active drive train control to improve energy capture...

ACTIVE DRIVE TRAIN CONTROL TO IMPROVE ENERGY CAPTURE OF WIND

TURBINES

By

Nathaniel Haro

A thesis

submitted in partial fulfillment

of the requirements for the degree of

Masters of Science in Mechanical Engineering

Boise State University

April 2007

The thesis presented by Nathaniel Haro entitled ACTIVE DRIVE TRAIN CONTROL

TO IMPROVE ENERGY CAPTURE OF WIND TURBINES is hereby approved:

__________________________________________ John Gardner Date Advisor

__________________________________________ Joe Guarino Date Committee Member

__________________________________________ John Chiasson Date Committee Member

__________________________________________ John R. (Jack) Pelton Date

Dean of the Graduate College

iii

ABSTRACT

Continuously variable transmissions have been increasingly used in the

automotive industry to eliminate shift shock and improve vehicle efficiency. This thesis

evaluates the effectiveness of a differential continuously variable transmission (DCVT)

used in a different application: wind turbine generators. The Controls Advanced

Research Turbine (CART) was modeled utilizing Matlab, Simulink, and SymDyn. First

the CART was modeled using its normal fixed ratio transmission. It was then modeled to

incorporate a two stage planetary gear train differential, in which the speed of second

stage ring gear was controlled to achieve two operating goals: constant generator speed

and constant tip speed ratio. The results were then analyzed to determine the DCVT’s

effects on power production, and also on the torques and speeds associated with that

power, in an attempt to optimize the wind turbine generator system. It was discovered

that when the system was controlled to achieve a constant generator speed and constant

tip speed ratio there was an increase in power production. Though when the system was

controlled to maintain a constant tip speed ratio the amount of power used to control it

was approximately twice that of the constant generator speed model, while producing

larger torques then either of the other systems. Due the negative aspects associated with

the constant tip speed ratio system, and the fact the average power produced of the two

controlled systems were approximately equally to each other, the constant generator

speed appears to be the more promising of the two.

iv

ACKNOWLEDGEMENTS

I would first like to thank Dr. John Gardner for not only giving me the

opportunity to return to grad school full time, by providing me a temporary position

within the university, but for also tolerating my never ending supply of questions and

interruptions to his work day. Secondly, I would like to thank my loving wife Sherry for

all that she has done for me through out the last several years. I love you. I would also

like to thank my parents for their unending support through out my educational career.

Lastly I would like to thank Dr. Joe Guarino and Dr. John Chiasson for making the time

and commitment to be part of my committee.

Again, thank you all. It has been an adventure that will never be forgotten.

v

TABLE OF CONTENTS

ABSTRACT...................................................................................................................... III

ACKNOWLEDGEMENTS.............................................................................................. IV

LIST OF FIGURES ....................................................................................................... VIII

LIST OF TABLES............................................................................................................ XI

NOMENCLATURE ........................................................................................................XII

CHAPTER 1 – INTRODUCTION ..................................................................................... 1

CHAPTER 2 – LITERATURE REVIEW .......................................................................... 3

Brief Wind Turbine History............................................................................................ 3

Current State of the Art................................................................................................... 4

Variable Speed Wind Turbines................................................................................... 4

Continuously Variable Speed Transmission Wind Turbines...................................... 6

Differential Drive Train Wind Turbines ..................................................................... 6

CHAPTER 3 – DIFFERENTIAL PLANETARY GEAR TRAIN ANALYSIS ................ 8

System State Equations................................................................................................... 9

Kinematic Analysis of Planetary Gear System............................................................. 10

Torque and Power Analysis of Planetary Gears ........................................................... 12

CHAPTER 4 – TURBINE MODEL................................................................................. 13

Modeling Platform........................................................................................................ 13

Turbine Specifications .................................................................................................. 13

Wind Simulation ........................................................................................................... 16

vi

Constant Speed.......................................................................................................... 16

Step Input .................................................................................................................. 17

Variable Wind Data .................................................................................................. 17

CHAPTER 5 – INDUCTION GENERATOR AND CONTROLLER DESIGN ............. 19

Induction Generator Design.......................................................................................... 19

Control Methods ........................................................................................................... 20

Controller Design.......................................................................................................... 21

Constant Generator Speed Pole and Zero Location.................................................. 22

Constant Tip Speed Ratio Pole and Zero Location................................................... 24

CHAPTER 6 – SIMMULATION RESULTS................................................................... 25

Constant Wind Speeds .................................................................................................. 25

Step Input Wind Speed ................................................................................................. 30

System Power............................................................................................................ 30

System Torques......................................................................................................... 33

System Shaft Speeds ................................................................................................. 34

Variable Wind Input ..................................................................................................... 36

System Power............................................................................................................ 36

System Torques......................................................................................................... 38

System Shaft Speeds ................................................................................................. 40

CHAPTER 7– CONCLUSIONS ...................................................................................... 43

CHAPTER 8– FUTURE RECOMMENDATIONS ......................................................... 45

REFERENCES ................................................................................................................. 46

APPENDIX A................................................................................................................... 48

vii

Free Body Diagrams ..................................................................................................... 48

APPENDIX B ................................................................................................................... 51

Effects in Power by changing the Second Stage Gear Ratio ........................................ 51

APPENDIX C: .................................................................................................................. 58

Glossary of Matlab Function Blocks ............................................................................ 58

APPENDIX D................................................................................................................... 61

Simulation Models........................................................................................................ 61

APPENDIX E ................................................................................................................... 66

Simulation Files ............................................................................................................ 66

viii

LIST OF FIGURES

Figure 1 DCVT Flow Chart ................................................................................................ 2

Figure 2 typical Cp vs. λ curve........................................................................................... 5

Figure 3 (a) Automatically regulated CVT with two half pulleys, (b) CVT half pulley

system [7]................................................................................................................ 6

Figure 4 Planetary Gear Train............................................................................................. 7

Figure 5 Two Stage Planetary Gear Train .......................................................................... 9

Figure 6 SymDyn Flow Chart........................................................................................... 15

Figure 7 SymDyn Verification Results [12] ..................................................................... 15

Figure 8 Step Wind Input.................................................................................................. 17

Figure 9 Interpolated and Adjusted Wind Speed.............................................................. 18

Figure 10 Damped Constant Generator Speed System..................................................... 23

Figure 11 Damped Constant Tip Speed Ratio System...................................................... 24

Figure 12 Typical Constant Wind Speed Results ............................................................. 25

Figure 13 (a) Steady State Wind Cp vs. λ, Wind Speeds Varing from 10 to 30 m/sec... 26

Figure 14 Cp vs. Changing Steady State Wind................................................................. 28

Figure 15 Power Produced at 18 m/sec vs. Blade Pitch Angle......................................... 28

Figure 16 Effects on Cp Due to Changes in the Second Stage Gear Ratio ...................... 29

Figure 17 Power in the Rotor from Step Winds ............................................................... 31

Figure 18 Power in the Ring from Step Winds................................................................. 32

Figure 19 Power in the Generator from Step Winds......................................................... 33

ix

Figure 20 Torque on the Ring Gear from a Step Input ..................................................... 34

Figure 21 Shaft Speed of the Rotor from Step Winds ...................................................... 35

Figure 22 Shaft Speed of the Generator from Step Winds ............................................... 35

Figure 23 Shaft Speed of the Ring Gear from Step Winds............................................... 36

Figure 24 Power in the Rotor from Variable Winds......................................................... 37

Figure 25 Power in the Ring Gear from Variable Winds ................................................. 37

Figure 26 Power in the Generator from Variable Winds.................................................. 38

Figure 27 Torque in the Rotor from Variable Winds ....................................................... 39

Figure 28 Shaft Speed of the Rotor in Variable Winds .................................................... 41

Figure 29 Shaft Speed of the Ring Gear in Variable Winds............................................. 41

Figure 30 Shaft Speed of the Generator in Variable Winds ............................................. 42

Figure 31 Change in Rotor Power Due to Changes in the Second Stage Gear Ratio with

Step Input .............................................................................................................. 52

Figure 32 Change in Ring Gear Power Due to Changes in the Second Stage Gear Ratio

with Step Input...................................................................................................... 52

Figure 33 Change in Generator Power Due to Changes in the Second Stage Gear Ratio

with Step Input...................................................................................................... 53


Step Input .............................................................................................................. 53


with Step Input...................................................................................................... 54

Figure 36 Change in Generator Power Due to Changes in the Second Stage Gear Ratio

with Step Input...................................................................................................... 54

x


Variable Input ....................................................................................................... 55

Figure 38 Change in Ring Power Due to Changes in the Second Stage Gear Ratio with

Variable Input ....................................................................................................... 55

Figure 39 Change in Generator Rotor Power Due to Changes in the Second Stage Gear

Ratio with Variable Input...................................................................................... 56


Variable Input ....................................................................................................... 56


with Variable Input ............................................................................................... 57

Figure 42 Change in Generator Rotor Power Due to Changes in the Second Stage Gear

Ratio with Variable Input...................................................................................... 57

xi

LIST OF TABLES

Table 1 SymDyn Degrees-of-Freedom............................................................................. 14

Table 2. Constant Generator Speed Controller Coefficients ............................................ 23

Table 3. Constant Tip Speed Ratio ................................................................................... 24

Table 4 Average Power Flow of System Components (kW)............................................ 37

Table 5 Variable Wind System Torques (kNm) ............................................................... 39

Table 6 System Shaft Speeds due to Variable Wind ........................................................ 40

xii

NOMENCLATURE

AC..........................................................................................................Alternating Current

