Steady-State Performance of the Delft Offshore...

Master of Science Thesis

Steady-State Performance of the Delft

Offshore Turbine

A. Jarquin Laguna B.Sc.

24th August 2010

Faculty of Aerospace Engineering · Delft University of Technology

Steady-State Performance of the Delft

Offshore Turbine

Master of Science Thesis

For obtaining the degree of Master of Science in Sustainable Energy

Technology at Delft University of Technology

A. Jarquin Laguna B.Sc.

24th August 2010

Faculty of Aerospace Engineering · Delft University of Technology

Delft University of Technology

Copyright c© A. Jarquin Laguna B.Sc.All rights reserved.

Delft University Of Technology

Department Of

Wind energy

The undersigned hereby certify that they have read and recommend to the Faculty ofAerospace Engineering for acceptance a thesis entitled “Steady-State Performance of

the Delft Offshore Turbine” by A. Jarquin Laguna B.Sc. in partial fulfillment ofthe requirements for the degree of Master of Science.

Dated: 24th August 2010

Head of department:prof.dr. G.J.W. van Bussel

Supervisor:ir. N.F.B. Diepeveen

External supervisor:ir. P.S. Albers

Summary

The Delft Offshore Turbine (DOT) is a DUWIND research project that focuses on reduc-ing the cost of offshore wind energy by bringing a radical change in offshore wind turbinetechnology. The main concept is to centralize electricity generation using pressurizedseawater from individual wind turbine pumping systems. The idea behind the DOT isthat the high power to weight ratio from hydraulic drive systems gives the opportunityfor a reduced nacelle mass and increased reliability of components by eliminating the useof individual geartrains and generators.

This thesis presents a first evaluation of the overall performance of a single DOT us-ing a baseline rotor (from the NREL 5 MW offshore wind turbine) with a possible hightip-speed operation up to 120 m/s.

A physical modelling approach was used, where the main system subcomponents fromdifferent physical domains (mechanical, hydraulic and aerodynamics) were modelled andintegrated in a single environment with Matlab -Simulink. The steady-state responseof the system was obtained as a function of wind speed. The main advantage of theDOT with a high speed operation is the possibility to get more mechanical power fromwind speeds in the range of 12 to 17 m/s, with a power capture close to 8 MW . Anoverall system performance of 80% was obtained (including pump/motor efficiencies, fric-tion losses), with a final electrical power output of 6.4 MW (28% more than the 5 MWreference turbine). Taking into account the wind speed probability with a typical offshoreWeibull distribution for the North Sea, a gross annual energy production of 24, 991 MWhwas obtained (2% less with respect to the 5 MW reference turbine).

A 30 kW testbench was used to validate experimentally the performance of the hydraulictransmission used in the computational model, resulting in deviations of 2 to 8% of thepredicted performance.In general it can be foreseen that the success of an offshore hy-draulic turbine will not be dictated by the individual power performance but by thepotential reduced cost and increased reliability of the overall system.

v

Contents

Summary iv

List of Figures xiii

List of Tables xv

Nomenclature xvi

1 Introduction 1

1.1 Offshore wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Report set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Delft Offshore Turbines 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Hydraulic drives for power transmission . . . . . . . . . . . . . . . . . . . 5

2.3 Overview of the DOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 The rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 Closed-loop subsystem . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.3 Open-loop subsystem . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.4 Central generator platform . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Hydraulic diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Control overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

vii

viii Contents

3 Basics of wind energy and fluid power 15

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Wind energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Energy capture from the wind . . . . . . . . . . . . . . . . . . . . 15

3.2.2 Aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Hydraulic drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Steady state performance . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.2 Pump efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.3 Compressibility effect . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Fluid flow in pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Friction losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 Compressibility in long pipelines . . . . . . . . . . . . . . . . . . . 23

3.5 Hydraulic turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.1 Principles of operation . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.2 The Pelton turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Preliminary design 31

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Operational limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 High tip speed rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.2 Hydraulic drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.3 Seawater pipeline pressure . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.4 Pelton turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Dimensioning of components . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4.1 Radial pump dimensioning . . . . . . . . . . . . . . . . . . . . . . 35

4.4.2 Hydraulic motor and variable pump dimensioning . . . . . . . . . 35

4.4.3 Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.4 Hydraulic turbine . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 Natural frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.5.1 Closed-loop natural frequency calculation . . . . . . . . . . . . . . 40

4.6 Control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Computational model 43

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 The physical network approach overview . . . . . . . . . . . . . . . . . . . 43

5.3 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.2 Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3.3 Closed-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.4 Open-loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Contents ix

5.3.5 Controller and scopes . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4 How it works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4.3 Post processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Simulation results 51

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Steady-state operational parameters . . . . . . . . . . . . . . . . . . . . . 51

6.2.1 High speed rotor operation . . . . . . . . . . . . . . . . . . . . . . 51

6.2.2 Hydraulic drives operation . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3.1 Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3.2 Power curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3.3 Energy yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Experimental validation 59

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.3 Data aquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.4 Data post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8 Conclusions and recommendations 65

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

References 67

A Blade Element Momentum theory 71

A.1 Momentum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.2 Blade element theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B Hydraulic motor analysis 77

B.1 Steady-state performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

B.1.1 Rotational speed generated . . . . . . . . . . . . . . . . . . . . . . 77

B.1.2 Torque delivered to the load . . . . . . . . . . . . . . . . . . . . . . 77

B.2 Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.2.1 Volumetric efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.2.2 Mechanical efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B.2.3 Total efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

C Moody diagram 81

x Contents

D BEM code validation 83

E Pump parameters estimation 85

E.1 Determination of the coefficient Cs . . . . . . . . . . . . . . . . . . . . . . 85

E.2 Determination of the coefficient Cdamp . . . . . . . . . . . . . . . . . . . . 86

E.3 Determination of the coefficient Cf . . . . . . . . . . . . . . . . . . . . . . 86

F Simulation method 87

G Annual energy yield evaluation 89

G.1 Wind speed distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

H Performance models for the experimental set-up 91

H.1 Hagglunds CA50 pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

H.2 Bosch Rexroth motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

H.3 Bosch Rexroth pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

List of Figures

1.1 Wind offshore electricity production 2011-2020 (Fichaux & Wilkes) . . . . 1

2.1 Artemis variable speed transmission (Artemis Intelligent Power) . . . . . 6

2.2 ChapDrive concept for a variable speed transmission (ChapDrive) . . . . 6

2.3 Simplified schematic of the DOT (Diepeveen & Van der Tempel) . . . . . 7

2.4 Component comparsion for the nacelle of a typical wind turbine and theDOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Possible lay-out configurations for the DOTs connection to central platform 9

2.6 3D model view of the Gilge Gilbe Pelton Turbine (Reiner) . . . . . . . . . 10

2.7 Transformer platform for repair and maintenance crew on site (BARD Off-shore) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.8 Hydraulic diagram of the Delft Offshore Turbine . . . . . . . . . . . . . . 12

2.9 Functional diagram of the DOT subsystems . . . . . . . . . . . . . . . . . 12

2.10 Subsystem level block diagram of the DOT . . . . . . . . . . . . . . . . . 14

3.1 Typical Cp-lambda curve (Manwell et al.) . . . . . . . . . . . . . . . . . . 16

3.2 Efficiency curves for a fixed displacement pump . . . . . . . . . . . . . . . 20

3.3 Pump perfomance for variable displacement conditions . . . . . . . . . . . 21

3.4 Compression work done by one of the cylinders of an axial piston pump(Murrenhoff) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 Schematic of an ideal pump connected to a constant load by a long pipeline 24

3.6 Friction and compressibility effects for a long pipeline . . . . . . . . . . . 25

3.7 Hydraulic turbines topology (Pencho) . . . . . . . . . . . . . . . . . . . . 26

3.8 Operational envelope of several hydraulic turbines (Pencho) . . . . . . . . 28

3.9 Specific speed correlation for different hydraulic turbines (Pencho) . . . . 29

3.10 Cross section of a nozzle with jet deflector (Pencho) . . . . . . . . . . . . 29

xi

xii List of Figures

3.11 Efficiency comparison of a six jet Pelton turbine with a Francis turbine(Reiner) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 Radial pump dimensioning for different operational pressures . . . . . . . 35

4.2 Seawater pump and hydraulic motor size ratio for different operationalpressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Mean fluid velocity for different diameters of the closed-loop pipeline . . . 37

4.4 Pressure loss for different diameters of the closed loop pipeline . . . . . . 37

4.5 Pressure loss per km of the open-loop seawater pipeline . . . . . . . . . . 38

4.6 Maximum power for multiple jet nozzles of a Pelton turbine . . . . . . . . 38

4.7 Pelton runner diameter for different generator speeds and number of jets . 39

4.8 Simplified model for the closed-loop natural frequency calculation (Schmitz& Vatheuer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.9 Schematic of the control strategy for a variable speed operation . . . . . . 41

5.1 General model of a single DOT overview . . . . . . . . . . . . . . . . . . . 44

5.2 DOT subsystem interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Schematic of the Closed-loop subsystem . . . . . . . . . . . . . . . . . . . 46

5.4 Schematic of the Open-loop subsystem . . . . . . . . . . . . . . . . . . . . 47

5.5 Schematic of the scopes for different parameters of the simulation . . . . . 48

5.6 General procedure of the DOT simulation . . . . . . . . . . . . . . . . . . 48

5.7 Subroutines and Matlab files needed for the DOT simulink model . . . . 49

6.1 Power and Torque curves for a 5 MW turbine in both typical and high-speed operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Thrust curve for a 5 MW turbine in both typical and high-speed operation 52

6.3 Rotor speed and pitch angle comparison for a for a 5 MW turbine in bothtypical and high-speed operation . . . . . . . . . . . . . . . . . . . . . . . 52

6.4 Dimensionless coefficient comparisons for a 5 MW turbine in both typicaland high-speed operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.5 Steady-state pressure for the closed and open loop of the DOT . . . . . . 53

6.6 Steady-state flow for the closed and open loop of the DOT . . . . . . . . . 54

6.7 Steady-state values for the required volumetric displacement of the seawa-ter pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.8 Hydraulic motor steady-state mechanical characteristics . . . . . . . . . . 55

6.9 Efficiency of the main components of the DOT for steady-state conditions 55

6.10 DOT energy transfer performance . . . . . . . . . . . . . . . . . . . . . . 56

6.11 Power curve comparison of the DOT and the NREL 5 MW turbine . . . . 56

6.12 Annual energy yield comparison for the northwest Europe coast . . . . . . 57

6.13 Normalized annual energy yield comparison for a mean wind speed of 10.1m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.1 Experimental set-up, 30 kW bench test . . . . . . . . . . . . . . . . . . . 60

List of Figures xiii

7.2 High torque, low speed hydraulic drives used in the test bench . . . . . . 61

7.3 Time domain record of measured pressure at the hydraulic transmission . 61

7.4 Performance of the hydraulic transmission for different pressures . . . . . 63

7.5 Performance of the hydraulic transmission . . . . . . . . . . . . . . . . . . 64

A.1 Energy capture in a stream tube . . . . . . . . . . . . . . . . . . . . . . . 71

A.2 Velocity and pressure profile at the rotor . . . . . . . . . . . . . . . . . . . 72

A.3 Blade elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

D.1 Steady-state parameters validation . . . . . . . . . . . . . . . . . . . . . . 83

F.1 time domain simulation method . . . . . . . . . . . . . . . . . . . . . . . . 87

G.1 Weibull distribution for scale factor a=10 and shape factor k as a parameter 90

H.1 Performance of the HagglundsCA50 pump . . . . . . . . . . . . . . . . . . 93

H.2 Performance of the Bosh-Rexroth 71 motor . . . . . . . . . . . . . . . . . 93

H.3 Performance of the Bosch-Rexroth A4VSO 125 pump . . . . . . . . . . . 94

xiv List of Figures

List of Tables

3.1 Range of net heads for hydraulic turbines . . . . . . . . . . . . . . . . . . 26

3.2 Generator synchronization speed . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 Main characteristics of the NREL turbine . . . . . . . . . . . . . . . . . . 32

4.2 Nominal values for commercial hydraulic drives . . . . . . . . . . . . . . . 33

4.3 Specific speed range for a Pelton turbine . . . . . . . . . . . . . . . . . . . 34

4.4 Summary of the chosen hydraulic drives characteristics . . . . . . . . . . . 36

4.5 Physical properties for mineral oil and seawater at operating temperatures 36

4.6 Summary of the chosen pipelines . . . . . . . . . . . . . . . . . . . . . . . 38

4.7 Pelton turbine data for a 500 MW central platform at 150 bar . . . . . . . 39

4.8 Rotor frequency range operation . . . . . . . . . . . . . . . . . . . . . . . 40

7.1 Main components of the experimental bench test . . . . . . . . . . . . . . 60

H.1 Performance parameters for the Hagg CA50 . . . . . . . . . . . . . . . . . 92

H.2 Performance of the Bosh-Rexroth 71 motor . . . . . . . . . . . . . . . . . 94

H.3 Performance of the Bosch-Rexroth A4VSO 125 pump . . . . . . . . . . . 94

xv

xvi Nomenclature

Nomenclature

Latin Symbols

A transversal area [m2]

a induction factor [−]

b number of cams in a radial pump [−]

c chord length [m]

Cd drag coefficient [−]

Cf friction coefficient [−]

CH hydraulic capacity [m3/Pa]

Cl lift coefficient [−]

Co hydraulic stiffness [Nm/rad]

Cp power coefficient [−]

Cs slip coefficient [−]

Cdamp damping coefficient [−]

CDax axial force coefficient [−]

Cext external cross port leakage coefficient [m3/(Pa s)]

Cint internal cross port leakage coefficient [m3/(Pa s)]

D rotor diameter [m]

Dax axial force [N ]

E Bulk modulus or fluid stiffness [Pa]

e roughness [m]

f friction factor [−]

FD drag force [N ]

xvii

xviii Nomenclature

FL lift force [N ]

g gravitational acceleration [m/s2]

H hydraulic head [m]

J moment of inertia [kg m2]

K constant [−]

kHP Hagen-Poiseuille coefficient [m3]

L length [m]

N number of jets for a Pelton turbine [−]

n generator speed [rpm]

Nb number of blades [−]

Ns number of blade elements [−]

ns specific speed [rpm]

P power [W ]

p pressure [Pa]

Q volumetric flow rate [m3/s]

Qs leakage flow rate [m3/s]

R rotor radius [m]

r local radius [m]

Re Reynolds number [−]

T torque [Nm]

U wind speed [m/s]

V volumetric displacement [m3/rad]

V0 initial volume [m3]

Vk volume funding of the piston [m3]

vm average velocity [m/s]

VUT lower dead volume of the piston [m3]

WA useful work [J ]

Wk compression work [J ]

Greek Symbols

α angle of attack [rad]

η efficiency [−]

λ tip speed ratio [−]

µ dynamic viscosity [Pa s]

ν kinematic viscosity [m2/s]

ω rotational speed [rad/s]

ωo natural frequency [rad/s]

Nomenclature xix

φ inflow angle [rad]

ρ fluid density [kg/m3]

θ pitch angle [rad]

Subscripts

cl closed-loop

d disk

damp damping

f friction

hs high-speed shaft

hyd hydraulic

ls low-speed shaft

m motor

max maximum

mech mechanical

nom nominal

ol open-loop

opt optimum

p pump

r rotor

rel relative

sw seawater

tan tangential

tot total

var variable displacement

vol volumetric

Abbreviations

BEM Blade Element Momentum Theory

DOT Delft Offshore Turbine

DUWIND Delft University Wind Energy Research Institute

EWEA European Wind Energy Association

IFAS Institute for Fluid Power and Control, RWTH Aachen University

NREL National Renewable Energy Laboratory (US)

xx Nomenclature

Chapter 1

Introduction

1.1 Offshore wind

Offshore wind power is one of the largest indigenous energy resources in Europe, withan enormous potential capable to power Europe seven times over. The European WindEnergy Association (EWEA) has established a target of 40 GW of offshore wind capacityin the EU by 2020; this target would require an average growth of 28% over the comingyears, making offshore wind power a key element in Europe’s energy future (Fichaux &Wilkes).

