ACTIVE DRIVE TRAIN CONTROL TO IMPROVE ENERGY CAPTURE OF WIND TURBINES By
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
Committee Member
Date
__________________________________________
John Chiasson
Committee Member
Date
__________________________________________
John R. (Jack) Pelton
Dean of the Graduate College
Date
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. iii
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. iv
TABLE OF CONTENTS
Continuously Variable Speed Transmission Wind Turbines...................................... 6
CHAPTER 3 – DIFFERENTIAL PLANETARY GEAR TRAIN ANALYSIS ................ 8
v
CHAPTER 5 – INDUCTION GENERATOR AND CONTROLLER DESIGN ............. 19
CHAPTER 8– FUTURE RECOMMENDATIONS ......................................................... 45
vi
Effects in Power by changing the Second Stage Gear Ratio ........................................ 51
vii
LIST OF FIGURES
curve........................................................................................... 5
Figure 13 (a) Steady State Wind Cp vs.
Wind Speeds Varing from 10 to 30 m/sec ... 26
Figure 16 Effects on Cp Due to Changes in the Second Stage Gear Ratio ...................... 29
viii
Figure 31 Change in Rotor Power Due to Changes in the Second Stage Gear Ratio with
Figure 34 Change in Rotor Power Due to Changes in the Second Stage Gear Ratio with
ix
Figure 37 Change in Rotor Power Due to Changes in the Second Stage Gear Ratio with
Figure 38 Change in Ring Power Due to Changes in the Second Stage Gear Ratio with
Figure 39 Change in Generator Rotor Power Due to Changes in the Second Stage Gear
Figure 40 Change in Rotor Power Due to Changes in the Second Stage Gear Ratio with
Figure 42 Change in Generator Rotor Power Due to Changes in the Second Stage Gear
x
LIST OF TABLES
Table 4 Average Power Flow of System Components (kW)............................................ 37
xi
NOMENCLATURE
AC .......................................................................................................... Alternating Current
CART........................................................................ Controls Advanced Research Turbine
C p
................................................................................................Coefficient of Performance
CVT............................................................................ Continuously Variable Transmission
DC .................................................................................................................. Direct Current
DCVT......................................................Differential Continuously Variable Transmission
DOF........................................................................................................Degree-of-Freedom
D r
.................................................................................................Diameter of the Ring Gear
D s
.................................................................................................. Diameter of the Sun Gear
J rotor
....................................................................................... Rotor Polar Moment of Inertia k..................................................................................................Generator Torque Constant
K
D
....................................... Coefficient of the Derivative Component of the PI Controller
K
G
.................................................................................................................. Controler Gain
K
I
............................................ Coefficient of the Integral Component of the PI Controller
K
P
.......................................... Coefficient of the Constant Component of the PI Controller
N r
................................................................................... Number of Teeth on the Ring Gear
NREL .................................................................... National Renewable Energy Laboratory
N s
..................................................................................... Number of Teeth on the Sun Gear
P ...................................................................................... Planetary Gear Train Speed Ratio
PID ..................................................................................... Proportional Integral Controller xii
S ..................................................................................................................... Generator Slip
λ
.................................................................................................................. Tip Speed Ratio
τ aero
.................................................................. Aero Dynamic Torque Created by the Wind
τ
C
............................................................................................Torque Applied to the Carrier
τ diff
............................................................................. Differential Torque Seen at the Rotor
τ
R
...................................................................................... Torque Applied to the Ring Gear
τ
S
........................................................................................ Torque Applied to the Sun Gear
ω c
................................................................................ Angular Velocity of the Carrier Gear
ω gen
........................................................................................... Generator Angular Velocity
⎯ω gen
............................................................................ Desired Generator Angular Velocity
ω r
....................................................................................Angular Velocity of the Ring Gear
ω
S
..................................................................................Angular Velocities of the Sun Gear xiii
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.
