume of the buoyancy tanks, the tanks span from the aft end to the forward end
of the structure. The tanks are of cylindrical shape with half-spherical ends. The displaced volume
(Vbt) can be calculated by
Vbt =
π
4
d2
bt (LS − dbt) + 1/6πd3
bt , (7.11)
where LS is the length of the structure in the hulls and dbt the outer diameter of the buoyancy
tank. The outer surface of a single buoyancy tank (Sbt) is given by
Sbt = πdbt (LS − dbt) + πd2
bt . (7.12)
Ballast Tanks
The ballast tanks need to fit between the outer shell and the structure, maintaining the required
clearance to both. Since the ballast tanks are assumed to be pressurized to the same pressure as the
environment, their shape can be complex in comparison to the buoyancy tanks. Both ends of the
ballast tanks are flat, while the cross section is supposed to maximize the space between structure
and shell.
Due to the variety of combinations of dimensions of the structural members and the outer
diameter of the hull, the easiest approach is to approximate the cross-sectional area (Ablst) with
polygons. First, a circular segment is drawn with the outer diameter
dblst = dH − tshell − cshell , (7.13)
where dH is the outer diameter of the hull, tshell the thickness of the shell, and cshell the minimal
clearance between the shell and all internal components. The apothem of the cross section can be
calculated based on the node radius of the space frame and is defied by
ablst =
rn√
2
+ 1/2dII + cstruc , (7.14)
where rn is the node radius of the space frame of the hull, dII the outer diameter of the diagonal
members in the space fame of the hull, and cstruc the minimal clearance between the structure and
other components. From this circular segment, the space occupied by the longitudinal structural
members in the space frame of the hull is subtracted. Important for this step is the correct placement
of the longitudinal structural members and considering the structural clearance (cstruc) together
with their diameter (dI).
7.2.7 Mooring Winch
The mechanism for lifting the platform to the surface relies on automatic winches which are suitable
for under water operation. Suitable systems are fabricated by various suppliers such as InterO-
ceanSystems LLC and Okeanus. Both suppliers are able to fabricate the winches with required
7.3. DESIGN CRITERIA 97
dimensions and required power to haul in the aft mooring lines in acceptable time.
7.3 Design Criteria
7.3.1 Load Cases
To the best of the authors knowledge, no mandatory or recommended load cases explicitly for OCTs
have been made available to date. However, the IEC 62600-2 [54] defines load cases for tidal energy
converter (TEC), which are comparable to ocean current turbines in several regards. Given the
specific conditions of the installation site and the lack of data from a turbine provider, the design
of the mooring system is based on a subset of the load cases defined in the code.
The loads on the system are governed by the magnitude of the current and its direction. Wave
and wind conditions are defined; however, their effect on the mooring system is limited to three
load cases because the platform is submerged during operation.
The details on the environmental conditions and the relevant limit states are summarized in
table 7.2. A description of each group of load cases can be found in the following sections. For load
cases other that the first and the last, the last letter indicates the direction of the current, while
wind and waves are always assumed to come directly from the front (180°): A indicates a current
from 180°, B indicates a current from 150°, and C indicates a current from 210°.
LC1: Static Equilibrium The first load case considers the equilibrium condition of the mooring
system at design depth. Although this load cases is not explicitly defined in IEC 62600-2 [54], it
was added to the list as it gives useful insights in the basic behavior of the system.
LC2: Normal Operation This group of load cases describes the normal operation of turbine
and mooring system. The platform operates at its design depth with the 1-year current and 1-
year wind. While the wind comes directly from the front (180°), the ocean current varies between
180° (LC2A), 150° (LC2B), and 210° (LC2C). For these load cases the ULS, FLS, and SLS are
considered.
LC3: Parked and Survival For these load cases, the turbine is supposed to be in survival mode
with its brakes applied. The loads are calculated for a 100-year current, 50-year waves, and 5 year
wind and reviewed for the ULS, and SLS. Wind and waves come directly from the front while the
current changes between 180° (LC3A), 150° (LC3B), and 210° (LC3C).
LC4: Parked These load cases assume a parked turbine with its brakes applied, but in contrast
to group LC3 not in survival mode, resulting in the return levels of 1-year for the current, wind,
and the waves. The loads are evaluated according to the ULS, FLS, and SLS.
98 CHAPTER 7. SYSTEM DESIGN
Table 7.2: Complete load case matrix.
Depth Current Angle Wave Wind ULS FLS SLS ALS
m m s−1 ° m m s−1
LC1 50 – – – – • •
LC2A 50 1.38 180 – 12 • • •
LC2B 50 1.38 150 – 12 • • •
LC2C 50 1.38 210 – 12 • • •
LC3A 50 1.55 180 8 16 • •
LC3B 50 1.55 150 8 16 • •
LC3C 50 1.55 210 8 16 • •
LC4A 50 1.38 180 2 12 • • •
LC4B 50 1.38 150 2 12 • • •
LC4C 50 1.38 210 2 12 • • •
LC5A 50 1.55 180 3 16 •
LC5B 50 1.55 150 3 16 •
LC5C 50 1.55 210 3 16 •
LC6A 50 1.55 180 3 16 •
LC6B 50 1.55 150 3 16 •
LC6C 50 1.55 210 3 16 •
LC7A 50 1.55 180 3 16 •
LC7B 50 1.55 150 3 16 •
LC7C 50 1.55 210 3 16 •
LC8A 50 1.55 180 3 16 •
LC8B 50 1.55 150 3 16 •
LC8C 50 1.55 210 3 16 •
LC9 0 1.19 180 0.5 5 •
7.3. DESIGN CRITERIA 99
LC5: Loss of Stability/Leakage In these load cases, the platform is supposed to have lost one
of its starboard ballast tanks. While the main buoyancy tanks are protected by the structure, the
ballast tanks are close to the outer surface and might be damaged by external factors or not flooded
with air after reaching design depth. While the current is the 100-year return level, the wind and
the waves have a return period of 5-years. The loads are evaluated according to the ALS.
LC6: Loss of Stability/Leakage Similar to the load case group LC5, the platform is supposed
to have lost its port ballast tanks. The return level for the current in 100-years, and for wind and
waves 5-years, while the loads are evaluated according to the ALS.
LC7: Loss of Line Under these load cases, the effect of losing the forward starboard mooring
line is evaluated according to the ALS. While it is also possible to loose the starboard aft mooring
line (or bridled line), the forward mooring lines experiences a much greater load due to the current
coming from the front. Equally to the load cases LC5 and LC6, the current is set to its 100-year
return level, and wind and waves to their respective 5-year return level.
LC8: Loss of Line Equally to a loss of the starboard mooring line in LC7, the loss of the
forward port mooring line is considered under the conditions of the ALS. Therefore, the platform
experiences the 100-year current, 5-year waves, and 5-year wind.
LC9: Maintenance The last load case considers maintenance of the platform at the water
surface under the ULS. The turbine is supposed to have its brakes applied with the current, wind,
and waves for maintenance. Transportation of the platform can happen in very different conditions,
in case the platform is towed to the construction site, the maximum current is expected to not exceed
1.19 m s−1, which is equal to the conditions expected for maintenance. In case the construction site
is at a considerable distance of the port it would be far more feasible to load the platform on a
vessel instead of towing it to the construction site.
7.3.2 Additional Environmental Conditions
Aside from the primary environmental conditions defined in section 7.3.1, corrosion and ma-
rine growth have to be taken into account. In case no specific data are available, corrosion of
0.3 mm year−1 should be considered for the suspended part of mooring lines in tropical waters,
as stated in DNVGL OS-E301 [32]. For a design life of 20 years this results in a total corrosion
allowance of 6 mm. According to PEMEX NRF-003:2000 [83], a thickness of the marine growth
of 35 mm should be applied to all structural members. A typical density for marine growth is
1325 kg m−3 [32]. Due to its specific shape, the mass of the marine growth on chains is twice the
value obtained with the nominal diameter.
100 CHAPTER 7. SYSTEM DESIGN
Table 7.3: Partial load safety factors for ULS and a safety level of SL2.
Load case Design category Partial load safety factor
LC1 Normal 1.35
LC2A Normal 1.35
LC2B Normal 1.35
LC2C Normal 1.35
LC3A Extreme 1.35
LC3B Extreme 1.35
LC3C Extreme 1.35
LC4A Normal 1.35
LC4B Normal 1.35
LC4C Normal 1.35
LC9 Transport 1.5
7.3.3 Requirements of Limit States
The criteria for evaluating the system’s performance depend on the analyzed limit state. More
information on the limit states can be found in section 3.2.4. The default safety level of a marine
structure should be SL2 [54]. The risk for humans due to failure of the structure or the mooring
system is very low, as the platform is supposed to not be boarded, neither during operations nor
during service of the platform. The environmental damage could be considered as small during an
accident, as only small amounts of lubricants are required in the winches and the turbine itself.
However, the loss of the structure leads to economic and political consequences wherefore the safety
level is not lowered to SL3 but maintained at SL2.
Ultimate Limit State The partial load safety factor which is applied to the loads defined in the
ULS depend on both, the design category and the source of the loading. Since in the simulations
loads of different sources are applied at the same time, the maximum partial load safety factor is
chosen for each design category. The safety factors for each load case are summarized in table 7.3.
For evaluating the stresses and buckling of the structure, a material factor of 1.1 was selected.
According to IEC 62600-10 [53], the analysis of the mooring system should be based on a safety
factor of 1.67 for loads estimated with a dynamic mooring analysis. For loads estimated with a
quasi-static mooring analysis a safety factor of 2.0 should be considered. The evaluation of the
anchor holding capacity is based on intact conditions and redundancy checks. ULS assumes an
intact system wherefore the safety factor for anchor piles under axial loading is 2.0 and under
lateral loading is 1.6 [53].
Accidental Limit State According to IEC 62600-2 [54], the partial load safety factor for the
ALS should be taken as 1.0 for analyzing accidental loads on the structure, while the material factor
7.3. DESIGN CRITERIA 101
is given by 1.1.
The mooring lines should be evaluated with a safety factor of 1.25 when based on a dynamic
mooring analysis, and with a safety factor of 1.43 when based on a quasi-static analysis, as mentioned
in IEC 62600-10 [53]. The holding capacity of anchor piles under accidental loads (i.e. redundancy
check) should be evaluated with a safety factor of 1.5 for axial loading and 1.2 for lateral loading,
as given in IEC 62600-10 [53].
Fatigue Limit State For estimating the fatigue life of offshore steel structures, ISO 19902 [55]
introduces a fatigue damage design factor and a local experience factor. The fatigue damage design
factor should be 2.0 for inspectable components that are non-critical and 5.0 for inspectable and
critical components. The local experience factor takes into account the experience with structures
of similar type or in similar conditions and is 1.0 if no evidence suggests otherwise.
According to the requirements of API RP-2SK [5], the fatigue life of the mooring system of
floating offshore structure should be designed with a safety factor of at least 3. The typical design
life of renewable offshore structure such as offshore wind turbines is 20 years. Assuming the same
design life for the MCEC, the mooring system should withstand the cyclic loads of 75 years.
Serviceability Limit State The conditions for serviceability are typically defined by the oper-
ator of the platform and the manufacturer of the turbine. In general, the platform should avoid
hitting other objects that are in proximity, not overstretch the power cable and avoid extreme
rotations. To keep the offset of the platform within a reasonable range, the platform should stay
within 10 m of the original position. Heel of the platform is not as critical as the other rotations
since it is in line with the rotational axis of the turbine. In order to avoid excessive bending of the
power cable, 20° will be considered as the limit for heel, while trim and heading should be limited
to 10° as their impact on the turbine’s performance is greater.
With the exception of resonance, accelerations are not of a concern for the platform as no humans
will board the structure. Furthermore, the platform is supposed to operate out of the influence of
wind and waves, which are typical environmental factors that introduce high accelerations.

Chapter 8
Optimization of Design Concept
The optimization process of the design of the position-keeping system is an iterative process due
to recursive dependencies. For instance, adjusting the platform’s buoyancy increases the diameter
of both hulls which itself changes the geometry of the internal space frame. Any changes in the
geometry, such as the position of the turbine, can significantly impact the equilibrium position
of the platform, potentially changing the loads in the mooring lines. To tackle this problem, a
genetic optimization algorithm and coupled mooring analysis and structural finite element analysis
were used. Since genetic optimization algorithms rely on evaluating a great number of different
individuals, it is crucial to have a time-efficient analysis of the different designs. The mooring
systems were analyzed based on quasi-static principles using straight, linear-elastic mooring lines,
while the structural deformations and stresses were calculated via a finite element analysis based on
beam elements. This approach enabled the optimization of both, the platform and mooring system
simultaneously.
The different parameters and their allowed range of values which formed part of the optimization
process are described in section 8.1 while a description of the numeric models used si given in
section 8.2. The automated optimization process by means of the genetic algorithm can be found
in section 8.3 and the subsequent manual fine-tuning in section 8.4.
Performance The performance of the individuals is evaluated via the mass of the whole system,
the achieved safety factor, the maximum angle, and the net buoyancy deviation. The achieved
safety factor (SF ) is calculated by
SF =
1/γMRk
γfFk
, (8.1)
where Fk is the characteristic load, and Rk the characteristic resistance of the used material, γf the
load factor, and γM the material factor. For the mooring lines, the load is the tension within the
line and the resistance the MBL as estimated with the formulae in Ma et al. [68], see equation (4.9).
In the case of the structure, the load is the stress in the structural members and the resistance their
yield strength. For buckling, the load is set to the buckling factor as calculated by Abaqus. Lastly,
103
104 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
D
H
Lballastxballast L2
LS | Lbuoyancy
Outer shell
Structural sections
Structure
1 2 3 4 5 6 7 8 9 10
Ballast tank
CL
Figure 8.1: Top view of the port side of the platform concept with the outer shell marked in black,
the structure in blue, and the different segments of the structure in green. The exemplary position
of one ballast tank per side is shown in orange. Not to scale.
the minimum of all three factors is taken. The maximum angle is determined by the highest value
among heel, trim, or heading of the platform at equilibrium. The net buoyancy deviation is the
difference between the design net buoyancy of 50 kg and the calculated net buoyancy. Any design
proposals with net buoyancy smaller than the minimum net buoyancy of 20 kg are discarded. The
net buoyancy (B̃) is calculated by
B̃ = (VP L − nBLLSTVBLLST + VMLs)ρw − (mP L +mMLs) , (8.2)
where VP L denotes the displaced volume of the complete platform, VBLLST the displaced volume
of a single ballast tank, nBLLST the number of ballast tanks, VMLs the displaced volume of all
mooring lines, ρw the density of the water, mP L the total mass of the platform, and mMLs the
mass of all mooring lines.
8.1 Parameter Subject to Optimization
8.1.1 Principals
The main parameter changed by the algorithm is the outer diameter of the hull. Several variables
depend on this diameter, such as the length of both hulls, the volume of the buoyancy tanks, and
the maximum volume available to the ballast tanks. Another parameter that can be changed is
the number of the hull segment, the strut is connected to (see figure 8.1). Since the strut spans
over two segments, the parameter refers to the aftmost segment of both. In the example of the
figure, the strut would be connected to segment 4. The number of hull segments is fixed to 10 as
it was found that this maximizes the available space inside the hull for a given hull diameter and
fixed aspect ratio of these segments. Therefore, the strut is allowed to connect to every structure
between the 1st to the 9th section, including both boundaries.
8.1. PARAMETER SUBJECT TO OPTIMIZATION 105
D
H
1
PortStarboard
Buoyancy tank
2
3
α
D
B
CL
Outer shellStructure
Figure 8.2: Front view of the platform concept with the outer shell marked in black, the structure
in blue, and the position of the tanks in orange. The ballast tanks are numbered on the port side.
8.1.2 Mooring System
The mooring system consists of three and four mooring lines respectively, that connect each of
the four corners of the platform to three or four anchors, depending on the system configuration
(see figure 8.3). The main part of the mooring lines consists of steel wire, while the last two
meters connecting to the anchor are steel chains. The diameter of both segments can be changed
independently by the algorithm, however, the dimensions are the same for all mooring lines to
reduce the complexity of the resulting system. The initial strain of the wire rope can be adjusted
for the forward mooring lines and for the aft mooring line(s) independently. Though, both forward
mooring lines have the same initial strain, which is equally true for the aft mooring line(s). The
strain of the forward mooring lines is not necessarily the same as the strain of the aft mooring
line(s). Additionally, the mooring footprint (χm) can be adjusted in the range of 0.8 to 2. The
mooring footprint is given as the ratio between the platform’s height above the sea bed at its initial
position (LML,v, see figure 8.3) and the horizontal distance of the anchor from the corresponding
fairlead (LML,h):
χML =
LML,h
LML,v
(8.3)
In the case of the mooring system with three mooring lines, the aft mooring line is bridled at the
top to connect to both aft ends of the platform (figure 8.3 b)). The length of the bridled section
(Lbridle) is variable and is defined as the fraction of the length of a straight line connecting the
anchor to the center point between both aft fairleads (L′
ML,aft, see figure 8.3 b)). The relative
length (χML,bridle) was changed between 0.1 and 0.5 and was defined by
χML,bridle =
Lbridle
L′
ML,aft
. (8.4)
The position of the bridle point (~pbridle) can be found by
~pbridle = χML,bridle