Figure 1.1: Wind offshore electricity production 2011-2020 (Fichaux & Wilkes)

With a growing market for offshore wind, a large trend for larger turbines (currentlyin the 5 MW range) is being developed in order to take more advantage of the windresource with fewer foundations. New generation of offshore wind turbines is dedicated

1

2 Introduction

to the offshore environment, with designs that are aimed at addressing some of the ma-jor challenges, such as corrosion, reliability and maintainability. However most of thistechnology is mainly based on incremental improvements from the basic concepts of windturbine systems onshore and modified to the offshore conditions.

Thus, the main driver for offshore wind technology is economic efficiency rather thantechnological efficiency. Furthermore, as the accessibility of offshore wind farms for repairand maintenance is lower than onshore, special attention is addressed regarding turbinereliability. This brings an opportunity to make significance reductions in the cost of en-ergy by developing innovative concepts with the objective to make offshore turbines assimple and robust as possible.

1.2 Problem description

The Delft Offshore Turbine (DOT) is a DUWIND research project that focuses on reduc-ing the cost of offshore wind energy by bringing a radical change in offshore wind turbinetechnology. The main concept is to centralize electricity generation using pressurizedseawater from individual wind turbine pumping systems. The idea behind the DOT isthat the high power to weight ratio from hydraulic drive systems gives the opportunityfor a reduced nacelle mass and increased reliability of components by eliminating the useof individual gear trains and generators (Diepeveen & Van der Tempel).The first steps have to be taken from the conceptual to the preliminary design, startingwith the proper dimensioning of the system and performance evaluation; hence a firstoverview and guideline of the performance evaluation of the DOT is given in this thesiswork.

1.3 Objective

“Determine the steady-state performance of a single Delft Offshore Turbine”

Some of the research questions are:Which are the operational conditions and component requirements?What is the system performance for steady-state conditions?Identify the main design challenges in terms of performance for a 5 MW system

1.4 Report set-up

This report is structured in the following way: Chapter two presents the conceptual ideaand a general description of the Delft Offshore Turbine, afterwards the basic principlesof wind energy and fluid power are given in Chapter three. It is not the intention of this

1.4 Report set-up 3

work to describe in detail each subject (for which more exhaustive and specific literatureis available), but to describe the main physical principles and give the basic equationsthat are necessary to integrate both subjects.

A preliminary design for a single DOT based on the conceptual idea and presented equa-tions is described in Chapter four. In this part of the work an initial dimensioning ofthe components is given based on the different design and operational requirements. Abaseline 5MW three bladed wind turbine rotor will be used as the starting point for thegeneral design of the system and although the goal of the DOT is to use a two bladedrotor, the existing rotor will provide a baseline for comparison of results.

Afterwards, Chapter five describes the computational model built in the Matlab -simulink platform, which uses the physical network modeling approach for dynamic anal-ysis in the time domain. In this model the main subcomponents from different physicaldomains (mechanical, hydraulic and aerodynamics) are modeled and integrated in a sin-gle environment. This computational tool will become a straight forward and effectivemethod for an integrated design of the system, and will allow further independent designof subcomponents. Chapter six presents the evaluation of the system performance forsteady-state conditions.

Chapter seven concerns the experimental validation of the hydraulic transmission perfor-mance carried out at the Institute for Fluid Power Drives and control (IFAS) at RWTHAachen University. Finally the conclusions and recommendations of this work are givenin Chapter eight.

4 Introduction

Chapter 2

The Delft Offshore Turbines

2.1 Introduction

This chapter describes the main components and characteristics of the Delft OffshoreTurbine. The primary objective is to present the conceptual design and the main ideasbehind the DOT. First, an overview of the current fluid power technology developed forwind energy applications is presented. Afterwards the description of the working principleof the DOT is addressed; with this aim, the entire system is divided into the followingsubsystems: the rotor, a closed-loop transmission, an open loop transmission and a centralplatform.

2.2 Hydraulic drives for power transmission

In the wind energy sector, the use of hydraulic drives for power transmission is not new,several works have already addressed this possibility in the past (JERICO; Rademakers),and all pointed out that the key problem was the lack of components specifically designedfor the efficiency requirements and power scales of that time. However, with the increasein size of commercial wind turbines and the current developments in the fluid power in-dustry, the idea of using hydraulics as an alternative solution for power transmissions hasbecome particularly attractive because of their high power density and increased reliabil-ity of components. Other advantages comprise the possibility of damping torque impulsesand the good controllability with outstanding dynamic performance (Murrenhoff; Schmitz& Vatheuer).

Several companies are currently developing new drive train solutions for wind turbinesinvolving the use of hydraulic drives:Artemis is a Scottish company who has developed a radial pump with electronically con-trolled poppet valves to suit wind turbine applications. With this variable speed trans-mission they are allowed to use synchronous generators and the company claims a 20%

5

6 The Delft Offshore Turbines

mass reduction of the nacelle (Artemis Intelligent Power). The overview of their conceptcan be observed in figure 2.1.

Figure 2.1: Artemis variable speed transmission (Artemis Intelligent Power)

ChapDrive is a Norwegian company, who has also developed a hydraulic transmissionwith a variable speed control system. The principal characteristic of this concept is therelocation of the major components from the nacelle in the top of the turbine to a powerunit at the base of the foundation (ChapDrive). They have a 900 kW prototype and theyare already working on a 5 MW project, an illustration of this concept is shown in nextfigure.

Figure 2.2: ChapDrive concept for a variable speed transmission (ChapDrive)

2.3 Overview of the DOT

A rotor is devoted to the conversion of the harnessed wind energy into useful mechani-cal energy. The closed-loop subsystem fulfils the function of transferring the mechanicalenergy from the nacelle to the base of the turbine; in order to do this a hydraulic pumpand motor are used in a hydrostatic transmission.

2.3 Overview of the DOT 7

At the base of the turbine a variable displacement pump with servo-control will allowto pump seawater from individual turbines to a central generator platform at constantpressure. Finally, in the central generator platform a few hydraulic turbines will convertthe hydraulic power, coming from the different turbines in the wind farm, into useful workin the form of high voltage electricity. A schematic of the system is shown in figure 2.3.

Figure 2.3: Simplified schematic of the DOT (Diepeveen & Van der Tempel)


2.3.1 The rotor

The rotor is the mechanical device that converts the kinetic energy from the wind intouseful mechanical energy. For the DOT it consists of a horizontal axis, high speed, twobladed rotor. The intention of a two bladed configuration is to have lower productioncosts and easier assembly; furthermore, a high speed operation will allow a reduction inthe size of components (therefore mass and cost) associated to the reduced torque neededin the transmission. If the configuration of the rotor is downwind or upwind, will dependon a detailed structural analysis in which the final blade stiffness should be enough toallow a safe tower clearance in the case of an upwind configuration. On the other hand,if a downwind configuration is used, special attention should be addressed for the towershadow effects (variations in the airflow velocity components due to the presence of thetower).

2.3.2 Closed-loop subsystem

This subsystem works as the drive train in a conventional turbine; it consists of a hydraulictransmission which main function is to transfer the mechanical power from the rotorshaft to the base of the turbine. Directly coupled to the rotor shaft, a hydraulic pumptransforms the mechanical energy of the rotor into hydraulic energy by pressurizing ahydraulic fluid, and converts it back into mechanical energy with a hydraulic motor locatedat the base of the turbine. A pump with a radial piston design unit is the most suitablecandidate for the kind of applications required by the DOT (Diepeveen).With the use of this first closed-loop, the need for a mechanical gearbox and powerelectronics used in conventional wind turbines is eliminated from the nacelle; furthermorethe total weight of the nacelle is reduced by using fewer components as shown in figure2.4.

Figure 2.4: Component comparsion for the nacelle of a typical wind turbine and the DOT

The use of high pressures in this subsystem allows to have a high power density, howevermain energy losses will occur due to the mechanical and volumetric efficiencies of the

2.3 Overview of the DOT 9

hydraulic drive systems. Volumetric efficiencies are associated to the fluid leakages fromthe pump and motor, which have to be connected through a drainage line to a fluidreservoir. The reservoir not only has the function to storage the hydraulic fluid, butalso helps to dissipate the heat produced by the internal friction of the hydraulic drives,(associated to their mechanical efficiency). An auxiliary pump is used to add an extrapressure to the return line, and at the same time provide cooled fluid. This extra boostpressure prevents cavitation in the pump ensuring a proper functioning of the system.

2.3.3 Open-loop subsystem

This subsystem has the main function of transforming the mechanical power from theclosed-loop and rotor into hydraulic power by using seawater as a medium to transferenergy from the base of each turbine to the central generator platform. A variable axialpiston pump with servo control is needed in order to have a variable transmission able tomatch the aerodynamic torque with the torque requirements of the load for the varyingwind speeds. This variable transmission between the closed and the open-loop will allowa variable speed operation of the rotor.

The different DOTs in a wind farm should be able to provide hydraulic power despite theirindividual performance: it won’t be unusual that different turbines connected to the sameseawater pipeline would be performing under different operational conditions (one may bein the wake of another turbine or simply in a parked condition among others). Keepingthis in mind and the fact that we are looking for a centralized electricity generation, aconstant pressure in the system is required in order to connect one or more turbines tothe same pipeline, thus transmitting all the hydraulic power from several wind turbinesto the generator platform. The specific lay-out for the piping connection of the differentindividual pipelines has to be studied in more detail; some of the options are presentedin figure 2.5.

Figure 2.5: Possible lay-out configurations for the DOTs connection to central platform

Friction losses are of particular interest due to the large pipe distances from the base ofeach turbine to the central generator platform, nevertheless for the operational conditionsof high pressure and low volumetric flows, these losses can be minimized with the properpipeline design.


2.3.4 Central generator platform

The main goal of this subsystem is to convert the pressurized seawater from individualwind turbine pumping systems into useful work. In this central platform, the pressurizedseawater is distributed over several hydraulic turbines. One of the main characteristicsof the DOT is the idea of a centralized electricity generation. As the power scales ofoffshore wind approaches the output rating of conventional power plants it won’t becomeunusual to think in the possibility of having just one or two central generators instead ofthe several individual generators from all the turbines in a wind farm (i.e. for a 500 MWwind farm, 100 generators of 5 MW are required). A few central generators will facilitatecompliance with typical grid connection requirements for wind power plants (i.e. behaviorduring disturbances in the grid and control of the power quality).

Furthermore, it is known that depending on the ambient conditions and the lay-out ofthe wind farm, power losses will occur due to wake effects. These wake effects representtypically a power loss in the range of 2 to 7% of the power output (Bierbooms). Froma centralized generation point of view, this means that if these losses are properly takeninto account in a preliminary phase, a size reduction of the central generator is possible,leading not only to economical benefits but to higher capacity factors of the entire windfarm.

Figure 2.6: 3D model view of the Gilge Gilbe Pelton Turbine (Reiner)

Hydraulic turbines are a well explored technology that has allowed hydropower to achievethe highest operating efficiencies of all known generation systems with large capacitiesup to 1 GW per unit (Pencho). In a typical hydropower plant, the hydraulic turbinetransforms the high water potential energy into mechanical rotational energy. For theDOT, this idea is also applied for the central generator platform, with the advantagethat hydraulic turbines can be used without the need of large reservoirs or dams due tothe fact that the seawater is already pressurized from the individual pumping systems.

2.4 Hydraulic diagram 11

Another advantage is that high voltage is directly supplied from the hydraulic turbines.

A central generator platform also represents an opportunity for increased availabilityof the wind farm if considered as a strategic place where repair and maintenance canbe done. This idea of a central platform for offshore wind farms as a strategic place foroperation and maintenance (not yet for generation) is already a key component in theplanning of some wind offshore companies like BARD (BARD Offshore).

Figure 2.7: Transformer platform for repair and maintenance crew on site (BARD Offshore)

2.4 Hydraulic diagram

In order to have a better idea of the interaction between the different components ofthe DOT, a specific hydraulic diagram for the proposed concept, including some of theauxiliary equipment is shown in figure 2.8.

In addition, a general overview of the different subsystems and their main componentscan be presented in a functional diagram as shown in figure 2.9.


Figure 2.8: Hydraulic diagram of the Delft Offshore Turbine

Figure 2.9: Functional diagram of the DOT subsystems

2.5 Control overview 13

2.5 Control overview

One of the main objectives of a wind turbine is to maximize the energy capture from thewind taking account of safe operational restrictions. In order to achieve this goal, thereare several control strategies adopted in the wind turbine design, being the variable speed,pitch-controlled the most popular choice in commercial megawatt turbines. In general,the variable speed operation allows the turbine to perform at maximum aerodynamic ef-ficiency for wind speeds between the cut-in and rated wind speed, while the pitch-controlis used to limit the power between the rated and the cut-out wind speed (Bianchi et al.).