Wind
Model
v wind
Turbine Blade
Model
Constant Tip
Speed Ratio
Controller
τ
ω
Rot
Rot
Controller
Model
ω
Ring
CVT
Model
Constant
Generator
Controller
τ
Gen
ω
Gen
Generator
Model
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 variablevoltage, 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 (C p
), which in turn increases the overall power captured through the equation:
P out
=
1
ρπ
R
2 v
2
Equation 1
3
C p
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 2 4 6 8 10 12
λ
Figure 2 typical Cp vs.
λ
curve
14 16 18
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].
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.
6
(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, C p
, derived for a rated wind speed. In variable speed operation, a wind turbine can operate at the optimal C p
, which varies with the wind speed. This can
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 C p
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.
8
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 (J rotor
), aero dynamic torque created by the wind (
τ aero
), and the torque of the system transmitted back through the differential (
τ diff
). 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.
τ
τ
ω rotor aero diff
=
=
= f f
2
1
1
J
( rotor v w
,
ω
(
τ aero rotor
(
τ gen
,
ω rotor
)
−
,
ω
τ diff
R
)
)
Equation 2 System State Equations
9
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:
−
N
R
N
S
=
ω
ω
S
R
−
−
ω
ω
C
C
Equation 3
P
=
N
R
N
S
=
D
D
S
R where
ω
S
, ω
C
, and
ω
R
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 (N
R
/ N
S
) or the diameter of the sun gear to the ring gear (D
R
/ D
S
).
Solving equation 2 for each of the three possible output variables,
ω
S
, ω c
, and
ω r respectively, yields the following equations:
ω
S
− ω
C
= −
P
( ω
R
− ω
C
)
ω
S
= ω
C
−
P
( ω
R
− ω
C
)
ω
S
= ω
C
(
1
+
P
)
− ω
R
P
( a )
P
( ω
R
− ω
C
)
= − ω
S
+ ω
C
P
ω
R
−
P
ω
C
= − ω
S
+ ω
C
P
ω
R
+ ω
S
= ω
C
+
P
ω c
ω
C
(
1
+
P
)
=
P
ω
R
+ ω
S
ω
C
=
P
ω
R
1
+
+
P
ω
S
( b )
P
( ω
R
− ω
C
)
= − ω
S
+ ω
C
P
ω
R
−
P
ω
C
= − ω
S
+ ω
C
P
ω
R
= ω
C
+
P
ω
C
− ω
S
ω
R
= ω
C
P
⎞
1
⎠
−
ω
S
P
( c )
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.
ω
S
= ω
C
(
1
+
P
) ω
C
=
1
P
ω
R
+
P
( b )
ω
R
= ω
C
P
( c )
⎞
1
⎠
( a )
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)
ω s
increases to infinity; (b)
ω c
increases to one; and (c)
ω r
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:
ω
S 1
= ω
C 1
(
1
+
P
1
)
( a )
ω
C 2
= ω
S 1
( b )
ω
S 2
= ω
C 2
(
1
+
P
2
)
− ω
R 2
P
2
Equation 6 (a-c)
( c ) where the subscript numbers refer to which planetary gear train they belong to, the first or second. With this final set of equations
ω r2
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:
τ
C
= τ
S
(
P
+
1
)
τ
R
= τ
S
P where
τ
C
,
τ
S
,
τ
R
, 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+N b
Degrees-of-freedoms (DOF) associated with the wind turbine and its tower, where N b
is the number of blades on the turbine. The following eight DOF are as follows:
DOF reference number
1
2
Description
Tower for-aft deflection
Tower side-to-side deflection
6
7
Generator shaft position
Shaft torsional deflection
9 Blade-1 flap angle
… …
8+N b
*
Blade-N b
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
Generator Torque
15
(
ω gen
) and its position (
θ gen
). 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].
Output signals
Internal loads
Blade Pitch
Wind Model
Aerodynamic
Model
Forces
Structural
Model
ω gen
θ gen
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]
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 Cpl
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 C p
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.
21
20
19
18
17
16
15
0 5 10 15
Time (sec)
20
Figure 8 Step Wind Input
25 30
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.