~pA,aft −


1
0
1

 ~pF L,aft,stb

 , (8.5)
106 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
a) b) c)
αML
L M
L
,h
LML,h
L
M
L
,v
L
br
id
le
L
' M
L
,a
ft
α
ML
Wire
Anchor
Forward
Aft
Fairlead
Chain
Figure 8.3: Concept of the mooring systems for optimization in top view. The segment of the
mooring line which is composed of wire is shown in black, while the chain segment is shown in
blue. The anchors are marked with orange crosses while the fairleads are marked with orange dots.
For better orientation, the shadow of the platform is shown in light gray. The lengths in b) are
measured parallel to the line. a) Mooring system with four lines. b) Mooring system with three
lines and bridled section. c) Side view of a single mooring line. Not to scale
where ~pA,aft is the position vector of the aft anchor point and ~pF L,aft,stb the position vector of the
aft starboard fairlead. Since the platform is symmetric to the center line, the center point between
both aft fairleads can be calculated by simply dropping the y coordinate of one of both fairleads.
8.1.3 Structural
The type of frames used for the hulls and for the strut can be changed individually, with both hulls
sharing the same (but mirrored) frame type. For the hull, four different frame types are available
as displayed in figure 8.4. The two possible hull types for the strut are shown in figure 8.5. The
dimensions of all four types of structural members can be changed as well and is selected from a list
that is explained in more detail in section 8.1.3. A limitation is that the diagonal members cannot
have a higher serial number than the corresponding girder.
Hollow Steel Profiles for Space Frame
The algorithm selects profiles for the space frame according to the definitions outlined in the norm
EN 10210. The thickness range of the round profiles is shown in table 8.1, while the complete list
can be found in table B.1.
8.1. PARAMETER SUBJECT TO OPTIMIZATION 107
Type 1 Type 3Type 2 Type 4
Figure 8.4: Possible configurations of space frame of hulls for the optimization algorithm. The
forward end of the starboard hull is shown in the top view. The possible connection with the
structure of the strut is shown with dotted gray lines. Vertical lines (shown in black) indicate
structural members of type 1 and blue lines indicate structural members of type 2.
Type 1
Forward
Aft
Forward
Aft
Type 2
Figure 8.5: Possible configurations of space frame of the strut for the optimization algorithm, shown
in top view. The possible connection with the structure of the hull is shown with dotted gray lines.
Horizontal lines (shown in black) indicate structural members of type 3 and blue lines indicate
structural members of type 4.
108 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Table 8.1: Diameter and thickness range of structural members. All values are given in mm.
Diameter Thickness
range
Diameter Thickness
range
30.0 2.6 . . . 10.0 88.9 3.2 . . . 25.0
31.8 2.6 . . . 10.0 95.0 3.6 . . . 28.0
33.7 2.6 . . . 11.0 101.6 3.6 . . . 28.0
38.0 2.6 . . . 12.5 108.0 3.6 . . . 30.0
42.4 2.6 . . . 14.2 114.3 3.6 . . . 32.0
44.5 2.6 . . . 14.2 121.0 4.0 . . . 32.0
48.3 2.6 . . . 16.0 127.0 4.0 . . . 36.0
51.0 2.6 . . . 16.0 133.0 4.0 . . . 40.0
54.0 2.6 . . . 16.0 139.7 4.0 . . . 45.0
57.0 2.9 . . . 16.0 146.0 4.5 . . . 45.0
60.3 2.9 . . . 17.5 152.4 4.5 . . . 45.0
63.5 2.9 . . . 17.5 159.0 4.5 . . . 55.0
70.0 2.9 . . . 17.5 165.1 4.5 . . . 55.0
76.1 2.9 . . . 20.0 168.3 4.5 . . . 55.0
82.5 3.2 . . . 25.0
8.1.4 Ballast Tanks
The number of ballast tanks per hull can be modified between one and three (see figure 8.2),
resulting in two to six ballast tanks in total. Furthermore, the length and the longitudinal position
can be changed for all ballast tanks simultaneously. The length (LBST ) is given as a fraction of the
length of the buoyancy tanks (LBT ) or nominal length of the structure (LS) as
L′
BST =
LBST
LBT
. (8.6)
The position of the ballast tank (xBLLST ) is given as a fraction of the length difference between
the buoyancy tank and the ballast tank (see figure 8.1) by
x′
BST =
xBLLST
LBT − LBST
, (8.7)
where x′
BST is the relative position of the ballast tanks in the x-direction and which is controlled
by the algorithm.
8.2. NUMERICAL MODEL 109
8.2 Numerical Model
8.2.1 Preliminary Estimations
The principal dimensions together with mass and buoyancy are reviewed before analyzing the
mooring system and the structure of the platform to avoid unnecessary calculations. If the model
is unable to float, there is no need to estimate the forces in the mooring line or stresses in the
structure of the platform.
The overall length of the hulls (LOA) is defined by the fineness ratio (χh) and the outer diameter
of the hull (dH):
LOA = 2χhdH (8.8)
Since the segments of the space frame are supposed to be quadratic, the node radius (rn) has to be
of the length of each segment. In this work, the number of segments is set to 10, wherefore the node
radius is indirectly defined by the hull’s outer diameter (dH). Accounting for the thickness of the
structural elements of type I, a given clearance between shell and structure, and the thickness of the
hull itself, the resulting, required node radius is found numerically with the function root_scalar.
This function is part of the scipy.optimize-module of Python 3 that was used in version 1.8.0 [120].
This function determines two positions along the x-axis of the hull where the radius of the tail
section and the nose section is equal, with the distance between these positions being 10 times the
resulting node radius.
The masses of the buoyancy tanks and the shell are estimated by multiplying the (outer) surface
area with the respective thickness and density of the material used. The surface area of the hull is
defined in equation (7.10), while the surface area of the buoyancy tank is defined by equation (7.12).
For the structural members, mass and volume are calculated using the corresponding cross-
section and the distance between nodes. Since this method does not account for overlapping of the
members, it results in slightly overestimating both volume and mass.
8.2.2 Quasi Static Mooring Analysis
Dynamic mooring analyses are computationally expensive. For the mooring analysis during genetic
optimization, a quasi-static mooring analysis was performed with the Python package OffshorePy.
This module was developed at the Institute of Engineering, National Autonomous University of
Mexico, and is described in more detail together with verification and validation in chapter 4.
The platform is modeled with a RigidBody, its mass and displaced volume are set to the values
calculated beforehand. Fairleads and anchor points are placed directly on the structure and on the
sea bed, respectively, and connected via segmented (and bridled, in the case of the three mooring
line configuration) mooring lines. For each connection between two segments, a neutrally buoyant
float is added, while the line composition is according to the concepts described in section 7.2.3.
The environmental conditions are then adjusted for each load case and the system is solved with
a maximum of 250 iterations. If no solution could be found after 250 iterations, but the current
maximum angle is within the range of −75° to 75°, the solving is resumed for another 1000 iterations.
110 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.6: Plot of exemplary three mooring line configuration of the system. Note that the sea
floor is displayed at the depth directly below the platform and not inclined as indicated by the
anchor points.
Figure 8.7: Plot of exemplary four mooring line configuration of the system. Note that the sea floor
is displayed at the depth directly below the platform and not inclined as indicated by the anchor
points.
The system is always solved with the option zero_matrix set to solve to avoid intermediate states
of the system with no restoring forces and thus an empty stiffness matrix. The design proposal is
rejected in case an EquilibriumError was raised during solving the system, i.e., no solution could
be found in either 250 or 1250 iterations. Exemplary plots for the three mooring line configuration
and the four mooring line configuration can be seen in figures 8.6 and 8.7, respectively. Note that
the sea floor is not displayed inclined as it is assumed for the calculations, but displayed horizontally
at the depth directly below the platform’s initial position.
8.2.3 Abaqus
The model for the structural analysis was created by using the scripting interface of Abaqus in
Python. A script template was created to generate a generic platform. For each design proposal and
each load case, a copy of the template is created and the necessary variables such as dimensions, rota-
tions, and loads are adjusted. For each load case, a separate working directory was created. Abaqus
8.2. NUMERICAL MODEL 111
was then executed from Python using the command abaqus cae nogui=<path_to_script>, where
<path_to_script> describes the absolute path to the python script.
The origin of the coordinate system was placed at the very aft of the platform’s structure, with
y and z centered at the center line. The x-axis was facing forward, the y-axis facing to port, and
the z-axis facing upwards.
Mesh The nodes were numbered consecutively starting with the starboard hull, then the port
hull, and lastly the strut. For each of the three elements, the numbering started at the aftmost
plane. Within each cross-section of the hulls, the numbering started with the lowest node on the
center line and proceeded with the node towards the center line. In the case of the strut, the
numbering of the nodes started at the starboard hull and ended at the port hull.
Depending on the shape of the structure, the nodes were then connected accordingly with wire
elements and organized into four different wire sets, each set for one type of structural member.
The cross-sectional profiles of the beams were created with element profiles by using the outer
radius and the thickness of the material. To each wire element, the corresponding pipe profile was
assigned. For their orientation, the n1 vector in positive y-direction was used for all elements on
the hulls and the n1 vector in positive x-direction was used for all elements of the strut. For the
steel used, the linear elastic isotropic material model was chosen with a density of 7850 kg m−3, a
Poisson ratio of 0.3, and a Young’s modulus of 210 GPa. To account for the buoyant force on the
elements, the gravitational acceleration was later adjusted to match the correct submerged weight
of the structure.
The mesh was then generated over the wire elements with B32 beam elements of size 0.05 m
and a deviation factor of 0.1.
Loads and Boundary Conditions The mooring forces of the starboard and port forward moor-
ings were applied to the foremost bottom node on each hull. The force of the buoyancy tanks was
applied to the nodes directly above the tank except for the foremost and sternmost nodes. The
turbine force is applied to the foremost node of the strut on the center line. The displacement of
the aftmost node on the center line was fixed in the y-direction, while the attachment points of the
forward mooring lines were fixed in the x-direction, and the z-movement was inhibited at all four
attachment points of the mooring lines. The gravity was set to 9.81 m s−2.
Load Steps The structure of the platform is evaluated for two different load steps. The first step
was the buckling analysis while the second step was used to evaluate the load-bearing capacity of
the structure. During the buckling analysis, the first 10 eigenvalues were calculated with 50 vectors
and a maximum of 100 iterations.
112 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.8: Performance of complete gene pool of mooring concepts. a)-c) Three-mooring line
configuration; d)-f) four-mooring line configuration. The complete gene pool is represented by
the blue dots, while the orange crosses indicate design proposals that have been filtered regarding
maximum mass, maximum inclination, and minimum safety factor.
8.3 Automated Optimization
The optimization was carried out for both, the three-mooring line configuration and the four-
mooring line configuration. The results of the process for both configurations can be found in
figure 8.8. The design proposals have been filtered to have a maximum mass of 5000 kg, a maximum
inclination of 5° in either direction, and a minimum safety factor of 1.2. The performance of
all filtered design proposals can be found in table 8.2 for the three-mooring line configuration
and in table 8.3 for the four-mooring line configuration. The number following the G in the
name of a design proposal indicates its generation and the number following the I indicates the
design proposal within its generation. 35 design proposals with these qualities were found with a
three-mooring line configuration and 21 design proposals with a four-mooring line configuration.
Qualitatively, there was no big difference in the complete gene pool between the three and four-
mooring line configurations. The maxima were in the same order of magnitude: The maximum
mass was found to be 14 277 kg and 15 774 kg, the maximum safety factor was 4.21 and 4.28, and the
net buoyancy deviation peaked at 14 528 kg and 14 667 kg, respectively. The former value correlates
to the three-mooring line configuration while the latter value correlates to the four-mooring line
configuration. The maximum angle peaked at around 70° since this was a condition for the algorithm
to consider this design proposal valid. The filtered design proposals, however, did present slight
differences, especially regarding the safety factor. For the three-mooring line configuration, a few
8.4. MANUAL OPTIMIZATION 113
Table 8.2: Filtered design proposals and their performance for the three-mooring line configuration.
SF: Safety factor; A: Maximum pitch, list, or yaw angle; B: Net buoyancy deviation.
Mass SF A B Mass SF A B
G026I000 3150.80 1.72 4.95 59.96 G109I024 3900.33 1.29 1.95 1676.21
G028I022 4890.93 1.60 3.90 3101.98 G115I021 4765.96 1.75 1.92 811.61
G028I023 4033.93 1.35 1.38 1797.70 G116I021 4746.97 2.41 3.98 92.40
G028I029 4585.51 2.19 3.05 1454.42 G125I031 3747.02 1.94 4.98 1991.58
G029I022 4082.77 1.69 4.68 92.41 G126I012 4132.73 1.37 2.24 2483.97
G030I002 3172.10 1.49 4.38 891.89 G127I014 3691.51 1.77 4.25 2106.10
G033I016 4127.67 1.26 4.03 3777.51 G129I003 3535.02 1.81 4.35 2248.65
G033I020 4588.84 1.30 4.61 374.50 G129I004 4383.11 1.59 4.85 832.33
G037I028 4227.08 1.29 1.71 3340.04 G129I012 3114.80 1.35 2.74 2610.75
G042I030 4811.44 1.57 4.42 2839.74 G132I015 4238.88 1.79 4.07 1334.22
G043I015 3627.72 1.29 2.21 2565.40 G134I000 3886.98 1.22 3.10 4982.99
G044I005 4217.75 1.35 3.90 1761.15 G139I031 4573.55 1.96 4.92 462.05
G062I015 4306.97 1.27 3.82 3420.90 G142I029 4553.19 2.42 3.86 476.47
G062I026 3760.22 1.25 2.76 1291.57 G144I020 3680.95 1.50 3.98 2380.66
G081I005 4158.72 1.38 4.46 1344.46 G146I007 3783.28 2.28 4.88 1707.39
G093I026 4515.65 1.24 3.49 135.52 G146I028 4040.68 1.67 4.52 2882.30
G108I012 4334.50 1.24 4.20 1271.85 G149I028 3772.75 1.68 4.74 2524.39
G109I002 4852.75 1.64 2.15 893.52
design proposals with a safety factor greater than 2 were found.
From the filtered design proposals, G111I007 (Four mooring lines) presented a good balance
between all four metrics. The mass of the system was high and the net buoyancy deviation was
also at the higher end. The maximum inclination of the platform, however, was relatively low.
Despite the safety factor being low in comparison to the factors found for the three-mooring line
configuration, it was among the highest for the four-mooring line configuration. Another good
candidate for further optimization was G146I007 (Three mooring lines) due to its much higher
safety factor and lower net buoyancy deviation.
8.4 Manual Optimization
For manual optimization, good knowledge of the influence of the parameter is key. In a paramet-
ric study, most of the parameters used in the automated optimization were investigated. Each
parameter was changed within a reasonable range and increment. Based on the findings, certain
parameters were updated and the resulting individual was evaluated after each change in the pa-
rameter. The parametric study was performed for each parameter individually, ignoring the fact
114 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Table 8.3: Filtered design proposals and their performance for the four-mooring line configuration.
SF: Safety factor; A: Maximum pitch, list, or yaw angle; B: Net buoyancy deviation.
Mass SF A B Mass SF A B
G002I021 4079.46 1.63 4.84 852.03 G051I022 4328.09 1.29 4.57 2009.24
G019I007 3236.80 1.23 4.33 1401.57 G060I027 4993.51 1.21 4.37 2963.45
G022I003 3240.73 1.49 4.94 1047.13 G111I007 4663.74 1.52 2.20 2034.12
G022I014 2863.36 1.26 4.29 1392.51 G132I021 3331.61 1.23 4.81 299.64
G034I011 4945.88 1.56 4.08 6938.49 G137I000 3511.37 1.27 3.82 2270.18
G035I011 4614.93 1.22 2.75 2353.10 G142I002 4374.23 1.77 4.90 1016.94
G035I018 3536.90 1.55 4.01 2855.94 G145I014 4024.44 1.71 4.30 231.34
G036I030 3651.00 1.29 2.53 1246.35 G150I011 4642.01 1.43 4.46 830.84
G041I022 4724.12 1.33 4.27 2887.71 G161I023 3238.37 1.22 3.31 4794.10
G042I019 3541.02 1.27 2.78 2975.88 G162I009 4313.40 1.58 3.42 3011.61
G047I009 4863.75 1.53 3.52 3208.63
that the parameters do not change the design proposal independently. After a few iterations, a new
study had to be conducted to see the effect on other parameters. After several repetitions, a design
was found as no improvement could be found without or at a reasonable cost.
In the following, the process for the design based on G146I007 in a three-mooring line configu-
ration and based on G111I007 in a four-mooring line configuration are described in more detail.
8.4.1 Three-Mooring Line Configuration
The results of the initial parametric study of the design proposal G146I007 for the three-mooring
line configuration can be found in figures 8.9 to 8.11. Despite the mass and net buoyancy deviation
changing a lot for the hull diameter, their variability was typically very low in comparison to
the adjusted safety factor and the maximum inclination. For the sake of simplicity, both, the
mass of the system and the net buoyancy deviation, were ignored during the manual optimization.
Increasing the relative length of the ballast tanks to 1.0 increased the adjusted safety factor but
more importantly lowered the maximum inclination by a quite significant amount (see figure 8.10
b) and c)). While the same held true for the relative position of the ballast tanks, this parameter
lost its significance once the relative length of the ballast tanks approaches 1.0 (see figure 8.10 f)
and g)). Via the mooring footprint it was also possible to increase the adjusted safety factor while
reducing the maximum inclination (see figure 8.10 j) and k)). By increasing the initial strain of the
aft lines figure 8.11 f) and g)) and by decreasing the wire diameter of the mooring lines figure 8.11
j) and k)), the same effect could be achieved.
The history of the parameters which were changed can be seen in table 8.4, where each changed
parameter is underlined in the next version. In the first iteration of the manual optimization process,
the mooring footprint was changed from 1.42 to 1.8 and the relative length of the ballast tank was
8.4. MANUAL OPTIMIZATION 115
Figure 8.9: Parametric study on G146I007 in three-mooring line configuration.
116 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.10: Parametric study on G146I007 in three-mooring line configuration (cont.).
increased from 0.86 to 1.0. In the next iteration, the mooring footprint was increased again from
1.8 to 2.5, the mooring angle adjusted from 24° to 25°, and the relative length of the bridle from
0.13 to 0.18. During the third iteration, the strut angle was modified from −10° to −4°, and the
mooring angle again from 25° to 32.5°, to finally change the wire diameter in the last iteration from
0.0114 m to 0.0185 m.
8.4. MANUAL OPTIMIZATION 117
Figure 8.11: Parametric study on G146I007 in three-mooring line configuration (cont.).
118 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.12: Parametric study on manually optimized G146I007 in three-mooring line configuration.
8.4. MANUAL OPTIMIZATION 119
Figure 8.13: Parametric study on manually optimized G146I007 in three-mooring line configuration
(cont.).
120 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.14: Parametric study on manually optimized G146I007 in three-mooring line configuration
(cont.).
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Table 8.4: Parameter changed during manual optimization of three-mooring line configuration
Parameter Basis I1V1 I1V2 I2V1 I2V2 I2V3 I3V1 I3V2 Final
Hull diameter 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4
Position of strut 2 2 2 2 2 2 2 2 2
Frame type of hull 3 3 3 3 3 3 3 3 3
Frame type of strut 2 2 2 2 2 2 2 2 2
Strut angle -10.0 -10.0 -10.0 -10.0 -10.0 -10.0 -4.0 -4.0 -4.0
Structural member 1 131 131 131 131 131 131 131 131 131
Structural member 2 180 180 180 180 180 180 180 180 180
Structural member 3 430 430 430 430 430 430 430 430 430
Structural member 4 138 138 138 138 138 138 138 138 138
Ballast tanks: number 1 1 1 1 1 1 1 1 1
Rel. length 0.86 0.86 1.0 1.0 1.0 1.0 1.0 1.0 1.0
Rel. position 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Mooring footprint 1.42 1.8 1.8 2.5 2.5 2.5 2.5 2.5 2.5
Mooring line angle 24.0 24.0 24.0 24.0 25.0 25.0 25.0 32.5 32.5
Initial forward strain 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033
Initial aft strain 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009 0.0009
Length of bridle 0.13 0.13 0.13 0.13 0.13 0.18 0.18 0.18 0.18
Mooring wire diameter 0.0114 0.0114 0.0114 0.0114 0.0114 0.0114 0.0114 0.0114 0.0185
Mooring chain diameter 0.0180 0.0180 0.0180 0.0180 0.0180 0.0180 0.0180 0.0180 0.0180
122 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
The analysis of the final design proposal can be found in figures 8.12 to 8.14. As can be
seen, mass, safety factor, and net buoyancy deviation did change within a reasonable range, with
the exception of the hull diameter, which had a great influence on both, mass and net buoyancy
deviation (see figure 8.12 a) to d)). However, for the chosen value mass, the adjusted safety factor,
and net buoyancy deviation were quite close to their optimum while the maximum inclination was
minimized. This was also true for the other parameter except for the relative bridle length. This
parameter still had some potential to be optimized by lowering the maximum inclination while
increasing the adjusted safety factor. Ultimately the relative length of the bridle was not decreased,
in part due to the fact, that the parameter was increased beforehand. Additionally, the performance
of the design was satisfactory as the adjusted safety factor is relatively high and the gain in the
maximum inclination would have been minimal. Therefore, the iterative process was stopped at
this point and the design proposal was analyzed in more depth in chapter 9.
8.4.2 Four-Mooring Line Configuration
The result of the initial parametric study of the design proposal G111I007 for the four-mooring line
configuration can be found in figures 8.15 and 8.16. Similar to the three-mooring line configuration,
the impact on the mass and the net buoyancy deviation was negligible in contrast to the adjusted
safety factor and the maximum inclination. For the manual optimization, those two metrics were
ignored, concentrating solely on increasing the adjusted safety factor and minimizing the maximum
inclination. Increasing the strut angle to 10° would have lead to a negligible change in the safety
factor but to a significant reduction in the maximum inclination (see figure 8.15 r) and s)). The
same held true for changing the relative position of the ballast tank to 0.6 (see figure 8.16 f) and
g)), the mooring angle to 20° (see figure 8.16 n) and o)), and the initial strain of the forward lines
to 0.2 (see figure 8.16 r) and s)). Before a new parameter study was performed, the strut angle was
changed from −2° to 10°, the relative position of the ballast tank from 0.45 to 0.6, the mooring
wire diameter from 0.0085 m to 0.0125 m, the mooring footprint from 1.38 to 1.6, and the mooring
angle from 24° to 20°. During the second iteration, the strut angle was changed from 10° to −8°
and the hull diameter from 1.539 m to 1.639 m. In the last and third iteration, the relative length
of the ballast tank was changed from 0.75 to 0.9, the mooring angle was changed again from 20° to
27.5° and finally, the mooring footprint was increased from 1.6 to 1.7.
The parametric study of the final version can be seen in figures 8.17 and 8.18, for which an
acceptable balance between the four parameters, but especially the safety factor and the maximum
inclination were found. Both are located either at an extreme value (e.g., figure 8.17 g)) or a
transition point (e.g., figure 8.17 c)).
8.4. MANUAL OPTIMIZATION 123
Figure 8.15: Parametric study on G111I007 in four-mooring line configuration.
124 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.16: Parametric study on G111I007 in four-mooring line configuration (cont.).
8.4. MANUAL OPTIMIZATION 125
Figure 8.17: Parametric study on manually optimized G111I007 in four-mooring line configuration.
126 CHAPTER 8. OPTIMIZATION OF DESIGN CONCEPT
Figure 8.18: Parametric study on manually optimized G111I007 in four-mooring line configuration
(cont.).
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Table 8.5: Parameter changed during manual optimization of four-mooring line configuration
Parameter Basis I1V1 I1V2 I1V3 I1V4 I1V5 I2V1 I2V2 I3V1 I3V2 Final
Hull diameter 1.539 1.539 1.539 1.539 1.539 1.539 1.539 1.639 1.639 1.639 1.639
Position of strut 3 3 3 3 3 3 3 3 3 3 3
Frame type of hull 4 4 4 4 4 4 4 4 4 4 4
Frame type of strut 2 2 2 2 2 2 2 2 2 2 2
Strut angle -2.0 10.0 10.0 10.0 10.0 10.0 -8.0 -8.0 -8.0 -8.0 -8.0
Structural member 1 226 226 226 226 226 226 226 226 226 226 226
Structural member 2 279 279 279 279 279 279 279 279 279 279 279
Structural member 3 353 353 353 353 353 353 353 353 353 353 353
Structural member 4 235 235 235 235 235 235 235 235 235 235 235
Ballast tanks: number 3 3 3 3 3 3 3 3 3 3 3
Rel. length 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.9 0.9 0.9
Rel. position 0.45 0.45 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6
Mooring footprint 1.38 1.38 1.38 1.38 1.6 1.6 1.6 1.6 1.6 1.6 1.7
Mooring line angle 24.0 24.0 24.0 24.0 24.0 20.0 20.0 20.0 20.0 27.5 27.5
Initial forward strain 0.0319 0.0319 0.0319 0.0319 0.0319 0.0319 0.0319 0.0319 0.0319 0.0319 0.0319
Initial aft strain 0.0251 0.0251 0.0251 0.0251 0.0251 0.0251 0.0251 0.0251 0.0251 0.0251 0.0251
Mooring wire diameter 0.0085 0.0085 0.0085 0.0125 0.0125 0.0125 0.0125 0.0125 0.0125 0.0125 0.0125
Mooring chain diameter 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159 0.0159