CART........................................................................ Controls Advanced Research Turbine

Cp................................................................................................Coefficient of Performance

CVT............................................................................ Continuously Variable Transmission

DC.................................................................................................................. Direct Current

DCVT......................................................Differential Continuously Variable Transmission

DOF........................................................................................................Degree-of-Freedom

Dr.................................................................................................Diameter of the Ring Gear

Ds.................................................................................................. Diameter of the Sun Gear

Jrotor....................................................................................... Rotor Polar Moment of Inertia

k..................................................................................................Generator Torque Constant

KD....................................... Coefficient of the Derivative Component of the PI Controller

KG.................................................................................................................. Controler Gain

KI............................................ Coefficient of the Integral Component of the PI Controller

KP .......................................... Coefficient of the Constant Component of the PI Controller

Nr................................................................................... Number of Teeth on the Ring Gear

NREL .................................................................... National Renewable Energy Laboratory

Ns.....................................................................................Number of Teeth on the Sun Gear

P ...................................................................................... Planetary Gear Train Speed Ratio

PID ..................................................................................... Proportional Integral Controller

xiii

S .....................................................................................................................Generator Slip

λ .................................................................................................................. Tip Speed Ratio

τaero ..................................................................Aero Dynamic Torque Created by the Wind

τC ............................................................................................Torque Applied to the Carrier

τdiff ............................................................................. Differential Torque Seen at the Rotor

τR ...................................................................................... Torque Applied to the Ring Gear

τS ........................................................................................Torque Applied to the Sun Gear

ωc................................................................................ Angular Velocity of the Carrier Gear

ωgen ........................................................................................... Generator Angular Velocity

⎯ωgen ............................................................................ Desired Generator Angular Velocity

ωr ....................................................................................Angular Velocity of the Ring Gear

ωS ..................................................................................Angular Velocities of the Sun Gear

1

CHAPTER 1 – INTRODUCTION

It is now commonly accepted that variable speed wind turbines can produce up to

20% more power than fixed speed wind turbines [1]. Another advantage to variable

speed wind turbines lies is in their torque-absorbing ability which increases the

operational life of the mechanical components. These advantages are currently

accomplished by allowing the generator and rotor to rotate at varying speeds as the wind

speed changes. The disadvantage to this approach is that the variable electricity produced

must be rectified and inverted before being added to the grid. These components increase

the overall system cost and can reduce total energy delivered by up to 10% due to heat

dissipation of the power electronics[2].

In this study a two stage planetary gear train, which will be know as a differential

continuously variable transmission or DCVT, will be used to control the effective gear

ratio between the rotor and generator to improve power captured. With this configuration

the ring gear of the second stage planetary gear train will be used to add or remove power

in an attempt to keep the system dynamically stable during changes in wind speed. It is

anticipated that this configuration will also improve overall turbine performance.

To effectively determine the overall effectiveness of the two systems, the power

produced, torque transients and required transmission speeds will all be evaluated

between them by modeling these dynamic systems. Simulink [3], which runs in

conjunction with Matlab [4], will be used to model all components of the wind turbine.

2

There are five major components to this model: wind, turbine blades, two stage planetary

gear transmission, transmission controller, and generator, as shown in Figure 1.

Constant Tip Speed Ratio Controller

Constant Generator Controller

CVT Model

Generator Model

Turbine Blade Model

Wind Model

Controller Model

ωRotvwind ωGen

τRot τGen

ωRing

Figure 1 DCVT Flow Chart

In the following pages several thing will be covered, beginning with a literature

review of relevant papers. This section will provide a brief history of wind turbines and

discuss the current state of their art. Chapter three will discuss differential planetary gear

trains. This chapter will be followed by an explanation of how the blade aerodynamic

model and wind input model will be implemented within this thesis. The last portion of

the model to be explained is the generator, in chapter 5. The final chapters will discuss

the results of the differential continuously variable transmission modeled herein and the

conclusions derived from those results.

3

CHAPTER 2 – LITERATURE REVIEW

Brief Wind Turbine History

According to Carlin [5], horizontal axis wind turbines may have been invented as

early as the twelfth century. These turbines were used for several operations, such as

milling wheat into flower, and would be driven at a variable rate. In the 19th century

wind turbines were largely used across the United States to pump water, charge batteries,

and run farm equipment [5]. In fact, in 1888, Brush Wind Turbines in Cleveland, Ohio

was producing up to 12 kW of direct current (DC) power for charging batteries [5]. Up

to this point in time, for the majority of uses for wind turbines, varying speed would only

impact the rate at which work would be accomplished, or the voltage produced. Even

though one can easily produce DC power from turbines rotating at a variable speed, due

to the high voltage that is needed to efficiently distribute DC power through long

transmission lines, it was very difficult to transport this power to where it was needed

most, in cities. Due to this fact, alternating current (AC) quickly became the choice for

power distribution and was standardized at 60 Hz within the United States.

With the use of AC power, the simple variable speed wind turbine could not

directly connect to the grid, because the erratic wind wouldn’t produce the constant 60

Hz now required to power all the new time saving home appliances quickly appearing

[5]. One of the first turbines designed to over come this 60 Hz obstacle was Palmer

Putnam’s grandpa’s knob machine [5]. Advanced for its day, this machine had full-span

pitch control, an active yaw drive, and was rated at 1.25 MW. The Smith-Putman turbine

4

circumvented the issues of variable speed wind turbines by fixing the rotational velocity

of the air foils and then directly connecting the generator to the electrical grid [5].

Though this eliminated the problem of variable speed generators, it has its draw backs.

With its fixed speed, the turbine was limiting its collection efficiency, as well as adding

substantial voltage spikes caused by erratic wind gusts. This constant speed approach

began to change in the early 1970’s. During this time wind turbines produced variable-

voltage, variable-frequency outputs, known as wild energy, which was tamed using diode

bridges to rectify the power in to DC then using an inverter to change it back to AC at the

required 60 Hz. This is one of the main methods to clean up wield energy and is still

employed in many wind turbines today.

The advantages of using a variable speed wind turbine design versus a fixed speed

wind turbine are now commonly accepted. Depending on location and wind profile a

variable speed wind turbine can produce up to 20 % more electrical power and increase

the life of its mechanical components [1]. In order to achieve the variable speed

operation several solutions have been devised, almost all of which deal with power

electronics.

Current State of the Art

Variable Speed Wind Turbines

Variable speed wind turbines use a variable speed generator and a fixed ratio

transmission that have a rated wind speed at which they operate [2]. Below their rated

wind speed, the turbines are controlled by adjusting the generator torque. This allows the

turbine blades to accelerate to a more favorable tip speed ratio (λ) thereby increasing the

5

power coefficient (Cp), which in turn increases the overall power captured through the

equation:

pout CvRP 32

21 ρπ=

Equation 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 2 4 6 8 10 12 14 16

λ

Cp

18

Figure 2 typical Cp vs. λ curve

When a turbine starts to operate above its rated wind speed, excess power is shed by

controlling the blade pitch, which then sheds power [6]. This shedding occurs because

the turbine blades are rotated along their own axes decreasing the lift they generate,

slowing the turbine down. One large disadvantage to this operation is that when the

turbine blades are stalled, potential power generation is lost. Another disadvantage to

these types of variable speed generators is that their output is generally not at the required

60 Hz, necessary to match the grid frequency. To solve this problem the generator output

is rectified to a DC voltage than inverted to match the grid frequency. This solution is

some what counterproductive due to the high cost of the electronics required to clean up

the power. This process also creates a 5 to 10 % loss of overall generated power [2].

6

Continuously Variable Speed Transmission Wind Turbines

An alternative to the rectification method is to use a varying mechanical

transmission to control the shaft speed of the rotor. Originally developed for automotive

uses, these continuously variable transmissions (CVT) are making their way into wind

turbines. One such design incorporates two adjustable v-belt half-pulleys to change the

gear ratio as the wind speed changes in order to maintain an optimum tip speed ratio.

These half-pulleys, located behind a fixed gear ratio transmission referred to as the over

gear, are used to change the gear ratio by changing the distance between the two halves,

thereby changing the diameter of the pulley that the v-belt rides on. By changing the

diameter of one or both of the pulleys, the output gear ratio can be controlled to maintain

a constant generator speed, eliminating the need for expensive power electronics.

(a)

(b)

Figure 3 (a) Automatically regulated CVT with two half pulleys, (b) CVT half pulley system [7].

Differential Drive Train Wind Turbines

An alternative to the v-belt design is that of a planetary gear train. A planetary

gear train transmission, shown in Figure 4, is a gear train in which two independent

7

coaxial gears, the sun and ring gears, are meshed with one or more gears or gear

assemblies, know as planet gears, mounted on an intermediate shaft, the carrier [8]. If the

ring gear is held in place and the sun gear is rotated, then each planet gear is forced to

rotate around its own axis as well as the axis of the sun gear. This, in turn, causes the

carrier to rotate. If each of the ring gear, sun gear, and carrier are held in position a,

different constant gear ratio will be produced. A planetary gear train differential is very

similar to a planetary gear train transmission, though unlike the planetary transmission

which holds one component still a planetary differential allows all components to rotate.

By controlling differentials ring gear speed , the effective overall gear ratio can be varied

and controlled to produce speeds favorable to the design criteria.

Figure 4 Planetary Gear Train

In the past, a single stage and a dual stage planetary gear train have been proven

to increase power output while simultaneously reducing the torques produced by the

variable wind [1, 2, 9]. Each of these studies controlled their systems using a set value

for the power coefficient, Cp, derived for a rated wind speed. In variable speed operation,

a wind turbine can operate at the optimal Cp, which varies with the wind speed. This can

be seen in Figure 2 above.

