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The Dissertation Committee for John Francis Hall certifies that this is the approved version of the following dissertation: Design and Control of a Variable Ratio ...
Copyright by John Francis Hall 2012

The Dissertation Committee for John Francis Hall certifies that this is the approved version of the following dissertation:

Design and Control of a Variable Ratio Gearbox for Distributed Wind Turbine Systems

Committee: Dongmei Chen, Supervisor Raul G. Longoria Glenn Y. Masada Siddharth B. Pratap Alfred E. Traver

Design and Control of a Variable Ratio Gearbox for Distributed Wind Turbine Systems

by John F. Hall, B.S.; M.S.E.

Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

The University of Texas at Austin August 2012

Dedication

I lovingly dedicate this dissertation to my wife, my four children, and my parents who supported me each step of the way.

Acknowledgements

My thanks and appreciation to Dongmei (Maggie) Chen for guidance and persevering with me to complete this research and write the dissertation. The members of my dissertation committee, Glenn Masada, Raul Longoria, Al Traver, and Sid Pratap, have generously given their time and expertise to better my work. I thank them for their contribution and their good-natured support. I am grateful to Richard Crawford, Glenn Masada, Dongmei Chen, and Mitch Pryor for financial support.

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Design and Control of a Variable Ratio Gearbox for Distributed Wind Turbine Systems John Francis Hall, Ph.D. The University of Texas at Austin, 2012

Supervisor: Dongmei Chen

Wind is one of the most promising resources in the renewable energy portfolio. Still, the cost of electrical power produced by small wind turbines impedes the use of this technology, which can otherwise provide power to millions of homes in rural regions worldwide. To encourage their use, small wind turbines must convert wind energy more effectively while avoiding increased equipment costs. A variable ratio gearbox (VRG) can provide this capability to the simple low-cost fixed-speed wind turbine through discrete operating speeds. The VRG concept is based upon mature technology taken from the automotive industry and is characterized by low cost and high reliability. A 100 kW model characterizes the benefits of integrating a VRG into a fixed-speed stallregulated wind turbine system. Simulation results suggest it improves the efficiency of the fixed-speed turbine in the partial-load region and has the ability to limit power in the full-load region where pitch control is often used. To maximize electrical production, mechanical braking is applied during the normal operation of the wind turbine. A strategy is used to select gear ratios that produce torque slightly above the maximum amount the generator can accept while simultaneously applying the mechanical brake, so that full-load production may be realized over greater ranges of the wind speed. Dynamic programming is used to establish the VRG ratios and an optimal control design. vi

This optimization strategy maximizes the energy production while insuring that the brake pads maintain a predetermined service life. In the final step of the research, a decisionmaking algorithm is developed to find the gears that emulate the ratios found in the optimal control design. The objective is to match the energy level as closely as possible, minimize the mass of the gears, and insure that tooth failure does not occur over the design life of the VRG. Recorded wind data of various wind classes is used to quantify the benefit of using the VRG. The results suggest that an optimized VRG design can increase wind energy production by roughly 10% at all of the sites in the study.

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Table of Contents List of Tables ......................................................................................................... xi List of Figures ...................................................................................................... xiii Chapter 1: Introduction ...........................................................................................1 1.1 Objective ..................................................................................................1 1.2 Motivation ................................................................................................1 Chapter 2: Literature Review ..................................................................................7 2.1 Methods of Wind Energy Conversion .....................................................8 2.2 Components of Wind Turbines ................................................................9 2.3 Aerodynamic Considerations .................................................................10 2.4 Electrical Power Generation ..................................................................12 2.5 Gearbox Issues .......................................................................................15 2.6 Variable Ratio Gearboxes ......................................................................17 2.7 Proposed Wind Turbine System ............................................................18 2.8 Methods of Control ................................................................................18 2.8.1 Control Mechanisms ..................................................................19 2.8.2 Control Algorithms ....................................................................20 Chapter 3: Wind Turbine Drivetrain Model .........................................................22 3.1 Aerodynamic Modeling .........................................................................23 3.2 Disc Brake Modeling .............................................................................25 3.2.1 Limitations Due to Heat .............................................................26 3.2.1.1 Heat Produced at the Brake............................................26 3.2.1.2 Heat Rejected by the Brake............................................27 3.2.2 Limitations Due to Pad Pressure ................................................28 3.3 Gears and Shafts Modeling ....................................................................29 3.4 Electric Generator Modeling..................................................................30 3.5 Model Verification .................................................................................31 3.6 Wind Data Construction ........................................................................32 viii