The main objective of the variable speed principle is to control the rotational speed ofthe rotor in such a way that the captured power is maximized for all the operating windspeeds below rated conditions. This can be achieved by varying the rotational speed ofthe rotor proportionally to the wind speed. It is the generator torque control, providedby the power electronics, what makes this approach possible in typical turbines.Pitch control for power limitation at high wind speeds is needed to avoid overloading ofthe turbine components and keep a constant power above rated wind speed. It consists onmodifying the pitch angle in such way that the leading edge of the blades is aligned withthe wind (pitch to feather), reducing in this way the angle of attack and consequently theaerodynamic performance. The blade pitching technology is currently based on electricor hydraulic actuators.

Because one of the final goals of the DOT is to minimize the cost of the supplied energy,the energy capture of the wind should be as efficient as possible, thus a variable-speedstrategy will be a suitable option. However, the variable speed operation differs from thetypical turbine due to the absence of the power electronics and the individual generatorthat make this possible. Instead, by adjusting the volumetric displacement in one of thehydraulic drives, the transmitted torque is modified in such a way that the rotationalspeed of the rotor can be controlled to the desired value. From the conceptual idea, thiscontrol is done in the open-loop subsystem, requiring a variable displacement seawaterpump. The control of the volumetric displacement of the pump through the servo mech-anism should be good enough to give an accurate and quick response for the dynamicbehavior of the rotor. A complete block diagram of the system is observed in figure 2.10.


Figure 2.10: Subsystem level block diagram of the DOT

Chapter 3

Basics of wind energy and fluid

power

3.1 Introduction

The main objective of this section is to present the basic principles of wind energy andfluid power. Although there is several literature available regarding both aspects sep-arately, an overview of the basic calculations is presented in order to understand themathematical description behind the DOT model presented in this work. For a moredetailed explanation of both topics, the reader is invited to consult the next references(Manwell et al.; Merritt).

3.2 Wind energy

3.2.1 Energy capture from the wind

The main purpose of a wind turbine is to convert the kinetic energy of the wind into usefulmechanical energy. For a wind stream flowing through a transversal area, the availablepower in the wind is expressed as:

Pwind =1

2ρ A U3 (3.1)

Where ρ is the air density, A is the transversal area and U is the wind speed.

Rotors from conventional wind turbines are designed in such a way that they are able toextract maximum power from the wind. Hence a way of characterizing the ability of arotor to capture wind energy is the power coefficient, which is defined as the ratio of theextracted power to the available wind power:

15

16 Basics of wind energy and fluid power

Cp =Protor

Pwind=

Protor12 ρ A U3

(3.2)

The theoretical maximum value of Cp, known as the Betz limit, is derived from the mo-mentum theory with a maximum value of 16/27 = 0.593

The power coefficient is a value inherent to the specific design of the blades, which istypically given as a function of a parameter called the tip speed ratio λ, and the pitchangle of the blade. The tip speed ratio represents the ratio of the blade tip speed to thefree stream wind velocity:

λ =ω R

U(3.3)

A typical Cp-lambda curve is shown in figure 3.1

Figure 3.1: Typical Cp-lambda curve (Manwell et al.)

To capture maximum power at every wind speed, the rotation speed of the rotor shouldbe changed in order to keep a constant value of the tip speed ratio where the power co-efficient is at its maximum.The total mechanical power extracted by the rotor is expressed in terms of an aerody-namic torque Tr, times the rotational speed of the rotor ωr:

Protor = Tr ωr (3.4)

Since the power extracted by the rotor is proportional to the third power of the windspeed, and taking into account that an optimal aerodynamic efficiency implies a linearincrease in the rotational speed of the rotor with respect to wind speed, it is found thatthe mechanical torque in the rotor is proportional to the second power of the wind speed.

3.3 Hydraulic drives 17

3.2.2 Aerodynamic loads

When the turbine rotor is rotating due to the wind, the rotor shaft experiences a torqueas well as an axial force. Due to its relatively high computational efficiency (Sant), theBlade Element Momentum (BEM) theory is the most common method to calculate aero-dynamic loads. The BEM method makes use of the momentum theory and the Bladeelement theory to estimate the axial force coefficient and induced velocities (velocity re-duction at the plane of the rotor from the free wind speed), from which calculation of theaerodynamic forces exerted on the rotor can be done. A more detailed description of themethod is found in Appendix A.

3.3 Hydraulic drives

Hydraulic pumps and motors are used to convert mechanical energy into hydraulic energyand viceversa, respectively. The hydraulic power obtained from a pump is given by theproduct of the volumetric flow Q and the pressure difference across the pump ∆p:

Phyd = ∆p Q (3.5)

Pumps are normally characterized by the flow obtained at a certain shaft speed, and theratio of these two parameters is known as the volumetric displacement Vp, this value istypically expressed in terms of a volume of fluid per revolution.

Vp =Q

ωp(3.6)

Hence, for an ideal pump analysis, the hydraulic parameters are obtained in terms of themechanical parameters, such that:

∆p =Tr

Vp(3.7)

Q = Vp ωp (3.8)

Pmech = Phyd =

(

Tr

Vp

)

(Vp ωp) (3.9)

A hydraulic pump is not an ideal machine, there are some considerations that should betaken into account like friction, pressure losses and leakage losses among others, so inorder to obtain an optimal efficiency of the system there should be certain relationshipbetween its parameters.


3.3.1 Steady state performance

A model is proposed based on well established theory of positive displacement pumps(Dasgupta & Mandal; Dasgupta et al.; Merritt; Murrenhoff) where some of the termsinvolve constant loss coefficients, and although some discussion has been addressed dueto their validity for a narrow range of operation, for the purpose of this work it givesenough insight of the pump performance in terms of the desired parameters like fluidviscosity, and bulk modulus. More detailed and complex models are found in (Mandal &Dasgupta), where losses are lumped in suitable resistive elements, however they are onlyvalid for mineral oil as a hydraulic fluid.The steady-state continuity equations for a piston hydraulic pump can be derived for thefollowing assumptions:

• Fluid inertia neglected.

• The effect of the pressure and temperature on the properties of the fluid is notconsidered.

• Leakages other than those from the pump are ignored.

• Laminar leakage flow.

• Return pressure neglected (for interest of uniformity of data presentation and sim-plicity of analysis).

Although there are many assumptions, none of them is unduly restrictive. The analysisthat will be described is centered in pumps; however an analogous description is madefor motors. Equations for the steady state analysis of the motor are found in Appendix B.

Generated flow

The net generated flow obtained from the pump is expressed as:

Q = Vp ωp −Qs (3.10)

Where Qs represents the slip flow due to leakages, this flow is usually laminar and there-fore inversely proportional to the viscosity of the fluid µ. The slip flow is also presentedas a function of a dimensionless coefficient of slip Cs:

Qs =

(

CsVp

µ

)

∆p (3.11)

Cs =µ

Vp(Cint + Cext) (3.12)

Cint is the internal or cross-leakage coefficient and Cext is the external leakage coefficient.The net flow is then expressed as:

Q = Vp ωp −(

CsVp

µ

)

∆p (3.13)


Generated pressure

In the same way as the generated flow, not all the torque obtained from the rotor isconverted into pressure. There is a torque that is needed to shear the fluid in the smallclearances between mechanical elements in relative motion, and a torque loss due to thebearing friction; both torques are expressed in terms of a dimensionless damping coeffi-cient Cdamp and a dimensionless friction coefficient Cf respectively.

Vp ∆p = Tr − Tdamp − Tfriction (3.14)

Vp ∆p = Tr − (Cdamp Vp µ ωp)− (Cf Vp ∆p) (3.15)

Hence, the pressure difference obtained from the rotor torque is expressed as:

∆p =1

1− Cf

(

Tr

Vp− Cdamp Vp µ ωp

)

(3.16)

3.3.2 Pump efficiencies

Volumetric efficiency

It is defined as the ratio of flow obtained from the pump to the flow in pump speed (idealflow). The volumetric efficiency describes internal and external leakages and the lossesthrough the compression work

ηvol =Q

Vp ωp(3.17)

Replacing from eq 3.13, the volumetric efficiency becomes:

ηvol = 1− Cs ∆p

µ ωp(3.18)

Mechanical-hydraulic efficiency

It is defined as the ratio of the net torque useful to generate pressure, to the mechanicaltorque supplied to the pump. The hydraulic mechanical efficiency describes the frictionand pressure losses.

ηmech =Vp ∆p

Tr(3.19)

By substituting eq 3.15, the torque efficiency becomes:

ηmech =1

1 + Cdampµ ωp

∆p + Cf(3.20)


Total pump efficiency

The overall or total pump efficiency is defined as the ratio of hydraulic power obtained tothe mechanical power supplied:

ηtot =Phyd

Pmech=

Q∆p

ωp Tr(3.21)

ηtot =

(

Q

Vp ωp

)

(

Vp ∆p

Tr

)

= ηvol ηmech (3.22)

Thus the overall efficiency is simply the product of the volumetric and mechanical effi-ciencies. Using eq 3.18 and eq 3.20, the overall efficiency becomes

ηtot =1− Cs ∆p

µ ωp

1 + Cdampµ ωp

∆p + Cf(3.23)

Therefore, the static performance of a pump is defined by the dimensionless parametersCs, Cdamp, Cf ; the properties of the fluid µ and the operational conditions ∆p and ωp. Agraph of the performance of a pump with constant volumetric displacement is observedin figure 3.2.

∆p/(µω) [−]

η

ηvol

ηmech

ηtot

Figure 3.2: Efficiency curves for a fixed displacement pump

In order to obtain the maximum efficiency in the mechanical to hydraulic power conver-sion, it is desirable to operate in the maximum point of the total efficiency curve. Oncethe pump parameters are established, the maximal theoretical efficiency and the requiredconditions for a steady-state performance are obtained.

Is important to notice from the efficiency curve that there is only one point where max-imum efficiency is achieved for a constant volumetric displacement. Modern pumps can


operate with variable volumetric displacements and are thus able to operate with differentefficiency curves, this is shown in figure 3.3.

∆p/(µω) [−]

η tot

0.1 Vp

0.2 Vp

0.4 Vp

0.6 Vp

0.75 Vp

1.0 Vp

Figure 3.3: Pump perfomance for variable displacement conditions

From last figure it is observed that the total efficiency of the pump increases for highervolumetric displacements, this can be justified by the fact that for larger pistons thereare lower relative leakage losses for the same pump characteristics. It is also observedthat the maximum efficiency for each displacement corresponds to a different value of∆pµ ω . Hence it is easily seen that for every operating value of this parameter, there is onlyone volumetric displacement at which maximum efficiency is reached.

3.3.3 Compressibility effect

No fluid is truly incompressible; when pressure is applied, the air molecules that are en-trained in the fluid can be compressed leading to a change in volume. Hence when acompression work takes place, there is only one part of the total work that is convertedinto useful work. In general, the decrease in volume due a change in pressure is calculatedas:

∆V = −V0 ∆p

E(3.24)

Where ∆V refers to the change in volume, V0 is the initial volume, ∆p is the change inpressure and E is the bulk modulus or stiffness of the fluid.

For a positive displacement pump, the compression work done by one of the cylindersover the column of liquid is observed in the diagram of figure 3.4, where the total work isthe sum of the useful work and the compression work. The compression work is directlyrelated to the compressibility of the fluid (inverse of the fluid bulk modulus or stiffnessE), which in this case is represented by a linear decrease of the volume with respect to


the pressure.

Figure 3.4: Compression work done by one of the cylinders of an axial piston pump (Mur-renhoff)

The compression work is obtained as the area under the right triangle from the diagramsuch that

WK =VUT ∆p2

2E(3.25)

On the other hand the useful work is represented by the area under the left rectangle

WA =

(

VK − VUT ∆p

E

)

∆p (3.26)

Thus the decrease in efficiency due to the compression work is expressed as

∆η =WK

WA=

∆p

2E(

VK

VUT− ∆p

E

) (3.27)

The final generated flow has to take into account the stiffness of the fluid by means ofthis compressibility efficiency, which will depend on the type of fluid. Thus one of theadvantages of using seawater for the open-loop of the DOT is its higher stiffness whencompared to mineral oil which is normally used in hydraulic machines, hence makingseawater a suitable hydraulic fluid in terms of compressibility effects.

3.4 Fluid flow in pipelines 23

3.4 Fluid flow in pipelines

3.4.1 Friction losses

As a fluid is flowing through a pipeline, it experiences some energy losses which are theresult of friction against the wall and viscous dissipation due to internal friction of flow.In a practical way energy losses are traduced as a pressure drop as it flows through thepipeline. The pressure drop in a pipeline ∆p is estimated according to (Pencho) as:

∆p = f

(

L

D

)

1

2ρ v2m (3.28)

Where f , the friction factor is a dimensionless number, L is the total length of the pipe,D the internal pipe diameter, ρ is the fluid density, and vm the average velocity. Thevalue of the friction factor will depend on whether the flow regime is laminar or turbulentaccording to the Reynolds number.

Re =vm D

ν(3.29)

With ν equal to the kinematic viscosity of the fluid. If Re < 2300 the flow is laminar andthe friction factor is computed as:

f =64

Re(3.30)

According to eq 3.30 the friction factor in laminar flow is independent of the wall roughnessand inversely proportional to the Reynolds number. When Re >> 2300 the flow isturbulent and the friction factor can be calculated according to the Colebrook and Whiteexpression and is not only dependent of the Reynolds number but also dependent of thewall roughness e:

1√f= −2log

(

e/D

3.7+

2.51

Re√f

)

(3.31)

It is clear that last expression is difficult to solve by hand, so an iterative calculationon the friction factor is generally the best way to compute a result. A graphical chartderived by Moody shows the friction factor as function of the Reynolds number and theroughness of the pipe, see Appendix C.

3.4.2 Compressibility in long pipelines

Due to the large piping distances, compressibility effects are particularly important for thedynamic behavior of the system in which the stiffness plays a key a factor. However, forsteady-state calculations, compressibility only plays a role in the initial transient responseof the system and doesn’t influence the final energy transfer. This initial time needed toreach a steady state is estimated by calculating the time required for a certain pumpto supply the extra volume of fluid needed to compensate for fluid compressibility givenin eq 3.24. To illustrate the effects of both friction and compressibility, the case of an


ideal pump with constant flow connected to a constant load through a long pipeline ispresented; a schematic is shown in figure 3.5.