8

CHAPTER 3 – DIFFERENTIAL PLANETARY GEAR TRAIN ANALYSIS

In this study a two stage planetary gear DCVT was used to control the effective

gear ratio between the rotor and generator in order to improve the power captured from

the wind while taking in to consideration the change in the optimum Cp value. This was

achieved by fixing one of the components of the first stage planetary gear in place, either

the ring gear, sun gear, or carrier. Then using one of the components in the second stage

planetary gear train to control the system so that when the wind velocity is at slower

speeds the servo motor will rotate one of the components, say the ring gear, in what will

be known as the negative direction, adding power into the system. As the speed of the

turbine blades increase, the servo motor compensates by decreasing its speed to maintain

a constant generator output shaft velocity or constant λ. The power that is being fed

through the system by the servo motor, when they are used to increase the speed of the

ring gear in the negative direction, is routed back onto the grid by the main generator

with an efficiency loss of 0.7 % of the total power [2]. At higher wind speeds the ring

gear is allowed to rotate in the positive direction. Here the servo motor operates as a

generator, capturing additional power from the wind, improving system efficiency.

9

Figure 5 Two Stage Planetary Gear Train

System State Equations

The system described above can be expressed using the system equations listed in

Equation 2. The acceleration of the rotor, which can then be integrated to produce the

rotor’s angular velocity, is a function of three variables: the rotor’s polar moment of

inertia (Jrotor), aero dynamic torque created by the wind (τaero), and the torque of the

system transmitted back through the differential (τdiff). The aero dynamic torque is

calculated by SymDyne and is a complex equation dictated by wind speed, rotor speed,

blade pitch angle, and other factors and is described further below. The differential

torque, seen by the rotor, is calculated through the kinematic equations described in the

following section and are driven by the torque associated with the generator, rotor speed

and the speed at which the controller drives the second stage ring gear. The operation of

this controller is discussed in chapter 5.

( )( )( )Rrotorgendiff

rotorwaero

diffaerorotor

rotor

fvf

J

ωωττωτ

ττω

,,,

1

2

1

==

−=

Equation 2 System State Equations

10

Kinematic Analysis of Planetary Gear System

In using a planetary gear configuration there are three possible locations where

power can be added or removed from the system; sun gear; ring gear; or carrier. Since

we will be using this as a differential to increase angular speed, decreasing torque, it is

first necessary to determine which configuration will produce the largest gains. Starting

with the equation for a planetary gear system:

S

R

S

R

CR

CS

S

R

DD

NNP

NN

==−−

=−ωωωω

Equation 3

where ωS , ωC , and ωR are the angular velocities of the sun, carrier, and ring gears. P is

the planetary gear train speed ratio , defined as the ratio of the number of teeth on the ring

gear to the sun gear (NR/ NS) or the diameter of the sun gear to the ring gear (DR/ DS).

Solving equation 2 for each of the three possible output variables, ωS , ωc , and ωr

respectively, yields the following equations:

( ) ( ) ( )

( )

( )

( )

)(

)(1

111)(

1

b

cP

PPP

PPa

PPPPPP

PPPPP

PPP

SRC

SCRSRC

SCCRcCSRRCS

CSCRCSCRCRCS

CSCRCSCRCRCS

++

=

−⎟⎠⎞

⎜⎝⎛ +=+=+

−+=+=+−+=

+−=−+−=−−−=

+−=−+−=−−−=−

ωωω

ωωωωωω

ωωωωωωωωωωω

ωωωωωωωωωωωω

ωωωωωωωωωωωω

Equation 4(a-c)

For the first planetary gear set we know that one of the three elements will be fixed, since

it is a fixed differential, or transmission. Also knowing that since the ring gear has to be

11

larger than the sun gear P is always greater than one. Knowing this, one is now able to

determine that the following are the largest possible increases in speed for the three

equations above.

( )

)()()(

111

1

cba

PPPP CR

RCCS ⎟

⎠⎞

⎜⎝⎛ +=

+=+= ωωωωωω

Equation 5(a-c)

If we take the limit of each of these equations as P goes to infinity it is easy to

determine that: (a) ωs increases to infinity; (b) ωc increases to one; and (c) ωr decreases to

one. Since it is necessary to have a larger output shaft speed than input, the best choice is

to have the carrier as the input and the sun gear as the output.

For the second stage planetary gear system we will use the same configuration as

the first, with the exception of a servo motor being added to the ring gear in order to

control the system. The three equations that are used to calculate the final output shaft

speed are as follows:

( )

( ) )(1

)(

)(1

22222

12

111

cPP

b

aP

RCS

SC

CS

ωωω

ωω

ωω

−+=

=

+=

Equation 6 (a-c)

where the subscript numbers refer to which planetary gear train they belong to, the first or

second. With this final set of equations ωr2 is governed by goals of the control system.

12

Torque and Power Analysis of Planetary Gears

Creating a free body diagram of each of the components and then evaluating the

equations derived (Appendix A) yields the following equations governing the torque of

the system:

( )1+= PSC ττ

PSR ττ =

where τC, τS, τR, are the torques applied to the carrier, sun gear and ring gear,

respectively. In these calculations it was assumed that the mass of each component of the

transmission, as well as the compliance of the gears, are negligible. This is due to the

fact that the mass of the rotor is so much larger that it dominates the dynamic response of

the system therefore the mass of each gear can be neglected.

Now that the generator torque is able to be tracked back through the two planetary

gear systems we can include the aerodynamic torque generated on the turbine blades by

the wind.

13

CHAPTER 4 – TURBINE MODEL

Modeling Platform

This simulation was implemented in Simulink, which uses Matlab as its

foundational operating program. “Simulink is a platform for multi-domain simulation

and model-based design for dynamic systems. It provides an interactive graphical

environment and a customizable set of block libraries, and can be extended for

specialized applications.” Matlab is an interactive environment that quickly performs

mathematical operations [3].

Turbine Specifications

In order to accurately model this system, a turbine was selected for the simulation.

It was determined that the Controls Advanced Research Turbine, or CART, would be

used. The CART was chosen because of the abundance of easily accessible data;

stemming from the National Renewable Energy Laboratory (NREL) research associated

with it. The CART is a 600-kW, horizontal-axis wind turbine, measuring 36.6 m tall at

its axis of rotation. It supports two upwind blades capable of full span blade pitch, with a

rotor diameter of 43.3 m [10]. Originally installed at Kahuku Point on the island of

Oahu, Hawaii, in 1996 it was moved to the National Wind Technology Center in Golden,

Colorado [11]. Another advantage for choosing the CART is that it facilitates in the use

of SymDyn.

SymDyn is an aeroelastic code written by the National Wind Technology Center.

It was developed to be used in the simulation, control, and design analysis of horizontal-

14

axis wind turbine systems, and was designed to run with in Simulink [12]. SymDyn uses

equivalent hinge modeling assumptions for the representation of flexible turbine

components such as the tower, drivetrain, and blades. This program has the ability to

calculate 8+Nb Degrees-of-freedoms (DOF) associated with the wind turbine and its

tower, where Nb is the number of blades on the turbine. The following eight DOF are as

follows:

DOF reference number Description

1 Tower for-aft deflection 2 Tower side-to-side deflection 3 Tower twist 4 Nacelle yaw 5 Nacelle tilt 6 Generator shaft position 7 Shaft torsional deflection 8 Hub teeter 9 Blade-1 flap angle … …

8+Nb* Blade-Nb flap angle

Table 1 SymDyn Degrees-of-Freedom

For each of the DOF listed SymDyn calculates reaction forces and moments along each

axis, producing sixty reaction forces for a simple two blade wind turbine. Since we are

only interested in the speed of the generator shaft we simply turn the other eight DOF off.

SymDyn also uses a subroutine known as AeroDyn to calculate the aerodynamic loads

created by the turbine blades. This subroutine was written in Fortran then compiled and

linked to Matlab interface code and uses blade-element-momentum theory with models

for dynamic stall, induced inflow, and tip/hub losses.

SymDyn requires several inputs in order to operate: blade pitch angle, wind

speed, generator torque, as well as initial conditions for the generators’ angular velocity

15

(ωgen) and its position (θgen). With these it first calculates the forces on the rotor blades

created by the current wind speed, with in the aerodynamic model. Those forces then get

passed to the structural model which uses them to calculate all specified DOF, listed in

Table 1. For additional information on the capabilities of AeroDyn, as well as how it

operates, can be found in its user’s guide [13].

Figure 6 SymDyn Flow Chart

Limited studies have been conducted in order to confirm the modeling

assumptions made in SymDyn. One of theses studies used ADAMS to verify the

responses of identical models. The only differences between the two simulations were

the equations of motion and how the integrations were carried out. Figure 7 shows a

comparison of the two models.

Figure 7 SymDyn Verification Results [12]

Forces

Internal loads

ωgen

θgen Structural Model

Aerodynamic

Model Wind Model

Generator TorqueOutput signals

Blade Pitch

16

Using SymDyn will allow us to easily calculate the rotational speed and torques

of the wind turbine blades without actually having to model the complex geometry of the

blades or knowing the Cp-l curves for each wind speed in the given wind spectrum, both

of which are difficult to measure and also hard to obtain from the manufacturer.

There are also several important things that need to be mentioned about SymDyn.