1.5
1
0.5
0
-0.5
-1
-1.5
3
2.5
2
-2
0 5 10 15
Time (sec)
20 25
Figure 9 Interpolated and Adjusted Wind Speed
30
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:
T
= k
(
ω − gen
ω gen
)
Equation 7 where k is a modeling constant,
ω gen
is the angular velocity of the generator, and
⎯ω gen
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,
ω gen and S, the of slip:
S
=
(
ω − gen
ω
ω gen gen
)
Equation 8
T
≈ kS
ω gen
Equation 9
19
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: k
≈
P
δω
2 gen
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.
⎡
⎢
⎣
θ
•
ω
• r r
⎤
⎥
⎦
=
⎡
⎢
-
1
0.4989
0
0
⎤
⎥
⎡
⎢
θ
ω r r
⎤
⎥
+
⎡
⎢
0
0.0578
⎤
⎥
λ =
[
0 1 .
167
] ⎡
⎢
θ
ω r r
⎤
⎥
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
λ =
0 .
0963 s
+
0 .
4989
ω r
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:
λ =
⎛
⎜⎜
K
D s
2 +
K
P s
+
K
I s
⎞
⎠ ⎝
0 .
0963 s
+
0 .
4989
Equation 13
ω r
In the preceding equation K
P
, K
I
, and K
D
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
20
K
G s
Equation 14
At this point the controller gain, K
G
, 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
K
P
K
I
79
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
0 .
75
K
G s
Equation 15 producing the following root locus plot and controller coefficients.
Root Locus
0.1
0
-0.1
-0.2
-0.3
-0.4
0.5
0.4
0.3
0.2
System: mysys
Gain: 1.87
Pole: -1.18 - 1.63e-008i
Damping: 1
Overshoot (%): 0
Frequency (rad/sec): 1.18
-0.5
-1.5
-1 -0.5
Real Axis
Figure 11 Damped Constant Tip Speed Ratio System
0
Table 3. Constant Tip Speed Ratio
Speed Controller Coefficients
K
K
I
P
1.87
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
caused by tower shadow. This oscillation was removed by taking the last one thousand
P
=
τω
Equation 16
7.5
7
8 x 10
5 Generator Power
6.5
6
5.5
5
4.5
4
0 5 10 15
Time (sec)
20 25
Figure 12 Typical Constant Wind Speed Results
30 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, C p
,
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 C p
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.
0.13
0.12
0.11
0.1
0.09
0.08
0.07
No Controller
Constant
ω gen
Constant
λ
0.06
2 2.5
3 3.5
4 4.5
5
Tip Speed Ratio (
λ
)
5.5
6 6.5
7
Figure 13 (a) Steady State Wind Cp vs.
λ,
Wind Speeds Varing from 10 to 30 m/sec
27
0.13
0.12
0.11
0.1
0.09
0.08
0.07
No Controller
Constant
ω gen
Constant
λ
0.06
2 2.5
3 3.5
4 4.5
5
Tip Speed Ratio (
λ
)
5.5
6 6.5
7
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 C p
. 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 C p
’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
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
14 26
No Controller
Constant
ω gen
Constant
λ
28 30 16 18 20 22 24
Wind Speed (m/sec)
Figure 14 Cp vs. Changing Steady State Wind
15 x 10
5
Constant Generator
Contorller off
Constant
λ
10
5
0
0 5
Blade Tip Angle (deg)
10
Figure 15 Power Produced at 18 m/sec vs. Blade Pitch Angle
15
It is also important to note that the C p
’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 C p
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 C p
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.
0.14
0.12
0.1
0.08
0.06
0.04
0.02
3:1
5:1
7:1
0
10 12 14 16 18
Wind Speed (m/sec)
20 22
Figure 16 Effects on Cp Due to Changes in the
Second Stage Gear Ratio
24
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
ω gen
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
ω gen
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
ω gen
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
7
6
5
4
3
2
1
0
10 x 10
5
9
8
5 10
Uncontrolled
Constant.