Chapter 9
Review of Final Design
After optimizing the design concepts, one design proposal was found for the three-mooring line
configuration and one for the four-mooring line configuration. Both proposals looked promising
since they comply with the requirements for the load cases they were checked against. These limit
states, however, resemble solely a subset of the required load cases as defined in IEC 62600-2 [54].
Furthermore, the mooring analysis is based on a quasi-static analysis. While it is possible to design
the mooring system based on this type of analysis, it is recommended to perform a dynamic mooring
analysis to gain more detailed knowledge of the stresses and positions involved. Furthermore, there
are load cases that cannot be simulated with the quasi-static mooring analysis tool used.
In this chapter, the three-mooring line design proposal is analyzed in more depth with the
proprietary mooring analysis software ProteusDS. The results were used to verify the compliance of
the design with the design codes. A free decay test was used to determine the natural frequency of
the structure for surge, sway, and heave, a fatigue analysis of the mooring system was performed,
and a suitable pile anchor for weak rock was designed. The applied design criteria such as allowable
mooring offset and partial safety factors, as well as the load cases considered, can be found in
chapter 7.
9.1 Methodology
The dynamic mooring analysis was run based on the load cases defined in section 7.3.1. While
for the optimization process, only a few load cases were selected to speed up the process, in this
dynamic analysis all defined load cases are used.
9.1.1 Dynamic Analysis of Mooring System
A base model with all required objects and features was created in ProteusDS. The platform was
modeled by a RigidBody connected to four DCabels. Each cable was attached to the platform at the
position of the respective fairlead and coupled via a connection object of type RigidBodyDCable-
129
130 CHAPTER 9. REVIEW OF FINAL DESIGN
PointConnection. In the case of the forward mooring lines, the remaining end was fixed in space as
it is supposed to be connected to the anchor at the sea floor, while the two aft mooring lines of the
bridled section were connected to the center aft mooring line via a DCableDCablePointConnection.
Similar to the forward mooring lines, the lower end of the aft center mooring line was fixed in space
to model the anchor. The segments of each mooring line were defined in segment mode 1 according
to the specifications. The properties of the used materials were later changed int the lib.ini-file
to the exact values of the analyzed design.
The shape of the outer shell was created with openSCAD. The result was exported to .stl and
then converted to a wavefront .obj file which is readable by ProteusDS. For converting the .stl
file to an .obj file, the positions of all vertices were rounded to 1 × 10−3 m. Vertices with the same
coordinates were joined and the polygons thus reduced to lines were deleted. Hydrodynamic and
aerodynamic loading were switched on for the outer shell, but hydrostatic loads were ignored as the
buoyancy is created by the tanks and not the outer shell itself. The tanks themselves were modeled
by simple cylinders with same volume and length but different diameter. They were placed at the
centroid of the corresponding tank. As these tanks are protected by the outer shell, they only
contributed to hydrostatic loading while all dynamic loading was switched off. The buoyancy of all
other features such as the structure, the turbine, and the outer shell, were modeled by cuboids of
corresponding volume, placed at the position of the feature.
For the first load case LC1, the final position of the corresponding quasi-static mooring analysis
was loaded to reduce time for settling of the platform’s position. For all subsequent load cases,
load case LC1 from the dynamic mooring analysis was loaded and afterwards adjusted to the
requirements by the load case.
Environment The current was set to a power law profile applied to the lower 50% of depth.
The exponent was 0.143, with heading and magnitude of current as defined in the respective load
case. While the water depth was set to the mean depth of 80 m, the inclined bottom was modeled
with a custom wavefront file. Waves and wind were only simulated for the load case LC9 at a
depth of 0 m, as their effect does not extend down to design depth of the platform. The waves were
calculated as Stokes waves of second order with the wave height according to equation (6.8). A
constant uniform wind profile was applied with the wind driven current profile as defined in DNV
RP C205 [23], reaching down to 20 m of depth. The environmental transition time was set to 5 s for
current, waves, and wind with all other environmental parameter were left at their default value.
Simulation The simulation was run for 35 s in simulated time at an output interval of 0.1 s with
the standard 4th/5th Runge-Kutta integrator. The truncation error was set to 0.001 and the time
step was limited to 1 × 10−15 s to 1 s and initially set to 0.001 s.
9.1.2 Estimation of Inertia
The inertia of the platform was calculated with respect to the origin of its local reference frame.
For irregularly shaped objects such as the shell, the tanks, or the water trapped inside the platform,
9.1. METHODOLOGY 131
the inertia was calculated numerically. Therefore stl files were created with openSCAD for surfaces
with additive inertia and surfaces with subtractive inertia. The mass, the centroid, and the inertia
of the surfaces were calculated by means of the numpy-stl library (version 2.17.1) for Python [118].
Depending on the type of surface, the reference of the inertia was shifted to the platforms origin
with the parallel axes theorem and added to or subtracted from the total inertia. Validation of
simple geometric shapes showed, that the numerical solution is equal to the analytical solution with
negligible error.
The inertia of the space frame was calculated analytically by calculating the inertia of each
structural member, shifting the inertia by means of the parallel axes theorem and tilting the refer-
ence axes according to the inclination of the member. The principal moments of inertia of a tubular
member are given by [47]
J1 = 1/2m
(
ro
2 + ri
2
)
(9.1)
J2 = J3 = 1/4m(ro
2 + ro
2 + 1/3l2) , (9.2)
where J1 is the moment of inertia about its longitudinal axis, J2 and J3 about its radial axes, m
is the mass of the member, ro its outer radius, ri its inner radius, and l its length. To adjust the
inertia matrix according to the inclination of the member, the angles between the principal axis
and the axis of the reference frame were computed. The entries of the inertia matrix (Jij) in the
reference frame of the platform are defined by [47]
Jxx = J1 cos2 αx1 + J2 cos2 αx2 + J3 cos2 αx3 (9.3)
Jyy = J1 cos2 αy1 + J2 cos2 αy2 + J3 cos2 αy3 (9.4)
Jzz = J1 cos2 αz1 + J2 cos2 αz2 + J3 cos2 αz3 (9.5)
Jxy = J1 cosαx1 cosαy1 + J2 cosαx2 cosαy2 + J3 cosαx3 cosαy3 (9.6)
Jxz = J1 cosαx1 cosαz1 + J2 cosαx2 cosαz2 + J3 cosαx3 cosαz3 (9.7)
Jyz = J1 cosαy1 cosαz1 + J2 cosαy2 cosαz2 + J3 cosαy3 cosαz3 , (9.8)
where αji is the angle between the i-th principal axis and j-th axis of the reference frame.
9.1.3 Structural Analysis
The structural analysis was performed similar to the analysis performed during optimization, which
is described in chapter 8. Additionally to the effects considered, the effect of the trapped water
and inertia forces of the platform were incorporated. The trapped water was modeled with point
masses at each node, together with a point force to counteract their weight. To consider inertia of
the platform, both, the linear and rotational, accelerations were adjusted to the values estimated
by ProteusDS for the rigid body. To cope with stresses at the connection between strut and hull,
the connections with the hull at the corresponding nodes were fitted with stiffener plates. The
plates were of trapezoidal shape, had a thickness of 0.03 m, a base length of 0.8878 m and an angle
132 CHAPTER 9. REVIEW OF FINAL DESIGN
at either side of 45°. These values are given with respect to the center line of the beams.
9.1.4 Design of Pile Anchor
The relevant design codes for marine applications define various methodologies for designing piles
and pile anchors in sand or clay. Methodologies for piles in weak rock, however, are hardly available.
A methodology for pile anchors in weak rock, similar to which can be found in the Cozumel Channel,
has been proposed by Bañuelos-García et al. [8]. The estimation of the pull-out and lateral capacity
were enhanced in this work regarding the aforementioned paper and are described in the following.
Guidance on the design of ground anchors and soil nails is given by Juran and Elias [60], as they
touch on durability considerations, performance criteria, and approaches to ensure the stability of
the different components. A method for estimating the shaft friction of anchor piles driven in weak
rock has been presented by Terente et al. [110]. The method is based on pile testing data which
suggested that current methods significantly underestimate the holding capacity of driven piles.
Serrano and Olalla [100] calculated the pull-out resistance of anchors in different materials based
on the failure criterion of Hook and Brown. Xu et al. [123] reviewed the state of the art of shear
mechanism of rock-socketed piles and estimation of the shaft friction. Pullout forces due to skin
friction of pre-bored anchor piles were evaluated in the study of Park et al. [81] and compared to
different design standards. Different methods for estimating the unit skin friction were compared
to measurements by McVay et al. [73]. They found, that estimating the unit skin friction based on
the unconfined compressive strength and the split tensile strength gives the best results. In case
only the unconfined compressive strength is available, results based on the method of Gupton and
Logan gave the best results, while calculations based on standard penetration tests gave the worst
results. A similar work on different methods for estimating the skin friction of piles in limestone
based on the unconfined compressive strength has been published by Alshenawy et al. [3].
The design of anchors is typically divided into different – more or less – independent tasks:
Vertical and horizontal capacity of the anchors. The common assumption of uncoupled behavior
between vertical and horizontal load capacity of the soil i.e. uncoupled interaction between soil
and structure in horizontal and vertical direction leads to this separation in the design process [88].
Typically, the design of anchors is divided into issues due to installation and issues due to operation,
which both have to be considered for evaluating structural integrity [88]. As the pile anchor in rock
is typically installed in pre-drilled holes in the rock and then grouted in place, the stresses during
installation are neglected.
For a given diameter of the pile (da), the minimum wall thickness (ta) can be calculated based
on API RP 2A-WSD [4] by
ta = 6.35 × 10−3 +
da
100
. (9.9)
Note that in contrast to the formula given in API RP 2A-WSD [4] the diameter should be given in
m and not in mm. The embedded length (Le) is then calculated following Poulos and Davis [85]
9.1. METHODOLOGY 133
by
Le = 4
(
EsI
nH
)1/5
, (9.10)
where Es is the Young’s modulus of the steel used, I is the second moment of area of the pile’s
cross-section, and nH the horizontal coefficient of subgrade reaction [8].
Structural Capacity
The structural capacity of the pile anchor is analyzed according to Bañuelos-García et al. [8].
However, the material factors are taken according to IEC 62600-2 [54]. Since pile anchors for floating
structure are only subject to tension and bending forces, buckling is not part of the expected failure
modes and therefore not analyzed. The structural capacity to axial loads is calculated by
Qsa =
1
γM
σyAc , (9.11)
where γM is the material factor corresponding to the load case and required safety level, σy the yield
strength of the material used, and Ac the cross-sectional area of the anchor pile. The structural
capacity to withstand lateral loads (Qsl) can be estimated by
Qsl =
σcrAc
2γM
, (9.12)
where the critical stress for shear loads (σcr) is defined by the maximum of
σcr =
1.6Es(
da
t′
a
)5/4
√
Le
da
≤ 0.6σy , (9.13)
and
σcr =
0.78Es(
da
t′
a
)3/2
≤ 0.6σy , (9.14)
where t′a is the adjusted wall thickness. In case the pile is welded via electric resistance, the adjusted
wall thickness is calculated by t′a = 0.93ta, otherwise the adjusted wall thickness is equal to the
(nominal) wall thickness of the pile (ta).
Axial Holding Capacity
The vertical holding capacity or pull-out capacity (Qha) of a pile in rock is based on the skin friction
(Qf ) and the submerged weight of the anchor, calculated by
Qha = Qf +WP . (9.15)
134 CHAPTER 9. REVIEW OF FINAL DESIGN
Different methods are available to estimate the pull-out capacity of piles in soft rocks such as
limestone, yet they all rely on the following relation:
Qf = τfAs , (9.16)
where τf is the resistance of the shaft to friction andAs the outer surface area of the shaft. According
to the guideline ROM 0.5-05 [86], the resistance of the shaft to friction for rocks is calculated by
τf = 0.1pvs , (9.17)
with the vertical pressure for sinking in rock (pvs) defined by
pvs = 3 × 10−3√
qu,rfDfA ≤ 15 MPa , (9.18)
where qu,r is the compressive strength of the rock (in Pa), fD a factor to consider the rock quality
designation (RQD), and fA a factor to consider alteration of the rock due to weathering. The design
code ROM 0.5-05 [86] includes also a factor for considering inclination of the load, which, however,
should be taken as 1 for the cases considered in this work. The factor considering RQD should be
the minimum of
fD = 2
√
s
da
< 1 (9.19)
and
fD = 0.2
√
χRQD
da
< 1 , (9.20)
where da is the diameter of the anchor, s the lithoclase spacing, and χRQD the RQD in percent.
The factor for alteration of the rock due to weathering (fA) shall be taken as 1.0 for weathering of
type I, as 0.7 for weathering of type II, and as 0.5 for weathering of type III. For more information
on the types of rock weathering see [86, chapter 2.2.9.7].
Additionally to the holding capacity of the anchor in the rock, the maximum allowable axial
forces for the connection between steel and grout shall be reviewed. This analysis can be performed
according to relevant design codes such as API RP 2A-WSD [4]. For plain connections between steel
and grout (i.e. without shear keys), the maximum allowable axial stress is limited to 0.138 MPa
and 0.184 MPa, for operation environmental conditions and design environmental conditions, re-
spectively. In case shear keys are used, they should be either a semi circular welded bead, a flat bar
with fillet welds, or a round bar with fillet welds. The maximum allowable axial stress is calculated
by
τf = 138 × 103 + 0.5qu,g
h
s
(9.21)
for operation environmental conditions, and
τf = 184 × 103 + 0.67qu,g
h
s
(9.22)
9.1. METHODOLOGY 135
for design environmental conditions, where τf is the maximum allowable load transfer stress, qu,g
the unconfined compressive strength of the grout used, h the outstand of the shear key, and s the
spacing of the shear keys or their pitch height if they are arranged in a continuous helix. The
unconfined compressive strength of the grout used must satisfy the following condition:
17.25 MPa ≤ qu,g ≤ 110 MPa
For more requirements on the design of shear keys for grouted connections see API RP 2A-WSD
[4, chapter 7.4.4.c].
Lateral Holding Capacity
The lateral holding capacity of piles is typically analyzed by means of p-y curves based on the
approach first introduced by Winkler. The p-y curves for weak rock such as limestone and the
differential equation to model interaction between the pile and the soil are described by Reese [89].
To avoid issues with adhesion under axial loads, the rock is supposed to be loaded until yielding
of the rock is reach and the elastic region of the corresponding p-y curve is left. The differential
equation describing the pile as beam and its interaction with soil via linear elastic springs is solved
with the finite difference method and is given by
EI
d4y
dz4
+ Fv
d2y
dz2
+ kiry = 0 , (9.23)
where E is the Young’s modulus of the used steel, I the second moment of inertia of the pile’s
cross-section, y the lateral deflection of the pile, z the depth below groundline, Fv the vertical load
on the pile (if analyzed at the same time), and kir the initial linear stiffness of the rock at depth z
according to the corresponding p-y curve.
At the tip of the pile, the shear force and the bending moment are supposed to be zero, while
at the pile’s top end at the groundline the shear force is equal to the horizontal load (Fh) and the
bending moment equal to the applied moment (M0). The moment at the groundline is calculated
by
M0 = speFh , (9.24)
where spe is the height above groundline of the padeye. Considering these boundary conditions and
the distance between nodes of the uniform grid (h), the nonzero entries of the stiffness matrix (K)
can be derived. Analyzing the axial and lateral capacity of the anchor pile separately as suggested
by Randolph and Gourvenec [88] (i.e. Fv = 0 for this analysis) leads to the equation system given
136 CHAPTER 9. REVIEW OF FINAL DESIGN
by
i = 0 :
(
2
EI
h4
+ kir
)
y0 − 4
EI
h4
y1 + 2
EI
h4
y2 =
2Fh
h2
(1 + spe) (9.25)
i = 1 : −2
EI
h4
y0 +
(
5
EI
h4
+ kir
)
y1 − 4
EI
h4
y2 +
EI
h4
y3 = −Fhspe
h2
(9.26)
2 ≤ i ≤ n− 2 :
EI
h4
yi−2 − 4
EI
h4
yi−1 +
(
+
EI
h4
+ kir
)
yi − 4
EI
h4
yi+1 +
EI
h4
yi+2 = 0 (9.27)
i = n− 1 :
EI
h4
yn−3 − 4
EI
h4
yn−2 +
(
5
EI
h4
+ kir
)
yn−1 − 2
EI
h4
yn = 0 (9.28)
i = n : 2
EI
h4
yn−2 − 4
EI
h4
yn−1 +
(
−2
EI
h4
+ kir
)
yn = 0 , (9.29)
where kir is the stiffness of the rock at the corresponding depth, i is the number of the node, n the
number of nodes, and h the distance between nodes, defined as
h =
Le
n
. (9.30)
According to Reese [89], the stiffness of weak rock (kir) can be calculated based on the initial
Young’s modulus of the rock (Eir) and the depth below the ground line (z) by
kir =



(
100 + 400z
3da
)
Eir , for 0 ≤ z ≤ 3da
500Eir , for z > 3da
, (9.31)
where da is the outer diameter of the anchor pile. The ultimate resistance of the rock is given by
pur =