The first being that SymDyn has a standard transmission incorporated into its model, but

by setting the gear ratio to one we can integrate SymDyn into a larger simulation that

contains a gearbox model. By doing this the shaft speed of the rotor is returned instead of

the speed generator’s shaft, enabling the DCVT and controller to be connected externally

to SymDyn, but still allowing SymDyn to be utilized in calculating the turbine speed and

torque. A full description of SymDyn can be found in [12] and [14].

Wind Simulation

Three types of wind inputs used to drive the system are: a constant speed input, a

step input, and an input extrapolated from measured wind data. Each of these wind

inputs are described below.

Constant Speed

The simplest of the three different wind models is that of the constant wind. This

wind speed was particularly useful in determining shaft speeds, shaft torques and power

requirements at a particular wind speed. It was also used in order to verify that the

system was operating as expected, such as checking to make sure that the net power in

and out of the system was zero. Once the system was stabilized this wind type was used

to determine how the individual systems Cp were responding at different constant wind

speeds, and how they correlated to each other.

17

Step Input

A step input was used to study how the system responded to changes in wind,

primarily looking at the response time and torque transients of each of the models. This

input is also the one which will produce the largest torque transients. It is also when the

generator will need to make the largest change is speed in order to compensate for the

sudden change in wind speed.

0 5 10 15 20 25 3015

16

17

18

19

20

21

Time (sec)

Win

d S

peed

(m/s

ec)

Figure 8 Step Wind Input

Variable Wind Data

In order to determine the effects of the DCVT as closely as possible,

approximated variable wind data was also used in each of the two models. These data

were obtained by recreating a plot found in [10], in one second intervals and then slightly

modifying it. These data, whose original average was 13.3 m/sec, needed to be shifted so

that its average lies at 0 m/sec. To achieve this, 13.3 was simply subtracted from each

data point, shifting them in the negative direction. This will allowed the data to be used

with multiple wind magnitudes by simply adding the desired average wind speed back

18

into the data, thus shifting it to desired speed. Though not completely accurate it will

suffice until a point when data can be obtained in the appropriate wind speed. Currently,

the majority of wind data is collected in ten minute averages, which is too large of a gap

for our current studies.

0 5 10 15 20 25 30-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Time (sec)

Win

d S

peed

(m/s

ec)

Figure 9 Interpolated and Adjusted Wind Speed

19

CHAPTER 5 – INDUCTION GENERATOR AND CONTROLLER DESIGN

Induction Generator Design

The torque generated by an induction generator can be modeled as being

proportional to the difference between the shaft speed and synchronous, and can be

represented as:

( )gengenkT ωω −= Equation 7

where k is a modeling constant, ωgen is the angular velocity of the generator, and⎯ωgen is

the synchronous speed of the generator, which is determined by the number of poles in

the generator and the grid frequency. In order to model this equation we must first learn

what the value of k is. To do this, it is necessary to first rewrite the torque equation in a

more useful form, in terms of k, ωgen and S, the of slip:

( )gen

gengenSω

ωω −=

Equation 8

genkST ω≈ Equation 9

20

Knowing that power produced by the generator is the product of its torque and its

rotational speed, and assuming that at its peak output it will produce 600 kW of power,

the nameplate capacity for the CART, the generator which runs at 1800 rpm with a ten

percent slip, and using the derived equation:

2gen

Pkδω

≈

Equation 10

k is found to be 169.76 Nm, for a 600 kW generator.

Control Methods

Since a portion of this project is to maintain it at a constant generator speed as

well as maintaining a constant tip speed ratio it was necessary to have two different

control methods. This stems from the difference in dynamic response time of each

system. The first of these will be looking at the reactions of the system when the

generator speed is held at a constant speed. In order to accomplish this, the generator

shaft is monitored. Its speed is then compared to the value of the desired generator speed

plus a ten percent slip, 1980 rpm. The error between the two is then fed into a

proportional integral (PI) controller which is used to adjust the speed of the ring gear

from second stage planetary gear system. The second control method is very similar to

the first, except that the wind speed and rotor speed are monitored in the attempt to

maintain a constant λ. The error between the actual λ and the desired is once again used

to control the second stage ring gear via a PI controller. Since the system is now

optimizing the rotor speed it operates much slower then the previous system, due to the

large inertia of the rotor which causes the controller values to differ.

21

Controller Design

A standard proportional integral derivative controller was chosen to aid in

controlling the system. In order to determine appropriate controller constants, the system

was first modeled in state-space form. To aid in this transformation Matlabs linmod

function was utilized [4]. This function “extracts the continuous- or discrete-time linear

state-space model of a system around an operating point [4].” By using linmod to control

the ring gear of the second stage planetary gear train around the operating point, the tip

speed ratio, the following state-space model was returned.

[ ] ⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡•

•

r

r

r

r

r

r

ωθ

λ

ωθ

ω

θ

167.10

0.05780

0 0.4989-01

Equation 11

From this set of equations the output λ is independent of its position and is only

dependent on the input speed of the ring gear. This is what should be expected since the

power generated by a wind turbine is a function of its rotational speed and torque rather

than the position of the blades. However, this position is used within the model by

SymDyn when calculating the torque generated by the turbine blades, taking into

consideration the blade position when determining tower shadow. Tower shadow affects

how the wind flows around the turbine tower, usually slowing down the wind in front of

it, which then in turn reduces the torques associated with the blade in front or behind it.

Since these are the state-space matrices for the system without the PI controller

the matrices will be the same if one was to use the generator torque as the input. From

these equations λ for the undamped system can now be represented as:

22

rsωλ

4989.00963.0

+=

Equation 12

which is a first order system with a single pole. Once we add the PI controller, an

additional pole, located at the origin, and an additional zero are added to the system. It is

the placement of the additional zero that determines how the PI controller will operate.

When adding a PI controller into the model Matlab the signal is modified and can be

represented in the form of:

rIPD

ssKsKsK

ωλ ⎟⎠⎞

⎜⎝⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛ ++=

4989.00963.02

Equation 13

In the preceding equation KP, KI, and KD are the constants associated with the PI

controller.

Constant Generator Speed Pole and Zero Location

The PI controller for the constant generator speed model was designed so that the

variation in the generator speed was minimized with a change in wind speed. This was

accomplished by placing the location of the controller defined zero far to the left of the

two poles, along the real axis. This position produces the root locus plot shown in Figure

10. The point where the loci return to the real axis is the point where, for the given zero

location, the system produces the quick response time as well as reducing the oscillations

produced by the controller. It was determined that a zero located at -20+0i was adequate

for this system, producing minimal variance in the generator speed, and yielding an

equation for the PI controller in the following form:

23

⎟⎠⎞

⎜⎝⎛ +

ssKG

20

Equation 14

At this point the controller gain, KG, was read off of the root locus plot of the controlled

system, shown to be 79 in Figure 10 below.

Figure 10 Damped Constant Generator Speed System

Comparing the coefficients of Equation 13 and Equation 14 would then yield the values

for the controller coefficients. In the case used in the system, and shown above they were

as follows:

Table 2. Constant Generator Speed Controller Coefficients KP 79 KI 1580

This process was repeated several times, implementing the new PI controller values for

each zero location until the results produced a controller that was favorable, which are the

results shown above.

24

Constant Tip Speed Ratio Pole and Zero Location

In a similar fashion, the pole location for the constant λ controller was placed at

⎟⎠⎞

⎜⎝⎛ +

ssKG

75.0

Equation 15

producing the following root locus plot and controller coefficients.

Root Locus

Real Axis

Imag

inar

y Ax

is

-1.5 -1 -0.5 0-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

System: mysysGain: 1.87Pole: -1.18 - 1.63e-008iDamping: 1Overshoot (%): 0Frequency (rad/sec): 1.18

Figure 11 Damped Constant Tip Speed Ratio System

Table 3. Constant Tip Speed Ratio

Speed Controller Coefficients KP 1.87 KI 1.4

It is relevant to mention that the PI controller for the constant λ is slower then the

controller for the constant generator model. Since the dynamic response of the rotor is

much slower than that of the generator, due to its large mass, a controller with the same

response time is not necessary. In fact, if a controller with a similar response time is used

the controller is then used to increase and decrease the rotor shaft speed, instead of

allowing the wind to do so.

25

CHAPTER 6 – SIMMULATION RESULTS

Constant Wind Speeds

As mentioned earlier, each model was evaluated using a constant wind speed over

a range of wind speeds. For each of these tests, results were produced in which the

power produced by system, calculated by using Equation 16, stabilized around some

constant value wind speed, Figure 12. At this point the only oscillation in the system was

caused by tower shadow. This oscillation was removed by taking the last one thousand

τω=P Equation 16

0 5 10 15 20 25 304

4.5

5

5.5

6

6.5

7

7.5

8x 105 Generator Power

Time (sec)

Pow

er (W

atts

)

Figure 12 Typical Constant Wind Speed Results

data points, when the system has reached a stead state value, and then averaging their

values. For each of the three systems these averages were then used to calculate the tip

speed ratio, λ, which was then plotted against their coefficients of performance, Cp,

26

which are displayed below in Figure 13 (a). The first thing to note is that one of the

models, the constant tip speed ratio system, only appears to display as two points. In

reality, these two points are multiple points plotted on top of each other. This is due to

the fact that in this model we are driving λ to a constant value. In order to properly

compare each system, the value of λ will be changed while maintaining a constant wind

speed of 18 m/sec. This curve is visible in Figure 13 (b) while being plotted against the

other two systems. The second thing to note is that since each of the models respond to

the same wind speed in a different manner, each curve resides in a different region of the

graph. Though each of these systems operates within the same general Cp region, they

each pass through it at different wind speeds. This is visible when the wind speed is

plotted against the coefficient of performance, in Figure 14.