3.7 Control Framework Establishment ........................................................34 3.8 Control Problem Formulation ................................................................36 Chapter 4: Rule Based Control Design .................................................................38 4.1 Gear Ratio Application ..........................................................................40 4.1.1 Normal Operation ......................................................................43 4.1.2 Gear-Changing Events ...............................................................43 4.2 Model Analysis ......................................................................................44 Chapter 5: System Performance............................................................................50 5.1 Gear Ratio Selection Methodology........................................................53 5.2 Energy Production Results .....................................................................55 5.3 Energy Production Gain in Varied Wind Conditions ............................63 5.4 Real Time Efficiency Gain by Using VRG ...........................................66 Chapter 6: Optimal Control Design ......................................................................72 6.1 Model Reconfiguration for Optimization ..............................................74 6.2 Brake Pad Life .......................................................................................75 6.3 Optimization Problem Formulation .......................................................76 6.4 Control Design .......................................................................................78 6.4.1 Gear Ratio Design Space ...........................................................79 6.4.2 Preliminary Programming ..........................................................80 6.4.3 Dynamic Programming ..............................................................83 6.4.3.1 NREL Wind Data Sets ...................................................83 6.4.3.2 Programming Technique ................................................83 6.5 Case Study Based on A Real Wind Site ................................................88 6.6 Wind Energy Increase across Several Wind Classes .............................91 Chapter 7: Gear Train Design and Analysis .........................................................95 7.1 Methodology ..........................................................................................97 7.1.1 Procedure .................................................................................100 7.1.1.1 Building combinations .................................................105 7.1.1.2 Energy production ........................................................109 ix

7.1.1.3 VRG Gearset Mass Calculation ...................................111 7.1.1.4 Fatigue Failure .............................................................112 7.2 Results ..................................................................................................116 References ............................................................................................................122 Vita .....................................................................................................................137

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List of Tables Table 2.1:

Common wind turbine configurations. ...............................................7

Table 3.1:

Classes of wind power density at 50 m [140]. ..................................33

Table 3.2:

National Renewable Energy Laboratories (NREL) wind turbine sites used in study. ....................................................................................34

Table 3.3:

Operating modes. ..............................................................................35

Table 4.1:

Data used in look-up table for 100 kW VRG-enabled wind turbine.41

Table 5.1:

Comparison of VRG to single speed turbine. ...................................59

Table 5.2:

VRG Gear ratios. ..............................................................................60

Table 5.3:

Performance results for cross section of class 1 - 7 wind profiles. ...69

Table 6.1:

Gear combinations generated. ...........................................................80

Table 6.2:

Simulation results for cross section of class 1 - 7 wind profiles. ......92

Table 7.1:

Characteristics for first group having at least six gearsets. .............101

Table 7.2:

Allowable range of VRG ratios for combination building. ............105

Table 7.3:

Allowable range of relative ratios for combination building. .........107

Table 7.4:

Example of available gearsets for set , subset . ............................108

Table 7.5:

Characteristics of the top five Pareto combinations for normal wind power density VRG design used at sites 1 though 18. ....................118

Table 7.6:

Characteristics of the top five Pareto combinations for high wind power density VRG design used at sites 19 and 20. ..................................118

Table 7.7:

Optimal gearsets selected through multiobjective function for normal power density design at sites 1 through 18. ....................................119

Table 7.8:

Optimal gearsets selected through multiobjective function for high power density design at sites 19 and 20. .........................................119 xi

Table 7.9:

Performance results for VRG wind turbine following gearset selection. .........................................................................................................120

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List of Figures Figure 1.1: Generalized wind turbine power curve. ..............................................3 Figure 2.1: Top view of VAWT (left) and HAWT (right) showing motion of blades in relation to wind. ............................................................................10 Figure 2.2: Turbine rotor blade pitch angle. ........................................................11 Figure 2.3: Sequence for synchronous electromagnetic field. ............................13 Figure 2.4: AC to DC to AC power smoothing by electronic conversion. .........15 Figure 2.5: Power coefficient as a function of speed ratio and blade pitch angle.19 Figure 3.1: VRG-enabled wind turbine configuration. .......................................22 Figure 3.2: Wind turbine model. .........................................................................23 Figure 3.3: The effect of speed ratio and pitch angle on the power coefficient. .24 Figure 3.4: Components of the disc brake. ..........................................................25 Figure 3.5: Diagram of drivetrain dynamics model. ...........................................29 Figure 3.6: Verification of model to 100 kW power curve. ................................32 Figure 3.7: Control problem formulation. ...........................................................37 Figure 4.1: Comparison of VRG-enabled production to original production, with traditional operating regions shown. .................................................38 Figure 4.2: Trend for VRG-enabled when used with mechanical brake, shown with new operating regions. ......................................................................39 Figure 4.3: General wind speed operating ranges for gears in 100 kW VRG-enabled model.................................................................................................40 Figure 4.4: Decision making structure implemented during normal operation...42 Figure 4.5: Decision making structure for changing gears..................................44