Figure 3.5: Schematic of an ideal pump connected to a constant load by a long pipeline

For an ideal pump with a constant flow rate, there is a transient behavior in which thefluid is compressed and pressure is built-up; both friction and compressibility effects havea non-linear dependency and it takes some tenths of seconds before the pressure is built tothe required level (a pressure slightly above the required load pressure to overcome frictionlosses). On the other hand, from the side of the load, although a constant pressure ismaintained, it also takes some tenths of seconds before any flow can really take place; theprocess described before can be observed in figure 3.6a and 3.6b. The volume of liquidrequired to pressurize the system to the desired level is given by the integration of theflow-rate against time, before positive value of flows are obtained.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200Friction and compressibility effect (5 MW D=0.25 m, L=1 km)

Time [s]

Pre

ssur

e ∆p

[ba

r]

PumpLoad

(a) Pressure

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−15

−10

−5

0

5x 10

4 Friction and compressibility effect (5 MW D=0.25 m, L=1 km)

Time [s]

Flo

w R

ate

Q [l

pm]

PumpLoad

(b) Flow rate

Figure 3.6: Friction and compressibility effects for a long pipeline

3.5 Hydraulic turbines 25

3.5 Hydraulic turbines

3.5.1 Principles of operation

Hydraulic turbines are the prime movers that transform the energy content of the waterinto mechanical energy and whose primary function is to drive an electric generator. De-pending on their working principle hydraulic turbines are classified into:

Impulse turbines: in which the driving energy is supplied by the water only in kineticform. The water pressure is converted into kinetic energy in the form of a high speed jetthat hits the runner.Reaction turbines: in which the driving energy is supplied partly in kinetic and partlyin pressure form. Water pressure can apply a force on the face in the runner blades anddecreases as it proceeds through the turbine. A schematic of different turbines is shownin figure 3.7.

(a) Pelton (b) Francis

(c) Kaplan

Figure 3.7: Hydraulic turbines topology (Pencho)

The first criteria for a turbine selection is the net head (or pressure) available. Table 3.1shows the operating head range for different kind of turbines.

With respect to the design value for the rated flow, it is necessary to know the flow regime


Table 3.1: Range of net heads for hydraulic turbines

Turbine type Head range [m]

Kaplan and Propeller 2 < H < 40Francis 10 < H < 350Pelton 50 < H < 1300Michelli−Banki 3 < H < 250Turgo 50 < H < 250

given by the Flow Duration Curve of the site (this is a specific method that gives theproportion of time during which the discharge equals or exceeds certain values); howeverthis method has a completely different interpretation from normal hydro to wind powerapplications where this flow has to be determined by the hydraulic performance and powerduration originated from the wind, which in addition is site dependent.

The main characteristics of a hydraulic turbine are defined as follows:

Efficiency:

ηturbine =PT

Phyd=

T ω

ρ g HT Q(3.32)

Where the hydraulic power is the product of the fluid density ρ, the gravitational accel-eration g, the total net head HT , and the flow rate Q.

Specific speed:

ns = n

√1000 P

H5/4(3.33)

The specific speed corresponds to a turbine that for a head of 1 meter produces 1 kW ofgenerated power. This parameter is modified by changing the rotor speed n, the turbinehead H, or by splitting the total power in multiple turbine rotors or injectors.

A general overview for turbine selection is observed in figure 3.8, where the operationalenvelopes for different hydraulic turbines is shown for different conditions of rated flowand net head.

The specific speed is a better indicator than the head range for the proper selectionof the type of turbine for a specific application, from literature (Pencho), a correlationbetween the type of turbine, characteristics speed and net head has been obtained; suchrelationship is observed in figure 3.9.


Figure 3.8: Operational envelope of several hydraulic turbines (Pencho)


Figure 3.9: Specific speed correlation for different hydraulic turbines (Pencho)

3.5.2 The Pelton turbine

From table 3.1 it is seen that for high head applications, the Pelton turbine is the mostsuitable. In this type of turbine, the high pressure water is converted into high velocitywater jets by a set of fixed nozzles. Mechanical torque is produced at the turbine shaftwhen the high-speed water jets hit the bowl shaped buckets placed around the turbinerunner. The needle is actuated by the turbine governor with a servomotor, and in the caseof a sudden load reduction, the water jet is deflected from the buckets by a jet deflector.

Figure 3.10: Cross section of a nozzle with jet deflector (Pencho)

An important advantage of the Pelton turbine is the ability to perform efficiently notonly at rated and partial loads, but over a wide range of flow rates starting from 10% ofthe rated flow. Thus Pelton turbines are a likely candidate for wind power applications,an example of their efficiency is shown in figure 3.11.For a Pelton turbine, once the characteristic speed is obtained, the general dimensionsof the turbine are obtained from relationships between the velocity leaving the nozzle


Figure 3.11: Efficiency comparison of a six jet Pelton turbine with a Francis turbine (Reiner)

and the tangential velocity needed in the pelton runner for optimum efficiency (Pencho).Hence the diameter in meters D, of the circle describing the buckets centre line are ob-tained as function of the total net head H, and the generator speed n in rpm:

D = 38.567

√H

n(3.34)

The range of speeds for a synchronous generator can be obtained according to the electricfrequency and the number of poles as shown in table 3.2 (Boldea).

Table 3.2: Generator synchronization speed

Number Frequency Number Frequencyof poles 50 Hz 60 Hz of poles 50 Hz 60 Hz

2 3000 3600 16 375 4504 1500 1800 18 333 4006 1000 1200 20 300 3608 750 900 22 272 32710 600 720 24 250 30012 500 600 26 231 27714 428 540 28 214 257

Chapter 4

Preliminary design

4.1 Introduction

The final objective of this chapter is to get an initial dimensioning of the principal compo-nents based on the design requirements of the system and the presented equations of lastchapter. According to the conceptual design and working principle of the DOT, thereare specific design requirements that the system should be able to comply. Moreover,operational limits for different components have to be taken into account for a properdimensioning of the system.

4.2 Design requirements

The design requirements are established as follow:

1) Wind turbine rotor:For the purposes of this work, the rotor is the starting point of the design. It is importantto mention that although the final goal of the DOT is to use a two bladed rotor, thedetailed design of this particular rotor is beyond the scope of this work; thus an existinghorizontal axis, upwind, three bladed rotor was chosen as the starting point. Moreoverthis rotor will provide a baseline for comparison of results with conventional turbines. Inaddition, the possibility of using this baseline rotor in a high-speed operation will also beanalyzed.

The NREL turbine is a conventional three-bladed upwind variable-speed variable blade-pitch-to-feather controlled turbine, which has been accepted by UPWIND as the offshore5 MW baseline wind turbine in order to asses offshore wind technology. This turbine willbe used as a reference to quantify the benefits of the DOT concept. Some of the grossproperties are observed in table 4.1, more detailed information can be found in (Jonkmanet al.).

31

32 Preliminary design

Table 4.1: Main characteristics of the NREL turbine

Wind regime IEC 61400-3(Offshore) Class 1B/ Class 6 wind

Rotor orientation Upwind

Hub Height 90 m

Rotor diameter/ Hub diameter 126 m/ 3m

Power 5 MW

Total Weight 872 ton

Cut in wind speed Vci 4 m/s

Rated wind speed Vrated 11.4 m/s

Cut out wind speed Vco 25 m/s

Overhang/ Tilt/ Precone 5 m/ 5o/ 2.5o

Control Variable speed/ Collective pitch

Minimum rotor speed 6.9 rpm

Maximum rotor speed 12.1 rpm

Maximum tip speed 80 m/s

2) Seawater pipeline at constant pressure:From the idea of having a centralized electricity generation, this requirement is imposedin order to have several pipelines connected to the central platform where hydraulic powerfrom the different pumping systems is converted into electricity. Because long pipelinesare required in order to collect the hydropower from the different wind turbines in thewind farm, friction losses play an important role; nevertheless with the proper design ofthe pipelines this problem can be minimized.

3) Availability of hydraulic drives components:Although the existence of hydraulic drives is not yet commercially available for multiMW applications, it appears that there is no technical reason for the required upscaling.Taking this into account the current work will assume the possibility of hydraulic driveswith rated capacities of at least 5 MW .

4.3 Operational limits

Beside the design requirements, operational limitations are imposed by the current tech-nology of the different components. It has to be taken into account that these restrictionscould vary in the future years, not only modifying the proposed design but giving oppor-tunity for new or improved concepts and proposals.

4.3 Operational limits 33

4.3.1 High tip speed rotor

As mentioned in section 4.2 the rotor of the NREL offshore 5 MW wind turbine is used,nevertheless a new concept regarding high tip speed operation for this rotor is also ana-lyzed.

Acoustic noise emission has restricted the design tip-speed of land base turbines, withvalues up to 80 m/s for a variable speed operation. For wind turbines designed specifi-cally for offshore, the noise emission is no longer a restriction, allowing a possibility forhigh speed rotors. Several studies have concluded that high speed operation is feasibleregarding power performance for operating tip speeds up to 120 m/s for a typical rotorof a 5 MW turbine (Jamieson; Knauer & Hanson).

A typical variable speed rotor will perform optimally below rated-speed, reaching thetip speed limit around 80 m/s very closely to this rated wind speed. However if the samerotor with the same parameters is used (same radius, tip-speed ratio) for a high speedoperation, the range of the operational rotor speed can be extended beyond rated windspeed while keeping a constant torque to keep mechanical loads at the design values. Thesame pitching control strategy is used to keep the rotor torque constant.

Is important to mention that this high-speed rotor concept is not inherent to the DelftOffshore Turbine, but can be generalized to the current wind turbine technology since themain design driver in the electrical subsystem is determined by the generator maximumtorque. It is beyond the scope of this work to give a detailed insight of other designdrivers that could limit the feasibility of this option (i.e. fatigue analysis or ultimateload analysis); nevertheless the potential benefits of a high-speed rotor configuration areconsidered for the final results of this work.

4.3.2 Hydraulic drives

For the hydraulic drives operational conditions were taken from (Hagglunds Drives) andsummarized in table 4.2.

Table 4.2: Nominal values for commercial hydraulic drives

Parameter Value Unit

Max pressure 350 [bar]Charge pressure 15 [bar]Mechanical efficiency* 0.98 [−]Volumetric efficiency 0.95 [−]Temperature range -35 to 70 [oC]Oil viscosity 20-40-1000 [cSt]

*Not valid for start-up conditions


4.3.3 Seawater pipeline pressure

In general, high power density of hydraulic drives is mainly possible due to the high oper-ating pressures. Because hydraulic power is the product of a pressure times a flow, highpressures are desired in order to minimize the flow velocity and reduce friction losses.However, due to the lower viscosity of seawater when compared to hydraulic oil, higherleakages through the seals and internal clearances of the hydraulic components are morelikely to occur when operating at high pressures, see eq 3.11. On the other hand a constantpressure (or constant head) requirement at the side of the hydraulic turbine is alreadyimposed. As mentioned before in chapter two, the required head for a Pelton turbine isin the range of 1000− 2000 m (100− 200 bar) (Pencho).

At this early stage, a value of 150 bars was chosen as a suitable value for the requiredpressure in the seawater pipeline, which will be confirmed afterwards in the preliminarydimensioning of components.

4.3.4 Pelton turbine

A single jet Pelton turbine generally operates with a specific speed in the range of 12and 26 rpm, for a head of 2000 and 100 meters respectively; the specific speed increaseswith the square root of the number of jets, with a maximum value of 60 rpm (Pencho).The max number of jets used in a Pelton turbine in existing turbines is six as stated by(Reiner). Hence, the range for the specific speeds is summarized in table 4.3. In terms ofrated output, the largest Pelton turbines have capacities up to 423 MW per unit (Kecket al.).

Table 4.3: Specific speed range for a Pelton turbine

No of jets ns,min ns,max

[−] [rpm] [rpm]

1 12 262 17 373 21 454 24 525 27 586 29 60

4.4 Dimensioning of components 35

4.4 Dimensioning of components

4.4.1 Radial pump dimensioning

The initial point of design is given by the aerodynamic torque from the wind. Dimen-sioning of the pump is based on the rotor torque and the maximum operational pressureof the pump. From eq 3.19 the required size of the pump is given by:

Vp =

(

Tr

pmax − pcharge

)

ηmech (4.1)

From last equation it is observed that as the allowed pressure increases, the size of thepump will reduce. In the same way, a smaller pump will be needed for smaller torquerequirements, for example in the case of a two bladed rotor as stated by (Jamieson). Fora max pressure of 350 bar, a pump of 784 L/rev is required as shown in figure 4.1.

100 200 300 400 500 600 7000

500

1000

1500

2000

2500

3000

3500

Pump operating pressure [bar]

Pum

p vo

lum

etric

dis

plac

emen

t [L/

rev]

Figure 4.1: Radial pump dimensioning for different operational pressures

4.4.2 Hydraulic motor and variable pump dimensioning

The dimensioning of both components is directly related depending on the operating pres-sure in the closed-loop subsystem and the pressure in the seawater pipeline, from eq 3.19this relationship is obtained:

Vp var

Vm=

(

pmax − pchargepseawater line

)

ηmech p ηmechm (4.2)

As shown in figure 4.2, the ratio of sizes between the hydraulic motor and the seawaterpump will increase for higher operational pressures in the closed-loop transmission anddecrease for higher seawater pressures. For an operating pressure in the closed-loop of350 bar, and for the operating pressure of the Pelton turbine of 150 bar, the size of thevariable pump will be approximately the double of the hydraulic motor. The hydraulicmotor with the characteristics of the Hagglunds series CB − 400 will be used (HagglundsDrives). A summary of the selected hydraulic drives is shown in table 4.4.


100 200 300 400 500 600 7000

1

2

3

4

5

Pump operating pressure [bar]

Rat

io o

f pum

p−m

otor

siz

es [−

]

p

seawater= 150 bar

pseawater

= 100 bar

pseawater

= 200 bar

pseawater

= 300 bar

Figure 4.2: Seawater pump and hydraulic motor size ratio for different operational pressures

Table 4.4: Summary of the chosen hydraulic drives characteristics

Nominal parametersHydraulic type V n pnom ν ηvol ηmech

Drive [L/rev] [rpm] [bar] [cSt] [%] [%]

Radial pump fixed 784 12.1 350 40 95 98Radial motor fixed 25.1 378 350 40 95 98Axial pump variable 53.84 378 150 1.2 95 98

4.4.3 Pipelines

Once the size of the hydraulic drives are obtained, the diameter of the hoses and pipelinesare calculated in such way that friction losses are minimized for the nominal volumetricflow at rated conditions. Physical properties of mineral oil and seawater are shown intable 4.5.