ω gen
Constant
λ
15
Time (sec)
20 25
Figure 17 Power in the Rotor from Step Winds
30
32
0
-5
-10
-15
5 x 10
4
Uncontrolled
Constant.
ω gen
Constant
λ
-20
0 5 10 15
Time (sec)
20 25
Figure 18 Power in the Ring from Step Winds
30 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
ω gen
system remains constant, which will aid in connecting the power produced by the generator directly to the grid.
33
8
6
12 x 10
5
10
4
2
0
0 5 10
Uncontrolled
Constant.
ω gen
Constant
λ
15
Time (sec)
20 25
Figure 19 Power in the Generator from Step Winds
30
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
2
1.5
3 x 10
4
2.5
1
0.5
0
0 5 10
Uncontrolled
Constant.
ω gen
Constant
λ
15
Time (sec)
20 25
Figure 20 Torque on the Ring Gear from a Step Input
30
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 C p
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.
5
4.5
4
3.5
0 5 10
Uncontrolled
Constant.
ω gen
Constant
λ
15
Time (sec)
20 25
Figure 21 Shaft Speed of the Rotor from Step Winds
220
30
215
210
205
200
195
190
0 5 10
Uncontrolled
Constant.
ω gen
Constant
λ
15
Time (sec)
20 25
Figure 22 Shaft Speed of the Generator from Step Winds
30
36
2
0
-2
-4
-6
-8
-10
-12
0 5 10
Uncontrolled
Constant.
ω gen
Constant
λ
15
Time (sec)
20 25
Figure 23 Shaft Speed of the Ring Gear from Step Winds
Variable Wind Input
30
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
ω gen
.
These power averages can be seen in Table 4.
Table 4 Average Power Flow of System Components (kW)
Uncontrolled
ω gen
Constant
λ
Ring 0.00 -63.84 -120.52
8
7
6
5
10 x 10
5
9
4
3
2
5 10 15
Time (sec)
20
Uncontrolled
Constant.
ω gen
Constant
λ
25 30
Figure 24 Power in the Rotor from Variable Winds
0.5
x 10
5
0
-0.5
-1
-1.5
-2
Uncontrolled
Constant. ω gen
Constant λ
10
-2.5
5 15
Time (sec)
20 25 30
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.
9
8
7
6
12 x 10
5
11
10
5
4
3
2
5
Uncontrolled
Constant.
ω gen
Constant
λ
25 30 10 15
Time (sec)
20
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)
Constant
Constant
ω
λ gen
Uncontrolled
Max
Average
Min
Max
Average
Min
Max
Average
Min
197.02 22.82 4.56
159.82 18.51 3.70
123.07 14.26 2.85
138.13 16.00 3.20
138.12 16.00 3.20
138.11 16.00 3.20
167.37 19.39 3.88
125.49 14.54 2.91
79.42 9.20 1.84
2 x 10
5
1.8
1.6
1.4
1.2
1
0.8
0.6
5
Uncontrolled
Constant.
ω gen
Constant λ
25 30 10 15
Time (sec)
20
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.
Constant
Constant
ω gen
λ
Uncontrolled
Table 6 System Shaft Speeds due to Variable Wind
Rotor Ring Generator rads/sec RPM rads/sec RPM rads/sec RPM
Max 4.2 -64.9 215.4 2056.8
Average 4.0 38.3
-7.4 -70.9 210.3 2008.3
Min 3.8 -83.3 205.3 1960.4
Max 4.7 -0.6
Average 4.3 41.2
-4.2 -40.3 207.3449 1979.9984
Min 3.9 -7.7
Average 4.8 45.5
0 0 211.3 2018.1
0 0 205.6 1963.5
0 0 199.3 1903.5
5
4.8
4.6
4.4
4.2
4
3.8
3.6
5
Uncontrolled
Constant.
ω gen
Constant
λ
25 30 10 15
Time (sec)
20
Figure 28 Shaft Speed of the Rotor in Variable Winds
-4
-6
-8
0
-2
-10
-12
-14
5 10 15
Time (sec)
20
Uncontrolled
Constant.