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 70.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Tip Speed Ratio (λ)

Cp

No ControllerConstant ωgen

Constant λ

Figure 13 (a) Steady State Wind Cp vs. λ, Wind Speeds Varing

from 10 to 30 m/sec

27

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 70.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Tip Speed Ratio (λ)

Cp


Constant λ

Figure 13 (b) Modified λ, Wind Speeds Varing from 10 to 30 m/sec

With all three tip speed ratios plotted against each other it is easy to tell that when

the generator is maintained at a constant speed, by adding additional or removing excess

power from the system through the second ring gear, the overall efficiency of the system

is increased by approximately two percent, at its maximum Cp. What is not apparent

from these curves is at what wind speed this maximum is reached. This can be seen by

plotting the wind speed against the coefficient of performance, shown in Figure 14. By

doing this we can tell that the system without a controller reaches its max efficiency at

higher wind speeds then either of the other two systems. It is also apparent that at lower

wind speeds each system that utilizes controllers operates at significantly higher Cp’s.

This is a large advantage since, depending on the area the turbine is placed, lower wind

speeds are often more predominate. This in turn equates to larger energy production.

28

14 16 18 20 22 24 26 28 300.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Wind Speed (m/sec)

Cp


Constant λ

Figure 14 Cp vs. Changing Steady State Wind

0 5 10 150

5

10

15x 105

Blade Tip Angle (deg)

Pow

er (W

)

Constant GeneratorContorller offConstant λ

Figure 15 Power Produced at 18 m/sec vs. Blade Pitch Angle

It is also important to note that the Cp’s for each of the systems reach a maximum at

approximately 0.123. This is significantly less then the expected values, ranging from

about 0.35 to about 0.45. This decrease in Cp is caused by the root blade pitch angle.

Currently an angle of fifteen degrees is being used to calculate all aerodynamic torques.

29

This aerodynamic torque, created on the blades by the wind, is dependant upon the wind

speed as well as the blade pitch. At small blade pitch angles each system is more

efficient but the range of wind speeds at which the blades will produce power is smaller,

due to stall at higher wind speeds. Typically the blade pitch angle would be controlled to

produce optimum power. Since this study is to focus on the comparison between

controlling a two stage planetary gear train with different controller algorithms, the

change in blade pitch angle was held constant to better visualize the effects of each

system.

Another way that the Cp of the constant generator model can be affected is to

change the overall gear ratio of the variable transmission, when the second stage ring

gear is held in place. This forces the curve to shift, changing the optimal wind speed as

well as the range to efficiently operate the turbine which can be seen in Figure 16 below.

Even thought this result is also being achieved by changing the rotational speed of the

ring gear, which also forces these curves to shift, by calibrating the system for a given

spectrum of wind will reduce the amount of power required to achieve the same results.

10 12 14 16 18 20 22 240

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Wind Speed (m/sec)

Cp

3:15:17:1

Figure 16 Effects on Cp Due to Changes in the

Second Stage Gear Ratio

30

Step Input Wind Speed

System Power

In evaluating the effectiveness of the three systems when a step wind speed input

was applied, several things were discovered. One of which was that when the system

was controlled to achieve a constant tip speed ratio of 4.5, the energy captured was two to

thirty percent larger than that of the uncontrolled system. Though it ranged from being

nineteen percent more efficient than the constant generator system to ten percent less.

Since the ring gear either increases or decreases the power that flows through the

generator from the system, the total power being removed from the wind can best be seen

by looking at the power in the rotor (Figure 17). At this point a comparison can be made

on how effective all three systems are at removing power from the wind, since the power

added or removed by the ring gear does not directly show up in the rotor. From this

same figure, it is also apparent that the power produced by the constant λ system is not

only the lowest, but also fluctuates the most.

It is pertinent to note that within the first five seconds of each simulation there is a

large sudden increase in power, most visible in the constant λ model. This is caused by

the value of the initial conditions at which each system starts at. When evaluating each

of the systems, in the step wind input as well as in the variable wind input that follows,

the first five seconds have be disregarded, as this is the time when the transient response

due to the initial conditions have decayed from the system.

In order to maintain either a constant ωgen of 207 rads/sec (1980 rpm), or a

constant λ of 4.5, power is routed through the ring gear and can be seen in Figure 18.

When looking at each of these systems it is visible that for the majority of the time power

31

is being added into the system. It is also noticeable that in order to keep the system stable

at a λ of 4.5, significantly more power than that of the constant generator model must be

pulled from the grid. For the constant ωgen system, power is being removed for all but a

very short interval. This indicates that for the current range of wind speeds, this

generator is currently operating with a gear ratio lower then what is required by the

system, and should be increased so that at a midrange wind speed the ring gear would be

at a zero velocity. This causes the constant ωgen curve in the figure above to shift up or

down, depending on how the gear ratio is changed. In order to achieve this, the size of

gears contained with in system should be changed, allowing it to maintain its same

functionality while requiring less additional power to be added into the system. For

comparison reasons the standard gear ratio of 43.2:1 was used in all three systems, 6.194

0 5 10 15 20 25 301

2

3

4

5

6

7

8

9

10x 105

Time (sec)

Pow

er (W

atts

)

UncontrolledConstant. ωgen

Constant λ

Figure 17 Power in the Rotor from Step Winds

32

0 5 10 15 20 25 30-20

-15

-10

-5

0

5x 104

Time (sec)

Pow

er (W

atts

)


Constant λ

Figure 18 Power in the Ring from Step Winds

for the first stage and 5.0 for the second stage. Plots of how changing the second stage

gear ratio affects the power through out the system can be seen in Appendix B.

By using the ring gear to control each of the two systems, the power output from

the generator is greatly different. Again, with the constant λ the power in the generator

has the largest power fluctuations, and these large power spikes create stresses in the

power electronics, which could become a source of failure. The uncontrolled system, like

the constant λ system, has an increase in power associated with the wind steps, but does

not include the large sudden spikes associated with the model driven to maintain a

constant tip speed ratio. The steps in power associated with the uncontrolled system,

caused by the same change in wind speed, are also smaller then those of the constant λ.

On the other hand, the power produced by the constant ωgen system remains constant,

which will aid in connecting the power produced by the generator directly to the grid.

33

0 5 10 15 20 25 300

2

4

6

8

10

12x 10

5

Time (sec)

Pow

er (W

atts

)


Constant λ

Figure 19 Power in the Generator from Step Winds

System Torques

How each of the different controllers affects the torques in each of the three

systems directly relates to the cost of the system, as well as its life span. The larger the

torque spikes in the system the more likely it is to fail, and the more expensive the

hardware becomes. When comparing each of the three systems it can be seen that the

one controlled to achieve a constant generator speed virtually eliminates all torque

transients caused by fluctuations in wind speed. In fact, for the two meter per second

increase in wind input the torque associated with the constant generator system changed

less then one tenth of a percent. Though only the torques associated with the ring gear

are shown, the rotor torques and generator torques all follow almost identical curves, just

at different magnitudes.

34

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3x 10

4

Time (sec)

Torq

ue (N

m)


Constant λ

Figure 20 Torque on the Ring Gear from a Step Input

System Shaft Speeds

The most visible place to see how the power added into the ring gear, of the constant λ

system, affects the power captured from the wind by the turbine is to look at the shaft

speed of the rotor. Here you are able to see how the power is affected by the ring gear

speed as each step in the wind speed occurs. When there is an increase in wind speed,

then the power added into the system, necessary to maintain the turbine blades at the

desired tip speed ratio, is decreases by slowing down the ring gear. On the other hand

when the wind speed decreases the required power increases, so at speeds that are below

the optimal speed for the turbine, a larger amount of power is necessary to maintain it at

its most efficient Cp level. Like with the constant generator system, the amount of power

added or removed from the system can be affected by changing the over all gear ratio,

which is also visible in Appendix B, and has been left for further studies. By observing

the reaction of the ring gear in the constant generator model caused by the change in wind

35

speeds it is apparent that the ring gear efficiently removes any changes in the rotor,

holding the generator shaft at a constant speed. In the constant λ model, it is evident that

the overall power removed from the wind is higher than either of the other two systems.

0 5 10 15 20 25 303.5

4

4.5

5

Time (sec)

Sha

ft S

peed

(rad

s/se

c)


Constant λ

Figure 21 Shaft Speed of the Rotor from Step Winds

0 5 10 15 20 25 30190

195

200

205

210

215

220

Time (sec)

Sha

ft S

peed

(rad

s/se

c)


Constant λ

Figure 22 Shaft Speed of the Generator from Step Winds

36

0 5 10 15 20 25 30-12

-10

-8

-6

-4

-2

0

2

Time (sec)

Sha

ft S

peed

(rad

s/se

c)


Constant λ

Figure 23 Shaft Speed of the Ring Gear from Step Winds

Variable Wind Input

System Power

The power removed from the wind by the wind turbine rotor when a variable

wind input is applied is, in one way, similar to that of the step, but in others significantly

different. Like with the step input, the power in the rotor of the model controlled to

achieve a constant tip speed ratio, λ, has power fluctuations that are almost identical to

those of the uncontrolled system, just at larger amplitudes. From this model it is also

apparent that with the physical parameters set as they are, gear ratio and blade pitch

angle, the power gained from the system is very similar to that of the uncontrolled

system. On the other hand, the system driven to maintain a constant generator speed has

very little power fluctuation, even with large changes in the wind speed. Examining the

power that is being routed through the ring gear for each of the two controlled systems

demonstrates how much grid power is being added into each system. What is clearly

visible is that during almost the entire operation the constant λ system requires

37

significantly more power to be added into the system. In fact, on average the constant λ

system required approximately twice the power input then that of the constant ωgen.