Table 4.5: Physical properties for mineral oil and seawater at operating temperatures

Parameter Unit Mineral oil Seawater

Temperature [oC] 54 15Density [kg/m3] 843 1030Kinematic viscosity [cSt] 44 1.17Bulk modulus [GPa] 1.54 2.34

Closed-loop pipelines

Because mineral oil is used, a first estimate of the required pipe diameter is done with thestandard ISO 4413 (Albers) in which a maximum flow velocity in the pressure pipe is setto 5 m/s. Using the selected pump size of 784 L/rev and the rotational speeds of 12.1and 18.2 rpm for normal and high-speed operation respectively, the mean flow velocity isobtained for different diameters of the pipe as shown in figure 4.3.

4.4 Dimensioning of components 37

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

8

10

Closed loop hose internal diameter [m]

Mea

n F

luid

Vel

ocity

[m/s

]

n=12.1 rpmn=18.2 rpmV

max ISO 4413

Figure 4.3: Mean fluid velocity for different diameters of the closed-loop pipeline

From last figure an initial diameter of 0.25 m will be used as starting point of the design.A total length of 110 m including the equivalent length of local resistances (due to bends,fittings, inlet/outlet losses, etc) is estimated taking into account the hub height of 90 m(Jonkman et al.), resulting in a pressure loss of roughly 1 bar as seen in figure 4.4.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

Closed loop hose internal diameter [m]

Pre

ssur

e lo

ss [b

ar]

n=12.1 rpmn=18.2 rpm

Figure 4.4: Pressure loss for different diameters of the closed loop pipeline

Seawater pipeline

The selection of the pipe diameter will be based directly in the resulting pressure loss.As it is seen in eq 3.28, these losses are directly proportional to the total length of thepipe; therefore a total length of 1100 m will be used as an indication of pressure drop perkm of pipeline. A diameter of 0.35 cm was selected for a pressure loss of 5 bar per km asit is shown in figure 4.5. A summary of the parameters for the chosen diameters of thepipelines is observed in table 4.6.


0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

20

Seawater pipeline internal diameter [m]

Pre

ssur

e lo

ss [b

ar/k

m]

n=12.1 rpmn=18.2 rpm

Figure 4.5: Pressure loss per km of the open-loop seawater pipeline

Table 4.6: Summary of the chosen pipelines

Nominal parametersPipeline Hydraulic Dint e vmean Re ploss

Fluid [m] [mm] [m/s] [−] [bar/km]

Closed loop mineral oil 0.25 0.0016 4.8 2.7e4 10Open loop seawater 0.35 0.0016 5.3 1.6e6 5

4.4.4 Hydraulic turbine

The number of total turbines needed in the central platform, will be determined by therated output of a single Pelton turbine. According to eq 3.33, the rated output can beexpressed in terms of the total head and rotational speed of the generator and turbine.Adjusting the equation to the number of jets used N :

P =1000

N

(

ns

nH1.25

)2

(4.3)

Taking into account a constant head of 1500 m, and the proper operating range of speedratios, rated power is obtained for different number of jets as shown in figure 4.6.

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

Speed ratio (n/ns) [−]

Rat

ed p

ower

[MW

]

N=1N=2N=3N=4N=5N=6

Figure 4.6: Maximum power for multiple jet nozzles of a Pelton turbine

4.5 Natural frequency analysis 39

From eq 3.34 the diameter of the runner of the turbine is obtained for the specific head(or pressure from the seawater line):

D = 38.567

√H

n(4.4)

The diameter of the runner of the turbine is computed for different speeds of the generatoras shown in figure 4.7.

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

Generator speed [rpm]

Pel

ton

runn

er d

iam

eter

[m]

Figure 4.7: Pelton runner diameter for different generator speeds and number of jets

From previous graphs and equations, a central platform of 500 MW and a net head of1484 m (for 150 bar of seawater pressure) would need only 1 or 2 Pelton turbines, themain data is listed in table 4.7.

Table 4.7: Pelton turbine data for a 500 MW central platform at 150 bar

Parameter Unit Option 1 Option 2

Rated output (per unit) [MW ] 250 500No. of units [−] 2 1No. of jets (per unit) [−] 5 5Outer diameter of runner [m] 2.5 3.5

4.5 Natural frequency analysis

Due to the harmonic excitations caused by the rotational frequency of the rotor and therotor blade passing frequency, the calculation of the natural frequencies of the hydraulicsubsystems are essential for the preliminary analysis of the system. If the natural fre-quency of the closed or open loop transmission lies within the operating frequency range ofthe turbine, resonance would occur resulting in undesired high forces and displacements.The frequency operation range for the chosen baseline turbine is seen in table 4.8.


Table 4.8: Rotor frequency range operation

Frequency range [Hz]

NREL rotor High-speed operation

Rotor speed range (1P region) 0.115− 0.202 0.115− 0.303Blade passing frequency range (3P region) 0.345− 0.605 0.345− 0.909

It is important to remark that for a two bladed turbine (which is the final intention ofthe DOT) the frequency range will change in both the rotational speed and passing bladefrequency.

4.5.1 Closed-loop natural frequency calculation

From basic dynamic calculations, the natural frequency of the closed loop subsystem willbe given by the rotational oil spring stiffness Co of the transmission and the mass momentof inertia of the system Jr such that:

ωo cl =

√

Co

Jr(4.5)

According to (Schmitz & Vatheuer) a simplified model is used to calculate analytically thedifferent parameters. The model only takes into account the rotor’s mass of inertia, andthe pump and oil volume that acts as a spring in the high pressure line. The simplifiedmodel is shown in figure 4.8.

Figure 4.8: Simplified model for the closed-loop natural frequency calculation (Schmitz &Vatheuer)

For a radial piston pump the oil volume has to be corrected according to the number ofcams of the pump. Resulting equations are as follow:

Co =2V 2

p

CH(4.6)

CH =πVp

b + Vpipe

Eoil(4.7)

4.6 Control strategy 41

Where Vpipe is the volume of fluid contained in the pressure line, and b is the number ofcams of the pump. For the closed-loop subsystem all the required values were alreadyobtained through this chapter; considering eight cams in the radial pump (HagglundsDrives) and a rotor inertia of 38,779,900 kg m2 (Jonkman et al.), the resulting naturalfrequency is:

ωo cl = 0.4761[

rads

]

= 0.0758 [Hz]

The obtained value of the natural frequency is below the rotor frequency ranges dueto the high inertia of the rotor and low stiffness caused by the high volume of hydraulicfluid in the pressure line; therefore no resonance problems are predicted.

4.6 Control strategy

In order to obtain a variable speed operation of the turbine, and thus optimal aerody-namic efficiency, the volumetric displacement of the seawater pump has to be continuouslyadjusted. An overview of the proposed controller system is observed in figure 4.9.

Figure 4.9: Schematic of the control strategy for a variable speed operation

By means of measuring the wind speed and the rotational speed of the rotor, a deviationfrom the optimal rotor speed can be calculated. The magnitude of this error, togetherwith the measurements of the rotor speed and pressure across the variable pump, will becomputed by the servo controller which will give a signal to control the displacement ofthe pump. The displacement of the pump will determine the torque transmitted to theclosed-loop and rotor and thus will be able to accelerate or decelerate the rotor to thedesired optimum speed.

From eq 3.3 it can be seen that for maintaining a constant tip speed ratio and thus


maximize energy capture below rated conditions, the rotor speed must change propor-tionally to the wind speed:

ωopt =λopt U

R(4.8)

The required torque of the rotor according to the characteristic torque-speed curve canbe obtained from eq 3.2, eq 3.3 and eq 3.4 such that:

Topt =1

2λ3opt

ρ π R5 CPmax ω2opt = K1 ω

2opt (4.9)

This torque has to be transmitted to the rotor by the variable pump by means of adjust-ing its volumetric displacement. From eq 3.19, the displacement of the pump needed totransmit the required torque is derived such that:

Vp var =Tm

∆pvar pηmech var p =

K1 ω2opt

∆pvar p

(

Vm

Vp

)

ηmech var p (4.10)

Therefore,

Vp var = K2

ω2opt

∆pvar p(4.11)

Last equations will allow a feedback control on the rotor speed for a variable-speed oper-ation by means of hydraulic drives.

Chapter 5

Computational model

5.1 Introduction

Due to the complexity and nonlinearity of the equations described in chapter three, acomputational model has been developed to simulate the performance of the Delft Off-shore Turbine. The computer model works under the Matlab -simulink platform, whichitself allows to perform time-based simulations for dynamic systems (MATLAB).

The objective of this section is to describe the structure of the model and describe itsmain characteristics and limitations. First an overview of the method used for the con-struction and solution of the equations is presented, followed by a description for thedifferent subcomponents.

5.2 The physical network approach overview

The computational model used in this work employs the physical network approach whichis particularly suited to simulate systems that consist of real physical components. In thisapproach each system is represented as consisting of functional elements that interact witheach other by exchanging energy through their ports. These connection ports are bidirec-tional and they mimic physical connections between elements, in other words blocks inthe model are connected in the same way that real physical components are connected.Each energy flow is characterized by a ’through variable’ and an ’across variable’, whoseproduct is usually an energy flow. For example, for mechanical translational systems,these variables are force and velocity; for mechanical rotational systems, torque and an-gular velocity; for hydraulic systems, flow rate and pressure; for electrical systems, currentand voltage.

A physical network is constructed based on the following principles:

43

44 Computational model

• Conserving ports directly connected have the same values for all their across vari-ables (such as pressure or angular velocity).

• Any through variable (such as flow rate or torque) transferred along a physical con-nection line is divided among the multiple components connected by the branches.For each through variable, the sum of all its values flowing into a branch pointequals the sum of all its values flowing out.

A very important advantage of the physical network approach is the possibility for anincremental modeling; this means that it allows substituting models of different levels ofcomplexity without introducing any changes to the schematic. For example, in an earlystage a pipeline can be only represented by a hydraulic resistance block, which accountsonly for friction loses. At a later stage, it may be important to take fluid compressibilityinto account, then the ’Hydraulic resistance’ block can be substituted by a ’Hydraulicpipeline’ block.

5.3 Model description

5.3.1 General

The general model consists of a single hydraulic turbine connected to a central platformthrough a single pipeline. Although the Delft Offshore Turbine implies the connection ofseveral turbines, the evaluation of a single turbine will give enough level of detail for afirst evaluation and comparison with other conventional turbines. At the central platformthe load characteristic is defined in terms of a required constant pressure. The behaviorand performance of the system can be evaluated as function of a uniform wind flow thatis defined by the user. The general model is observed in figure 5.1.

Figure 5.1: General model of a single DOT overview

As explained in the last chapters, each individual turbine consists of a rotor, a closed-loop

5.3 Model description 45

an open-loop transmission, and a controller. In the model this subsystems are intercon-nected between each other as observed in figure 5.2; it is important to mention thatstraight lines represent physical connections between the subsystems, while arrowed linesrepresent signals coming from different sensors.

Figure 5.2: DOT subsystem interaction

The density, kinematic viscosity and the stiffness of the different hydraulic fluids are as-sumed to take a constant value for the operating temperatures and will be a design choiceaccording to the fluid of interest, in this case mineral oil for the closed-loop and seawaterfor the open-loop.

5.3.2 Rotor

Given a certain wind speed and the parameters of a variable speed wind turbine, aero-dynamic loads, power and pitch angle are calculated through a simplified Blade ElementMomentum Method (BEM) code. For steady-state calculations and simplicity of analysisthe next assumptions were made:

• Uniform flow (i.e. wind speed constant over rotor plane); no yawed flow, wind shearor tower shadow).

• No wake rotation (i.e. no tangential induction factor).

• No blade tip loss factor.

• Just one annular section (the total rotor plane).

Although several assumptions were made, the code gives enough level of detail for theintended purposes of this work; validation of the results obtained are compared withthose of GH Bladed, a commercial wind simulation program (Bossanyi), see Appendix D.Furthermore this code allowed a straight forward modification in the control strategy inorder to simulate the proposed high-speed configuration of the rotor.


5.3.3 Closed-loop

This subsystem in the model is built with a fixed displacement pump, a hydraulic motor,two hydraulic pipelines to account for the pressure and return lines, and a charge pumpwhich is able to deliver a constant pressure to the return line. The schematic of the closeloop is shown in figure 5.3. Sensors are also shown in green color.

Figure 5.3: Schematic of the Closed-loop subsystem

As it was seen in chapter three, the performance of hydraulic drives was defined by thedimensionless parameters Cs, Cdamp and Cf . However for the required operational condi-tions of volumetric and mechanical efficiencies, these parameters were obtained in termsof the nominal values of pressure, rotational speed, and viscosity at rated conditions. Thederivation of these relationships is found in Appendix E.

Other assumptions for the hydraulic drives used in the model are:

• Fluid compressibility is neglected.

• No loading on the pump shaft such as inertia, friction, spring and so on is considered.

• Leakages are assumed to be linearly proportional to its pressure differential.

5.3 Model description 47

• For the piping lines, the flow is assumed to be fully developed along the pipe lengthand fluid inertia is not taken into account.

5.3.4 Open-loop

The open-loop subsystem for a single turbine is modeled by variable displacement pump,whose displacement is controlled by the signal provided by the controller. Flow rate andpressure sensors are also included. The assumptions for the variable pump are the sameas for the fixed pump used in the closed-loop. An overview of this subsystem is shown infigure 5.4.

Figure 5.4: Schematic of the Open-loop subsystem

5.3.5 Controller and scopes

All the sensors and actuators are considered to have an ideal behavior. An ideal controlwas established according to the equations derived in the proposed control strategy; asaturation limit on the pressure sensor was used in order to avoid division by zero in thecontroller.

After the simulation is finished, results for the different operational parameters are ob-served graphically within the same simulink model through a ‘scope’ subsystem as shownin figure 5.5.


Figure 5.5: Schematic of the scopes for different parameters of the simulation

5.4 How it works

The general procedure of the programme is illustrated in figure 5.6.

Figure 5.6: General procedure of the DOT simulation

5.4.1 Initialization

This part of the programme loads the directories where all the required files are locatedand reads the input data for the simulation. The input data is given in a Matlab

file (m-file), and it contains all the required parameters for the different components in

5.4 How it works 49

the model, as well as physical properties to be used. Data specified in the Input file:DOT parameters.m

• Physical properties .

• Rotor parameters.

• Pumps and motor parameters.

• Pipeline parameters.

• Load parameters.

5.4.2 Simulation

Based on the input data, a physical network is built for the model from which equationsare constructed. Initial conditions are checked and solution of the equations is performed.A more detailed description of the simulation method can be found in Appendix F.Main simulation is done through a simulink model file. In order to perform a successfulsimulation, a series of Matlab files are required in the model; these files correspond todifferent sub-routines needed for the calculations of the aerodynamic loads. A schematicis shown in figure 5.7.