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Figure 4.6: Simulated performance of each VRG ratio across operating spectrum. ...........................................................................................................45 Figure 4.7: Brake characteristics during operation..............................................47 Figure 4.8: Electricity production gains (top) and the corresponding gear ratio vs. wind speed (bottom). ........................................................................49 Figure 5.1: Power curve changes in response to varied gear ratio. .....................51 Figure 5.2: Effect of gear ratio on speed ratio (left), power coefficient (middle), and torque (right). ....................................................................................52 Figure 5.3: Design algorithm used to develop VRG. ..........................................54 Figure 5.4: Gear ratios used in design of VRG. ..................................................56 Figure 5.5: Power curve for a four-speed VRG. .................................................57 Figure 5.6: Power curve for a five-speed VRG. ..................................................58 Figure 5.7: Power curve for a six-speed VRG. ...................................................58 Figure 5.8: Additional area added to power curve by using a VRG. ..................61 Figure 5.9: Amount of power (top), power coefficient (middle), and gear applied in (bottom) relation to wind speed for a VRG-enabled wind turbine. ..63 Figure 5.10: Histograms showing distribution of wind of select wind profiles. ...65 Figure 5.11: Plot showing wind speed (top), and corresponding production, (bottom), occurring in 24-hour period for site 31266 (set no. 9). .....................67 Figure 5.12: Plot showing wind speed (top), and corresponding production, (bottom), occurring in 24-hour period for site 31266 (set no. 9). .....................68 Figure 5.13: Trends in the increase in efficiency for a VRG-enabled wind turbine.70 Figure 6.1: Power curve for VRG when used with brake to limit torque in Region 3'. ...........................................................................................................73 Figure 6.2: Modified rules used with program optimization. ..............................74 xiv

Figure 6.3: Wind turbine decision-making model. ..............................................79 Figure 6.4: Decision-making structure for preliminary programming. ...............81 Figure 6.5: Example of a valid (top) and an invalid (bottom) VRG combination.82 Figure 6.6: Dynamic programming algorithm for computing the cost at step . 84 Figure 6.7: Plot showing total energy produced (top), brake wear (middle), and cost (bottom) for site 16577 (set no. 16) for weight,

= 0.960. ..............85

Figure 6.8: Technique for finding optimal VRG combination, and weight variable, , that will be used to operate the wind turbine................................86 Figure 6.9: Surface plot showing total cost (top) and total energy (bottom) as a function of

for each of the 575,413 gear combination for site 16577

(set no. 16). .......................................................................................87 Figure 6.10: Plot showing wind speed (top), additional energy produced by the VRG (bottom, left axis) and cumulative brake pad used (bottom, right axis) over a three year period, for site 21291 (set no. 14), programmed with a weight coefficient,

= 0.980. ...........................................................89

Figure 6.11: Results for site 21291 (set no. 14) with weight,

= 0.980, showing VRG

power curve (top), gear data used for control (upper-middle), power dissipated (lower-middle), and turbine speed (bottom). ...................90 Figure 6.12: Optimal VRG ratios for each set of wind data. .................................93 Figure 7.1: Gear configuration for six-speed VRG. ............................................96 Figure 7.2: Typical trends for module 3 gearsets, showing the ratio (top left), total mass (top right), total teeth (bottom left), and ratio granularity (bottom right). .................................................................................................99 Figure 7.3: Procedure for finding valid gearset combinations for a given set of wind sites. ................................................................................................102 xv

Figure 7.4: Objective function data, Pareto curve, and optimal solution of second VRG design designated for sites 19 and 20. ...................................104 Figure 7.5: Sorted ratios used to build subgroups of combination sets. ............107 Figure 7.6: Illustration of energy data used to predict combination performance in VRG. ...............................................................................................110 Figure 7.7: Endurance limit based on ratio of applied to allowable loading. ....115 Figure 7.8: Mean energy loss and gearset mass data for combinations considered in objective function for sites 1 through 18 (top) and sites 19 and 20 (bottom)...........................................................................................117