These power averages can be seen in Table 4.

Table 4 Average Power Flow of System Components (kW) Uncontrolled Constant ωgen Constant λ

Generator 521.94 663.49 717.23 Ring 0.00 -63.84 -120.52 Rotor 521.94 599.65 596.71

5 10 15 20 25 302

3

4

5

6

7

8

9

10x 105

Time (sec)

Pow

er (W

atts

)


Constant λ

Figure 24 Power in the Rotor from Variable Winds

5 10 15 20 25 30-2.5

-2

-1.5

-1

-0.5

0

0.5x 105

Time (sec)

Pow

er (W

atts

)


Constant λ

Figure 25 Power in the Ring Gear from Variable Winds

38

The total power that is being routed through the main generator is equal to the

total into the system via the rotor minus that being added or subtracted by the controlling

ring gear. In both controlled systems this power is being added into the system which

causes the power being transferred through the generator to be a little misleading. When

looking at Figure 26 below it appears that the constant λ model is out producing both

systems but in reality it is the second most productive system. This is shown, in Table 4,

by summing the power flowing through the generator and the ring gear and then

averaging their values.

5 10 15 20 25 302

3

4

5

6

7

8

9

10

11

12x 105

Time (sec)

Pow

er (W

atts

)


Constant λ

Figure 26 Power in the Generator from Variable Winds

System Torques

Like the torques created on the constant tip speed ratio system when using a step

input, the torques associated with variable winds are once again greater than the other

two systems. Unlike the step input, the large torque spikes associated with the λ system

are no longer present. Even so, the fluctuations in torques created in order to achieve an

39

optimal tip speed ratio are larger than that of the uncontrolled system, but they both

follow similar curves, just at different magnitudes. When the averages of the torques are

compared you can easily see that not only are the torques of the λ system greater but also

have a larger range, that of 127 kNm, compared to the uncontrolled system with a range

of 88 kNm. Once again the model controlled for a constant generator speed has virtually

no fluctuations in any part of the system, shown in Table 5 below.

Table 5 Variable Wind System Torques (kNm) Rotor Ring Gear Generator

Max 197.02 22.82 4.56 Average 159.82 18.51 3.70 Constant λ Min 123.07 14.26 2.85 Max 138.13 16.00 3.20 Average 138.12 16.00 3.20 Constant ωgenMin 138.11 16.00 3.20 Max 167.37 19.39 3.88 Average 125.49 14.54 2.91 Uncontrolled Min 79.42 9.20 1.84

5 10 15 20 25 300.6

0.8

1

1.2

1.4

1.6

1.8

2x 105

Time (sec)

Torq

ue (N

m)


Constant λ

Figure 27 Torque in the Rotor from Variable Winds

40

System Shaft Speeds

As the ring adds and removes power to and from the system, it behaves in

different manners. In the constant generator model the rotor speed is allowed to rotate at

whatever speed it’s capable of, so long as the rotation of the ring gear maintains a

constant generator speed. In the constant λ model, the blades are sped up or down in

order to maintain the most efficient tip speed ratio. This means that the shaft speed of the

λ system needs to speed up and slow down at a much faster rate. Since accelerating or

decelerating the large mass of the turbine blades takes a large amount of energy, it is easy

to see why the power flowing through the ring gear is so much larger then the other two

models. In this case power is added into the system primarily to slow down the turbine

blades to hold λ at value of 4.5.

Table 6 shows the speeds at which each of the components rotate. Comparing

theses values it’s noticeable that the speed of the constant generator model remains at

almost exactly 1980 RPM, 1800 plus 10% slip, with a variance between its maximum and

minimum speeds of 0.0082 RPM. The speed of each of the turbine elements are shown

in Figure 28 through Figure 30.

Table 6 System Shaft Speeds due to Variable Wind Rotor Ring Generator

rads/sec RPM rads/sec RPM rads/sec RPM

Max 4.2 39.8 -6.8 -64.9 215.4 2056.8 Average 4.0 38.3 -7.4 -70.9 210.3 2008.3 Constant λ

Min 3.8 36.2 -8.7 -83.3 205.3 1960.4 Max 4.7 45.2 -0.6 -5.6 207.3459 1980.0071

Average 4.3 41.2 -4.2 -40.3 207.3449 1979.9984Constant ωgen Min 3.9 37.3 -7.7 -73.6 207.3440 1979.9889

Max 4.9 46.8 0 0 211.3 2018.1 Average 4.8 45.5 0 0 205.6 1963.5 Uncontrolled

Min 4.6 44.1 0 0 199.3 1903.5

41

5 10 15 20 25 303.6

3.8

4

4.2

4.4

4.6

4.8

5

Time (sec)

Sha

ft S

peed

(rad

s/se

c)


Constant λ

Figure 28 Shaft Speed of the Rotor in Variable Winds

5 10 15 20 25 30-14

-12

-10

-8

-6

-4

-2

0

Time (sec)

Sha

ft S

peed

(rad

s/se

c)


Constant λ

Figure 29 Shaft Speed of the Ring Gear in Variable Winds

42

5 10 15 20 25 30198

200

202

204

206

208

210

212

214

216

Time (sec)

Sha

ft S

peed

(rad

s/se

c)


Constant λ

Figure 30 Shaft Speed of the Generator in Variable Winds

43

CHAPTER 7– CONCLUSIONS

For this study, using a constant rotor pitch angle of 15° and a gear ratio of 43.165

for all three systems, it’s apparent that when the generator is held at a constant value of

1980 RPM, several benefits are realized. With this constant speed the high costs

associated with the power electronics can be greatly reduced. This is because only ten

percent of the power in the system will need to flow through the controller

motor/generator, which would then need to be rectified and inverted, or possibly stored

for later use by the controller. The rest of the power, flowing through the main

generator, can then be directly added on to the grid at the required frequency of 50 Hz or

60 Hz. By maintaining a constant generator speed, on average, more power is also

removed from the system while adding less power into it via the control motor/generator,

as compared to the constant λ system. This same system also removes all large torque

transients which could potentially lead to a longer life span, possibly being the optimal

system to implement.

It was anticipated that when using the ring gear to control the system to achieve a

constant tip speed ratio more power would be produced. This did happen but not at the

levels that were desired. What did greatly increase were the transient torques associated

with each member of the system, by approximately twenty seven percent. With these

large spikes in torque came large fluctuations in power which will require larger and

more expensive power electronics in order to connect to the grid, defeating the purpose of

implementing a DCVT. These large torques also decrease drive train life, while also

44

increasing operation and maintenance costs, proving to be counter productive to the

system.

Another important thing discovered during this thesis was how an over controlled

λ system operates. When the controller is designed to operate with a rapid response and

the wind speed fluctuates the controller tries to rapidly increase or decrease the rotors

angular velocity in order to maintain a constant λ. This in turn not only required large

amounts additional power, but also greatly increased the torques through the system.

45

CHAPTER 8– FUTURE RECOMMENDATIONS

Several things still need to take place in order to completely verify the

effectiveness of using a two stage planetary continuously variable transmission in a wind

turbine, as well as to determine which control system will produce optimal power with

minimal side effects. To determine this, optimization of the blade pitch angle to the

wind speed range at which the turbine will be operating within, should be completed.

The gear ratio for the two controlled systems should also be set so that the

motor/generator is primarily removing power from the system, instead of adding to it.

This detailed analysis will produce results that will show how each of the optimized

systems compare to each other.

46

REFERENCES

1. Idan, M. and D. Lior, Continuous Variable Speed Wind Turbine: Transmission

Concept and Robust Control. Wind engineering, 2000. 24(3): p. 151-767.

2. Idan, M., D. Lior, and G. Shaviv, A Robust Controller for a Novel Variable Speed

Wind Turbine Transmission. Wind engineering, 2000. 24(3): p. 151-167.

3. Simulink 2006, The MathWorks, Inc.: Natic, MA.

4. MATLAB. 2006, The MathWorks, Inc.: Natic, MA.

5. Carlin, P.W., A.S. Laxson, and E.B. Muljadi, The History and State of the Art of

Vaiable-Speed Wind Turbine Technology. Wind Energy, 2003. 6: p. 129-151.

6. Iqbal, M.T., A. Coonick, and L.L. Frerris, Dynamic Control Options for Variable

Speed Wind Turbines. Wind engineering, 1994(1): p. 1-12.

7. Mangialardi, L., and Mantriota, G., Dynamic Behavior of Wind Power Systems

Equipped with Automatically Regulated Continuously Variable Transmission.

Renewable energy, 1996. 7(2): p. 185-203.

8. Paul, B., Kinematics and Dynamics of Planar Machinery. 1979: Prentice-Hall.

9. Zhao, X. and P. Maisser, A Novel Power Splitting Drive Train for Variable Speed

Wind Power Generators. Renewable energy, 2003. 28(13): p. 2001-2011.

10. Johnson, K., L. Fingersh, and A. Wright, Controls Advanced Research Turbine:

Lessons Learned During Advanced Controls Testing. 2005, National Renewable

Energy Laboratory.

47

11. Fingersh, L. and K. Johnson, Controls Advanced Research Turbine (CART)

Commissioning and Baseline Data Collection. 2002, National Renewable Energy

Laboratory.

12. Stol, K.A. and G.S. Bir, SymDyn,

http://wind.nrel.gov/designcodes/simulators/symdyne/, Editor. Last modified 26-

May-2005, accessed 26-May-2005, NWTC Design Codes.