Figure 5.7: Subroutines and Matlab files needed for the DOT simulink model

5.4.3 Post processing

Once the simulation process has finished, the post processing of the results is done throughanother Matlab file. Plots are obtained for the different operational characteristics ofthe system like pressures, torques, flows or rotational velocities, therefore also efficiencies


and performance for every subsystem can be displayed.

The powercurve of the system, together with the wind data through a Weibull distri-bution is used to estimate and compare the gross annual energy yield for the DOT anda typical 5 MW turbine. Examples of the output graphs and results are shown in nextchapter.

Chapter 6

Simulation results

6.1 Introduction

This chapter shows the simulation results for the steady-state parameters of a single DOT.Operational conditions for the different subsystems are presented and the performanceof the system is compared with a 5 MW reference turbine. Finally an estimation of theannual energy yield is given for typical wind conditions of the North Sea.

6.2 Steady-state operational parameters

6.2.1 High speed rotor operation

The first results that will be analyzed are those regarding the high-speed operation of therotor as proposed in the preliminary design. Steady-state parameters were obtained withthe BEM method as function of the wind speed.

0 5 10 15 20 250

2000

4000

6000

8000

10000

Wind Speed [m/s]

Mec

hani

cal P

ower

[kW

]

Power typicalPower High−speed

0 5 10 15 20 250

2000

4000

6000

8000

10000

Tor

que

[kN

m]

Torque

Figure 6.1: Power and Torque curves for a 5 MW turbine in both typical and high-speedoperation

51

52 Simulation results

As observed in figure 6.1 there is a significant modification of the power curve for thehigh speed operation, with a potential gain of power of more than 2 MW for wind speedsabove 17 m/s. This result gives an enormous potential for increased energy productionespecially for higher wind speeds.

Regarding the loads, the maximum values of the thrust and torque are kept to the samelevel, which means that for steady-state results, a high speed operation is possible withoutexceeding the structural requirements. The pitch rate needed for normal operation is inthe range of 0 to 25 degrees, while for a high speed operation it is reduced to around16 degrees as observed in figure 6.3, hence a relaxation of the required pitch rate in thecontroller can also be achieved.

0 5 10 15 20 250

100

200

300

400

500

600

700

800

Wind Speed [m/s]

Thr

ust f

orce

[kN

]

Thrust typicalThrust High−speed

Figure 6.2: Thrust curve for a 5 MW turbine in both typical and high-speed operation

0 5 10 15 20 250

5

10

15

20

25

Wind Speed [m/s]

Rot

or s

peed

[rpm

]

Speed typicalHigh−speed

0 5 10 15 20 250

5

10

15

20

25P

itch

angl

e [d

eg]

Pitch typicalPitch High−speed

Figure 6.3: Rotor speed and pitch angle comparison for a for a 5 MW turbine in both typicaland high-speed operation

For wind speeds above 11.4 m/s (from which torque is kept constant) the high speedoperation implies a smoother reduction of the aerodynamic parameters when compared

6.2 Steady-state operational parameters 53

to the typical operation as shown by the aerodynamic coefficients in figure 6.4.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Wind Speed [m/s]

Coe

ffici

ents

[−]

Cd typicalCd High−speedCp typicalCp High−speed

Figure 6.4: Dimensionless coefficient comparisons for a 5 MW turbine in both typical andhigh-speed operation

Although from a steady-state point of view the high-speed rotor concept is plausible,the final feasibility of this option will de determined by other design drivers like fatiguedamage and ultimate load requirements.

6.2.2 Hydraulic drives operation

The operational results for the flow rates and pressures of the hydraulic drives in theclosed and open loop was obtained as a function of wind speed. As observed in figure6.5, for the closed-loop pressures above 100 bar are observed above wind speeds of 6 m/s,until the final operational value of 350 bar is reached at rated wind speeds.

0 5 10 15 20 250

50

100

150

200

250

300

350

400

Wind Speed [m/s]

Pre

ssur

e [b

ar]

Closed−loopOpen−loop

Figure 6.5: Steady-state pressure for the closed and open loop of the DOT

For the open-loop a nearly constant operational pressure is observed; an increase in pres-sure in the order of a few bars is observed for higher wind speeds due to higher volumetricflows, which result in increased friction losses. When comparing the hydraulic parame-ters, it is important to point out the difference in hydraulic fluids used in the closed andopen loop (mineral oil and seawater respectively).


0 5 10 15 20 250

0.5

1

1.5

2

2.5

3x 10

4

Wind Speed [m/s]

Flo

w r

ate

[lpm

]

Closed−loopOpen−loop

Figure 6.6: Steady-state flow for the closed and open loop of the DOT

According to the proposed control strategy, a variable speed operation can be obtainedby modifying the volumetric displacement of the seawater pump of the open-loop. Thetheoretical required value according to the steady-state operational results is shown infigure 6.7 for different wind speeds.

0 5 10 15 20 250

10

20

30

40

50

60

Wind Speed [m/s]

Sea

wat

er A

xial

pum

pvo

lum

etric

dis

plac

emen

t [L/

rev]

Figure 6.7: Steady-state values for the required volumetric displacement of the seawaterpump

Because the closed-loop works only as a fixed transmission (both the radial pump andmotor have a fixed displacement), the speed and torque characteristics of hydraulic motorare similar to the rotor with an increase in speed and decrease in torque as shown in figure6.8.

6.3 Performance 55

0 5 10 15 20 250

75

150

225

300

375

450

525

600

Wind Speed [m/s]

Spe

ed [r

pm]

speed

0 5 10 15 20 250

20

40

60

80

100

120

140

160

Tor

que

[kN

m]

torque

Figure 6.8: Hydraulic motor steady-state mechanical characteristics

6.3 Performance

6.3.1 Efficiencies

The efficiency of an individual DOT can be defined in terms of energy transfer betweeneach subsystem, hence from the captured energy of the wind to the central generatorplatform where a final useful energy flow is obtained as electricity.After energy capture from the wind, mechanical power is transmitted from the rotor tothe base of the turbine through the closed-loop transmission, where leakage and torquelosses of the hydraulic transmission have to be taken into account. In what it respectsto the open-loop transmission, friction losses and efficiency of the variable pump are themain factors that affect the total energy transfer. A final efficiency regarding the hy-draulic turbine and generator are considered in the central generator platform. Fromsteady-state parameters the energy transfer efficiency of the main components of the sys-tem is observed in figure 6.9.

0 5 10 15 20 250.7

0.75

0.8

0.85

0.9

0.95

1DOT Performance

Wind Speed [m/s]

Effi

cien

cy [−

]

FixPumpMotorVarPumpSewater PipelineTurbine−gen

Figure 6.9: Efficiency of the main components of the DOT for steady-state conditions

From the last figure it is seen that the hydraulic drives of the closed-loop operate at highefficiencies between 94 and 96% for wind speeds between 4 and 25 m/s. For the seawatervariable pump, a lower efficiency from the rest of the components is observed, where the


slow asymptotic increase is dictated by the change in volumetric displacement imposed bythe control strategy. Because of a constant pressure load in the seawater pump, leakagelosses are of great significance especially at low wind speeds; therefore although seawaterpipeline losses are relatively small, there is only an effective flow rate for wind speedsabove 4 m/s.

The consequence of each of these efficiencies in the total energy transfer can be observedas a series of power curves for a single turbine resulting in an overall efficiency of 80% asshown in figure 6.10.

0 5 10 15 20 250

2000

4000

6000

8000

10000DOT Performance

Wind Speed [m/s]

Pow

er [k

W]

Mech PowerClosed−Loop mechPowerHydr Power at baseHydr Power central platformElect Power

Figure 6.10: DOT energy transfer performance

6.3.2 Power curves

Once the performance for the different subsystems is obtained, a final power curve is com-pared with that of a typical wind turbine, in this case the NREL 5 MW turbine whichwas used as the starting point of the design. The resulting steady-state power curve forthe different wind speeds is shown in figure 6.11.

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000Steady−State Power Curves

Wind Speed [m/s]

Pow

er [k

W]

DOTNREL

Figure 6.11: Power curve comparison of the DOT and the NREL 5 MW turbine

6.3 Performance 57

The performance of a single DOT is lower for wind speeds below 14 m/s mainly dueto the efficiencies of the hydraulic drives, however it is the high speed operation of therotor what makes possible a modification of the power curve with a potential gain of1.5 MW with respect to the typical turbine for wind speeds above 17 m/s. Total en-ergy production will depend not only on the power curve but also on the prevailing windconditions at a specific location.

6.3.3 Energy yield

A better evaluation of the DOT turbine is done by estimating the energy production ina year. Estimation for the gross annual energy yield is done by combining the resultingpower curves with the wind speed statistical distribution of a particular site. Wind statis-tics are usually described by the Weibull probability distribution function, which dependson two parameters, a scale and a shape factor. More detailed information about energyyield calculations and wind distribution are given in Appendix G.

A comparison of a single DOT and the NREL turbine is done by plotting the grossannual energy yield for different mean wind speeds at hub height, see figure 6.12. Forcomparison purposes, a value of 2.2 for the shape factor was assumed as a representativevalue of the northwest Europe coast (Bierbooms).

7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 1210

15

20

25

30

35

Mean Wind Speed at hub height [m/s]

Gro

ss a

nnua

len

ergy

yie

ld [G

Wh/

a]

DOTNREL

Figure 6.12: Annual energy yield comparison for the northwest Europe coast

Last figure shows that the DOT becomes particularly attractive in terms of energy pro-duction for mean wind speeds above 10.5 m/s, where a higher gross annual energy yieldis obtained than the NREL turbine; for lower wind speeds, the lower performance shouldbe compensated by other potential advantages of the DOT, such as reliability as well assize and weight reduction of components.

In order to evaluate the influence of the different subsystems performance in the an-nual energy yield of the DOT, a normalized energy production with respect to the NRELturbine is obtained, see figure 6.13. For this specific case a mean wind speed of 10.1 m/swas used as a representative value for remote offshore at the North Sea (Bierbooms),resulting for the DOT in a gross annual energy production of 24, 991 MWh.


DOT NREL0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4N

orm

aliz

ed G

ross

ann

ual

ener

gy p

rodu

ctio

n [−

]

Elect PowerHydr Power central platformHydr Power base turbineClosed−Loop mechPowerMech power

Figure 6.13: Normalized annual energy yield comparison for a mean wind speed of 10.1 m/s

It is observed that for this particular wind conditions, the high speed operation of theDOT allows to increase in around 17% the aerodynamic energy production when com-pared to the normal operation. On the other hand, due to the different cumulative effi-ciencies of the DOT subsystems, the total electrical energy production is 2% less than theconventional NREL turbine. Major losses occur in the closed-loop transmission and theseawater pump, while relatively small losses occur due to friction in the seawater pipeline.

Chapter 7

Experimental validation

7.1 Introduction

The purpose of this section is to validate the steady-state performance obtained in thecomputational model with real experimental data. The experimental part was carried outat the Institute for Fluid Power Drives and control (IFAS) at RWTH Aachen Universitywithin the frame of the IDEA League Research Grant and was possible thanks to thecollaboration with TU Delft through the IDEA League network.

7.2 Experimental setup

Since the current work assumed the availability of hydraulic drives in the MW range thatare not yet commercially available, experimental validation was done with available com-ponents with the correct down-scaling. Taking this into account, a 30 kW wind energytest bench at IFAS was used for validating experimentally the closed-loop transmissionperformance for different operational pressures of the system. Due to the fact that thetest bench doesn’t count with a real wind turbine rotor able to deliver mechanical power,an electric motor was used as the main power source. The test bench was designed insuch way that all the power delivered to the hydraulic transmission is fed back to theelectric motor, therefore the motor only has to provide enough power to overcome thelosses throughout the system.

A schematic of the experimental setup is shown in figure 7.1. The synchronous electricmotor (2) is used to drive a variable axial piston pump (1) at constant speed (1500 rpm).Hydraulic power generated from this pump is used to drive the hydraulic motor (5), wherethe mechanical power produced represents the mechanical power delivered by the windturbine rotor (high torque, low speed characteristic). The hydraulic transmission is thencomprised by the hydraulic pump (4) and motor (3). Finally, the shaft of the hydraulicmotor (3) is coupled to the electric motor (2) in order to feed-back the transmitted power.

59

60 Experimental validation

Figure 7.1: Experimental set-up, 30 kW bench test

Table 7.1: Main components of the experimental bench test

Item number Component

1 Pump Bosch Rexroth A4VSO 125cc2 Electric motor 30kW, 1500 rpm3 Pump Bosch Rexroth 71cc4 Pump Hagglunds CA50 3140cc5 Motor Hagglunds CA50 3140cc6 Torque transducer7 Rotational speed sensor8 Pressure sensor9 Cooling system

Due to the variable displacement characteristic of pump (1), different operational condi-tions of torque and speed at the low speed shaft can be represented, however in order toavoid high pressure rises, a pressure control strategy was implemented. Thus, the systemcould be operated at different defined pressures either by letting the controller actuate orby adjusting manually the displacement of the pump.

7.3 Data aquisition

Suitable sensors and instruments were used to obtain torque, pressure and rotationalspeed measurements as shown by components (6), (7) and (8) in figure 7.1, with a totalnumber of seven input channels.

A torque transducer (6) was mounted at the high speed shaft, with an output signal of

7.3 Data aquisition 61

Figure 7.2: High torque, low speed hydraulic drives used in the test bench

+/ − 5 V olts for a rated torque of 500 Nm. Two rotational speed sensors (7) weremounted at the low and high speed shafts; each sensor gives an increment in the rotaryposition of the shaft, from which rotational speed was calculated by taking the derivativeof the measured values. Four pressure sensors (8) were used for the hydraulic drivescoupled at the low speed shaft: two for the low and high pressure lines of the pump, andtwo for the low and high pressure lines of the hydraulic motor. Each sensor provided anoutput signal of 0 to 10 V olts for a range of 0 to 16 and 0 to 400 bar for the low and highpressure lines respectively.

Figure 7.3: Time domain record of measured pressure at the hydraulic transmission

The test bench was run for 15 different operational pressures; for each defined pressureand once steady state conditions were achieved, measurements were taken for 5 seconds


with a sample rate of 1/1000 seconds. Figure 7.3 shows a measured time series of pressureat the high pressure line of the hydraulic transmission.

7.4 Data post-processing

In order to obtain the performance of the system, other parameters of the transmissionthat couldn’t be measured directly were estimated with help of the theory explained insection 3. On the other hand, due to the stochastic variations around the mean value ofthe measured data, a moving average of the previous 0.1 seconds was used to ’smooth’each input channel.