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Chapter 1: Introduction 1.1 OBJECTIVE The purpose of this body of research is to establish a design and control framework to demonstrate that a variable ratio gearbox (VRG) can improve wind energy capture of distributed wind turbines. The VRG provides a low-cost mechanical method to increase the economic viability of the overall turbine system. A simulation model of a 100 kW fixed-speed wind turbine with a VRG will be presented. The model relies on time-dependent wind data to characterize the benefits of the VRG at a variety of actual wind turbine sites. The study uses dynamic programming (DP) to find the optimal gear ratios for the VRG, and prescribes a method for controlling it in conjunction with a mechanical brake. Using the brake extends operation at maximum generator torque, and thus full-load production, over broader ranges of wind speed. In order to reduce the computation expense associated with the DP method, an algorithm is developed to constrain the DP searching envelop through finding effective combinations of gears and assessing the resulted gain in energy production. The ultimate goal of the control design is to achieve an optimal tradeoff between system design and dynamic performance. 1.2 MOTIVATION The United States presently relies on fossil fuels to supply up to 70% of the 9.9x1016 BTU of energy it consumes each year [1], however, concerns over national energy security and greenhouse gas emissions are steering this demand toward renewable resources such as wind [2]. Energy derived from wind currently supplies only 24.7 GW [3] of the electrical capacity, which is small when considering that U.S. land-based

Some portions of this chapter have previously appeared in publications.

1

resources could provide an estimated 800 GW of electrical power [4]. This potential is recognized by the United States Department of Energy, which has a goal of reaching a wind capacity level of 304 GW, or 20% of the projected electrical power supply by the year 2030 [5]. Research and development efforts have primarily focused on, and benefitted, large wind turbines that are used for mass production of energy at wind farms. Still, the Department of Energy is relying on distributed wind turbines to make a contribution towards the 20% capacity objective [6] and recommends these types for agricultural communities, public lands, schools, rural homeowners, and private businesses [2]. As a stand-alone unit the distributed wind turbine does not require a power grid or the electrical transmission lines, which are costly to install, and otherwise impede the development of large wind farms [5].

Although the federal government offers tax

incentives for small wind development [7], this group of turbines still has a greater initial cost, which falls in the range of $3 to $6 per watt [8,9], versus the $1.92 for large wind turbines [7]. This disparity has hindered distributed wind development and points to a need for research directed at low cost concepts that can boost efficiency of turbines in this class [10,11]. Distributed wind turbines commonly use a fixed-speed power generator because of the low cost and high reliability. The weakness to this configuration is the inability to control and adjust the rotor speed in relation to the variable wind speed, which can otherwise be used to improve wind energy conversion efficiency and reduce dynamic loads [12,13].

While variable-speed operation is the preferred technique for large

commercial wind turbines, the lack of interest in small wind turbines is likely responsible for a dearth of power conversion electronics [14]. To overcome this problem small wind manufacturers implement photovoltaic power converters that have been unreliable and 2

costly to replace for this unintended application [14–16].

Hence, fixed-speed wind

turbines that are only optimal for a particular narrow range of speed [12,17], end up being the practical choice for small wind applications. The lower efficiency of fixed-speed turbines is evident in the partial-load zone, referred to as Region 2 shown in Fig. 1.1, where the system operates below the rated wind speed. Here there is a potential to extract more power from the wind than what is currently obtained, but the fixed-speed of the turbine rotor limits the aerodynamic efficiency. 1.2

Region 2 Partial load operation

Region 3 Full load operation

power [p.u.]

1 0.8 0.6 0.4 cut-in

0.2 0

0

rated speed

5

10 15 wind speed [m/s]

cut-off

20

25

Figure 1.1: Generalized wind turbine power curve. In addition to limitations in partial-load operation, distributed wind turbines are also at a disadvantage during-full-load operation, shown as Region 3 in Fig. 1.1, when the turbine must be able to limit the power to the generator. Large wind turbines are more likely to achieve this through pitch control [18–21], where motorized rotor blades can be oriented to work as aerodynamic brakes that restrict the rotor speed. However, small 3