αrqu,rda
(
1 + 1.4 z
da
)
, for 0 ≤ z ≤ 3da
5.2αrqu,rda , for z > 3da
, (9.32)
where αr is the strength reduction factor (see Reese [89]), and qu,r the compressive strength of the
rock. The elastic limit deflection of the rock is defined as
yA =
(
pur
2 (rrmda)0.25
kir
)4/3
, (9.33)
where rrm is a constant describing the properties of the rock, ranging between 0.0005 and 0.00005
(see Reese [89]).
To find the horizontal load associated to a deflection reaching the elastic limit of the rock, the
equation system is solved for an arbitrary horizontal load which then is used to extrapolate (or
interpolate) the maximum load. The horizontal holding capacity can then be calculated by
Qhl = yA,0
Fh
y0
, (9.34)
9.1. METHODOLOGY 137
where yA,0 is the elastic limit deflection of the rock at groundline, Fh is the arbitrary horizontal load
for which the system was solved, and y0 the lateral deflection at groundline due to the arbitrary
force Fh.
9.1.5 Fatigue Life of Mooring Lines and Components
Estimating the annual fatigue damage per environmental state of mooring lines or mooring com-
ponents is typically based on either T-N curves (i.e. in API RP-2SK [5]) or on S-N curves (i.e.
DNVGL OS-E301 [32]). Following the guidance of API RP-2SK [5], the T-N curves for tension-
tension fatigue are defined by
nRm = k , (9.35)
where n is the number of load cycles leading to failure, R the ratio of tension range to MBL, and
m and k are the slope and intercept of the T-N curve, respectively. When calculating the fatigue
life of mooring line components, stress concentration factors should be considered. The damage is
typically calculated as the combined damage of so called low frequency and wave frequency tensions
which are computed under mean loading due to other environmental conditions such as wind and
currents. API RP-2SK [5] suggests four methods for combining damage due to low frequency
and wave frequency tensions. Simple summation (Method 1) gives acceptable estimates for the
fatigue life when either damage due to wave frequency or damage due to low frequency tensions are
dominant. This is the case when the ratio between the wave frequency standard deviation tension
and the low frequency standard deviation tension satisfies one of the following conditions:
σwf
σlf
≥ 1.5 ;
σwf
σlf
≤ 0.5
Method 2 is based on a combined spectrum of low frequency and wave frequency tensions and
gives conservative results which tend to significantly overestimate the fatigue damage. To adjust
the combined spectrum method, a narrow-banded correction factor was introduced for Method
3. The most accurate yet computationally very expensive method is Method 4 which is based on
time domain cycle counting. Since in this work and under operating conditions no wave frequency
tensions are present, the method of simple summation is used. Ignoring the damage due to waves,
the annual fatigue damage due to environmental state i is defined by
DF,i =
ni,lf
k
(√
2Ri,lf
)m
Γ (1 + m/2) , (9.36)
where ni,lf is the number of low frequency cycles per year for environmental state i, Ri,lf the ratio
between low frequency standard deviation tension range and MBL, and Γ the gamma function. The
number of low frequency cycles per year for environmental state i is calculated by
ni,lf = fi,lfTi , (9.37)
138 CHAPTER 9. REVIEW OF FINAL DESIGN
where fi,lf is the zero up-crossing frequency of the tension spectrum in environmental state i, and
Ti the time per year spent in environmental state i. With 3.155 76 × 107 seconds per year, the time
per year spent in environmental state i can be obtained by
Ti = 3.155 76 × 107 Pi , (9.38)
where Pi is the probability for environmental state ii [5].
Low Frequency RMS Tension Range
As stated, for the fatigue calculations only the low frequency fatigue damage is considered. This
damage is typically caused by low frequency wave loads, as wind and currents are considered by
the mean load, which in turn has influence on the stiffness of the mooring system. However, at
design depth of the platform, no wave or wind loads are present, wherefore the low frequency fatigue
damage is calculated based on the cyclic current range and the tensions caused by this range in the
mooring lines. To find the low frequency component of the current, empirical data are analyzed
with a discrete Fourier transform (DFT) and the resulting amplitudes are compared. In case the
available data were not obtained with a sufficient sample frequency, the root mean square (RMS)
of the difference in current speed between consecutive peaks is evaluated as well. Additionally,
the natural frequencies of the platform for surge, sway, and yaw motions should be considered, if
supported by the data and available in the result of the DFT. The frequency which causes the most
damage is then selected and used for the fatigue analysis.
For the analysis, the following procedure is applied. The natural frequency (fn) of the complete
system is determined by a free decay test simulated with ProteusDS (see section 9.3). For the
empirical data of the current for the construction site, the DFT is computed and the frequencies
are selected which cause the most damage. The current amplitude (ûk) corresponding to frequency
f of the DFT can be calculated by
ûk =
2
n
|Xk| , (9.39)
where n is the number of samples and Xk the k-th coefficient of the DFT. The current amplitude
(ûk) based on the difference in current speed between constecutive samples is given by
ûk =
ui+1 − ui
2
, (9.40)
where ui is the current for sample i and ui+1 the current for sample i+ 1. The mooring system is
then analyzed for a higher current (uh) and a lower current (ul), resulting in two tensions per load
case and mooring line The currents are defined by
uh = uLC,i + ûk and ul = uLC,i − ûk , (9.41)
where (uLC,i) is the current corresponding to the load case for which the fatigue damage is estimated.
For each mooring line the low frequency RMS tension range (Ri,lf ) is calculated as the absolute
9.2. DYNAMIC MOORING ANALYSIS 139
difference of both resulting tensions in the line by
Ri,lf = |Tlf,h − Tlf,l| , (9.42)
where Tlf,h is the tension corresponding to uh, and Tlf,l the tensions corresponding to ul. This
analysis is performed for each mooring line, note, that the higher tension is not necessarily caused
by the higher current.
The probability for each load case (Pi) is determined based on the return value of the current’s
magnitude and the probability for the direction. For each magnitude of current, different angles are
analyzed. To determine the probability of the direction, the mode of the direction is determined
with a histogram for the empirical data and with a bin width of 0.5°. A second histogram is
created with a bin width of 5°, centered around the mode. The sum of the probability for the
analyzed directions of the current is then adjusted to 100% while maintaining the relation between
the directions.
The approach is based on quasi-static calculations. However, as the current is typically assumed
to be constant for low frequency fatigue damage estimations the error is expected to be negligible.
9.2 Dynamic Mooring Analysis
A dynamic mooring analysis was run for the three-mooring line configuration. The design proposal
resulting from the automated and manual optimization was analyzed with ProteusDS for all 23
load cases as defined in section 7.3.1. For more information on ProteusDS the reader is referred
to section 3.3.1 or the manual [35]. The simulations were not able to find static or quasi-static
equilibrium within a reasonable simulation time. Therefore, the mean value (in case of position and
rotation) and maximum value (in case of tensions) of the last 20 s was selected. For all load cases
analyzed, the last 20 s of simulation time do include several periods of oscillation around the mean
value. The results of the calculation can be found in the following figures.
Mooring Offset The displacement from its initial position (i.e., mooring offset) can be found in
figure 9.1. Noticeable is the altering yet expected y-displacement in figure 9.1. Load cases with
the current coming from starboard (ending with B) showed a pronounced deflection in positive
y-direction, while the load cases with the current coming from port (ending with C ) showed a
deflection in the negative y-direction of the same magnitude, yet marginally larger in absolute
terms. An exception from this behavior was LC5B which showed with 1.06 m a significantly higher
deflection in y-direction than its direct counterpart LC5C with −0.44 m. This is believed to be the
result of the missing starboard ballast tank, which leads to a greater deflection in port direction.
This is supported by the fact, that the deflections in the load cases with missing port ballast tank
showed a qualitatively equal, but opposite behavior. Thus, in LC6B the deflection resulted to
0.40 m and in LC6C to −1.06 m. The deflections during the load cases with the current coming
from the front showed noticeable deflection in the opposite direction of the missing ballast tank,
140 CHAPTER 9. REVIEW OF FINAL DESIGN
Figure 9.1: Mooring offset of platform in three mooring line configuration.
as for LC5A a deflection of 0.30 m and for LC6A a deflection of −0.30 m was estimated. For the
load cases with the starboard mooring line missing (LC7A through LC7C), the platform naturally
deflected to port with its maximum offset in the y-direction for LC7B, where it saw current coming
from starboard. Analogously, in the load cases with the port mooring line missing (LC8A through
LC8C), the platform was pushed to starboard, peaking for LC8C with the current originating from
the platform’s port side.
Except for the load cases LC7A through LC8C (line loss), the platform was pulled against the
current. Despite the constant offset from its initial position, the platform was hardly moved in
the x-direction by the presence of the current. For load case LC1 without current a deflection of
0.191 m in the x-direction was estimated while for LC2A a deflection of 0.18 m was determined, both
measured from the initial position. The influence of the current was visible yet barely of importance
as it was in the order of magnitude of a centimeter. For the load cases with a missing tank and
the current coming from the side of the missing tank, a higher forward offset of the platform was
detected. For LC5B the platform moved 0.28 m ahead of its initial position, whereas a movement
of 0.29 m was calculated for LC6C.
The z-offset of the platform was also slightly above its initial position with 0.24 m for LC1.
Outliers could be detected for the cases with a missing ballast tank, especially for the current
coming from the same direction. For those cases, the z-offset got below 0 (−0.13 m for LC5B and
−0.17 m for LC6C). For the load cases with missing lines, the platform rose to 5.47 m (LC7C) and
8.62 m (LC8A). Notably is the fact, the minimum offsets in each group were achieved for the cases
with the current coming from the side with the intact mooring line (LC7C and (LC8B).
Platform Rotation The rotation of the platform for each load case can be found in figure 9.2.
In synchronization with the maximum y-offsets, the heel angle of the platform peaks as well. For
the load cases with the current coming from starboard, the platform lowered its starboard side, and
vice versa for the load cases with the current coming from port. Expectably this was even more
pronounced for LC5B (missing starboard ballast tank with current from the same direction) and
9.2. DYNAMIC MOORING ANALYSIS 141
Figure 9.2: Mean rotation of platform in three mooring line configuration.
LC6C (missing port side ballast tank with current from the same direction). The heel angles for
the cases with a missing mooring line were very high.
The pitch angle of the platform was rather small, with its peaks found for the load cases with
missing lines (LC7A through LC8C) and the cases LC5B and LC6C. Yet, the maximum pitch angle
could be found for LC9 when the platform was at the water surface for maintenance. The angle of
66.0° was very high which suggests unfavorable conditions for maintaining the platform. However,
as service vessels are required, the platform can be moored to those, hence avoiding extreme motions
and especially relative movement between platform and service vessel.
The yaw angle of the platform was relatively close to 0° for the load cases with the current
coming from the front. Exceptions were the load cases with missing tanks (LC5A and LC6A),
the load cases with missing lines, and the load case for maintenance. For the load cases with the
current coming from the sides, the platform tended to yaw towards the other side, as can be seen
by the altering yaw angle for load cases B and load cases C. The missing ballast tanks seemed to
have shifted the yaw angled towards the opposite side of the missing tank. Thus, in LC5B the yaw
angle (4.1°) was much greater than the angles found for the load cases LC2B, LC3B, and LC5B.
Similarly, for LC6C, the absolute yaw angle (−3.7°) was greater than for comparable load cases.
Tensions in the mooring lines he maximum tensions in mooring lines are depicted in figure 9.3.
As can be seen, the maximum tension could be found in the aft center mooring line for the load
cases LC1 through LC6C. For the load cases with missing lines, the remaining forward line was the
most loaded, while for the load cases under maintenance, the mooring lines were generally unloaded
with the forward port side mooring line seeing the greatest load. For conditions with no current,
the aft center mooring line was exposed to nearly double the load to the two forward-facing mooring
lines (107.7 kN for aft center, 62.7 kN for forward starboard, and 63.5 kN for forward port). This is
expected, as the force by the two forward lines has to be compensated by the aft center mooring
line while the mooring angle of the forward lines slightly increases their load.
Due to the current in the load cases LC2A, LC3A, and LC4A, the load on the forward lines
142 CHAPTER 9. REVIEW OF FINAL DESIGN
Figure 9.3: Maximum tension in mooring lines of platform. fwd refers to the mooring lines connected
to the forward end of the platform, prt to the port side mooring lines, stb to the mooring lines on
starboard side, and ctr to the one at center line.
increased while the load on the aft mooring lines on port and starboard decreased. For the corre-
sponding load cases with the current coming from the side, the mooring line on the same side saw
an increase in its load.
The conditions for the load cases with missing mooring lines (LC7A through LC8C) lead to
a pronounced increase in the load on the remaining forward line, with a peak in the load for the
load cases with the current coming from the side of the missing line. The load in the forward
port side line peaked at 158.9 kN (LC7B) slightly higher than the load in the forward starboard
line with 143.1 kN (LC8C). For the four remaining load cases in the scenarios, the peak loads were
comparable for the load case with the current coming from the front and for the load cases with
the current coming from the same side as the missing line. The loads were estimated to 103.3 kN
and 105.3 kN for the load cases LC7A and LC8A, respectively. For the load cases LC7C and LC8B,
the loads were calculated to be slightly lower with 96.5 kN and 97.3 kN, respectively.
Ultimate Limit State
The maximum load for the relevant load cases was found to be 109.45 kN. This load was detected
in the aft center mooring line during the load case LC1. The ratio between the MBL of 360 kN and
the load is 3.289. Since this ratio is significantly higher than the required adjusted safety factor of
2.171, the mooring design satisfies the demands of IEC 62600-10 [53] for the ULS.
The maximum stress was determined to 296 MPa and the minimum buckling factor to 4.8.
Considering the corresponding partial load safety factor and material factor, steel of type S450 or
similar is sufficient to comply with the design codes.
Serviceability Limit State
The SLS is evaluated for the load cases LC1 through LC4C. Regarding the absolute mooring offset
(i.e, displacement from initial position) the platform stayed well within 0.6 m, with the maximum
9.3. NATURAL FREQUENCY 143
deflections of 0.56 m and −0.58 m detected in the y-direction for LC3B and LC4C, respectively (see
figure 9.1 a)).
The maximum inclinations of the platform for the load cases evaluated were 9.8° and −9.2°.
Both heel angles were detected for the load cases LC3B and LC3C, respectively. They are well
within the defined limits, however, as these angles are relative to the rotational axis of the turbine
they are expected to have negligible influence on the turbine’s performance. The maximum heading
was estimated to 2.9° and −2.8° for LC3B and LC3C, respectively, and showed a significantly lower
deviation. The maximum pitch angle was determined to −1.4° for LC3C.
Accidental Limit State
For the accidental limit state, the maximum load was determined to 162.51 kN for the forward port
mooring line. In this condition (LC7B), the forward starboard mooring line was missing with the
current flowing in the direction of 150° (i.e. from starboard). The maximum stress was determined
to 397 MPa, while the minimum buckling factor was found to be 3.2, for which steel of type S450
or similar is sufficient to satisfy the design requirements.
9.3 Natural Frequency
A free decay test simulated in ProteusDS was used to determine the natural frequency of the
platform for surge, sway, and heave. From its initial position the linear velocity of the platform was
changed to 3 m s−1 for each direction separately. The deflection from its mean position for each
direction are shown in figure 9.4 a), c), and e). A discrete Fourier transformation was then performed
on the deflection to find the magnitude of the individual components and the peak frequency. The
fast Fourier transformation-algorithm was used as provided by the Python-module numpy in version
1.22.4 [48]. As depicted in figure 9.4 b), d), and f), the magnitudes of the components have a clear
peak at around 14.8 s for surge, at 16.8 s for sway, and at 15.1 s for heave.
9.4 Fatigue Life
The fatigue life of the mooring system and the platform is calculated following the methodology
outlined in section 3.2.3, which is based on relevant design codes such as IEC 62600-10 [53] and
API RP-2SK [5].
9.4.1 Low Frequency Tension Range
The low frequency tension range is based on changes in the current speed since the influence of
wind and waves is negligible for the design depth of the system.
Current Range Since empirical data for the current are only available at a sample period of
30 min and 60 min (see section 6.1), the RMS for the difference in current speed between consecutive
144 CHAPTER 9. REVIEW OF FINAL DESIGN
Figure 9.4: Results of free decay test to determine natural frequency of the system. a), c), e)
show the signal for each direction; b), d), f) show the magnitude of the components of the signal,
obtained through discrete Fourier transformation.
peaks is calculated as well as the most common time span between two peaks. The current amplitude
based on measurements by the CANEK-project with 30 min sample period can be seen in table 9.1.
The amplitude for a period of 60 min was obtained as half the height difference of consecutive
peaks while the remaining values were obtained from the DFT. Since the current amplitude with
a period of 60 min is significantly higher than all other values, a low frequency current range of
8.476 × 10−2 m s−1 with a period of 60 min is used.
Tension Range The tension range was calculated based on the same quasi-static model as de-
scribed in section 8.2. For each load case, the mooring system was analyzed at the reference current
speed, half the current range above the reference speed, and half the current range below the ref-
erence speed. The tensions obtained for each mooring line and material type can be found in
tables 9.2 to 9.6. The tension denoted with high corresponds to the higher current speed and is
not necessarily the higher obtained tension. Vice versa, tensions denoted with low were caused by
the lower current speed. Especially for the forward mooring lines under current coming from the
front, the higher tension was caused by the lower current speed. The current direction is reported
in the global reference frame of the numerical model with 180° corresponding to the peak current
direction in the earth’s reference frame. The maximum ratio of mean tension to MBL was 0.31 for
the wire.
9.4. FATIGUE LIFE 145
Table 9.1: Current amplitudes for low frequency fatigue life analysis.
Period Current
Amplitude
in days in m s−1
1/12 4.238 × 10−2
0.5 4.51 × 10−3
1.0 7.16 × 10−3
2.0 8.13 × 10−3
7.0 1.38 × 10−2
14.0 1.18 × 10−2
28.0 2.44 × 10−2
Table 9.2: Tensions for fatigue life analysis of forward starboard mooring line.
Tensions
LC Mat. Speed Dir. Mean High Low Range
in m s−1 in ° in N in N in N in N
LC2A Wire 1.38 180 1.10 × 104 1.08 × 104 1.11 × 104 3.04 × 102
LC2A Chain 1.38 180 9.26 × 103 9.10 × 103 9.42 × 103 3.18 × 102
LC2B Wire 1.38 150 1.55 × 104 1.56 × 104 1.54 × 104 2.52 × 102
LC2B Chain 1.38 150 1.40 × 104 1.41 × 104 1.38 × 104 2.60 × 102
LC2C Wire 1.38 210 7.37 × 103 7.03 × 103 7.72 × 103 6.87 × 102
LC2C Chain 1.38 210 5.42 × 103 5.03 × 103 5.79 × 103 7.63 × 102
LC: Load case; Mat.: Material; Dir.: Direction
Table 9.3: Tensions for fatigue life analysis of forward port mooring line.
Tensions
LC Mat. Speed Dir. Mean High Low Range
in m s−1 in ° in N in N in N in N
LC2A Wire 1.38 180 1.04 × 104 1.02 × 104 1.05 × 104 2.97 × 102
LC2A Chain 1.38 180 9.49 × 103 9.33 × 103 9.64 × 103 3.03 × 102
LC2B Wire 1.38 150 6.64 × 103 6.28 × 103 6.99 × 103 7.05 × 102
LC2B Chain 1.38 150 5.65 × 103 5.28 × 103 6.01 × 103 7.37 × 102
LC2C Wire 1.38 210 1.50 × 104 1.51 × 104 1.48 × 104 2.60 × 102
LC2C Chain 1.38 210 1.42 × 104 1.43 × 104 1.40 × 104 2.63 × 102
LC: Load case; Mat.: Material; Dir.: Direction
146 CHAPTER 9. REVIEW OF FINAL DESIGN
Table 9.4: Tensions for fatigue life analysis of aft center mooring line.
Tensions
LC Mat. Speed Dir. Mean High Low Range
in m s−1 in ° in N in N in N in N
LC2A Wire 1.38 180 2.62 × 104 2.65 × 104 2.59 × 104 6.60 × 102
LC2A Chain 1.38 180 2.54 × 104 2.58 × 104 2.51 × 104 6.63 × 102
LC2B Wire 1.38 150 2.56 × 104 2.59 × 104 2.53 × 104 6.06 × 102
LC2B Chain 1.38 150 2.48 × 104 2.51 × 104 2.45 × 104 6.10 × 102
LC2C Wire 1.38 210 2.56 × 104 2.59 × 104 2.53 × 104 5.90 × 102
LC2C Chain 1.38 210 2.48 × 104 2.51 × 104 2.45 × 104 5.93 × 102
LC: Load case; Mat.: Material; Dir.: Direction
Table 9.5: Tensions for fatigue life analysis of aft starboard mooring line.
Tensions
LC Mat. Speed Dir. Mean High Low Range
in m s−1 in ° in N in N in N in N
LC2A Wire 1.38 180 1.34 × 104 1.35 × 104 1.32 × 104 3.34 × 102
LC2B Wire 1.38 150 1.26 × 104 1.27 × 104 1.25 × 104 2.29 × 102
LC2C Wire 1.38 210 1.35 × 104 1.37 × 104 1.33 × 104 3.67 × 102
LC: Load case; Mat.: Material; Dir.: Direction
Table 9.6: Tensions for fatigue life analysis of aft port mooring line.
Tensions
LC Mat. Speed Dir. Mean High Low Range
in m s−1 in ° in N in N in N in N
LC2A Wire 1.38 180 1.43 × 104 1.45 × 104 1.41 × 104 3.51 × 102
LC2B Wire 1.38 150 1.44 × 104 1.46 × 104 1.42 × 104 3.97 × 102
LC2C Wire 1.38 210 1.35 × 104 1.36 × 104 1.34 × 104 2.45 × 102
LC: Load case; Mat.: Material; Dir.: Direction
9.4. FATIGUE LIFE 147
Table 9.7: Adjusted probabilities for load cases for fatigue life analysis.
Load Case Direction Observations Adjusted
Probability
in ° – –
LC2A 34.75 10538 9.984 × 10−1
LC2B 64.75 12 1.137 × 10−3
LC2C 4.75 5 4.737 × 10−4
Table 9.8: Annual fatigue damage estimation of forward starboard mooring line.
LC Mat TTR Probability n cycles Damage Total Damage
LC2A Wire 9.876 × 10−4 9.984 × 10−1 8751.9 5.594 × 10−11
5.671 × 10−11LC2B Wire 8.190 × 10−4 1.137 × 10−3 10.0 2.962 × 10−14
LC2C Wire 2.229 × 10−3 4.737 × 10−4 4.2 7.409 × 10−13
LC2A Chain 1.034 × 10−3 9.984 × 10−1 8751.9 1.150 × 10−7
1.159 × 10−7LC2B Chain 8.441 × 10−4 1.137 × 10−3 10.0 7.131 × 10−11
LC2C Chain 2.478 × 10−3 4.737 × 10−4 4.2 7.516 × 10−10
LC: Load case; Mat: Material; TRR: Tension range ratio R
9.4.2 Probabilities of Load Cases
Since the effect of the environmental conditions at design depth of the platform is the same for LC2
and LC4 (see section 7.3.1), fatigue life analysis was only performed for LC2A, LC2B and LC2C.
The direction, the number of observations, and the adjusted probability can be seen in table 9.7. As
can be seen, the current direction was nearly always 34.75° (i.e. flowing in direction of north-east
to north) while current hardly flows in the other two directions.
9.4.3 Fatigue Life of System
According to API RP-2SK [5], the intercept parameter of the T-N curves for wire rope depends
on the ratio between mean load and MBL, which was found to be 0.13 at maximum. For the wire
rope, the T-N curve defined by the intercept k = 687 and the slope m = 4.09 was used, while for
the chain, the T-N curve defined by k = 316 and m = 3.0 was used. The number of cycles (in
the tables denoted with n cycles) was calculated according to equations (9.37) and (9.38). The
annual damage per load case and total annual damage per mooring line segment are defined by
equations (3.9) and (9.36). The results of the calculations per mooring line and load case are shown
in tables 9.8 to 9.12. As can be seen, the segment most susceptible to fatigue damage is the chain
segment of the aft center mooring line. Considering a safety factor of 3, the expected fatigue life of
this mooring line is 320 204 years. Hence it is safe to consider that the mooring lines are unlikely
148 CHAPTER 9. REVIEW OF FINAL DESIGN
Table 9.9: Annual fatigue damage estimation of forward port mooring line.
LC Mat TTR Probability n cycles Damage Total Damage
LC2A Wire 9.639 × 10−4 9.984 × 10−1 8751.9 5.066 × 10−11
5.266 × 10−11LC2B Wire 2.289 × 10−3 1.137 × 10−3 10.0 1.982 × 10−12
LC2C Wire 8.434 × 10−4 4.737 × 10−4 4.2 1.392 × 10−14
LC2A Chain 9.835 × 10−4 9.984 × 10−1 8751.9 9.907 × 10−8
1.008 × 10−7LC2B Chain 2.392 × 10−3 1.137 × 10−3 10.0 1.624 × 10−9
LC2C Chain 8.538 × 10−4 4.737 × 10−4 4.2 3.075 × 10−11
LC: Load case; Mat: Material; TRR: Tension range ratio R
Table 9.10: Annual fatigue damage estimation of aft center mooring line.
LC Mat TTR Probability n cycles Damage Total Damage
LC2A Wire 2.144 × 10−3 9.984 × 10−1 8751.9 1.331 × 10−9
1.333 × 10−9LC2B Wire 1.967 × 10−3 1.137 × 10−3 10.0 1.068 × 10−12
LC2C Wire 1.914 × 10−3 4.737 × 10−4 4.2 3.979 × 10−13
LC2A Chain 2.152 × 10−3 9.984 × 10−1 8751.9 1.038 × 10−6
1.041 × 10−6LC2B Chain 1.980 × 10−3 1.137 × 10−3 10.0 9.207 × 10−10
LC2C Chain 1.924 × 10−3 4.737 × 10−4 4.2 3.520 × 10−10
LC: Load case; Mat: Material; TRR: Tension range ratio R
Table 9.11: Annual fatigue damage estimation of aft starboard mooring line.
LC Mat TTR Probability n cycles Damage Total Damage
LC2A Wire 1.083 × 10−3 9.984 × 10−1 8751.9 8.163 × 10−11
8.170 × 10−11LC2B Wire 7.436 × 10−4 1.137 × 10−3 10.0 1.997 × 10−14
LC2C Wire 1.190 × 10−3 4.737 × 10−4 4.2 5.689 × 10−14
LC: Load case; Mat: Material; TRR: Tension range ratio R
Table 9.12: Annual fatigue damage estimation of aft port mooring line.
LC Mat TTR Probability n cycles Damage Total Damage
LC2A Wire 1.140 × 10−3 9.984 × 10−1 8751.9 1.008 × 10−10
1.010 × 10−10LC2B Wire 1.290 × 10−3 1.137 × 10−3 10.0 1.902 × 10−13
LC2C Wire 7.950 × 10−4 4.737 × 10−4 4.2 1.093 × 10−14
LC: Load case; Mat: Material; TRR: Tension range ratio R
9.5. ANCHOR DESIGN 149
to fail due to fatigue.
9.5 Anchor Design
For anchoring MCECs various types of anchors may be suitable, depending on the exact conditions
of the construction site and the loads exerted by the mooring lines. For rocky sea floor, as it is
expected for the region analyzed in this work, pre-bored pile anchors and gravity anchors are well
suited. However, the slope reduces the holding capacity of gravity anchors. Furthermore, the slope
is exposed to sediment movement from the higher regions to the lower regions [77, 98].
In this section, the structural capacity and holding capacity of pre-bored pile anchors is analyzed
according to the methodology described in section 9.1.4. Based on the dynamic simulations, the
design loads considering the individual load factors for each load case were calculated and compared
against the capacities.
The design loads (Fd) for analyzing the structural integrity of the anchor are computed according
to equation (3.1). In the case of the geotechnical design, the characteristic load was modified with
the safety factor and design factor as stated in equation (3.7), with both factors according to IEC
62600-10 [53]. The design loads corresponding to each load case are summarized in table 9.13. As
can be seen for this specific mooring arrangement, the anchor design is governed by the load case in
equilibrium conditions under no current (LC1). The maximum load in the aft center mooring line
is reduced under current. Considering the appropriate factors for the corresponding limit state, all
four design loads are superior of this load case in comparison to all other analyzed load cases.
Due to the lack of characterization of the soil in the Channel of Cozumel, characteristics of the
assumed limestone are taken from measurements on the mainland. Approximate relations between
different values as they can be found in literature. According to Santiago Sierra [94], the density
of the limestone was found to be 2110.2 kg m−3 with a uniaxial unconfined compressive strength of
27.73 MPa and a minimum Young’s modulus of 4.3185 GPa. The tensile strength of limestone was
estimated to 6.907 02MPa following equation (6.10), as published by Nazir et al. [80]. The rock
quality was assumed to be 50% with type I weathering. The strength reduction factor (αr) was
taken as 1, and the rock constant (rrm) as 5 × 10−5.
Considering these values, two pile anchors were designed, one with shear keys and one without.
For the steel, a Young’s modulus of 210 GPa, a density of 7850 kg m−3, and a yield stress of 235 MPa
was assumed. The used grout had a shear stress of 0.138 MPa and a compressive strength of
17.25 MPa. To satisfy the design requirements, the anchor without shear keys required a diameter
of 115 mm, a thickness of 13 mm, and an embedded length of 1.714 m. Considering shear keys
with a height of 2.4 mm and a pitch of 24.0 mm, the required diameter was reduced to 60 mm, the
thickness to 12 mm, and the required embedded length to 1.083 m.
Following the methodology described in section 9.1.4, the structural capacity in axial direction
of the anchor without shear keys was determined to Qsa = 889.956 kN, the structural capacity
in lateral direction was estimated to Qsl = 266.987 kN, the holding capacity in axial direction to
Qha = 85.505 kN, and the holding capacity in lateral direction to Qhl = 903.873 kN. For the anchor
150 CHAPTER 9. REVIEW OF FINAL DESIGN
Table 9.13: Loads for the design of a suitable pile anchor.
Structural Geotechnical
axial lateral axial lateral
LC1 44.060 kN 115.624 kN 84.856 kN 178.146 kN
LC2A 33.142 kN 86.146 kN 63.830 kN 132.729 kN
LC2C 32.814 kN 86.162 kN 63.198 kN 132.753 kN
LC2B 34.340 kN 88.476 kN 66.135 kN 136.318 kN
LC3A 33.420 kN 86.792 kN 64.365 kN 133.725 kN
LC3C 33.283 kN 88.808 kN 64.100 kN 136.830 kN
LC3B 33.890 kN 87.411 kN 65.270 kN 134.677 kN
LC4A 34.785 kN 90.482 kN 66.993 kN 139.410 kN
LC4C 33.990 kN 89.659 kN 65.463 kN 138.142 kN
LC4B 35.574 kN 91.781 kN 68.513 kN 141.411 kN
LC5A 20.582 kN 53.939 kN 40.136 kN 84.145 kN
LC5C 5.814 kN 14.939 kN 11.337 kN 23.305 kN
LC5B 24.713 kN 67.382 kN 48.190 kN 105.115 kN
LC6A 20.688 kN 54.302 kN 40.342 kN 84.711 kN
LC6C 33.833 kN 87.446 kN 65.974 kN 136.415 kN
LC6B 21.918 kN 57.227 kN 42.739 kN 89.274 kN
LC7A 21.291 kN 52.820 kN 41.518 kN 82.399 kN
LC7C 39.758 kN 86.206 kN 77.529 kN 134.481 kN
LC7B 9.174 kN 22.559 kN 17.889 kN 35.192 kN
LC8A 27.835 kN 70.054 kN 54.278 kN 109.285 kN
LC8C 11.903 kN 30.304 kN 23.211 kN 47.274 kN
LC8B 29.269 kN 74.349 kN 57.075 kN 115.984 kN
LC9 15.351 kN 16.192 kN 29.566 kN 24.948 kN
9.6. REQUIRED MOORING EQUIPMENT 151
with shear keys, the structural capacity in axial direction was calculated to Qsa = 386.587 kN, the
structural capacity in lateral direction to Qsl = 115.976 kN, the holding capacity in axial direction
to Qha = 161.083 kN, and the holding capacity in lateral direction to Qhl = 219.690 kN. It was
found, that for the anchor with shear keys, the holding capacity in axial direction was limited by
the structural capacity in lateral direction, while for the anchor without shear keys, the adhesion
to the grout was the limiting factor.
9.6 Required Mooring Equipment
Aside from the components analyzed above, additional equipment is required. To join the anchor
chain to the wire rope, one wire rope socket and a shackle are required. For connecting the forward
wire rope to the structure, a joining shackle, an anchor shackle, a swivel, and a wire rope socket are
required. At the lower end, the mooring chain requires an anchor shackle to be connected to the
anchor pile. Each shackle requires a matching cottor pin. The upper ends of the aft wire ropes are
directly connected to the mooring winch. To connect all three aft mooring lines a master link (or
sling link) is required together with six joining shackles, three swivels, and three wire rope sockets.
If no suitable master link is available, a triangular mooring plate can be used to which the joining
shackles are connected. Considering the maximum detected load and the required ASF, all mooring
components should be rated for 260 kN. To lower the platform to it’s design depth, the winches
must be able to store 50 m of wire rope and provide a minimum pull of 11 kN.