13. Laino, D., AeroDyn, http://wind.nrel.gov/designcodes/simulators/aerodyn/, Editor.

Last modified 05-July-2005, accessed 05-July-2005, NWTC Design Codes.

14. Stol, K.A. and G.S. Bir, User's Guide for SymDyn Version 1.2. 2003, National

Wind Technology Center.

http://wind.nrel.gov/designcodes/simulators/symdyne/

http://wind.nrel.gov/designcodes/simulators/aerodyn/

48

APPENDIX A Free Body Diagrams

49

Resulting Forces

Planetary Gear System

Sun Gear

Ring Gear

S

SS

SSS

RF

RFM

JJM

ττ

ω

=

−=

=

=

=

∑

∑•

00

0

R

RR

RRR

RF

RFM

J

JM

ττ

ω

=

−=

=

=

=

∑

∑•

00

0

Carrier

Planet Gear

C

CC

CCC

RF

RFM

JJM

ττ

ω

=

−=

=

=

=

∑

∑•

00

0

SRC

SRC

RS

PRPS

FFFFFF

FM

MaFFF

RFRFJ

JM

220

00

00

==++=

=

=

=

=+−=

=

=

∑

∑

∑•

ω

50

Resulting Torques

Carrier Torque Sun Torque Ring Torque

( )

RSC

SRCC

CCC

SRC

SCP

PSR

FFF

RRFRF

RRR

RRRRRR

222

2

2

==

+=

=

+=

−=+=

τ

τ

( )

( )

( )1

1

1

+=

+=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=

+=

P

PRR

P

RR

RF

RRF

CS

SC

S

R

S

RSC

S

SS

SRSC

ττ

ττ

ττ

ττ

( )

( )

PP

PP

RR

RF

RRF

SR

SR

CR

R

SRC

R

RR

SRRC

ττ

ττ

ττ

ττ

ττ

=

⎟⎠⎞

⎜⎝⎛ +

+=

⎟⎠⎞

⎜⎝⎛ +

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

=

+=

11

1

11

1

51

APPENDIX B

Effects in Power by changing the Second Stage Gear Ratio

52

Step Wind Input

Constant Generator Speed Controlled System

0 5 10 15 20 25

8.5x 105

304

4.5

5

5.5

6

6.5

7

7.5

8

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 31 Change in Rotor Power Due to Changes in the

Second Stage Gear Ratio with Step Input

0 5 10 15 20 25 30-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 32 Change in Ring Gear Power Due to Changes in the


53

0 5 10 15 20 25 306.6335

6.634

6.6345

6.635

6.6355

6.636

6.6365x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 33 Change in Generator Power Due to Changes in the


Constant Tip Speed Controlled System

0 5 10 15 20 25

14x 105

30

2

4

6

8

10

12

Time (sec)

Pow

er (W

atts

)

3:15:17:1



54

0 5 10 15 20 25 30-10

-8

-6

-4

-2

0

2

x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1



0 5 10 15 20 25 30

2

4

6

8

10

12

14

16

x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 36 Change in Generator Power Due to Changes in the


55

Variable Wind Input

Constant Generator Speed Controlled System

0 5 10 15 20 25

8.5x 105

304

4.5

5

5.5

6

6.5

7

7.5

8

Time (sec)

Pow

er (W

atts

)

3:15:17:1


Second Stage Gear Ratio with Variable Input

0 5 10 15 20 25 30-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 38 Change in Ring Power Due to Changes in the


56

0 5 10 15 20 25 306.6335

6.634

6.6345

6.635

6.6355

6.636

6.6365x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 39 Change in Generator Rotor Power Due to Changes

in the Second Stage Gear Ratio with Variable Input

Constant Tip Speed Controlled System

0 5 10 15 20 25

14x 105

30

2

4

6

8

10

12

Time (sec)

Pow

er (W

atts

)

3:15:17:1



57

0 5 10 15 20 25 30-10

-8

-6

-4

-2

0

2

x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1



0 5 10 15 20 25 30

2

4

6

8

10

12

14

16

x 105

Time (sec)

Pow

er (W

atts

)

3:15:17:1

Figure 42 Change in Generator Rotor Power Due to Changes

in the Second Stage Gear Ratio with Variable Input

58

APPENDIX C:

Glossary of Matlab Function Blocks

59

Block Name Block Icon Description

Clock

Outputs the current simulation time.

Constant

Outputs a constant value

Demux

Splits a vector signal into a scalar or smaller vectors.

From

Receives data from a Goto block with the same name, "A" in this case.

From Workspace

Outputs the value defined by the workspace variable.

Function

Runs input through a specified function or through a m-file function.

Gain

Multiplies the input value by a constant or gain.

Goto

Sends data to a From block with the same name, "A" in this case.

In

Provide an input port for a subsystem or model.

Integrator

Continuous time integration of input signal.

60

Lookup Table

Perform one dimension interpolation of input values using the specified table.

Mux

Creates vector signal from scalars or smaller vectors.

Out

Provide an output port for a subsystem or model.

PID Controler

Implements a PID controller in the form: P+I/s+Ds

Product

Multiplies or divides inputs.

Scope

Displays a graph on the input(s) vs. simulation time.

Sub System

Creates sub system.

Sum

Adds or subtracts inputs.

To Workspace

Writes input to specified variable.

Transfer Function

Processes the input signal through a specified transfer function

61

APPENDIX D

Simulation Models

62

Global Models

Constant Generator Model

63

Constant Tip Speed Model

64

Masked Models

Wind Model

65

DCVT Model

66

APPENDIX E

Simulation Files

67

M-Files called by DCVT Model

Shaft Speed Calculation

function Ws=GearRatio(v)

Wc = v(1); % rads/sec P = v(2); % m or # of teeth Wr = v(3); % rads/sec Ws=Wc*(1+P)-Wr*P; % rads/sec

Torque Calculation

function T=Torque(v)

Ts=v(1); % Nm (torque of carrier) P=v(2); % m or # of teeth Tc=Ts*(P+1); Tr=Ts*P; T(1)=Tc; T(2)=Tr;

M-Files called by SymDyn

Inputsim

% inputsim.m: Contains initialization data for analyses % SymDyn v1.20 7/15/03 % Assumes S.I. units % %% % General inputs %% active_dofs = [6]; % active degrees of freedom from the list [1,2,3,4,5,6,7,8,9,...8+Nb] aero_flag = 1; % aerodynamics flag (1 = aero on, 0 = aero off) usewindfile_flag = 0; % flag for use of AeroDyn wind file (1 = yes, 0 = no) g = 9.81; % gravity acceleration [m/s^2] %% % Joint structural damping [N.m.s/rad] %%

68

Cjoint(1) = 0; % tower fore-aft damping Cjoint(2) = 0; % tower side-to-side damping Cjoint(3) = 0; % tower twist damping Cjoint(4) = 0; % yaw joint damping Cjoint(5) = 0; % tilt joint damping Cjoint(6) = 0; % generator shaft dampin gCjoint(7) = 0; % shaft torsion damping Cjoint(8) = 0; % teeter joint damping Cjoint(9) = 0; % blade flap damping %% % Override SymDyn parameters if desired % Do not change Nb here (this must by done in inputprops.m and SymDynPP rerun) %% %tilt0 = 0; % zero tilt (example) %K4 = 1e6; % nonzero yaw stiffness (example) %K5 = 1e7; % nonzero tilt stiffness (example) %% % Initial conditions and prescribed displacements and velocities %% % Equilibrium position for joints, when spring torque is zero [radians] % Fixed tilt and precone values are already included q0 = zeros(1,8+Nb); % zeros % Initial conditions for joint angles [radians] % Fixed tilt and precone values are already included q_init(1) = 0; % tower fore-aft q_init(2) = 0; % tower lateral q_init(3) = 0; % tower twist q_init(4) = 0; % yaw q_init(5) = 0; % tilt q_init(6) = 0; % azimuth q_init(7) = 0; % shaft compliance q_init(8) = 0; % teeter q_init(9) = 0; % flap of blade #1 q_init(10) = 0; % flap of blade #2 %q_init(11) = 0; % flap of blade #3 - uncomment for 3-bladed rotor % Initial conditions for joint velocities [radians/s] qdot_init(1) = 0; % tower fore-aft rate qdot_init(2) = 0; % tower lateral rate qdot_init(3) = 0; % tower twist rate qdot_init(4) = 0; % yaw rate qdot_init(5) = 0; % tilt rate qdot_init(6) = 4.5 %41*pi/30; % generator speed qdot_init(7) = 0; % shaft compliance rate qdot_init(8) = 0; % teeter rate qdot_init(9) = 0; % flap rate of blade #1 qdot_init(10) = 0; % flap rate of blade #2