The torque at the low speed shaft was calculated with the pressure differences acrossthe motor and pump; the torque transmitted by a hydraulic motor or pump is defined by

For a motor:

Tm = ∆pm Vm ηmechm (7.1)

For a pump:

Tp =∆pp Vp

ηmechm(7.2)

Because both hydraulic drives are connected to the low speed shaft (Tm = Tp), and bothhave the same volumetric displacement (Vp = Vm), (they are the same hydraulic machine,the only difference is their operation as a motor or pump), it was supposed that they alsohave the same performance. Thus, by substituting the mechanical efficiency from eq 7.2into eq 7.1, the torque at the low speed shaft is obtained:

Tls = Vp

√

∆pp ∆pm (7.3)

With the last result, mechanical power is calculated in both low and high speed shafts(rotational speeds and the torque in the high speed shaft were directly measured). Hencethe total efficiency of the hydraulic transmission is obtained as the ratio of the outputmechanical power (at the high speed shaft) and the input mechanical power (at the lowspeed shaft):

ηtot =Phs

Pls=

Ths ωhs

Tls ωls(7.4)

Mechanical efficiency of the transmission can also be derived from the torques of the lowand high speed shafts by taking into account the transmission ratio n such that:

ηmech =Ths n

Tls=

(

Ths

Tls

) (

Vp

Vm

)

(7.5)

7.5 Experimental results 63

Once the total and the mechanical efficiencies are known, the volumetric efficiency is sim-ply the ratio between both:

ηvol =ηtotηmech

(7.6)

7.5 Experimental results

From each data set, mean values for the different parameters were obtained and com-pared with the results of the computational model. Parameters for the computationalmodel were taken from the specified technical data of product manuals for the differentcomponents, see Appendix H.

The performance of the system is summarized in the next figures, where results of thecomputational model are shown with a continuous line, and results of experiments areshown with different markers. In general a vey good correlation between the computa-tional model and experimental data is observed in figure 7.4.

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pressure [bar]

Effi

cien

cy [−

]

ηtotal

ηmech

ηvol

Figure 7.4: Performance of the hydraulic transmission for different pressures

The maximum efficiency of the hydraulic transmission obtained experimentally was 0.82,and 0.84 with the computational model; the main variation is in the volumetric efficiencywhere the computational model predicts a better performance (meaning less leakage).

Figure 7.5 shows that the mechanical efficiency differs from experimental data in terms ofthe mechanical power that is transferred by the hydraulic transmission; this conditions isobserved mainly for partial loads below 20 kW where the computational model predicts


8% higher performance than in reality.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Transmitted Power [kW]

Effi

cien

cy [−

]

ηtotal

ηmech

ηvol

Figure 7.5: Performance of the hydraulic transmission

Despite the relationships obtained in the model it is important to point out that thereare factors that could influence the results of the experimental data that may be difficultto estimate, an example is a change in the physical properties of the fluid due to anincrease or decrease of temperature which is directly associated to the losses and coolingcapacity of the system. However the computational model shows a good correlation withthe experimental performance of the system, which makes it a suitable tool for a firstevaluation of different design proposals under several operational conditions.

Chapter 8

Conclusions and recommendations

8.1 Conclusions

This thesis work gives a first estimate of the performance and steady-state conditionsof a single Delft offshore turbine using the rotor of the NREL 5 MW baseline turbine.Despite the fact that a high speed operation of the rotor is plausible with an increasedpotential in energy capture, the different energy transfers that occur in the main subsys-tems, make the total efficiency of the system to be lower than a typical turbine with anoverall efficiency around 80%. Main losses of energy occur in the seawater pump, wherethe variable-speed operation of the rotor reduces the efficiency of the pump due to thehigher leakages at low wind speeds. For typical offshore conditions of the North Sea, thetotal energy production of the DOT is relatively lower than a typical turbine; neverthelesshigher energy productions become possible for locations with mean wind speeds above10.5 m/s.

The computational model used to obtain the previous results was validated with experi-mental data for the closed-loop transmission resulting in performance deviations of 2 to8% (meaning that the computational model predicted a higher performance in this rangeof values with respect to experimental data), where mayor deviation occurs at partialload conditions. Thus the developed model constitutes a straight forward and effectivetool for an integrated design of the overall system with enough level of detail for a firstevaluation of the system performance.

8.2 Recommendations

Although this work gives a first step in the preliminary design of the proposed concept,only steady-state conditions were evaluated and analyzed for the performance of a singleturbine. However, because environmental conditions and therefore the response of thesystem vary in time, the dynamic analysis is essential to understand the behavior of the

65

66 Conclusions and recommendations

whole system. Furthermore, it is of particular interest to analyze the dynamic charac-teristics of hydraulic drives and long pipelines due to their influence in the stiffness andtherefore in the dynamic performance of the system.

With respect to the proposed high-speed operation of the NREL rotor, a detailed analysisregarding fatigue and ultimate loads is recommended to determine the feasibility and toindentify the main design drivers that limit the maximum tip speed for offshore turbines.

Regarding the idea of a centralized electricity generation scheme through hydraulic power,a more detailed model of the hydraulic turbines including their dynamic response for dif-ferent load characteristics would give a better insight of the overall performance of thewhole offshore wind farm, which would allow to quantify the possible advantages of theproposed concept concerning the integration with the electrical grid.

It is important to mention that the obtained results assume availability of hydraulicdrives in the MW range with design requirements and operational limits imposed bycurrent technology. Although these components are not yet commercially available, notechnical reason seem to be a limiting factor for the required upscaling, however it be-comes more complicated to make an estimation in terms of reliability and cost reduction.With all these different aspects in mind, further work is needed to quantify the potentialadvantages and therefore determine if the economical benefits of a ‘centralized electric-ity offshore wind farm’ outweigh the lower energy production of the individual hydraulicturbines, only then the success of the proposed concept will be foreseen.

References

Albers, P. (2010). Motion Control in Offshore and Dredging. Springer.

Artemis Intelligent Power. (2010). http://www.artemiswind.com/.

Bard Offshore. (2010). http://www.bard-offshore.de/.

Bianchi, F. D., De Battista, H., & Mantz, R. J. (2007). Wind Turbine Control Systems.In Principles,Modelling and gain scheduling design. Springer.

Bierbooms, W. (2009). Offshore wind farm design, lecture notes. Delft University ofTechnology.

Boldea, I. (2006). Synchronous Generators. Taylor & Francis Group.

Bossanyi, E. (2007). Gh Bladed User Manual [Manuel de logiciel]. Glasgow : GarradHassan & Partners Limited.

ChapDrive. (2010). http://www.chapdrive.com/.

Cornell, R. (1981). Dynamic Simulation of a Hydrostatically Propelled Vehicle. SAEpaper , Vol. 811253 , pp. .

Dasgupta, K., & Mandal, S. (2002). Analysis of the steady-state performance of a multi-plunger hydraulic pump. Proceedings of the Institution of Mechanical Engineers, PartA: Journal of Power and Energy , Vol. 216 , pp. 471 – 479.

Dasgupta, K., Watton, J., & Pan, S. (2006). Open-loop dynamic performance of aservo-valve controlled motor transmission system with pump loading using steady-statecharacteristics. Mechanism and Machine Theory , Vol. 41 , pp. 262 – 282.

Diepeveen, N. (2009). Design Considerations for a Wind-Powered Seawater Pump. DelftUniversity of Technology, European Offshore Wind Conference Proceedings.

Diepeveen, N., & Van der Tempel, J. (2008, December). Delft Offshore Turbines, thefuture of wind energy. Delft University of Technology.

67

68 References

Fichaux, N., & Wilkes, J. (2009, September). Oceans of Opportunity. European WindEnergy Association.

Hagglunds drives. (2010). http://www.hagglunds.com/.

Jamieson, P. (2009). Light-Weight, High-Speed Rotors for Offshore. Garrad Hassan &Partners Limited, European Offshore Wind.

JERICO. (1981, July). Hydraulic Wind Energy Conversion. Jacobs Energy Research.

Jonkman, J., Butterfield, S., Musial, W., & Scott, G. (2009, February). Definition of a5-mw Reference Wind Turbine for Offshore System Development (No NREL/TP-500-38060). NREL National Renewable Laboratory.

Keck, H., Vullioud, G., & Joye, P. (2000). Commissioning and Operation Experience withthe Worlds largest Pelton turbines Bieudron. VATECH hydro.

Knauer, A., & Hanson, D. (2007). High Tip Speed Operation for Offshore Wind Turbines.IFE Institute for Energy Technology, European Wind Energy Congress.

Mandal, S., & Dasgupta, K. (2009). Theoretical and experimental studies on the steady-state performance of low-speed high-torque hydrostatic drives. part 1: modelling. Pro-ceedings of the Institution of Mechanical Engineers, Part C: Journal of MechanicalEngineering Science, Vol. 223 , pp. 2663 – 2674.

Manwell, J., McGowan, J., & Rogers, A. (2002). Wind Energy Explained, Theory, Designand Application. England : John Wiley & Sons.

MATLAB. (2010). Simscape Users Guide [Manuel de logiciel]. Natick : The MathWorks,Inc.

Merritt, H. (1991). Hydraulic Control Systems. John Wiley & Sons.

Murrenhoff, H. (1999). Systematic approach to the control of hydrostatic drives. Journalof Systems and Control Engineering , Vol. 213 , pp. 333 – 347.

Murrenhoff, H. (2007). Grundlagen der Fluidtechnik, Teil 1:Hydraulik. Institute for FluidPower Drives and Control, RWTH Aachen University.

Pencho, C. (1998, June). Layman’s handbook on how to develop a small hydro site.European Small Hydropower.

Rademakers, L. (1988). Possibilities of Variable Transmissions in Wind Turbines. M.Sc.Thesis, University of Technology, Eindhoven, the Netherlands.

Reiner, M. (2006, May). Pelton turbine design of Gilgel Gibe ii. Voith Hydro.

Sant, T. (2007). Improving BEM-based Aerodynamic Models in Wind Turbine DesignCodes. Ph.D. Thesis, Delft University of Technology, Wind Energy Research InstituteDUWIND.

Schmitz, J., & Vatheuer, N. (2010). Development of a Hydrostatic Transmission for WindTurbines. Institute for Fluid Power Drives and Control, RWTH Aachen University.

References 69

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70 References

Appendix A

Blade Element Momentum theory

The BEM method combines the Momentum theory and the Blade element theory to ob-tain the aerodynamic loads exerted in the rotor.

A.1 Momentum theory

This theory applies the basic conservation laws to obtain a description of the axial flowof air that passes through the rotor. This air passing through the rotor will experience adecrease in velocity as the wind turbine extracts the kinetic energy from the wind. Therotor is represented by an infinitely thin and permeable disc, where the stream tube willexpand as it decreases its velocity in the rotor plane and in the far wake as shown infigure A.1.

Figure A.1: Energy capture in a stream tube

Usually the velocity at the disc is expressed in terms of an induction factor, which ex-presses the decreased wind speed passing through the actuator disc to the undisturbed

71

72 Blade Element Momentum theory

wind speed:

a =Uo − U∞

Uo(A.1)

Ud = Uo (1− a) (A.2)

The axial force exerted on the actuator disc can be obtained using the momentum theory,which states that the mass flow rate is constant along all the stream tube. This mass flowthrough the actuator disc is driven by a change in static pressure immediately before andafter the disc as shown in figure A.2, therefore the axial force developed by the actuatordisc on the incident flow is given by:

Dax =(

p+d − p−d

)

Ad = (Uo − U∞) ρ Ad Ud (A.3)

Figure A.2: Velocity and pressure profile at the rotor

By using Bernoulli’s equation, the pressure drop can be obtained. This equation statesthat as long as no work is done on the fluid, the total energy for the flow remains con-stant. This means that a relationship between the pressures and velocities upstream anddownstream can be obtained:

(

p+d − p−d

)

=1

2ρ(

U2o − U2

∞

)

(A.4)

Combining the last equations, the total axial force can be obtained in terms of the induc-tion factor and the upstream wind velocity

Dax = Ad1

2ρ U2

o 4a (1− a) (A.5)

The axial force can be expressed in terms of a dimensionless axial force coefficient definedas follows:

CDax =Dax

Fwind=

Dax

Ad12 ρ U2

o

(A.6)

A.2 Blade element theory 73

From the last two equations the axial force coefficient can b expressed in terms of theinduction factor such that

CDax = 4a (1− a) (A.7)

Although there is direct relationship between the axial force coefficient and the inductionfactor, none of these parameters is directly known. Thus the blade element theory willbe used to obtain other expressions for the axial force and induction factor; combinationof both theories will allow to obtain all the unknown parameters. It is important tomention that values of the induction factor larger than 0.5 (partial) flow reversal occurso momentum theory no longer can be applied; instead some empirical relation is used.

A.2 Blade element theory

This theory is based on the analysis of aerodynamic forces applied to a finite number ofblade elements as shown in figure A.3. The theory assumes that every blade element canbe treated independently (no aerodynamic interaction between elements) and that theforces exerted on each element are determined only by the lift and drag characteristics ofthe airfoil shape of the blade.

Figure A.3: Blade elements

Each blade element moves in the airflow at a relative velocity, which is caused by thecombination of the wind velocity on the rotor plane and the tangential velocity due tothe rotation of the blade.

Ud = Uo (1− a) (A.8)

Utan = ωr r (A.9)

The wind relative velocity the is obtained as the sum of velocities such that

Urel =√

U2d + U2

tan (A.10)


Airflow causes a pressure difference around the blade element, which creates a lift anddrag force in a perpendicular and parallel direction to the relative wind respectively.

FL = CL(α)1

2ρ U2

rel c∆r (A.11)

FD = CD(α)1

2ρ U2

rel c∆r (A.12)

(a) Relative wind velocity (b) Resulting force in x direction

Lift and drag forces are generally expressed in terms of the lift and drag coefficients, whichare function of the angle of attack alpha defined as the angle that the flow makes withthe chord

α = φ− θ (A.13)

Where φ is the angle of inflow, and θ is the pitch angle measured between the rotor planeand the chord.

φ = arctan

(

Ud

Utan

)

(A.14)

The lift and drag forces can be divided into axial and tangential components, in such waythat the resulting force for an element in the x direction can be determined.