Chapter 10
Conclusions
A tool for quasi-static analysis of submerged mooring systems has been presented in chapter 4. It
is provided as a Python 3 package and makes use of the widely used numpy package for handling
arrays, matrices, and the corresponding mathematical operations. The system is limited to lines that
can be approximated by straight lines and the model is unable to estimate hydrodynamic forces. The
only forces considered by the model are hydrostatic buoyancy and weight of all objects, forces due to
linear elastic strain of the mooring lines. and forces explicitly stated by the user. The calculation is
based on three main types of objects. Rigid bodies have six DOFs, while points provide three DOFs.
Point objects are divided into fairleads, that are coupled to the corresponding rigid body, anchor
points that are fixed in space, and floats that can freely move in space. Mooring lines are represented
by line objects which have to be connected to a point object, and thus do not contribute to the
global displacement vector. The package has been verified with simple hand calculations to ensure
the correctness of the implementation. Furthermore, two laboratory experimental series have been
performed to gain knowledge on the error that can be expected. The experiments took place in the
open circulation tank in the Institute of Engineering of the UNAM. A buoy was moored with four
lines at approximately half the water depth and the magnitude of the current was altered during
the experiments. It was measured with a Vectrino just upstream of the buoy while the position
of the buoy was tracked using a high-speed camera and later analyzed with ImageJ/Trackmate.
While the vertical deflection of the buoy could not be reliably reproduced by the numerical model,
the deflection in the x-direction (i.e., in direction of the current) and the trim of the buoy (i.e.,
rotation around the horizontal axis perpendicular to the current) showed good agreement with the
experiments. The errors found are believed to be a result of the experimental setup and the quality
of the current in the circulation tank, as high deviations from the mean velocity and unsteady
disturbance of the surface were reported. Additionally, the estimation of the tension in the mooring
lines was validated with full-scale experiments and numerical predictions with SSMOOR available
in the literature. The mean tensions in the mooring line were predicted with a relative error of
-7.3% and +4% respectively. Despite the fact of the error for the deviation of the buoy in the
laboratory experiments, the low error regarding the mean tensions in the mooring line proves that
153
154 CHAPTER 10. CONCLUSIONS
the presented Python package for quasi-static analysis of submerged mooring systems is a valuable
tool for evaluating this kind of systems in early design phases. Especially the low computational
effort required to find a solution sets it apart from commercial software for dynamic mooring
analysis, enabling fast review and iteration of design concepts.
Additionally, a genetic algorithm was developed (see chapter 5) to be used in conjunction with
the quasi-static mooring analysis tool. The algorithm can optimize a given function for single or
multiple objectives, the latter by searching the Pareto-optimal front. The template of the genome
is defined in an XML file by giving parameters, their possible range, and their type. Supported
types are integers, floating numbers with a defined number of decimal points, and a list of which
the parameter is set to a specific item. Additionally to the template, a function for evaluating the
individuals has to be provided to the algorithm, returning the objectives and whether the individual
satisfies or not the requirements. The algorithm has been tested with the Griewank-function and
showed quick and efficient behavior in finding the global minimum.
The Chanel of Cozumel has been analyzed regarding extreme values of wind, waves, and the
current, which is described in chapter 6. The analysis of the wind and the waves is based on
reanalysis data ERA5 database provided by the ECMWF. The wind speeds have also been analyzed
based on the data available in the HURDAT2 database, maintained by the NHC of the United States
of America. The current has been based on the HYCOM model and was analyzed at the surface and
a depth of 50 m. The methodology for analyzing extreme wind speeds due to hurricanes was taken
from available literature, while extreme currents, waves, and wind were estimated based on the
POT approach and the GPD with automated threshold selection. The methodology was derived
from methods available in the literature, e.g., the threshold selection was not developed by the
author of this work. The adapted method was presented in a scientific article and can be found in
appendix C. Additionally, a summary of available data in the literature on different environmental
conditions such as sea level rise, geotechnical properties of the region, tides, and storm surge is
given. A significant discrepancy between the extreme wind speeds due to hurricanes and due to
the wind speed at a height of 10 m was found. The data suggest wind speeds of 41.37 m s−1 due
to a hurricane with a return period of 10 years, while the 10-year wind speed based on ERA5-data
was estimated to 15.64 m s−1. However, the GPD excess model showed good agreement with the
(numerical) observations as between 98.3% and 100% of all observations were found within the
corresponding 95% confidence interval. For wind and wave an unbound excess model was found,
while for the current at surface and a depth of 50 m a bound model was predicted.
The different design considerations for the platform have been documented and a design concept
was elaborated, its description can be found in chapter 7. This design concept was then optimized
with the genetic optimization algorithm, while the evaluation of the design proposals was carried out
with the Python package for quasi-static mooring analysis and Abaqus for the structural analysis,
as can be seen in chapter 8. An optimal design proposal was then selected from the set of optimized
proposals and manually fine-tuned with the help of a parametric study. This procedure was carried
out for two design concepts with the main difference in the usage of three anchors for the first
concept and four anchors for the second concept. Both concepts have been evaluated regarding
10.1. FUTURE RESEARCH 155
their mass, the obtained safety factor, the maximum angle of the platform, and the net buoyancy.
The final parametric study of the three-mooring line concept showed interesting behavior. For
example, a small increase in the out diameter of the hulls has only a small effect on the safety
factor and the maximum angle while the effect on mass and net buoyancy is nearly linear. The
angle of the strut showed a small range to obtain a high safety factor, while its influence on net
buoyancy and mass is negligible over the analyzed domain. The mooring footprint showed a clear
advantage of larger footprints regarding achieved safety factor, maximum angle, and net buoyancy.
The angle between the forward mooring lines and the center line showed only a significant effect
on the maximum angle, for which an optimum angle could be determined. The three-mooring line
concept was finally analyzed based on a dynamic mooring analysis performed with ProteusDS (see
chapter 9). The fatigue life of the mooring lines was estimated and two different pile anchor designs
suitable for weak rock were designed. Due to the slope and the sediment transport from higher
to lower regions found in the Cozumel Channel, dead weight and drag embedment anchors are
not suitable. Suction and gravity installed anchors are typically not the best choice for rocky sea
bed and their installation is challenging as the slope and the predominant currents may deflect the
anchor and alter its ultimate capacity. The recommended anchors are proprietary rock anchors,
or (micro-) pile anchors. Therefore, an improved methodology pile anchors suitable for anchoring
OCT in regions with the sea floor made up of soft rock was presented. This methodology is
based on available scientific literature and international design codes and guidelines. It was found,
that the load relevant for the design of the anchor is encountered during equilibrium conditions
without current in the aft center mooring line. Current from the front tends to reduce the load
on the mooring line, while the safety factors according to the relevant design codes increase the
characteristic design load to exceed the loads experienced during line loss. The mooring system,
the platform, and the design anchor show good agreement with the relevant, international design
codes.
Despite the shorter and therefore stiffer mooring lines on the port side of the platform, no
significant effect on the platform was detected during normal operation, which would require its
mitigation. Even under heavy loads from the front, the heel of the platform was not significantly
influenced. In contrast to the concept with four mooring lines, for the concept with three mooring
lines a much higher inclination during line loss was reported. Neither of the concepts would be
operable under such conditions, wherefore the acceptability of such deflections depends on other
factors such as the permissible strain of the power cable or the stresses in the remaining mooring
lines. Due to the reduced number of anchors, the three-mooring line concept seems favorable. This
however might be subject to change, as a different construction site, updated parameters of the
installed turbine, or requirements of a complete array have great influence.
10.1 Future Research
At the time of writing this work, no turbine ready for deployment is able to efficiently harness the
current in the Cozumel Channel. The design of the platform is based on values that can be expected
156 CHAPTER 10. CONCLUSIONS
from a turbine. However, these values may change for the final design of the prototype which
requires analysis if the assumptions are still valid and could lead to adjusting several parameters of
the platform, the mooring system, or the anchor.
The analysis tool for subsurface mooring systems was only validated with laboratory experimen-
tal data and a single full-scale experiment. Further effort could be made with different scales and a
more controlled environment, to gain further knowledge on the relative error of the model regarding
deflections, rotations, and the involved tensions. Furthermore, the model relies on straight mooring
lines, yet could easily be extended by catenary mooring lines considering the drag of the current
at different depths. Simple hydrostatic calculations based on the theory of small angles and the
metacentric height could also be introduced to extend the tool to bodies floating on the surface.
More advanced hydrostatic calculations including effects such as the Froude-Krylov force could be
based on polygonal meshes provided by STL files or wavefront obj-files. This would additionally
open up the possibility to estimate the hydrodynamic drag by coupling the analysis to a potential
flow solver and estimate the friction based on empirical formulae.
The design methodology for the pile anchor in weak rocks it was not validated with laboratory-
scale or full-scale experiments and the properties on the rock are based on literature for the region,
based on mostly land based specimen. Independently on the use case of the anchor, common
recommendation when designing piles for rocks is to verify the assumptions made during design
with full-scale tests. For future research, extensive research on anchor design is recommended. This
should include taking rock samples from the specific site, ideally taken from under water, to analyze
them regarding the key properties for the methodology such as Young’s modulus, the unconfined
compressive strength or the rock quality designation. The local stratification such as thickness and
composition of the sediment layer on top of the rock should also be determined. After evaluation
of the specimen in laboratory, full-scale pull-out tests are recommended for validating the design
methodology and the laboratory experiments. Due to the small expected dimensions of the anchors,
laboratory experiments could be replaced by full-scale, land-based test close to the construction site.
In the example, two tubular pile anchors were designed. Especially for the case which is limited
by the structural capacity, a solid pile anchor (sometimes also referred to as micropile) could be
considered to decrease the outer diameter and therefore the effort which has to be made during
pre-drilling the rock. Another approach to pile anchor installation in rock is to equip the tip of
the pile with a drill head. Such a design might decrease the time required for installation which is
favorable if the current presents a significant challenge to the installation of pile anchors.
The platform was designed to house an OCT with its axis parallel to the ocean current and a
diameter of 5 m. For smaller diameters, the platform might be scaled down to achieve a more effec-
tive material usage. Turbines with significantly larger diameter, however, require the platform to be
adjusted in its size. Not only due to the reduced clearance between rotor and hulls, but also due to
possibly changed weight and therefore weigth distribution. The Institute of Engineering is currently
developing a cross-flow turbine for Cozumel Channel, following the Gorlov design. To mount such
a turbine to the current concept of the platform, two turbines might be mounted to the platform at
the position a parallel-axis turbine would be placed. One turbine would be pointing upwards and
10.1. FUTURE RESEARCH 157
the other turbine would be pointing downwards in a cantilever configuration. Such a configuration
would introduce significant stress in the rotational axes of the turbines, though. A mechanically
more favorable design for cross-flow turbines might be to connect the hulls with two struts, an
upper and a lower strut. Between both struts, two counter-rotating turbines could be mounted
side by side, reducing the mechanical requirements on the shaft and increasing serviceability on the
turbines.