wind turbines rely on less costly approaches such as changing the yaw angle (turning the rotor axis away from the wind) or by using blades that are designed to inherently stall in high wind speeds [12,13,22]. While these techniques keep the complexity of the system low, they do not have the response or efficiency of pitch control in large wind turbine applications [23]. An alternative method to using variable-speed generators and pitch control is developing a mechanical mechanism that varies the gearbox ratio. Multispeed gearbox design is well-refined and these devices have worked reliably in many existing industry applications [24]. In the automotive industry, gearboxes are mass produced and the technology has become low cost [25], unlike the power electronics required in variablespeed operations. A variable ratio gearbox in the wind turbine drivetrain has strong potential to increase the wind energy capture during partial-load operation at a competitive cost. If successful, this technology will promote the distributed wind turbine application nationwide. A continuously variable transmission (CVT) concept has recently been proposed to provide the gear ratio-changing capability to the wind turbine drivetrain [15,16]. However, automotive industry experience has proven that the CVT is more suitable for low-torque high-speed operating conditions. To convert wind energy, small turbines can operate at speeds up to 500 RPM [26], while those in the megawatt class can move as slow as 10 RPM [17,27]. The reason that operating speed decreases with an increase in size is because of the rotor diameter, which affects the power coefficient by way of the blade tip speed. Hence, one can expect that as the turbine size increases, so does the impossibility of using A CVT. Durability issues have indeed been a problem for the CVT in the wind turbine industry [28], and is the reason why this concept has failed to get traction despite appearing periodically in literature since as early as 1986 [29–34]. 4

Current developments in the automotive industry suggest the CVT, which transfers energy by frictional surfaces, is best suited for low-torque applications that deliver less than 200 N-m of torque [35]. Considering the limitation of the CVT, there is a gap in the techniques that can be applied to improve the performance of wind turbines that are greater than about 30 kW and smaller than those in the megawatt class where variable speed operation is used. This research proposes to design a VRG with distinct gear ratios to enable the wind turbine capturing more wind energy. In higher and lower wind speed regions, lower gear ratios can benefit the turbine system and modulate the power coefficient. This maximizes the generated power in these regions and provides the impetus for the VRG. Like the CVT, the VRG can be used in combination with a main gearbox. The gears provide more robust torque transfer and make the VRG a suitable device for the low-speed and hightorque operation of wind turbine systems. In variable-speed applications there may also be some benefit, such as reducing the number of poles in the generator, to using a VRG. However, the low-cost and reliability of a fixed-speed generator are more in line with the underlying theme of reducing small wind cost. The use of a VRG in a wind turbine application is a new concept and therefore more research is required to define the design, operation, and control strategies for this system. The dissertation is organized as follows. Chapter 2 is a review of the current literature in wind energy conversion. It discusses the state of the technology and points to the need for the VRG. Chapter 3 presents an example of a turbine drivetrain system for small wind turbines. This system is designed for 100 kW rated power, which can apply up to 600 N-m of torque at 80 RPM to the VRG. Through this approach the VRG is configured into a wind turbine model with an existing gearbox and a 100 kW squirrel cage induction generator. Stall-regulated blades are also used to be consistent with the 5

approach of selecting components that are simple, reliable, and low cost. Subsequently, a dynamic wind turbine model is developed for this system and provides detailed description of the system dynamics. A rule-based control technique is employed to investigate the VRG-enabled system performance needed to evaluate the proposed concept. Chapter 4 presents a technique to find a set of practical VRG gear ratios with the VRG only. These are used in the model to simulate the energy production at actual turbine sites using wind data provided by National Renewable Energy Laboratories (NREL). Chapter 5 presents strategy to have dual control authority over the VRG and the mechanical brake.

This enables the mechanical brake to maintain generator torque

(similar to the blades being used in a pitch-controlled system) to obtain full production for longer ranges of wind speed above the rated speed.

Dynamic programming is

employed in the decision-making process to find the optimal VRG combination that maximizes energy production while avoiding brake premature failure. Results from this analysis establish the VRG gear ratios and the maximum energy that can be extracted at the NREL recorded sites. In Chapter 6, the gear fatigue and the gearbox weight are considered as the design constraints in the optimization process. An optimization algorithm is developed to find the actual VRG gear ratios based on a trade-off of maximizing energy capture and the design constraints. The selected configurations are again evaluated using the NREL wind data and must have a life of 20 years or greater. In Chapter 7 inferences are made from the overall work and the future of this research is discussed.