69

%qdot_init(11) = 0; % flap rate of blade #3 - uncomment for 3-bladed rotor %% % Inputs for calculation of steady-state operating point (using calc_steady.m) % and for linearization (using calc_ABCD.m) % 'constant-speed' means azimuth position is not an active degree-of-freedom %% % parameters for steady-state only: trim_case = 1; % calc_steady.m case (1 = find gen torque, 2 = find coll. pitch) % - ignored for constant-speed case % parameters for steady-state and linearization: wdata_op = [18, 0, 0, 0, 0.2, 0, 0]; % operating pt hub-height wind data (delta in deg) theta_op = 12*pi/180; % operating pt collective blade pitch angle [rad] Tg_op = 152129; % operating pt gen. torque [Nm] - ignored for constant-speed case omega_des = 41*pi/30; % desired mean gen. speed [rad/s] - ignored for constant-speed case nsteps = 200; % number of time steps to save operating point and state matrices over % parameters for linearization only: torque_ctrl_swtch = 0; % Gen. torque control (0 = off, 1 = on) pitch_ctrl_swtch = 2; % Pitch control type (0 = no pitch, 1 = coll. pitch, 2 = individ. pitch) wdata_dist = [1]; % Elements of AeroDyn HH wind data for treatment as disturbance, from [1,...,7] load_meas = [9,10]; % List of load locations to define linear outputs, from [1,...,8+Nb] load_meas_compt = [0,0,0,0,0,1; % Boolean matrix for desired load components, one row for each 0,0,0,0,0,1]; % location in load_meas, representing [Fx,Fy,Fz,Mx,My,Mz] twr_sg_height = 9.3; % Height of tower strain gauge from base for tower load measurements %% % Simulation inputs (custom user variables for Simulink models) % Typical variables are wdata, theta, and Tg, but others may be appended % % wdata = [0.0, 18.0, 0, 0, 0, 0.2, 0, 0]; % custom wdata (steady wind) % wdata = [00.00, 18.0, 0, 0, 0, 0.2, 0, 0; %% custom wdata (step in wind speed) % 20.00, 18.0, 0, 0, 0, 0.2, 0, 0; % 20.01, 20.0, 0, 0, 0, 0.2, 0, 0;

70

% 25.00, 20.0, 0, 0, 0, 0.2, 0, 0]; wdata = [00.00, 0.0, 0, 0, 0, 0.2, 0, 0; 10.00, 0.0, 0, 0, 0, 0.2, 0, 0; 10.01, 2.0, 0, 0, 0, 0.2, 0, 0; 15.00, 2.0, 0, 0, 0, 0.2, 0, 0; 15.01, 0.0, 0, 0, 0, 0.2, 0, 0; 20.00, 0.0, 0, 0, 0, 0.2, 0, 0; 20.01, -2.0, 0, 0, 0, 0.2, 0, 0; 25.00, -2.0, 0, 0, 0, 0.2, 0, 0; 25.01, 1.0, 0, 0, 0, 0.2, 0, 0; 30.00, 1.0, 0, 0, 0, 0.2, 0, 0]; %wdata = [0.0, 16.0, 0, 0, 0, 0.2, 0, 0; % custom wdata (ramp in wind speed) % 30.0, 20.0, 0, 0, 0, 0.2, 0, 0]; theta = 12*pi/180*ones(Nb,1); % custom pitch angles Tg = 152129; % custom generator torque

Inputprops

% inputprops.m: Contains the input turbine properties for derivation of SymDyn parameters % via the SymDyn preprocessor (SymDynPP.m) % SymDyn v1.20 7/15/03 % Assumes S.I. units % ftitle = 'CART properties (6/02)'; % title for reference % Geometry and other constants Nb = 2; % number of blades rigid_hub = 0; % 0 = free teeter, 1 = locked teeter (for use in frequency matching) gearratio = 1; % gearbox gear ratio precone = 0; % blade precone, pos. moves blade tips downwind [deg] tilt0 = -3.77; % nominal tilt, pos. moves downwind end of nacelle up [deg] delta3 = 0; % teeter axis angular offset (ignored for locked teeter or Nb>2) [deg] omega0 = 42; % nominal rotor speed, pos. clockwise when looking downwind [rpm] dtheight = 34.862; % tower height [m] dtilt = 1.734; % height from tower top to tilt axis, pos. up [m] dshaft = 0; % dist. from tilt axis to shaft axis, normal to shaft, pos. down [m] dteeter = -3.867; % dist. from tilt axis to teeter axis, parallel to shaft, pos. downwind [m] dhradius = 1.381; % dist. from teeter axis to blade root, normal to hub centerline [m] dbroot = 0; % dist. from teeter axis to blade root, parallel to hub centerline [m] dblength = 19.9548; % blade length from root to tip [m]

71

% Center of mass locations cyoke = 0; % c.g. of nacelle yoke, measured up from tower top along yaw axis [m] cnx = 0; % c.g. of nacelle, measured down from tilt axis [m] cny = -0.402; % c.g. of nacelle, measured downwind from tilt axis [m] cHSS = 0; % c.g. of HSS + generator from tilt axis along shaft, pos. upwind [m] cLSS = -3.867; % c.g. of LSS from tilt axis along shaft, pos. downwind [m] chub = 0; % c.g. of hub from teeter axis, measured upwind along hub center [m] % Masses myoke = 0; % mass of nacelle yoke [kg] mnac = 23228; % mass of nacelle + nonrotating parts of generator and shaft bearings [kg] mHSS = 0; % mass of HSS + rotating generator parts [kg] mLSS = 5885; % mass of LSS [kg] mhub = 5852; % mass of hub [kg] % Moments of inertia (MOI's) Iyokex = 0; % MOI of nacelle yoke in [yoke] frame [kg.m^2] Iyokey = 0; % " Iyokez = 0; % " Inacx = 3.659e4; % MOI of nacelle and all nonrotating gen. parts in [nac] frame [kg.m^2] Inacy = 1.2e3; % " Inacz = 3.659e4; % " IHSSlat = 0; % lateral MOI of HSS + generator [kg.m^2] IHSSlong = 34.4; % longitudinal MOI of HSS + generator [kg.m^2] ILSSlat = 0; % lateral MOI of LSS [kg.m^2] ILSSlong = 0; % longitudinal MOI of LSS [kg.m^2] Ihubx = 1.5e4; % MOI of hub in [hub] frame [kg.m^2] Ihuby = 0; % " Ihubz = 1.5e4; % " % Joint and shaft stiffnesses kyaw = 0; % yaw joint torsional stiffness [N.m/rad] ktilt = 0; % tilt joint torsional stiffness [N.m/rad] kteet = 0; % teeter torsional stiffness (ignored for rigid hub or Nb>2) [N.m/rad] kLSS = 2.690e7; % LSS torsional stiffness (value <= 0 interpreted as rigid) [N.m/rad] kHSS = -1; % HSS torsional stiffness (value <= 0 interpreted as rigid) [N.m/rad] % Tower distributed properties

72

% [ x/height (m), mass-per-unit-length (kg/m), I/L (kg.m), GJ (N.m^2), EI (N.m^2) ] % % must contain at least two rows, one for x/height = 0.0 and one for x/height = 1.0 tdata = [ 0.000 1548 3444 3.06E+10 8.31E+10 0.066 1361 2311 2.05E+10 5.58E+10 0.197 1428 1277 1.13E+10 3.09E+10 0.262 1311 742 6.57E+09 1.80E+10 0.329 1311 742 6.57E+09 1.80E+10 0.430 1311 742 6.57E+09 1.80E+10 0.514 878 482 4.28E+09 1.17E+10 0.614 878 482 4.28E+09 1.17E+10 0.698 878 482 4.28E+09 1.17E+10 0.782 599 317 2.81E+09 7.65E+09 0.881 599 317 2.81E+09 7.65E+09 0.966 1311 742 6.57E+09 1.80E+10 1.000 1311 742 6.57E+09 1.80E+10 ]; mtop = 1610; % lumped mass at tower top (part of tower not nacelle, e.g. for yaw bearings) % Blade distributed properties % [ x/length (m), mass-per-unit-length (kg/m), Iy/L (kg.m), Iz/L (kg.m), ea_twist (deg), % EIy (N.m^2), EIz (N.m^2), chord (m), aero_twist (deg) ] % % must contain at least two rows, one for x/length = 0.0 and one for x/length = 1.0 bdata = [ 0.000 282.92 29.47 12.33 3.44 2.83E+08 1.65E+08 1.143 .44 0.022 290.24 33.11 11.97 3.37 3.18E+08 1.61E+08 1.196 3.37 0.053 261.88 34.19 10.57 3.27 3.28E+08 1.42E+08 1.268 3.27 0.114 201.28 31.97 7.35 3.08 3.07E+08 9.87E+07 1.411 3.08 0.175 186.52 35.48 5.82 2.88 3.40E+08 7.84E+07 1.555 2.88 0.236 169.1 35.67 4.41 2.69 3.42E+08 5.92E+07 1.699 2.69 0.300 149.28 29.02 3.38 2.45 2.78E+08 4.54E+07 1.637 2.45 0.364 133.19 24.71 2.54 2.21 2.37E+08 3.41E+07 1.575 2.21 0.427 111.74 17.58 1.86 1.91 1.69E+08 2.50E+07 1.494 1.91 0.491 96.86 14.34 1.33 1.61 1.38E+08 1.79E+07 1.412 1.61 0.554 78.57 9.81 0.92 1.24 9.40E+07 1.23E+07 1.331 1.24 0.618 65.03 7.54 0.61 0.86 7.25E+07 8.19E+06 1.250 0.86 0.682 49.68 4.87 0.38 0.38 4.67E+07 5.14E+06 1.168 0.38 0.745 37.59 3.40 0.23 -0.11 3.26E+07 3.02E+06 1.087 -0.11 0.809 25.01 1.98 0.12 -0.77 1.90E+07 1.62E+06 1.006 -0.77 0.873 16.01 1.35 0.06 -1.43 1.30E+07 8.68E+05 0.925 -1.43 0.936 10.73 0.92 0.03 -2.37 8.85E+06 4.68E+05 0.843 -2.37 1.000 6.02 0.71 0.02 -3.31 6.80E+06 2.09E+05 0.762 -3.31]; aero_elem_loc = 20; % list of AeroDyn element locations from the blade root as a fraction of blade length _OR_ an integer for the number of equilength elements per blade

active drive train control to improve energy capture...

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