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Appendices
169

Appendix A
Quasi-Static Mooring Analysis
A.1 Example for Usage of OffshorePy
In the following, two simple examples are presented to show the basic usage of OffshorePy and
some of its key features.
A.1.1 Simple Mooring System
In the first example, a very simple mooring system is modeled consisting of a buoy moored via a
single mooring line. The buoy is assumed to be completely submerged at all times, initially placed
at a depth of 10 m. The water depth is 80 m and the water density 1025 kg m−3.
At first, the module hast to be loaded as well as pi for converting degrees into radians and
calculating the volume of the spherical buoy:
1 import offshorepy
2 from numpy import pi
The object storing the mooring system model is created, note that the water depth only serves the
purpose to draw the sea floor in the figures. The default gravitational acceleration is 9.81 m s2 and
can be set via the argument g in case a different acceleration is required.
1 qs_system = offshorepy . MooringSystem ( water_depth =80 , water_density =1025)
Wire rope with a nominal diameter of 15 mm is used as material for the mooring line, with the
material properties estimated as given in Ma et al. [68]. The method add_material_from_library
adds a materials with the properties according to Ma et al. [68] to the model and returns the
material object. The name is used for internal reference, while the material_type determines
which equations are used to estimate the properties.
1 wire = qs_system . add_material_from_library ( material_type =’wire ’, nominal_diameter
=0.015 , name=’wire 15 ’)
The spherical buoy has a diameter of 2 m and a density 0.5 times the of water. If not explicitly
given, the COG and the COB are assumed to be at the origin of the body. A rigid body object is
171
172 APPENDIX A. QUASI-STATIC MOORING ANALYSIS
created with its position 10 m below the water surface and no rotation. Volume and mass of a rigid
body are set via the according property.
1 >>> buoy = qs_system . add_rigid_body (name=’buoy ’, position =[0 ,0 , -10 ,0 ,0 ,0])
2 >>> buoy. volume = pi /6*2**3
3 >>> buoy.mass = 0.5* buoy. volume *1025
Lines need to be connected to two points. Therefore, the anchor point on the sea floor and the
fairlead on the buoy are created. Anchor points are fixed in space and do not move, while the
positions of fairleads are linked to the corresponding rigid body object. Fairleads are created by
using the rigid body object and anchor points directly via the mooring system object.
1 buoy. add_fairlead (name=’fairlead ’, rel_position =[0 ,0 , -1])
2 qs_system . add_anchor_point (name=’anchor ’, position =[0 ,0 , -80])
Adding a fairlead or anchor point to the system returns the newly created object. However, all
objects can also be retrieved later via their name:
1 fairlead = qs_system . get_object (’fairlead ’)
Next, a line is created spanning from the fairlead to the anchor point. To refer to the points either
their object or their name can be passed to the method. The same is true for the material. The
newly created line is returned.
1 qs_system . add_line (name=’cable ’, point0 =fairlead , point1 =’anchor ’, material =wire)
The system can now be solved:
1 qs_system . solve ()
This function return three values: The first values is the maximum error, the second value the
number of required iterations and the third value a list of all indices of the iterations for which
an empty stiffness matrix was found. The default behavior is to raise an error on encountering
an empty stiffness matrix. This can be avoided by passing the argument zero_matrix=’solve’.
This way, on encountering an empty stiffness matrix (i.e. due to slack mooring lines) all objects
are moved in the direction of their resulting force and moment vector. The system can be plotted
and stored to a file with matplotlib by:
1 qs_system .plot( save_plot_to =’example1 .png ’, show_tensions = True )
The output can be seen in figure A.1, with the mooring system shown in a side view (figure A.1 a)),
front view (figure A.1b)), and top view (figure A.1 c)). The option show_tensions plots the lines in
a color according their maximum tensions. The local coordinate system of rigid bodies is indicated
with red (positive x-direction), green (positive y-direction), and blue (positive z-direction) lines.
The direction perpendicular to the figure’s plane is additionally marked with a dot (towards the
viewer) or a cross (away from the viewer). Tensions in the lines and residual forces (i.e. external
forces) in all objects can be retrieved via the corresponding properties:
1 line = qs_system . get_object (’cable ’)
2 print (line. tension )
3 print (line. residual [0])
A.1. EXAMPLE FOR USAGE OF OFFSHOREPY 173
Figure A.1: Resulting mooring system of single line example.
The output would be:
1 (21059.666353338114 , 20392.091353338114)
2 [0. , 0., 21059.66635334]
In case of lines, the first returned value corresponds to point 0 and the second value to point 1,
which are in this case the fairlead and the anchor point, respectively. The position of the buoy can
be retrieved by the property position:
1 print (buoy. position )
This returns a numpy array with six entries, resulting in the following output:
1 [ 0., 0., -9.92949747 , 0., 0., 0.]
The first three entries correspond to the position in x-, y-, and z-direction and the last three entries
are the corresponding rotations in radians.
A.1.2 Multi-line Mooring System with Segmented Lines
The second example consists of a multi-line-mooring system with 12 segmented mooring lines. The
lower most 10 m of each mooring line are composed of 8 mm-chain while for the rest of the mooring
line 15 mm-wire rope is used. The basic model is created equal to the mooring system in example
one:
1 from numpy import pi , sin , cos , radians
2 import offshorepy
3
4 qs_system = offshorepy . MooringSystem ( water_depth =80 , water_density =1025)
The required materials are loaded from the material library and based on the properties given in
Ma et al. [68]:
1 wire = qs_system . add_material_from_library ( material_type =’wire ’, nominal_diameter
=0.015 , name=’wire 15 ’)
2 chain = qs_system . add_material_from_library ( material_type =’chain ’,
nominal_diameter =0.008 , name=’chain 8’)
174 APPENDIX A. QUASI-STATIC MOORING ANALYSIS
A spherical buoy with a diameter of 2 m and a density 0.5 times the of water is created. A force of
10 000N in positive x-direction is added to the body. The first three components are the components
of the force, the last three the components of the moment. If no position of the force is given, the
force is applied at the origin of the rigid body.
1 buoy = qs_system . add_rigid_body (name=’buoy ’, position =[0 ,0 , -15 ,0 ,0 ,0])
2 buoy. volume = pi /4*3**2*6
3 buoy.mass = 0.5* buoy. volume *1025
4 buoy. add_external_force ( force =[1e4 ,0 ,0 ,0 ,0 ,0])
The fairleads are created at the angles defined below. The corresponding anchor points are 60 m
radially outwards. When no name is given while creating points or lines, the objects are named
automatically according to their type and a consecutive number (e.g. Fairlead0 for the first fairlead
and AnchorPoint2 for the third anchor point). In total, 12 Fairlead-Anchor pairs are created:
1 angles = [25 , 30, 35, 145 , 150 , 155 , 205 , 210 , 215 , 325 , 330 , 335]
2 for a in angles :
3 buoy. add_fairlead ( rel_position =[1.5* cos( radians (a)) ,1.5* sin( radians (a)) , -1.5])
4 qs_system . add_anchor_point ( position =[60* cos( radians (a)) ,60* sin( radians (a)) , -80])
The fairleads created are then connected to their corresponding anchor point via a segmented
mooring line. To add a segmented line, a method of the MooringSystem-object is available. By
passing two materials and two unstretched lengths, a mooring line with two segments is created.
Since the first unstretched length is 0 and no strain is given, this segment’s length is adjusted to
the remaining length.
1 for i in range (len( angles )):
2 qs_system . add_segmented_line ( point0 =’Fairlead {} ’. format (i), point1 =’AnchorPoint
{} ’. format (i), materials =[ wire , chain ], unstretched_lengths =[0 , 10])
Finally the system can be solved and the result plotted:
1 qs_system . solve ()
2 _ = qs_system .plot( save_plot_to =’example02 .png ’, show_tensions = True )
The resulting mooring system can be seen in figure A.2. In the front view (figure A.2 b)), the port
mooring lines (i.e. right side) show a different tension than the starboard (i.e. left side) mooring
lines, despite the example being symmetric to the x-z plane. Since the mooring lines are plotted
in the order as they were created, the aft port mooring lines cover the forward port mooring lines.
On the starboard side the forward mooring lines cover the aft mooring lines. The correct tensions
can be seen in the top view (figure A.2 c)).
A.1. EXAMPLE FOR USAGE OF OFFSHOREPY 175
Figure A.2: Resulting mooring system of multi line example.
A.1.3 Python Code to Generate OffshorePy model for Verification
In the following the Python code to Generate offshorepy model for Verification is shown.
1 import numpy as np
2
3 import offshorepy
4
5 # DATA
6 water_depth = 80
7 water_density = 1025
8 water_viscosity = 1.0023e -6
9
10 buoy_diameter = 1
11 buoy_density = 200
12 buoy_depth = 40
13
14 unstretched_length = 39.5
15
16 current_speeds = [0.0 , 0.5 , 1, 1.5 , 2]
17
18 for current_speed in current_speeds :
19 # DRAG FORCE
20 Rn = current_speed * buoy_diameter / water_viscosity
21 if current_speed == 0:
22 Cd = 0
23 else :
24 Cd = 24/ Rn
25 Cd += (2.6* Rn /5.0) /(1+( Rn /5.0) **(1.52) )
26 Cd += (0.411*( Rn /263000) **( -7.94) ) /(1+( Rn /263000) **( -8.0))
27 Cd += (0.25* Rn /1 e6)/(1+ Rn /1 e6)
28 Fd = 0.5* Cd* water_density * current_speed **2 * np.pi /4* buoy_diameter **2
29 Fd = np. array ([1* np.sign( current_speed ) ,0 ,0])*Fd
30
31 # NUMERICAL MODEL
176 APPENDIX A. QUASI-STATIC MOORING ANALYSIS
32 qs_system = offshorepy . MooringSystem ( water_depth = water_depth , water_density =
water_density )
33 qs_system . add_material_from_library (’wire ’, 0.02 , name=’wire ’)
34
35 # Create rigid body at the desired depth with no rotation
36 buoy = qs_system . add_rigid_body (name=’buoy ’, position =[0,0,- buoy_depth ,0 ,0 ,0])
37 buoy. volume = np.pi /6* buoy_diameter **3
38 buoy.mass = buoy. volume * buoy_density
39 # Apply external force to buoy according to its drag force . The moment is 0 and
the force is applied at the righi body ’s origin
40 buoy. add_external_force (np. hstack ((Fd , [0 ,0 ,0])), [0 ,0 ,0])
41
42 buoy. add_fairlead (’fairlead ’, rel_position =[0,0,- buoy_diameter /2])
43 qs_system . add_anchor_point ([0,0,- water_depth ], name=’anchor ’)
44
45 line = qs_system . add_line (’fairlead ’, ’anchor ’, material =’wire ’,
unstretched_length = unstretched_length )
46
47 # Solve the system with a maximum of 100 iterations and solve issues with empty
stiffness matrix
48 try:
49 qs_system . solve ( maxIter =100 , zero_matrix =’solve ’)
50 except qsma. errors . EquilibriumError as e:
51 print (e)
52
53 # OUTPUT
54 print (’Current speed : {:.1f} ms -1 ’. format ( current_speed ))
55 print (’ Drag: {:.3e} N’. format (Fd))
56 print (’ Buoy , position : ’+( ’{:.2e}m ’*3+ ’{:.2e}rad ’*3). format (* buoy. position ))
57 print (’ Line , stretched length : {:.3e} N’. format (line. stretched_length ))
58 print (’ Line , strain : {:.3e} N’. format (line. strain ))
59 print (’ Line , tension at fairlead : {:.3e} N’. format (line. tension [0]))
A.2 Complementary Data for Experiments
In the following tables, complementary data for the experiments of the experimental series A are
presented.
A.2. COMPLEMENTARY DATA FOR EXPERIMENTS 177
Table A.1: Statistical data on the current measured during the experiment series A. Given is the
mean, the standard deviation (σ), the 2.5th (q2.5) and the 97.5th quantile (q97.5), and the mean
correlation as reported by the Vectrino. Units in m s−1.
Experiment Mean σ q2.5 q97.5 Correlation
A01F1_A 0.000 0.026 -0.054 0.051 91.1
A02F1_B 0.003 0.020 -0.041 0.042 95.1
A03F1_C 0.007 0.014 -0.023 0.034 95.9
A04F2_A 0.108 0.018 0.075 0.140 95.1
A05F2_B 0.102 0.011 0.079 0.125 95.2
A06F2_C 0.110 0.011 0.087 0.130 95.0
A07F3_A 0.209 0.016 0.175 0.238 94.8
A08F3_B 0.211 0.012 0.185 0.234 95.3
A09F3_C 0.214 0.015 0.183 0.242 94.7
A10F4_A 0.322 0.019 0.286 0.360 94.1
A11F4_B 0.317 0.020 0.278 0.356 93.7
A12F4_C 0.316 0.019 0.278 0.354 93.2
A13F5_A 0.417 0.028 0.362 0.476 93.1
A14F5_B 0.424 0.027 0.372 0.480 92.4
A15F5_C 0.420 0.024 0.372 0.467 91.8
A16F6_A 0.536 0.032 0.478 0.600 95.1
A17F6_B 0.538 0.030 0.483 0.597 94.9
A18F6_C 0.540 0.034 0.475 0.609 94.7
A19F7_A 0.637 0.036 0.563 0.701 95.9
A20F7_B 0.631 0.033 0.569 0.695 96.0
A21F7_C 0.633 0.034 0.563 0.697 95.8
178 APPENDIX A. QUASI-STATIC MOORING ANALYSIS
Table A.2: Signed relative error of experimental series A.
Experiment Speed x-Deflection y-Rotation
in m s−1 in m in °
A01F1_A 0.000 -1.000 1.803 . . . -3.803 -1.000 -17.816 . . . 15.816
A02F1_B 0.000 -1.000 18.298 . . . -20.298 -1.000 17.199 . . . -19.199
A03F1_C 0.000 -1.000 -3.447 . . . 1.447 -1.000 220.294 . . . -222.294
A04F2_A 0.108 0.877 1.444 . . . 0.310 -0.011 0.553 . . . -0.576
A05F2_B 0.102 1.386 2.195 . . . 0.577 -0.181 0.346 . . . -0.708
A06F2_C 0.110 3.500 4.818 . . . 2.182 -0.235 0.188 . . . -0.658
A07F3_A 0.209 0.113 0.245 . . . -0.020 0.280 0.551 . . . 0.009
A08F3_B 0.211 0.115 0.246 . . . -0.015 0.288 0.555 . . . 0.021
A09F3_C 0.214 0.123 0.250 . . . -0.005 0.297 0.557 . . . 0.036
A10F4_A 0.322 0.127 0.208 . . . 0.046 0.348 0.516 . . . 0.180
A11F4_B 0.317 0.079 0.159 . . . -0.001 0.236 0.395 . . . 0.078
A12F4_C 0.316 0.074 0.155 . . . -0.006 0.263 0.427 . . . 0.099
A13F5_A 0.417 -0.332 -0.307 . . . -0.358 0.458 0.597 . . . 0.319
A14F5_B 0.424 -0.328 -0.303 . . . -0.353 0.531 0.672 . . . 0.389
A15F5_C 0.419 -0.380 -0.356 . . . -0.403 0.436 0.571 . . . 0.300
A16F6_A 0.536 -0.435 -0.383 . . . -0.486 0.493 0.700 . . . 0.287
A17F6_B 0.538 -0.436 -0.385 . . . -0.487 0.522 0.730 . . . 0.313
A18F6_C 0.540 -0.447 -0.397 . . . -0.496 0.500 0.704 . . . 0.297
A19F7_A 0.636 -0.462 -0.372 . . . -0.552 0.450 0.784 . . . 0.116
A20F7_B 0.631 -0.473 -0.384 . . . -0.563 0.424 0.758 . . . 0.091
A21F7_C 0.633 -0.483 -0.395 . . . -0.570 0.436 0.770 . . . 0.101
A.2. COMPLEMENTARY DATA FOR EXPERIMENTS 179
Table A.3: Statistical data on the current measured during the experiment series B. Given is the
mean, the standard deviation (σ), the 2.5th (q2.5) and the 97.5th quantile (q97.5), and the mean
correlation as reported by the Vectrino. Units in m s−1.
Experiment Mean σ q2.5 q97.5 Correlation
B01F1_A 0.008 0.036 -0.070 0.083 67.5
B02F1_B 0.010 0.030 -0.050 0.077 69.5
B03F1_C 0.002 0.039 -0.082 0.085 52.8
B04F2_A 0.055 0.009 0.035 0.075 81.7
B05F2_B 0.066 0.021 0.028 0.112 57.8
B06F2_C 0.075 0.022 0.035 0.126 55.5
B07F3_A 0.115 0.014 0.090 0.143 64.4
B08F3_B 0.125 0.015 0.098 0.159 64.0
B09F3_C 0.121 0.014 0.096 0.153 64.6
B10F4_A 0.174 0.014 0.147 0.204 71.7
B11F4_B 0.176 0.013 0.151 0.204 70.6
B12F4_C 0.176 0.015 0.147 0.205 72.7
B13F5_A 0.218 0.014 0.190 0.245 78.6
B14F5_B 0.220 0.014 0.192 0.249 78.0
B15F5_C 0.216 0.015 0.188 0.249 79.1
B16F6_A 0.271 0.016 0.239 0.302 81.3
B17F6_B 0.276 0.016 0.244 0.308 80.5
B18F6_C 0.271 0.017 0.237 0.303 80.0
B19F7_A 0.328 0.017 0.292 0.362 84.1
B20F7_B 0.322 0.019 0.282 0.356 83.6
B21F7_C 0.317 0.020 0.278 0.356 83.6
B22F8_A 0.378 0.021 0.336 0.417 86.3
B23F8_B 0.376 0.022 0.333 0.417 86.4
B24F8_C 0.381 0.023 0.337 0.429 87.9
180 APPENDIX A. QUASI-STATIC MOORING ANALYSIS
Table A.4: Signed relative error of experimental series B.
Experiment Speed x-Deflection y-Rotation
in m s−1 in m in °
B01F1_A 0.000 -1.00 -3.80 . . . 1.80 -1.00 -10.10 . . . 8.10
B02F1_B 0.000 -1.03 3.93 . . . -5.99 -0.99 -6.90 . . . 4.92
B03F1_C 0.000 -1.09 5.31 . . . -7.50 -1.02 2.57 . . . -4.60
B04F2_A 0.055 -1.00 -3.81 . . . 1.82 -1.00 -8.88 . . . 6.89
B05F2_B 0.067 0.09 -4.32 . . . 4.51 -0.65 -4.90 . . . 3.60
B06F2_C 0.076 -0.35 -2.13 . . . 1.43 -1.14 0.01 . . . -2.29
B07F3_A 0.115 -0.84 -1.18 . . . -0.51 -0.48 -3.83 . . . 2.88
B08F3_B 0.126 -0.68 -0.98 . . . -0.37 0.04 -2.91 . . . 2.99
B09F3_C 0.121 -0.62 -0.92 . . . -0.33 0.87 -3.55 . . . 5.29
B10F4_A 0.174 -0.75 -0.96 . . . -0.53 0.52 -3.35 . . . 4.38
B11F4_B 0.176 -0.49 -0.70 . . . -0.28 2.40 -1.83 . . . 6.64
B12F4_C 0.176 -0.46 -0.68 . . . -0.24 0.86 -1.39 . . . 3.11
B13F5_A 0.218 -0.66 -0.80 . . . -0.51 -0.53 -1.10 . . . 0.04
B14F5_B 0.220 -0.51 -0.64 . . . -0.37 -0.28 -0.85 . . . 0.29
B15F5_C 0.216 -0.48 -0.62 . . . -0.35 0.07 -0.76 . . . 0.90
B16F6_A 0.271 -0.66 -0.75 . . . -0.56 -0.35 -0.87 . . . 0.18
B17F6_B 0.276 -0.48 -0.57 . . . -0.39 -0.05 -0.54 . . . 0.43
B18F6_C 0.271 -0.45 -0.54 . . . -0.36 0.01 -0.49 . . . 0.52
B19F7_A 0.328 -0.63 -0.69 . . . -0.56 -0.40 -0.71 . . . -0.09
B20F7_B 0.322 -0.43 -0.50 . . . -0.37 -0.04 -0.37 . . . 0.30
B21F7_C 0.317 -0.46 -0.53 . . . -0.40 -0.06 -0.40 . . . 0.28
Appendix B
Dimensions of structural members
In the following, the complete list of structural members which was used during optimization of the
structure is shown. The list includes all 542 possible profiles, ordered by outer diameter and wall
thickness and with all dimensions given in mm.
Table B.1: Diameter and thickness range of structural members.
i Profile i Profile i Profile i Profile
0 30.0 x 2.6 1 30.0 x 2.9 2 30.0 x 3.2 3 30.0 x 3.6
4 30.0 x 4.0 5 30.0 x 4.5 6 30.0 x 5.0 7 30.0 x 5.6
8 30.0 x 6.3 9 30.0 x 7.1 10 30.0 x 8.0 11 30.0 x 8.8
12 30.0 x 10.0 13 31.8 x 2.6 14 31.8 x 2.9 15 31.8 x 3.2
16 31.8 x 3.6 17 31.8 x 4.0 18 31.8 x 4.5 19 31.8 x 5.0
20 31.8 x 5.6 21 31.8 x 6.3 22 31.8 x 7.1 23 31.8 x 8.0
24 31.8 x 8.8 25 31.8 x 10.0 26 33.7 x 2.6 27 33.7 x 2.9
28 33.7 x 3.2 29 33.7 x 3.6 30 33.7 x 4.0 31 33.7 x 4.5
32 33.7 x 5.0 33 33.7 x 5.6 34 33.7 x 6.3 35 33.7 x 7.1
36 33.7 x 8.0 37 33.7 x 8.8 38 33.7 x 10.0 39 33.7 x 11.0
40 38.0 x 2.6 41 38.0 x 2.9 42 38.0 x 3.2 43 38.0 x 3.6
44 38.0 x 4.0 45 38.0 x 4.5 46 38.0 x 5.0 47 38.0 x 5.6
48 38.0 x 6.3 49 38.0 x 7.1 50 38.0 x 8.0 51 38.0 x 8.8
52 38.0 x 10.0 53 38.0 x 11.0 54 38.0 x 12.5 55 42.4 x 2.6
56 42.4 x 2.9 57 42.4 x 3.2 58 42.4 x 3.6 59 42.4 x 4.0
60 42.4 x 4.5 61 42.4 x 5.0 62 42.4 x 5.6 63 42.4 x 6.3
64 42.4 x 7.1 65 42.4 x 8.0 66 42.4 x 8.8 67 42.4 x 10.0
68 42.4 x 11.0 69 42.4 x 12.5 70 42.4 x 14.2 71 44.5 x 2.6
72 44.5 x 2.9 73 44.5 x 3.2 74 44.5 x 3.6 75 44.5 x 4.0
76 44.5 x 4.5 77 44.5 x 5.0 78 44.5 x 5.6 79 44.5 x 6.3
continued on next page
181
182 APPENDIX B. DIMENSIONS OF STRUCTURAL MEMBERS
Table B.1: Diameter and thickness range of structural members.
i Profile i Profile i Profile i Profile
80 44.5 x 7.1 81 44.5 x 8.0 82 44.5 x 8.8 83 44.5 x 10.0
84 44.5 x 11.0 85 44.5 x 12.5 86 44.5 x 14.2 87 48.3 x 2.6
88 48.3 x 2.9 89 48.3 x 3.2 90 48.3 x 3.6 91 48.3 x 4.0
92 48.3 x 4.5 93 48.3 x 5.0 94 48.3 x 5.6 95 48.3 x 6.3
96 48.3 x 7.1 97 48.3 x 8.0 98 48.3 x 8.8 99 48.3 x 10.0
100 48.3 x 11.0 101 48.3 x 12.5 102 48.3 x 14.2 103 48.3 x 16.0
104 51.0 x 2.6 105 51.0 x 2.9 106 51.0 x 3.2 107 51.0 x 3.6
108 51.0 x 4.0 109 51.0 x 4.5 110 51.0 x 5.0 111 51.0 x 5.6
112 51.0 x 6.3 113 51.0 x 7.1 114 51.0 x 8.0 115 51.0 x 8.8
116 51.0 x 10.0 117 51.0 x 11.0 118 51.0 x 12.5 119 51.0 x 14.2
120 51.0 x 16.0 121 54.0 x 2.6 122 54.0 x 2.9 123 54.0 x 3.2
124 54.0 x 3.6 125 54.0 x 4.0 126 54.0 x 4.5 127 54.0 x 5.0
128 54.0 x 5.6 129 54.0 x 6.3 130 54.0 x 7.1 131 54.0 x 8.0
132 54.0 x 8.8 133 54.0 x 10.0 134 54.0 x 11.0 135 54.0 x 12.5
136 54.0 x 14.2 137 54.0 x 16.0 138 57.0 x 2.9 139 57.0 x 3.2
140 57.0 x 3.6 141 57.0 x 4.0 142 57.0 x 4.5 143 57.0 x 5.0
144 57.0 x 5.6 145 57.0 x 6.3 146 57.0 x 7.1 147 57.0 x 8.0
148 57.0 x 8.8 149 57.0 x 10.0 150 57.0 x 11.0 151 57.0 x 12.5
152 57.0 x 14.2 153 57.0 x 16.0 154 60.3 x 2.9 155 60.3 x 3.2
156 60.3 x 3.6 157 60.3 x 4.0 158 60.3 x 4.5 159 60.3 x 5.0
160 60.3 x 5.6 161 60.3 x 6.3 162 60.3 x 7.1 163 60.3 x 8.0
164 60.3 x 8.8 165 60.3 x 10.0 166 60.3 x 11.0 167 60.3 x 12.5
168 60.3 x 14.2 169 60.3 x 16.0 170 60.3 x 17.5 171 63.5 x 2.9
172 63.5 x 3.2 173 63.5 x 3.6 174 63.5 x 4.0 175 63.5 x 4.5
176 63.5 x 5.0 177 63.5 x 5.6 178 63.5 x 6.3 179 63.5 x 7.1
180 63.5 x 8.0 181 63.5 x 8.8 182 63.5 x 10.0 183 63.5 x 11.0
184 63.5 x 12.5 185 63.5 x 14.2 186 63.5 x 16.0 187 63.5 x 17.5
188 70.0 x 2.9 189 70.0 x 3.2 190 70.0 x 3.6 191 70.0 x 4.0
192 70.0 x 4.5 193 70.0 x 5.0 194 70.0 x 5.6 195 70.0 x 6.3
196 70.0 x 7.1 197 70.0 x 8.0 198 70.0 x 8.8 199 70.0 x 10.0
200 70.0 x 11.0 201 70.0 x 12.5 202 70.0 x 14.2 203 70.0 x 16.0
204 70.0 x 17.5 205 76.1 x 2.9 206 76.1 x 3.2 207 76.1 x 3.6
208 76.1 x 4.0 209 76.1 x 4.5 210 76.1 x 5.0 211 76.1 x 5.6
212 76.1 x 6.3 213 76.1 x 7.1 214 76.1 x 8.0 215 76.1 x 8.8
216 76.1 x 10.0 217 76.1 x 11.0 218 76.1 x 12.5 219 76.1 x 14.2
220 76.1 x 16.0 221 76.1 x 17.5 222 76.1 x 20.0 223 82.5 x 3.2
continued on next page
183
Table B.1: Diameter and thickness range of structural members.
i Profile i Profile i Profile i Profile
224 82.5 x 3.6 225 82.5 x 4.0 226 82.5 x 4.5 227 82.5 x 5.0
228 82.5 x 5.6 229 82.5 x 6.3 230 82.5 x 7.1 231 82.5 x 8.0
232 82.5 x 8.8 233 82.5 x 10.0 234 82.5 x 11.0 235 82.5 x 12.5
236 82.5 x 14.2 237 82.5 x 16.0 238 82.5 x 17.5 239 82.5 x 20.0
240 82.5 x 22.2 241 82.5 x 25.0 242 88.9 x 3.2 243 88.9 x 3.6
244 88.9 x 4.0 245 88.9 x 4.5 246 88.9 x 5.0 247 88.9 x 5.6
248 88.9 x 6.3 249 88.9 x 7.1 250 88.9 x 8.0 251 88.9 x 8.8
252 88.9 x 10.0 253 88.9 x 11.0 254 88.9 x 12.5 255 88.9 x 14.2
256 88.9 x 16.0 257 88.9 x 17.5 258 88.9 x 20.0 259 88.9 x 22.2
260 88.9 x 25.0 261 95.0 x 3.6 262 95.0 x 4.0 263 95.0 x 4.5
264 95.0 x 5.0 265 95.0 x 5.6 266 95.0 x 6.3 267 95.0 x 7.1
268 95.0 x 8.0 269 95.0 x 8.8 270 95.0 x 10.0 271 95.0 x 11.0
272 95.0 x 12.5 273 95.0 x 14.2 274 95.0 x 16.0 275 95.0 x 17.5
276 95.0 x 20.0 277 95.0 x 22.2 278 95.0 x 25.0 279 95.0 x 28.0
280 101.6 x 3.6 281 101.6 x 4.0 282 101.6 x 4.5 283 101.6 x 5.0
284 101.6 x 5.6 285 101.6 x 6.3 286 101.6 x 7.1 287 101.6 x 8.0
288 101.6 x 8.8 289 101.6 x 10.0 290 101.6 x 11.0 291 101.6 x 12.5
292 101.6 x 14.2 293 101.6 x 16.0 294 101.6 x 17.5 295 101.6 x 20.0
296 101.6 x 22.2 297 101.6 x 25.0 298 101.6 x 28.0 299 108.0 x 3.6
300 108.0 x 4.0 301 108.0 x 4.5 302 108.0 x 5.0 303 108.0 x 5.6
304 108.0 x 6.3 305 108.0 x 7.1 306 108.0 x 8.0 307 108.0 x 8.8
308 108.0 x 10.0 309 108.0 x 11.0 310 108.0 x 12.5 311 108.0 x 14.2
312 108.0 x 16.0 313 108.0 x 17.5 314 108.0 x 20.0 315 108.0 x 22.2
316 108.0 x 25.0 317 108.0 x 28.0 318 108.0 x 30.0 319 114.3 x 3.6
320 114.3 x 4.0 321 114.3 x 4.5 322 114.3 x 5.0 323 114.3 x 5.6
324 114.3 x 6.3 325 114.3 x 7.1 326 114.3 x 8.0 327 114.3 x 8.8
328 114.3 x 10.0 329 114.3 x 11.0 330 114.3 x 12.5 331 114.3 x 14.2
332 114.3 x 16.0 333 114.3 x 17.5 334 114.3 x 20.0 335 114.3 x 22.2
336 114.3 x 25.0 337 114.3 x 28.0 338 114.3 x 30.0 339 114.3 x 32.0
340 121.0 x 4.0 341 121.0 x 4.5 342 121.0 x 5.0 343 121.0 x 5.6
344 121.0 x 6.3 345 121.0 x 7.1 346 121.0 x 8.0 347 121.0 x 8.8
348 121.0 x 10.0 349 121.0 x 11.0 350 121.0 x 12.5 351 121.0 x 14.2
352 121.0 x 16.0 353 121.0 x 17.5 354 121.0 x 20.0 355 121.0 x 22.2
356 121.0 x 25.0 357 121.0 x 28.0 358 121.0 x 30.0 359 121.0 x 32.0
360 127.0 x 4.0 361 127.0 x 4.5 362 127.0 x 5.0 363 127.0 x 5.6
364 127.0 x 6.3 365 127.0 x 7.1 366 127.0 x 8.0 367 127.0 x 8.8
continued on next page
184 APPENDIX B. DIMENSIONS OF STRUCTURAL MEMBERS
Table B.1: Diameter and thickness range of structural members.
i Profile i Profile i Profile i Profile
368 127.0 x 10.0 369 127.0 x 11.0 370 127.0 x 12.5 371 127.0 x 14.2
372 127.0 x 16.0 373 127.0 x 17.5 374 127.0 x 20.0 375 127.0 x 22.2
376 127.0 x 25.0 377 127.0 x 28.0 378 127.0 x 30.0 379 127.0 x 32.0
380 127.0 x 36.0 381 133.0 x 4.0 382 133.0 x 4.5 383 133.0 x 5.0
384 133.0 x 5.6 385 133.0 x 6.3 386 133.0 x 7.1 387 133.0 x 8.0
388 133.0 x 8.8 389 133.0 x 10.0 390 133.0 x 11.0 391 133.0 x 12.5
392 133.0 x 14.2 393 133.0 x 16.0 394 133.0 x 17.5 395 133.0 x 20.0
396 133.0 x 22.2 397 133.0 x 25.0 398 133.0 x 28.0 399 133.0 x 30.0
400 133.0 x 32.0 401 133.0 x 36.0 402 133.0 x 40.0 403 139.7 x 4.0
404 139.7 x 4.5 405 139.7 x 5.0 406 139.7 x 5.6 407 139.7 x 6.3
408 139.7 x 7.1 409 139.7 x 8.0 410 139.7 x 8.8 411 139.7 x 10.0
412 139.7 x 11.0 413 139.7 x 12.5 414 139.7 x 14.2 415 139.7 x 16.0
416 139.7 x 17.5 417 139.7 x 20.0 418 139.7 x 22.2 419 139.7 x 25.0
420 139.7 x 28.0 421 139.7 x 30.0 422 139.7 x 32.0 423 139.7 x 36.0
424 139.7 x 40.0 425 139.7 x 45.0 426 146.0 x 4.5 427 146.0 x 5.0
428 146.0 x 5.6 429 146.0 x 6.3 430 146.0 x 7.1 431 146.0 x 8.0
432 146.0 x 8.8 433 146.0 x 10.0 434 146.0 x 11.0 435 146.0 x 12.5
436 146.0 x 14.2 437 146.0 x 16.0 438 146.0 x 17.5 439 146.0 x 20.0
440 146.0 x 22.2 441 146.0 x 25.0 442 146.0 x 28.0 443 146.0 x 30.0
444 146.0 x 32.0 445 146.0 x 36.0 446 146.0 x 40.0 447 146.0 x 45.0
448 152.4 x 4.5 449 152.4 x 5.0 450 152.4 x 5.6 451 152.4 x 6.3
452 152.4 x 7.1 453 152.4 x 8.0 454 152.4 x 8.8 455 152.4 x 10.0
456 152.4 x 11.0 457 152.4 x 12.5 458 152.4 x 14.2 459 152.4 x 16.0
460 152.4 x 17.5 461 152.4 x 20.0 462 152.4 x 22.2 463 152.4 x 25.0
464 152.4 x 28.0 465 152.4 x 30.0 466 152.4 x 32.0 467 152.4 x 36.0
468 152.4 x 40.0 469 152.4 x 45.0 470 159.0 x 4.5 471 159.0 x 5.0
472 159.0 x 5.6 473 159.0 x 6.3 474 159.0 x 7.1 475 159.0 x 8.0
476 159.0 x 8.8 477 159.0 x 10.0 478 159.0 x 11.0 479 159.0 x 12.5
480 159.0 x 14.2 481 159.0 x 16.0 482 159.0 x 17.5 483 159.0 x 20.0
484 159.0 x 22.2 485 159.0 x 25.0 486 159.0 x 28.0 487 159.0 x 30.0
488 159.0 x 32.0 489 159.0 x 36.0 490 159.0 x 40.0 491 159.0 x 45.0
492 159.0 x 50.0 493 159.0 x 55.0 494 165.1 x 4.5 495 165.1 x 5.0
496 165.1 x 5.6 497 165.1 x 6.3 498 165.1 x 7.1 499 165.1 x 8.0
500 165.1 x 8.8 501 165.1 x 10.0 502 165.1 x 11.0 503 165.1 x 12.5
504 165.1 x 14.2 505 165.1 x 16.0 506 165.1 x 17.5 507 165.1 x 20.0
508 165.1 x 22.2 509 165.1 x 25.0 510 165.1 x 28.0 511 165.1 x 30.0
continued on next page
185
Table B.1: Diameter and thickness range of structural members.
i Profile i Profile i Profile i Profile
512 165.1 x 32.0 513 165.1 x 36.0 514 165.1 x 40.0 515 165.1 x 45.0
516 165.1 x 50.0 517 165.1 x 55.0 518 168.3 x 4.5 519 168.3 x 5.0
520 168.3 x 5.6 521 168.3 x 6.3 522 168.3 x 7.1 523 168.3 x 8.0
524 168.3 x 8.8 525 168.3 x 10.0 526 168.3 x 11.0 527 168.3 x 12.5
528 168.3 x 14.2 529 168.3 x 16.0 530 168.3 x 17.5 531 168.3 x 20.0
532 168.3 x 22.2 533 168.3 x 25.0 534 168.3 x 28.0 535 168.3 x 30.0
536 168.3 x 32.0 537 168.3 x 36.0 538 168.3 x 40.0 539 168.3 x 45.0
540 168.3 x 50.0 541 168.3 x 55.0