Fx = FL cosφ+ FD sinφ (A.15)

The total force exerted on the blade is the sum of the individual forces from all the blade

A.2 Blade element theory 75

elements and the number of blades. This results into a total thrust axial force

Dax = Nb

Ns∑

i=1

Fx i (A.16)

The axial force coefficient can be obtained

CDax =Dax

Fwind=

Dax

Ad12 ρ U2

o

(A.17)

Similarly to the calculation of the axial force, the resulting force in the rotating directioncan be obtained

Frot = FL sinφ− FD cosφ (A.18)

Hence the total torque that produces useful work is given by:

Tr = Nb

Ns∑

i=1

Ftot i ri (A.19)

It is important to notice that the total axial force coefficient and rotor torque dependsagain on the induced factor. Therefore by combining the expression for the total axialforce coefficient from both momentum and blade element theories, the axial force and theinduced factor can be obtained.

Appendix B

Hydraulic motor analysis

B.1 Steady-state performance

Although the analysis described so far has been centered in pumps, an analogous de-scription can be made for motors. The continuity equation for the flow and the torquebalance equation can be derived for steady-state conditions under the same assumptionsof section 3.

B.1.1 Rotational speed generated

The rotational speed of the motor will be generated by the net flow that can be used bythe motor, this is the flow delivered to the motor minus the leakage loses; the compress-ibility effect is also included:

Vm ωm =

(

Q− CsVm

µ∆p

)

(

1−[

2E

(

VK

VUT− ∆p

E

)]

−1

∆p

)

(B.1)

where the dimensionless coefficient of slip Cs is defined in the same way as for the pump interms of the internal and external leakage coefficients Cint and Cext of the motor. Takinginto account the volumetric displacement of the motor, the generated rotational speedcan be obtained according to

ωm =1

Vm

(

Q− CsVm

µ∆p

)

(

1−[

2E

(

VK

VUT− ∆p

E

)]

−1

∆p

)

(B.2)

B.1.2 Torque delivered to the load

The net torque delivered to the load can be derived from the ideal generated torque duethe pressure difference across the motor and its volumetric displacement; torque losses

77

78 Hydraulic motor analysis

due to friction and damping are also taken into account

Tm = Vm ∆p− (Cdamp Vm µ ωm)− (Cf Vm ∆p) (B.3)

B.2 Efficiencies

B.2.1 Volumetric efficiency

Is defined as the ratio of flow in motor speed, or ideal flow, to the flow delivered to themotor

ηvol =Vm ωm

Q(B.4)

Replacing Q the volumetric efficiency becomes:

ηvol =

(

1−[

2E

(

VK

VUT− ∆p

E

)]

−1

∆p

)

−1

+Cs ∆p

µ ωm

−1

(B.5)

B.2.2 Mechanical efficiency

Is defined as the ratio of the net torque delivered to the load, to the ideal torque deliveredby the motor

ηmech =Tm

Vm ∆p(B.6)

By substituting Tm, the torque efficiency becomes:

ηmech = 1− Cdampµ ωm

∆p− Cf (B.7)

B.2.3 Total efficiency

The total motor efficiency is defined as the ratio of mechanical power obtained to thehydraulic power supplied

ηtot =Pmech

Phyd=

ωm Tm

Q∆p(B.8)

ηtot =

(

Vm ωm

Q

)(

Tm

Vm ∆p

)

= ηvol ηmech (B.9)

ηtot =1− Cdamp

µ ωm

∆p − Cf(

1−[

2E(

VK

VUT− ∆p

E

)]

−1∆p

)

−1

+ Cs ∆pµ ωm

(B.10)

B.2 Efficiencies 79

In the same way as for a pump, the performance of a motor can be defined by the pa-rameters Cs, Cdamp, Cf , µ, E, ∆p and ωm.

80 Hydraulic motor analysis

Appendix C

Moody diagram

81

82 Moody diagram

Appendix D

BEM code validation

The validation of the used BEM code is done with the steady-state parameters of thecommercial software Garrad Hassan Bladed (Bossanyi).

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

Wind Speed [m/s]

MechPower [kW] BEMMechPower [kW] GH BladedTorque [kNm] BEMTorque [kNm] GH Bladed

(a)

0 5 10 15 20 250

100

200

300

400

500

600

700

800

Wind Speed [m/s]

Thr

ust f

orce

[kN

]

BEMGH Bladed

(b)

Figure D.1: Steady-state parameters validation

83

84 BEM code validation

0 5 10 15 20 250

5

10

15

20

25

Wind Speed [m/s]

Rotor speed [rpm] BEMRotor speed [rpm] GH BladedPitch angle [deg] BEMPitch angle [deg] GH Bladed

(c)

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Wind Speed [m/s]

Coe

ffici

ents

[−]

Cd BEMCd GH BladedCp BEMCp GH Bladed

(d)

Figure D.1: Steady-state parameters validation

From last figures it can be concluded that the steady-state parameters show relativelysmall deviation, which makes the used BEM code a straight forward and reliable tool fora first overview and comparison of results.

Appendix E

Pump parameters estimation

E.1 Determination of the coefficient Cs

The parameter Cs which is directly related to the leakage flow Qs , can be estimatedbased on the assumption that it is linearly proportional to the pressure difference acrossthe pump and can be calculated using the Hagen-Poiseuille formula

∆p =128 µ l

π D4Qs =

µ

kHPQs (E.1)

Where D4 and l are geometric parameters of the leakage path and kHP stands for theHagen-Poiseuille coefficient. In this formula the leakage flow is assumed to be laminarand therefore inversely proportional to the absolute viscosity of the fluid µ.

The leakage flow at ∆p = pnom and ω = ωnom can be determined from catalog dataof existing pumps such that

Qs = Vp ωnom (1− ηvol) (E.2)

Hence the Hagen-Poiseuille coefficient can be obtained as

kHP =Vp ωnom (1− ηvol)µnom

∆pnom(E.3)

Therefore the leakage flow can be obtained as

Qs =kHP

µ∆p = Cs

Vp

µ∆p (E.4)

The parameter Cs can be then defined merely in terms of operational values from com-mercial pumps

85

86 Pump parameters estimation

Cs =ωnom (1− ηvol)µnom

∆pnom(E.5)

E.2 Determination of the coefficient Cdamp

The value of this parameter is directly associated to the torque losses due to a dampingtorque required for starting conditions, therefore for a steady-state analysis this coefficientwill have a value of zero.

E.3 Determination of the coefficient Cf

The parameter Cf is associated to losses of torque due to friction between the mechanicalelements. If steady-state conditions are assumed, then the value of the coefficient Cf canbe obtained as a function of the desired mechanical efficiency

For a pump:

Cf =1− ηmech

ηmech(E.6)

For a motor:

Cf = 1− ηmech (E.7)

Appendix F

Simulation method

Figure F.1: time domain simulation method

Model validation: the software validates the model configuration and check data entries.

Network construction: the physical network is constructed based on the basic princi-

87

88 Simulation method

ples for Through and Across variables.

Equation construction: based on the constructed network and the values for differentparameters, equations are constructed for the model. These equations can contain twodifferent type of variables: ” Dynamic: Time derivative of this variable appear in equa-tions. Dynamic variables are the independent states for simulation. ” Algebraic: Timederivative of this variable does not appear in equations. Algebraic variables are alwaysdependent (on dynamic variables, other algebraic variables, or inputs).

Computing Initial Conditions: initial conditions are calculated only in the begin-ning of the simulation at t=0. They are computed by setting all dynamic variables to 0,and solving for all the system variables.

Performing Transient Initialization: after computing initial conditions for dynamicvariables, transient initialization allows to solve for algebraic variables and derivatives ofdynamic variables. A consistent set of initial conditions is provided for the next transientsolve phase.

Transient Solve: the continuous differential equations are integrated in time to computeall variables as a function of time. The simulation is performed according to the resultsof the transient solve until it encounters an event, such as a zero crossing or discontinuity.If an event is encountered, the solver returns to the phase of transient initialization, andthen back to transient solve. This cycle continues until the end of the simulation.

Appendix G

Annual energy yield evaluation

The evaluation of the annual energy production of a wind turbine is determined by thewind turbine power curve and the wind speed statistical distribution, according to (Man-well et al.).

Ew = T

∞∫

0

h(U) PPC(U)dU = TPw (G.1)

Where PPC is the analytical expression for the wind turbine power curve, h(U) is the prob-ability density function of the mean wind speed at hub height, T is the integration timeinterval (usually 1 year = 8760 h) and Pw is the average output power over the interval T .

For discrete distributions, as most practical cases, then:

Ew = TNv∑

i=1

Hi PPC(Ui) (G.2)

Where Hi is the discreet probability of wind speed bin i, Ui is the center of bin i and Nv

is the number of wind speeds bins used.

From the gross annual energy yield, more detailed losses have to be taken into accountfor a realistic estimate of the expected production; these losses can be included in anaggregate efficiency factor such that

Ewnet= η Ew (G.3)

The efficiency factor η has to account mainly for several losses: electrical losses along thewind farm, availability of the wind turbine, availability of the grid, power curve change(due to icing, aging, surface contamination) and accuracy of the wind potential assess-ment. These evaluations are more difficult to estimate for a preliminary analysis since

89

90 Annual energy yield evaluation

they are site specific; nevertheless one specific value to account for the overall efficiencycan be used for initial estimation and comparison purposes.

G.1 Wind speed distribution

Wind statistics are well described by the Weibull probability distribution function, whichis given by

For 0 ≤ U ≤ ∞

h(U) =k

a

(

U

a

)k−1

exp

(

−(

U

a

)k)

(G.4)

a =Uavg(year)

Γ(

1 + 1k

) (G.5)

As can be observed the probability density function depends on a shape parameter k, anda scale parameter a, where Uavg is the average yearly wind speed, and Γ is the Gammafunction.

An example of the Weibull distribution with the shape factor as a parameter and ascale parameter of a = 10 can be seen in the next figure.

Figure G.1: Weibull distribution for scale factor a=10 and shape factor k as a parameter

Appendix H

Performance models for the

experimental set-up

The performance models of the hydraulic drives can be obtained as function of two mainlosses: leakage losses qL and friction losses Tfr on the shaft. A general correlation forthese values can be expressed in terms of nominal conditions such that:

qL = V ω kL1

(

p

pnom

)kLP(

V

Vmax

)kLV(

ω

ωnom

)kLω

(H.1)

Tfr = V p kF1

(

p

pnom

)kFP(

V

Vmax

)kFV(

ω

ωnom

)kFω

(H.2)

Where pnom and ωnom are the nominal pressure and angular velocity, kL1 and kF1 are theleakage and friction proportionality coefficients, and kLP , kLV , kLω, kFP , kFV and kFω

are approximating coefficients.

The approximating coefficients are obtained from efficiency plots, usually provided bythe hydraulic drive manufacturer. Therefore the volumetric and mechanical efficiencycan be derived for a pump and motor as follows:

For a pump:

ηvol =V ω − qL

V ω= 1− kL1

(

p

pnom

)kLP(

V

Vmax

)kLV(

ω

ωnom

)kLω

(H.3)

ηmech =V p

V p+ Tfr=

1

1 + kF1

(

ppnom

)kFP(

VVmax

)kFV(

ωωnom

)kFω(H.4)

91

92 Performance models for the experimental set-up

For a motor:

ηvol =V ω

V ω + qL=

1

1 + kL1(

ppnom

)kLP(

VVmax

)kLV(

ωωnom

)kLω(H.5)

ηmech =V p− Tfr

V p= 1− kF1

(

p

pnom

)kFP(

V

Vmax

)kFV(

ω

ωnom

)kFω

(H.6)

The curve-fitting procedure is based on the comparison of the efficiency from the lastexpressions, and the data from the product manuals. One way of obtaining the approxi-mating coefficients by data fitting is solving the following optimization problem:

minx

F (x) x = [kL1, kLP , kLV , kLω]

F (x) =∑

i

∑

j

∑

k

(

ηexp (pi, Vj , ωk)−(

1− kL1

(

pipnom

)kLP(

Vj

Vmax

)kLV(

ωk

ωnom

)kLω

))2

(H.7)

Where i is the number of experimental pressure points, j is the number of experimentaldisplacement pumps and k is the number of experimental rotational speed points. Thismethod is the same used in (Cornell), and can be solved by using the Optimization Tool-box in Matlab . For the hydraulic drives used in the experimental model, the followingvalues were obtained according to previously described procedure with data from theproduct manuals.

H.1 Hagglunds CA50 pump

Table H.1: Performance parameters for the Hagg CA50

Volumetric efficiency

Parameter Value

kL1 0.0172kLP 1.5379kLV 0.0kLω 0.0

Resnorm = 3.9165e− 6V = 3140 [cc/rev]

Mechanical efficiency according to manual: 0.97

H.2 Bosch Rexroth motor 93

0

100

200

300

0

50

100

150

2000.98

0.985

0.99

0.995

1

Pressure [bar]Speed [rpm]

Vol

umet

ric e

ffici

ency

[−]

Figure H.1: Performance of the HagglundsCA50 pump

H.2 Bosch Rexroth motor

0100

200300

400

0

500

1000

1500

2000

25000.92

0.94

0.96

0.98

1


Vol

umet

ric e

ffici

ency

[−]

0100

200300

0500

10001500

20002500

0

0.2

0.4

0.6

0.8

1


Mec

hani

cal e

ffici

ency

[−]

Figure H.2: Performance of the Bosh-Rexroth 71 motor

94 Performance models for the experimental set-up

Table H.2: Performance of the Bosh-Rexroth 71 motor

Volumetric efficiency Mechanical efficiency

Parameter Value Parameter Value

kL1 0.0761 kF1 0.0583kLP 1.3554 kFP -0.6889kLV 0.0 kFV 0.0kLω 0.2055 kFω -0.4005

Resnorm = 1.3719e− 4 Resnorm = 0.0556V = 71 [cc/rev]

H.3 Bosch Rexroth pump

0

100

200

300

0500100015002000

0

0.2

0.4

0.6

0.8

1

Pressure [bar]

Speed [rpm]

Vol

umet

ric e

ffici

ency

[−]

0100

200300

400

0

1000

2000

30000.4

0.5

0.6

0.7

0.8

0.9

1


Mec

hani

cal e

ffici

ency

[−]

Figure H.3: Performance of the Bosch-Rexroth A4VSO 125 pump

Table H.3: Performance of the Bosch-Rexroth A4VSO 125 pump

Volumetric efficiency Mechanical efficiency

Parameter Value Parameter Value

kL1 0.0490 kF1 0.0381kLP 0.5415 kFP -0.1kLV 0.0 kFV 0.0kLω -1.7859 kFω -0.5779

Resnorm = 5.1084e− 4 Resnorm = 2.1537e− 18V = 125 [cc/rev]

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