ω gen
Constant
λ
25 30
Figure 29 Shaft Speed of the Ring Gear in Variable Winds
41
216
214
212
210
208
206
204
202
200
198
5 10 15
Time (sec)
20
Uncontrolled
Constant.
ω gen
Constant
λ
25 30
Figure 30 Shaft Speed of the Generator in Variable Winds
42
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
3.
4.
5.
1.
2.
REFERENCES
Idan, M. and D. Lior, Continuous Variable Speed Wind Turbine: Transmission
Concept and Robust Control.
Wind engineering, 2000. 24 (3): p. 151-767.
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.
Simulink 2006, The MathWorks, Inc.: Natic, MA.
MATLAB . 2006, The MathWorks, Inc.: Natic, MA.
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.
7.
Iqbal, M.T., A. Coonick, and L.L. Frerris, Dynamic Control Options for Variable
Speed Wind Turbines.
Wind engineering, 1994(1): p. 1-12.
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, . 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, 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.
APPENDIX A
Free Body Diagrams
48
Resulting Forces
Planetary Gear System
49
J
∑
=
M
0
=
J
0
∑
M
= τ
F
S
=
=
S
τ
−
S
R
S
0
F
S
•
ω
R
S
Sun Gear
J
∑
=
M
0
F
R
=
=
R
R
J
•
ω
0
∑
=
M
τ
R
=
−
0
F
R
R
R
τ
R
Ring Gear
J
∑
=
M
0
=
J
0
∑
=
M
τ
C
=
−
0
F
C
•
ω
R
C
F
C
=
τ
C
R
C
Carrier
J
∑
=
M
0
=
J
•
ω
0
F
S
∑
=
=
−
F
S
F
R
F
=
R
P
Ma
M
=
0
0
∑
=
F
F
C
=
F
C
=
0
+
2 F
R
F
R
=
+
+
F
R
R
P
F
S
2 F
S
Resulting Torques
Carrier Torque
R
R
=
R
S
+
2 R
P
R
P
R
C
=
=
R
C
R
R
−
R
S
+
R
S
2
τ
C
τ
C
F
C
=
=
=
F
C
F
C
2 F
S
R
C
(
R
R
+
R
S
=
2
2 F
R
)
τ
C
F
S
Sun Torque
=
F
S
(
R
R
+
R
S
=
τ
S
R
S
)
τ
C
= τ
S
⎛
⎜⎜
R
R
R
S
+
⎞
1
⎠
τ
τ
P
S
C
=
R
R
=
=
R
S
τ (
P
+
1
)
S
τ
C
(
P
+
1
)
τ
C
F
R
Ring Torque
=
F
R
(
R
R
+
R
S
=
τ
R
R
R
)
τ
C
⎞
⎟⎟
τ
R
τ
R
τ
R
= τ
R
⎛
⎜⎜ 1
+
R
S
R
R
=
=
τ
C
τ
S
1
1
(
P
P
+
1
)
1
1
P
= τ
S
P
50
APPENDIX B
Effects in Power by changing the Second Stage Gear Ratio
51
Step Wind Input
Constant Generator Speed Controlled System
8.5
x 10
5
8
7.5
7
6.5
6
3:1
5:1
7:1
5.5
5
4.5
0.5
0
-0.5
-1
-1.5
-2
-2.5
4
0 5 10 15
Time (sec)
20 25 30
Figure 31 Change in Rotor Power Due to Changes in the
Second Stage Gear Ratio with Step Input
2 x 10
5
1.5
1
3:1
5:1
7:1
-3
0 5 10 15
Time (sec)
20 25 30
Figure 32 Change in Ring Gear Power Due to Changes in the
Second Stage Gear Ratio with Step Input
52
6.6365
x 10
5
6.636
6.6355
6.635
6.6345
6.634
3:1
5:1
7:1
6.6335
0 5 10 15
Time (sec)
20 25 30
Figure 33 Change in Generator Power Due to Changes in the
Second Stage Gear Ratio with Step Input
Constant Tip Speed Controlled System
14 x 10
5
12
3:1
5:1
7:1
10
8
6
4
2
0 5 10 15
Time (sec)
20 25
Figure 34 Change in Rotor Power Due to Changes in the
Second Stage Gear Ratio with Step Input
30
53
0
-2
-4
-6
2 x 10
5
-8 3:1
5:1
7:1
30
-10
0 5 10 15
Time (sec)
20 25
Figure 35 Change in Ring Gear Power Due to Changes in the
Second Stage Gear Ratio with Step Input
10
8
14
12
16 x 10
5
3:1
5:1
7:1
6
4
2
0 5 10 15
Time (sec)
20 25 30