Appendix C
Publication in Frontiers in Marine
Science
187
Extreme Value Analysis of Ocean
Currents in the Mexican Caribbean
Based on HYCOM Numerical
Model Data
Michael Ring*, Paola Elizabeth Rodrı́guez-Ocampo, Rodolfo Silva and Edgar Mendoza*
Institute of Engineering, National Autonomous University of Mexico, Mexico City, Mexico
Comprehensive knowledge of extreme values is required for designing offshore structures
and ocean current turbines. However, data on the return levels of ocean currents are rarely
available. This is the case for the Mexican Caribbean, where enormous energy potential in
the ocean currents has recently been detected. In this study, long-term numerical data
from the Hybrid Coordinate Ocean Model for a depth of 50m was adjusted via linear
quantile regression to short-term empirical data for a depth of 49m. The error of the results
was estimated using simplified extreme value analysis. Based on the numerical data, a
comprehensive extreme value analysis was conducted using the peaks over threshold
method and fitting a Generalized Pareto Distribution to the data. This method relies on
filtering peaks with a moving time window and an automated threshold selection based on
a reparameterised scale parameter of the Generalized Pareto Distribution. The adjusted
numerical model is shown to underestimate the empirical data with the error converging to
almost 22% for rare events (return period > 10years). The method showed consistent
results in the domain, with some anomalies only at the boundaries of the underlying
numerical model. The methodology is suitable for estimating the return levels of ocean
currents provided by HYCOM, although further research is needed to reduce the error of
the numerical model.
Keywords: ocean current, return level, extreme value analysis, peaks over threshold, generalized pareto
distribution, Caribbean Sea, HYCOM
1 INTRODUCTION
In recent years, many projects have sought to harvest ocean energy from tidal currents. For instance,
in early 2021, Sustainable Marine launched the Pempa’q Instream Tidal Energy project to harvest the
tidal energy, using a 420 kW PLAT-I 6.4 platform, in the Bay of Fundy, Nova Scotia, Canada
(Sustainable Marine, 2021). Similarly, Orbital Marine Power launched their O2 platform in the
north of Scotland, UK. This platform has two turbines each with a diameter of 20m and a rated
Abbreviations: GPD, Generalized Pareto Distribution; HYCOM, Hybrid Coordinate Ocean Model; POT, Peaks over
threshold; CI95%, 95% confidence interval.
Frontiers in Marine Science | www.frontiersin.org June 2022 | Volume 9 | Article 8668741
Edited by:
Alvise Benetazzo,
Institute of Marine Science (CNR), Italy
Reviewed by:
Oyvind Breivik,
Norwegian Meteorological Institute,
Norway
Antonio Ricchi,
University of L’Aquila, Italy
*Correspondence:
Edgar Mendoza
EMendozaB@iingen.unam.mx
Michael Ring
MRing@iingen.unam.mx
Specialty section:
This article was submitted to
Physical Oceanography,
a section of the journal
Frontiers in Marine Science
Received: 31 January 2022
Accepted: 25 April 2022
Published: 13 June 2022
Citation:
Ring M, Rodrı́guez-Ocampo PE,
Silva R and Mendoza E (2022) Extreme
Value Analysis of Ocean Currents in
the Mexican Caribbean Based on
HYCOM Numerical Model Data.
Front. Mar. Sci. 9:866874.
doi: 10.3389/fmars.2022.866874
ORIGINAL RESEARCH
published: 13 June 2022
doi: 10.3389/fmars.2022.866874
power of 1 MW (Orbital Marine Power Ltd, 2021). The
successful deployment of platforms for such technologies
requires currents that are typically avoided by other industries
because they are too intensive. These technologies require
currents that are often too strong to be harnessed for other
uses. Consequently, there is limited knowledge about the exact
environmental conditions near the currents.
Fan et al. (2010) studied the currents obtained from the
Hybrid Coordinate Ocean Model (HYCOM) in the Gulf of
Mexico, and compared them against field measurements for
the same area. Their results show inconsistencies for low-
frequency motions, such as eddies, in the numerical model.
The model also tends to overestimate deeper currents during
loop current eddy events. Cetina et al. (2006) studied current
circulation in the same area, finding that the direction of the
currents may reverse for several weeks, mainly due to passing
eddies within the main current stream, near Chinchorro Bank,
south of Cozumel Island, in the Caribbean Sea. Other studies on
subinertial flows have been carried out using short-term
measurements, to characterize the currents at this site (Chávez
et al., 2003), the fluctuations of the current (Ochoa et al., 2005),
and their variability (Abascal et al., 2003). The strong correlation
between the flow through the Cozumel Channel and that at the
centre of the Yucatan Channel was found by Athie et al. (2011),
who compared simultaneous measurements in both channels
over 8 months in 2000 and 2001. The tidal currents in the
Yucatan Channel, which separates the northern tip of the
Yucatan Peninsula and the west coast of Cuba, were studied by
Carrillo González et al. (2007). From their measurements, they
found that the amplitude of the diurnal components of the tide is
about ten times greater than the semi-diurnal components.
In relation to the modelling and characterisation of currents
several relevant researches have been published. Jonathan and
Ewans (2013) reviewed the behaviour of extreme value modelling
for the characterization of ocean environments for the design of
marine structures. They summarized basic concepts and
modelling with covariates and multivariates. Extreme ocean
currents in the north west Atlantic were analysed by Oliver
et al. (2012), based on numerical data, using a Monte-Carlo
simulation for the integration of tidal and non tidal currents.
Standard statistical methods for extreme values were extended to
handle the temporal dependence, directionality, and tidal non-
stationarity of ocean current extremes, by Robinson and Tawn
(1997). They found that the tidal current and directionality in
non-extreme surge currents explain the strong directionality in
the speed of extreme ocean currents. Devis-Morales et al. (2017)
analysed extreme wind and wave events in the Caribbean,
applying the block model, peaks over threshold (POT) method,
and the individual storms method, to obtain estimates of extreme
values for the Colombian Caribbean coast.
Moeini et al. (2010) compared the quality of two sources of
surface winds for wave modelling in the Persian Gulf. They used
measurements of the wind and wind data generated by the
climatological model of the European Center for Medium
Range Weather Forecasts as data input for the wave model.
The waves were simulated with the SWANmodel (for Simulating
Waves Nearshore) and compared to empirical wave data
measured 20 km away from the meteorological station which
recorded the wind data. They performed extreme value analysis
(EVA) based on the measured and modelled wave data and
found that the wave data generated with the empirical wind data
matched the empirical wave data much better than the wave data
generated with the modelled wind. Niroomandi et al. (2018)
simulated waves in Chesapeake Bay and validated the results
with measurements. They performed an EVA comparing
generalized extreme value function and Generalized Pareto
Distribution (GPD). They also studied the effect of key
parameters, including threshold value, time span and data
length on the design wave heights. Park et al. (2020) used EVA
to obtain the return levels for wave, wind and currents for the
Barents Sea. Their analysis is based on hindcast data generated
with the Global Reanalysis of Ocean Waves 2012 model. They
based their EVA on the Gumbel distribution, and the 2- and 3-
parameter Weibull distribution and ultimately suggest using the
Weibull distribution for the wind speed and current speed.
Viselli et al. (2015) calculated extreme wind and waves in the
Gulf of Maine, USA, by applying the POT method with short
block lengths of 4 to 8 days to ensure the peaks were
independent. For each block, only the maximum peak was
selected, which also had to be over half a block length after the
previously selected peak. This method was adapted from Simiu
(2011) and aims to avoid serially related peaks. Liu et al. (2018)
used the average conditional exceedance rate method to estimate
extreme current speeds with multi-year return periods, based on
data obtained from a platform in the South China Sea. Bore et al.
(2019) used a marginal model to determine the statistical
extremes of current speed, by evaluating the signal in
deterministic and stochastic components. Qi and Shi (2009)
used the three-parameter Weibull distribution to estimate the
distribution of extreme winds, waves, and currents, using data
from 30-year hindcasts to which the Weibull distribution
was fitted.
Thompson et al. (2009) introduced a methodology for
automatic threshold selection based on statistical parameters as
described in Coles (2001). Their method was applied to extreme
wave height by increasing the threshold from the 50th percentile
upward, until a specific condition was satisfied. Similarly, Solari
et al. (2017) presented a methodology for automatic threshold
selection, defining possible thresholds by a list of peaks within a
moving time window. The parameters of a GPD are calculated
for each set of peaks, defined by threshold and the moving time
window. For each GPD the p-value was estimated employing the
right-tail weighted Anderson-Darling test. The threshold, which
minimizes one minus the p-value for the specific threshold, was
selected while its uncertainty is estimated using the bootstrap
technique. Liang et al. (2019) selected possible thresholds which
are uniformly distributed in the upper half of the data. For each
threshold, a GPD is fitted to the data, and the differences in
return periods values with increasing thresholds are plotted. A
stable region for the return period with an increasing threshold
indicates independence from the threshold, and the lower bound
of the area is selected as the final threshold. Coles and Simiu
Ring et al. Mexican Caribbean: Extreme Ocean Currents
Frontiers in Marine Science | www.frontiersin.org June 2022 | Volume 9 | Article 8668742
(2003) proposed the use of resampling schemes to measure
uncertainties caused by the relatively short length of the
numerical data of hurricane extreme values. They adapted a
bootstrap method and used empirical corrections to adjust the
bias in the distributions obtained. Morton and Bowers (1996)
studied the multivariate point process model in extreme value
analyses. As an example, they used a moored semi-submersible
and its response to wind and waves (i.e., bivariate analysis) and
estimated the 50-year mooring force and return period contours
for a 50-year combined wind-wave condition.
The Cozumel Channel in the Mexican Caribbean Sea has
significant potential for the harnessing of ocean currents
(Hernández-Fontes et al., 2019; Bárcenas Graniel et al., 2021).
The predominant current direction is in north east direction,
especially for the higher current speeds. Since the currents in this
region are mainly caused by global ocean currents, the direction
rarely changes (Alcérreca-Huerta et al., 2019). When it does, it is
usually caused by eddies within the ocean current that result in a
relatively low flow in the opposite direction. South east of
Cozumel Island the mean current speed was determined to be
0.9ms-1 with a standard deviation of 0.2ms-1. In the wake of the
Cozumel Channel the mean current speed was measured as
1.3ms−1 with a standard deviation of 0.3ms−1 (Cetina et al.,
2006). The oceanic climate, the biodiversity and intensive
tourism are the main reason why this region is unattractive for
conventional marine structures, however it is an area with great
potential for harvesting energy from ocean currents.
Long-term data, for at least 20 years (Devis-Morales et al.,
2017) are necessary to correctly design offshore structures that
take into consideration extreme events. The current
measurements are available for a depth of 49m in the Cozumel
Channel, but empirical data covers less than two years. On the
other hand, simulated data of high spatial resolution are available
for the current in the Cozumel Channel, although the degree of
error with respect to the real current in this region is not known.
This paper is based on measurements from the Canek project
2009/2010 in the Channel of Cozumel. It addresses such
shortcomings by comparing and adjusting the simulated data
with empirical data, and subsequently performing an EVA on the
numerical data. The analysis was applied to the northern part of
the Mexican Caribbean, marked in red in Figure 1. The study
area extends from the south of Cozumel Island to Cabo Catoche,
north of Cancún, and to the east of the continental shelf.
Although the numerical model overestimates the current, the
EVA results are expected to give valuable predictions for
extreme currents.
2 MATERIALS AND METHODS
2.1 Data Sources
Empirical and numerical data were used for the theoretical
analysis presented here. Both sources provide data on the
eastward and northward components of the ocean currents in
the study area for different depths and different temporal
resolutions. The Canek project, which has carried out similar
measurements in the past (Chávez et al., 2003), was responsible
for the measurements of the current in the Cozumel Channel.
The Canek research project, also known as the Estudio de la
circulacioń y elintercambio a traveś del Canal de Yucatań (Study
of circulation and exchange through the Yucatan Canal) has
been coordinated by the Centro de Investigacioń Cientıfíca y
Educacioń Superior de Ensenada since its foundation in 1996.
The data were obtained using a stationary, long-range acoustic
Doppler current profiler at N20°32.218′ W087°02.738′ [see
Figure 1B] anchored at a depth of approximately 400m, and
measuring every half hour, from 9th April 2009 to 14thMay 2011.
The data on depth were recorded in 16 cells, with the shallowest
cell at a depth of 49m. For the numerical data, the HYCOM was
chosen because of its good temporal range and resolution, and
excellent spatial resolution, compared to other products. In this
study, the data of the reanalysis model HYCOM + NCODA
GOMu0.04 experiment 50.1 are used, which are publicly
available in https://www.hycom.org/data/gomu0pt04/expt-
50pt1. The model provides the current components at 40
depths for the Mexican Caribbean (among other regions),
covering 1st January 1993 to 31st December 2012, at a temporal
resolution of three hours and a spatial resolution of 0.04 in both
eastern and northern directions. Numerical current data are
reported at a depth of 50m while the empirical data describe
the current at 49m.
2.2 Validating and Adjusting Numerical
Data With Empirical Data
Interpolation of HYCOM data to match the Canek data was
carried out using the griddata function, available in the SciPy-
module (version 1.6.1) for Python 3 (Virtanen et al., 2020). The
four nodes of the numerical model were used as input, which
surround the location of the measured field data. Due to the
different sampling frequencies of the data sources, the data with
higher frequency (i.e., the empirical data) had to be reduced. The
data provided by the Canek project were sampled every 30 min
and every hour, depending on the date. As the HYCOM
numerical data reports the instantaneous value every three-
hours, the empirical data were reduced by discarding every
time step which is not available in the numerical data.
A linear quantile regression was performed on the current
speed, using the quantreg model, as provided by the statsmodels
module (version 0.12.2) for Python 3 (Seabold and Perktold,
2010). The linear regression was estimated for the empirical data
proportional to the numerical data with the intercept set to zero.
To estimate the error produced by the numerical data, a
simplified EVA was performed for both data sets, the empirical
data in its original form and the numerical data in its adjusted
form but reduced to the temporal range of the empirical data set.
The analysis is described in detail in section 2.3. However, due to
the low number of observations available in both sets, the
methodology had to be modified. As a threshold, the 0.5th-
quantile was used in contrast to the suggested automated
threshold selection. However, the same range of possible
thresholds was used to estimate the confidence interval. The
(signed) relative error between empirical and numerical data is
Ring et al. Mexican Caribbean: Extreme Ocean Currents
Frontiers in Marine Science | www.frontiersin.org June 2022 | Volume 9 | Article 8668743
defined as
er   =   1  −
um
ue
,   (1)
where ue is the empirical current speed and um is the speed as
predicted by the numerical model. Besides the mentioned
modules for Python 3 (Van Rossum and Drake, 2009),
substantial parts of the data processing have been carried out
with the NumPy-module in version 1.20.1 (Harris et al., 2020)
and the pandas-module in version 1.2.2 (Wes McKinney, 2010).
2.3 Extreme Return Levels With Peaks
Over Threshold
The methodology used assumes that for a random variable (x)
the excess over a suitable threshold (u) can be modelled by a
GPD. Liang et al. (2019) define the cumulative density function
of the GPD as
F xð Þ =
1 − 1 + x
x−u
s

 
−1=xð Þ
for x ≠ 0
1 − exp  − x−u
s

 
for x = 0
    ,   with x ≥ u   ,
8
<
:
(2)
Where x represents the random variable, u the threshold, x the
shape parameter, and s the scale parameter.
The procedure can be summarized as follows, where the
automated threshold selection method is based on the work of
Thompson et al. (2009):
1. Selection of peaks using a moving time window.
2. Detection and filtering of outliers using the quartile method.
3. Identification of potential thresholds between the 25th and 98th
percentiles, or the 100th highest peak, whichever is less.
4. For each potential threshold uj:
(a) Fit a GPD through all peaks for which xi > uj.
(b) Determine a reparameterised scale parameter (s∗
i ) and its
difference (Ds∗
i ) tothe next higher threshold (uj+1).
(c) Fit the normal distributionwith zeromean through the difference
of the reparameterised scale parameter corresponding to the
current and all greater thresholds (Ds∗
i jui ≥ uj).
5. Selection of the lowest threshold for which the p-value of the
normal distribution through the difference of the reparameterised
scale parameter is greater than a significance level of 5%.
6. Estimation of the return levels based on the selected threshold.
To consider the phenomenon as random, the realization of
each variable should be independent. However, with the
temporal resolution provided, the data analysed in this study is
not random. To select only values independent of temporally
close values (later called peaks), a moving time window was used,
as suggested in Solari et al. (2017). The time window is of fixed
length, depending on the variable type, and moves consecutively
through the time series. If the value in the centre of the time
window is the maximum of that time window, this value is
selected as an independent peak.
Outliers may be present in the set of selected peaks, which
would alter the final excess model. For the automated and semi-
automated detection of outliers, a great variety of methods are
available (Hodge and Austin, 2004). One of the simplest methods
suitable for univariate data is based on quartiles and presented in
Laurikkala et al. (2000). The authors define an upper (uu) and
lower threshold (ul), beyond which all peaks are considered as
outliers and are consequently discarded. Both thresholds are
defined by
ul = q1 − 1:5 q3 − q1ð Þ (3)
uu = q3 + 1:5 q3 − q1ð Þ  , (4)
where q1 is the first quartile (25
th percentile) and q3 is the third
quartile (75th percentile).
From the previously selected peaks, potential thresholds are
selected, as suggested in Thompson et al. (2009). The potential
thresholds are equally spaced between 25th and 98th percentile. If
less than 100 peaks are found above the 98th percentile, the 100th
A B
FIGURE 1 | (A) Study area within the Mexican Caribbean and (B) position of the acoustic Doppler current profiler (ADCP).
Ring et al. Mexican Caribbean: Extreme Ocean Currents
Frontiers in Marine Science | www.frontiersin.org June 2022 | Volume 9 | Article 8668744
largest peak is selected as the upper limit of the range for
potential thresholds.
For each threshold, all peaks xi > uj are selected, and a GPD is
fitted through those peaks. The shape (xj) and scale (sj)
parameters of the GPD are determined by the function
genpareto.fit, which is part of the SciPy.stats-package. The
location parameter is held fixed to the corresponding threshold
u. The reparameterised scale parameter, which is defined by
s
∗
j =  sj − xjuj  , (5)
should be constant above a suitable threshold, following Coles
(2001). This relationship was extended by Thompson et al.
(2009) by fitting a normal distribution with a mean of zero
through the difference of the reparameterised scale parameter for
the current and all greater thresholds. This difference is
defined by
Ds
∗
j = s
∗
j+1 − s
∗
j    : (6)
The first threshold for which the corresponding normal
distribution has a p-value ≥ 0.05 is selected for calculating the
return levels. As a test for normality, the Kolmogorov-Smirnov
test is used, as implemented in the ks_1samp function of the
SciPy.stats-package. The return level Xm (Coles, 2001) can be
calculated as
xm =
u + s
x
mzuð Þx−1
 
for x ≠ 0
u + s log  mzuð Þ for x = 0
     :
8
<
:
(7)
The average number of peaksm during a return period (TB) is
defined by
m =
np
ny
 TB   , (8)
where np is the total number of peaks and ny the number of years
for which data is available.
The exceedance probability of threshold ẑ u, the complete
Variance-Covariance Matrix V and the variance of return level
Var (x̂m) are estimated, as stated in the following equations
(Coles, 2001), where the values with a hat indicate the
estimation of the corresponding value.
^
z u =
npot
np
   , (9)
where npot is the number of peaks over threshold
V =
ẑ u
np
1 − ẑ u
 
0 0
0 Var (ŝ ) Cov (ŝ , ^x )
0 Cov (^x , ŝ ) Var (^x )
2
6
6
6
6
4
3
7
7
7
7
5
   , (10)
Var (x̂m) = ∇x̂T
m V  ∇ x̂m     , (11)
with
∇x̂m = ∂ xm
∂ zu
, ∂ xm
∂s , ∂ xm
∂ x
h iT
=
smx
zu
x−1
x
−1 mzuð Þx−1
 
−sx
−2 mzuð Þx−1
 
+ sx
−1 mzuð Þx log  mzuð Þ
2
6
6
6
6
4
3
7
7
7
7
5
(12)
3 RESULTS
3.1 Numerical Data Accuracy
The empirical and numerical data used have different temporal
resolutions. Therefore, the empirical data from the Canek project
were downsampled by discarding all time steps that are
unavailable in the data provided by the numerical model. In
Figure 2 the unadjusted numerical data show a clear bias
towards overestimation. The numerical data were adjusted by
the linear regression model, which was estimated with a quantile
regression using the 0.5th-quantile, and is defined by
u0m = 0:791831um  , (13)
where u′
m is the adjusted current speed. The adjusted numerical
data reflect the empirical data much better. The effect of the
model adjustment is strongly reflected by the mean relative error,
that is reduced from −0.255 to 0.006. The mean absolute relative
error of 0.288 and the root mean squared relative error of 0.365
are reduced to 0.153 and 0.206, respectively. Of greater concern,
however, is the missing tail of the probability distribution of both
numerical data in Figurs 2A, as these are of great importance for
EVAs. Especially in the adjusted data, this leads to a pronounced
underestimation of higher current speeds (i.e., rare events) as
seen in the deviation from the diagonal in Figure 2B.
To quantify the effect of the missing tail on the extreme value
estimations, a simplified EVA was performed. Due to the short
time range, the length of the time window was reduced to 7 days
(i.e., 56 observations). Additionally, the 0.5th-quantile was
selected as the threshold rather than the proposed automated
threshold selection. As expected from the results presented in
Figure 2B , the adjusted numerical model shows an
underestimation of extreme values, as can be seen in
Figure 3A. Nevertheless, for rare events (return period > 10
years) the relative error converges to a value just below 0.22 (see
Figure 3B). The large 95% confidence interval (CI95%) in
Figure 3A is the result of the short temporal coverage of the
data used for the analysis. It should be noted that the CI in
Figure 3B is not CI95%; it is the maximum error estimated by the
upper and lower bounds of the CI95% in Figure 3A.
3.2 Extreme Value Analysis
The methodology was applied to several nodes in the Mexican
Caribbean, shown as grey dots in Figure 4. It should be noted
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that not every node contains data on the current, as some are on
land, or in waters of less than 50m depth. The four nodes marked
in red were selected as the results suggest that it is possible to
obtain a different behaviour with respect to the GPD fit. The
node at position P1 (20.520° N, 86.600° W) is where the current
is most concentrated off the east coast of Cozumel. The node at
position P2 (20.640° N, 86.960° W), is in the Cozumel Channel,
near a possible site for the installation of ocean current turbines
[see Alcérreca-Huerta et al. (2019)]. That at P3 (21.040° N,
86.560° W) is in the wake of the Cozumel Channel, off the coast
of Cancun, and the node at P4 (21.800° N, 86.480° W) is in the
Yucatan current northeast of Cancun.
To determine the optimal length of the time window, the
number of peaks identified in the windows was analysed for the
nodes at P1 to P4. In Figure 5A, there is a steady fall in the
number of peaks, but it remains above the critical number of 200.
At a length of 25 days, the number of peaks for all four nodes is
just below 250. The relative difference between the number of
A B
FIGURE 2 | Adjustment of the HYCOM numerical data to the Canek-project empirical data [from the ADCP shown in Figure 1 (B)]. (A) Histogram showing the
numerical data before and after the adjustment. (B) Q-Q plot of numerical data before and after the adjustment. The 0.01st 0.5th and 0.99th-quantiles are marked for
both data sets.
A B
FIGURE 3 | Results of the simplified EVA for the adjusted numerical and empirical data. The adjusted numerical data are limited to the temporal range of the
empirical data. (A) Return level plot. (B) Relative error of estimated return levels.
Ring et al. Mexican Caribbean: Extreme Ocean Currents
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peaks and the length of time window (see Figure 5B) shows a
decreasing trend, as the length of the time window increases.
From a 21 day length, the relative difference is less than 10%,
dipping briefly below the 5% mark at a length of 23 days. To
spread the number of peaks evenly within the time range
analysed, and to avoid having too few peaks, a time window of
23 days in length was chosen.
Figure 6 shows the statistical data of the GPD fit for each
node. While the north, east, and southern boundaries of the
domain are determined by the node selection, the western
boundary is a feature of the numerical data generated by
HYCOM for this site. The number of identified peaks seems to
be similar in the study region (see Figure 6A), with a slight
decrease towards deeper waters.
Figures 6B, C) show the selected threshold for each node and
the corresponding p-value for the automatic threshold selection,
respectively. The value of selected thresholds tend to increase in
the centre of the channel, and in the stream close to the east coast
of Cozumel Island that extends northward, along the Cancun
coast. This is expected, since the current becomes more intense at
these locations. As it was possible to find a suitable threshold for
all nodes with information on the current velocity, the p-value is
over 0.05 in significance, although some inconsistencies of above
0.1, and even 0.15, are found throughout the domain.
There is no clear trend in the shape parameter of the fitted
GPD in Figure 6D. However, it was estimated to be negative for
all nodes, producing a bounded GPD. For a few nodes at the
northwestern boundary, the shape parameter was estimated to be
very close to zero. The scale parameter in Figure 6E indicates a
slight increase off Cozumel Island and at the northeastern
boundary, which leads to a thicker tail for the GPD in those
regions (i.e., increased return levels).
Figure 6F shows the number of peaks above the threshold
which lie within the estimated CI95%. For nearly all the nodes, the
estimated CI95% covers 100% of the numerical observations. As is
to be expected, not all the observations are within the CI95% for
all the nodes. However, the number of nodes for which some
observations are outside the CI95% is small, while the minimum
share within the analysed region is still above 90%. Despite this
apparent overestimation of the CI95%, this suggests that the
methodology of GPD fit together with the estimation of the
CI95% is suitable and the results of the GPD for the given input
data is reliable.
Return levels for the selected return periods, on the
corresponding lower and upper boundaries of the CI95% are
shown in Figure 7. The expected return level (central column)
increases in the channel and the main current, which extends
northwards from the east of the Cozumel Island. This trend is
further pronounced in the case of the CI95% upper boundary
(right column of Figure 7), which is in agreement with the
results in Figures 6B, E. The region with higher shape
parameters at the northwest edge of the domain (see
Figure 6D) is not noticeable in the estimated return level
(centre column in Figure 7). However, in case of the CI95%-
FIGURE 4 | Location of the nodes for the numerical model which lie within the study area. The nodes marked in red are positions for which more details regarding
the GPD fit are presented.
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limits, that region stands out with lower return level for the
lower bound of the CI95% and higher return levels for the
upper limit, suggesting a much higher uncertainty. The
distribution over the rest of the domain is as expected,
see Figure 6.
The parameters for the GPD excess model for the four
nodes seen in Figure 4 are summarized in Table 1. Except for
the node at position P4, the shape parameters are negative,
with all of CI95% below zero. Compared to the standard error,
the shape parameter at P4 is small, giving a CI95% closely
centred around zero. However, this result could be due to an
error in the numerical model, as mentioned above. The
highest scale parameter is found at P1, which produces a
thicker tail to the probability distribution. Nevertheless, the
bound nature of the excess model (due to the negative shape
parameter) prevents high return levels for this node. The
number of peaks found for each node is similar, just above
the critical threshold of 200. Slightly more than 100 peaks
were found above the selected threshold. The number of
peaks, and peaks above the threshold, suggests that the
A B
D E F
C
FIGURE 6 | Statistical data for the GPD fit. (A) Number of identified peaks. (B) Selected thresholds. (C) p-value for the selected thresholds. (D) Shape factor for
GPD fit to POT. (E) Scale factor for GPD fit to POT. (F) Share of peaks within the CI95%.
A
B
FIGURE 5 | Relation between the number of independent peaks and the length of the time window for the nodes at P1 to P4. (A) Number of identified peaks.
(B) Relative difference in number of identified peaks to previous time window length.
Ring et al. Mexican Caribbean: Extreme Ocean Currents
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selected time span of 20 years is a bit short, but still sufficient
to perform the EVA.
In Figure 8, the peaks, POT, and the thresholds are shown
for the four nodes. None of these nodes have a cluster of peaks
(or lack thereof), suggesting that a 23 day time window is
sufficient. The distributions of peaks, together with the filtered
outliers, are shown for each node in Figure 9. The nodes at P1
to P3 show a standard distribution of peaks. The node at P4
has a multi-modal distribution, suggesting an error, and that
the conclusions drawn from the data might not be reliable.
No outliers at the upper end were found for the node at P2,
whereas at P1 and P4 there were one each, and at P3, two. At
the lower end, a few outliers were also detected and filtered
out, but due to the nature of POT methods, these tend to have
no significant effect on the outcome.
Figures 10–13 present the corresponding diagnostic plots
for the GPD excess model. For a detailed interpretation of this
type of plot, the reader is referred to Coles (2001). Despite
A B
D E F
G IH
J K L
M N
C
O
FIGURE 7 | Return levels for 50m depth for different return periods. All the figures in one row correspond to the same return period; (A–C) 2 years, (D–F) 5 years,
(G–I) 10 years, (J–L) 25 years, and (M–O) 50 years. The left column shows the lower bound of the CI95%, the right column the upper bound of the CI95%, and the
central column the predicted return level.
Ring et al. Mexican Caribbean: Extreme Ocean Currents
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slight deviations in the diagnostic plot for the node at P1 in
Figure 10, and especially the q-q plot in Figure 10B, 100% of
the empirical POT still lie within the CI95%, as seen in
Figure 10C and tab. 1. Both plots suggest an overestimation
of the GPD model. There are few peaks, especially visible in
the density plot (Figure 10D). However, the bound excess
model seems to give a good fit for the underlying numerical
data. The diagnostic plots for the node at P2 (Figure 11) show
some deviations between the numerical data and GPD excess
model in the p-p plot (Figure 11A). Around the 0.6 mark, the
GPD excess model shows a slight overestimation. This
deviation is also visible in the q-q plot (Figure 11B) at
speeds of about 1.45ms−1 and in the return level-plot
(Figure 11C ) a t the same speeds . Desp i t e the se
inconsistencies, the GPD excess model is a fits the data well.
The p-p plot presents some discrepancies at the 60%
percentile at P3 (Figure 12A). The excess model also differs
from the numerical data for higher speeds, as seen in
Figures 12B, C. However, all the observations are within the
estimated CI95% of the return level, suggesting that the GPD
excess model application is reliable.
The plots in Figure 13 bring into doubt whether this excess
model can be used to reliably estimate the extreme values of
the node at P4. Although the CI95% includes all the numerical
observations, the p-p plot (Figure 13A) and especially the q-q
plot (Figure 13B) looks unusual. A slight s-shaped deviation
is present, with considerable inconsistencies above 1.2ms−1 in
the q-q plot. Additionally, and as observed in Figure 7, the
CI95% in Figure 13C is quite large, while the density plot in
Figure 13D shows a reasonable fit to the data.
4 DISCUSSION
For adjustment of the numerical data, the empirical and
numerical data were filtered to match their time steps. The
high relative error of -25.5% was reduced to 0.6% by linear
quantile regression. However, the mean absolute relative error
and the root mean square relative error cannot be reduced in
the same way. This indicates that, despite the adjustment,
the numerical model is not able to accurately reproduce the
behaviour of the current in this region. Furthermore, the lack
of the tail in the numerical data histogram proves that there is
still room to improve the HYCOM numerical model. The
effect of the missing tail on extreme value predictions was
estimated with a simplified EVA. Despite the short time series,
A B
DC
FIGURE 8 | Distribution of peaks over time for the nodes at P1 to P4, as identified by means of a 23 days moving time window. The peaks under threshold are
marked with grey dots, the peaks over threshold with black dots, and the selected threshold by the blue line. (A) Node at P1, (B) Node at P2, (C) Node at P3, and
(D) Node at P4.
TABLE 1 | Values for GPD fit for four nodes.
Node at P1 Node at P2 Node at P3 Node at P4
Threshold (uo) in ms-1 1.2397 1.3267 1.0400 1.0784
Shape parameter (x) -0.3965 -0.3171 -0.3669 -0.0839
Corresponding CI95% -0.480 -0.423 -0.453 -0.280
… … … …
-0.313 -0.211 -0.280 0.112
Scale parameter (s) 0.1827 0.1501 0.1417 0.1088
Peaks (np) 222 239 242 249
POT (npot) 111 114 111 102
POT in CI95% of return level 100% 100% 100% 100%
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A B
DC
FIGURE 10 | Diagnostic plot for the GPD excess model fitted to 3-hourly current for the node at P1. (A) p-p plot, (B) q-q plot, (C) return level plot, and (D) density plot.
FIGURE 9 | Detected outliers and distribution of peaks for nodes at P1 to P4. The first and third quartiles are shown as solid lines and the median as a dotted line.
The filtered outliers are shown as blue dots and the selected threshold as a thin blue line.
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the error can be estimated at 22% underestimation for rare
events with a return period of > 10years. However, the error
presents very low variability for rare events, and the error
converges to a value close to 22%. This makes it easy to
account for in design processes. Nevertheless, these results
should be addressed in future research in order to accurately
identify the source of the error and to characterize it over a
larger area, instead of a single point.
As shown by the large CI95% of the simplified EVA, the
temporal coverage of the empirical data is not sufficient to
reliably estimate extreme values. However, the error between
the extreme value predictions of empirical and numerical data
is consistent. This error gives the necessary information to
have sufficiently detailed knowledge on the extreme value
predictions derived from the HYCOM model.
For most of the nodes, the EVA showed consistent
behaviour over the domain analysed. Some inconsistencies
were found, especially at the boundary of the numerical
domain and on the northwest edge of the continental shelf.
Besides the significant changes in the bathymetry in those
regions, the ordered grid of the numerical model might not be
fully capable of representing the nature of the current in
boundaries that are not aligned with the grid. As can be
seen for the node at P4, the estimated CI95% is large, and the
multimodal peak distribution suggests an unusual behaviour
of the HYCOM data for this area. The results for these nodes
may be unreliable, and it is suggested that the data obtained
for these locations is used with special care.
The other three nodes, which had more information on the
GPD fit, showed unremarkable results, as the EVA represents
a good fit for the numerical observations. For most nodes,
100% of the numerical observations were found to be within
the estimated CI95%. This share should be much closer to 95%,
indicating that the estimated CI95% is larger than it should be.
In contrast to the assumed symmetric distribution of the
CI95%, a log-likelihood profile could give better results and
might be investigated in future studies if the overestimation of
the CI95% represents an issue.
The extreme values found a reasonable distance from the
coast vary considerably. Therefore, it is important to carefully
select a region with similar behaviour in terms of GPD fit.
Basing design values on a region with heterogeneous
behaviour could lead to erroneous design choices. Besides
the variability, regions with similar return periods are found
on either side of Cozumel Island. In terms of extreme currents,
with reduced effort it may be possible to adapt energy
harvest ing devices des igned for Cozumel Channel
conditions to the conditions on the east coast of Cozumel.
A B
DC
FIGURE 11 | Diagnostic plot for the GPD excess model fitted to 3-hourly current for the node at P2. (A) p-p plot, (B) q-q plot, (C) return level plot, and (D) density plot.
Ring et al. Mexican Caribbean: Extreme Ocean Currents
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A B
DC
FIGURE 12 | Diagnostic plot for the GPD excess model fitted to 3-hourly current for the node at P3. (A) p-p plot, (B) q-q plot, (C) return level plot, and (D) density plot.
A B
DC
FIGURE 13 | Diagnostic plot for the GPD excess model fitted to 3-hourly current for the node at P4. (A) p-p plot, (B) q-q plot, (C) return level plot, and (D) density plot.
Ring et al. Mexican Caribbean: Extreme Ocean Currents
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5 CONCLUSIONS
It was found that the HYCOM model does not accurately
reproduce the current velocities in the Cozumel Channel.
Adjusting the model with a linear quantile regression
reduces the mean absolute relative error to 15.3%, but the
lack of a tail in the distribution of the numerical data leads to
an underestimation of extreme values of almost 22%.
Applied to a range of nodes within the Mexican Caribbean,
the methodology showed consistently – and to some extent
predictable – behaviour. In the Cozumel Channel and in the
main current, the threshold and the extreme values are
naturally higher than in regions with lower current
intensities. The difference in return levels can be explained
by the threshold and the scale parameter.
Despite the shortcomings of the numerical model, the
methodology presented for estimating extreme values of ocean
currents based on HYCOM data proves to be a valuable tool due
to the predictability of the error for extreme values.
DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article
are available from the corresponding author, MR, upon
reasonable request.
AUTHOR CONTRIBUTIONS
Conceptualization and methodology: PR-O and MR; data
processing, analysis and visualisation: MR; writing – original
draft preparation: PR-O and MR; writing – review and editing:
PR-O, MR, RS, and EM; supervision and project administration:
RS and EM; funding acquisition: EM; All authors contributed to
manuscript revision, read, and approved the submitted version.
FUNDING
This research was funded by the CONACYT-SENER-
SUSTENTABILIDAD ENERGÉTICA project: FSE-2014-06-
249795 “Centro Mexicano de Innovación en Energı ́a del
Océano (CEMIE Océano)”.
ACKNOWLEDGMENTS
The authors would like to thank Julio Candela and Julio
Sheinbaum for permission to use their empirical data and Jill
Taylor for reviewing the English language. Furthermore, the first
author is grateful for the financial support provided by the
CONACYT doctoral fellowship.
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Ring et al. Mexican Caribbean: Extreme Ocean Currents
Frontiers in Marine Science | www.frontiersin.org June 2022 | Volume 9 | Article 86687415