Figure 36 Change in Generator Power Due to Changes in the
Second Stage Gear Ratio with Step Input
54
Variable Wind Input
Constant Generator Speed Controlled System
8.5
x 10
5
8
7.5
7
6.5
6
5.5
5
4.5
3:1
5:1
7:1
4
0 5 10 15
Time (sec)
20 25
Figure 37 Change in Rotor Power Due to Changes in the
Second Stage Gear Ratio with Variable Input
30
-0.5
-1
-1.5
-2
-2.5
1
0.5
0
2 x 10
5
1.5
3:1
5:1
7:1
-3
0 5 10 15
Time (sec)
20 25
Figure 38 Change in Ring Power Due to Changes in the
Second Stage Gear Ratio with Variable Input
30
55
6.6365
x 10
5
6.636
6.6355
6.635
6.6345
6.634
3:1
5:1
7:1
6.6335
0 5 10 15
Time (sec)
20 25 30
Figure 39 Change in Generator Rotor Power Due to Changes in the Second Stage Gear Ratio with Variable Input
Constant Tip Speed Controlled System
14 x 10
5
12
3:1
5:1
7:1
10
8
6
4
2
0 5 10 15
Time (sec)
20 25
Figure 40 Change in Rotor Power Due to Changes in the
Second Stage Gear Ratio with Variable Input
30
56
0
-2
-4
-6
2 x 10
5
-8 3:1
5:1
7:1
30
-10
0 5 10 15
Time (sec)
20 25
Figure 41 Change in Ring Gear Power Due to Changes in the
Second Stage Gear Ratio with Variable Input
10
8
14
12
16 x 10
5
3:1
5:1
7:1
6
4
2
0 5 10 15
Time (sec)
20 25 30
Figure 42 Change in Generator Rotor Power Due to Changes in the Second Stage Gear Ratio with Variable Input
57
APPENDIX C:
Glossary of Matlab Function Blocks
58
Block Name
Clock
Constant
Demux
From
From Workspace
Function
Gain
Goto
In
59
Block Icon Description
Outputs the current simulation time.
Outputs a constant value
Splits a vector signal into a scalar or smaller vectors.
Receives data from a Goto block with the same name, "A" in this case.
Outputs the value defined by the workspace variable.
Runs input through a specified function or through a m-file function.
Multiplies the input value by a constant or gain.
Sends data to a From block with the same name, "A" in this case.
Provide an input port for a subsystem or model.
Integrator Continuous time integration of input signal.
Lookup Table
Mux
Out
PID Controler
Product
Scope
Sub System
Sum
To Workspace
Transfer Function
60
Perform one dimension interpolation of input values using the specified table.
Creates vector signal from scalars or smaller vectors.
Provide an output port for a subsystem or model.
Implements a PID controller in the form:
P+I/s+Ds
Multiplies or divides inputs.
Displays a graph on the input(s) vs. simulation time.
Creates sub system.
Adds or subtracts inputs.
Writes input to specified variable.
Processes the input signal through a specified transfer function
APPENDIX D
Simulation Models
61
Global Models
Constant Generator Model
62
Constant Tip Speed Model
63
Masked Models
Wind Model
64
DCVT Model
65
APPENDIX E
Simulation Files
66
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 g
Cjoint(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 3bladed 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-offreedom
%%
% 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