6

Chapter 2: Literature Review The conversion of wind energy to electrical power involves two stages. Initially, kinetic air acts on the turbine rotor blades to create rotating mechanical energy. This motion then conveys to an electric generator that produces electricity. Intermittent wind speed presents difficulties at the turbine level where the power must be generated at 60 Hz and with a voltage that is compatible with the grid and electrical appliances [17]. Improving this energy conversion process has been the goal of much research and is the target of this effort as well [10,11]. A second issue is that the grid has no storage capacity and therefore demand must always be equal to the production. It is desirable to integrate the wind turbine with a storage component so that dispatchable wind power can be sent to the grid. While this second issue is crucial for the wind energy sector, it is beyond the scope of this research and additional reading can be found in [10,36,37]. This research focuses on the variable speed operation of wind turbine systems.

Table 2.1:

  Partial  Full 

Common wind turbine configurations.

Some portions of this chapter have previously appeared in publications.

7

   

Allowable variation from grid synchronous speed

Passive stall Blade pitch angle

Power Conversion No No

Yaw Control

Yes Yes Yes No

Rotor Speed

Rotor speed

A B C D

Generator Type Squirrel Cage Induction Generator Wound rotor induction Generator Doubly-Fed Induction Generator Synchronous

Power control

Gearbox

Type

Drive train

No No  

   

+1% +10% -40 to +30% Independent

2.1 METHODS OF WIND ENERGY CONVERSION The energy transformation process is performed using either fixed or variablespeed wind turbines [17,38]. The former is the traditional approach most widely used and has the lowest cost [39–41]. Variable speed operation is more efficient, reduces drive train fatigue, and is the latest trend in wind turbine development [17,42–45]. There are numerous ways that these wind turbines can be configured, however the most common configurations have been identified as shown in Table 2.1 [38,41,46,47]. Types A, B, and C use induction generators, although with increased complexity in going from A to C that enables greater speed variation. Type A [39,40,48–50] uses a passive rotor which requires constant speed operation. Type B [51–55] employs a wound rotor that has controlled resistance, which allows greater variation than Type A. Nevertheless, the amount of variation is very limited, and hence these first two types are commonly lumped together into a single category that comprises the fixed-speed turbines [27,56,57]. The principle here is to operate the generator at a speed that is synchronized with the load-side frequency, thus coercing the variable input into a steady output. In moving to Types C and D, there is a considerable increase in the amount of variation, and hence this class of turbines is known as variable speed. If the generator is allowed to vary in relation to the wind speed, the output power is of varied frequency and voltage [17,58,59]. This mode of variable operation is the case for types C and D, which must employ electronics to convert this assortment of generator power to match the load. The generator used in type C [60–64] is doubly fed and produces a portion of power that is compatible with the demand frequency, while the remainder must be conditioned. A synchronous generator is used for type D [65–68], which is completely isolated from the 8

grid by a power converter. Hence this configuration provides the greatest flexibility in varying rotor speed. When moving from the fixed to variable speed class in Table 2.1, the amount of allowable variation in rotor speed increases along with the aerodynamic efficiency and equipment cost. There is also an appreciable rise in cost in going from type C to D where both the generator type and amount of load conditioning change.

The operating

principles behind all of these configurations and the associated components will be discussed later in this chapter. These configurations can be used with either a horizontal or vertical axis wind turbine [69].

However, with a few exceptions, most commercially available wind

turbines use the former concept, which is deemed to be less complex and lower cost [70,71]. Types A, B, and C normally employ a gearbox to increase the rotor speed to match what is required of the generator [27]. Type D avoids a gearbox and instead uses a customized large generator with a high number of poles to enable low speed power generation [47,72]. Literature discusses variations of this type that use a gearbox to reduce the size of the synchronous generator [72] or to enable the use of the commercially available induction generator [73]. All of the types in Table 2.1 can be found in the large wind turbine class, while distributed wind is less likely to make use of power converting equipment due to the cost [15,16]. 2.2 COMPONENTS OF WIND TURBINES A wind conversion system will have at least a turbine, generator, controller, and may have either a gearbox, frequency converter or a combination of these two [27,47]. There are two types of rotors referred to in wind turbines. One relates to the large turbine rotor central to the hub and blade assembly that the wind acts upon [74]. The second is 9

the generator rotor that spins within the electrical generator against the electromagnetic torque of the stator [75]. Another distinction to make is between direct driven and direct connect. The direct driven refers to an arrangement where the turbine directly drives the generator without a gearbox. The direct connect type implies a configuration with a generator that connects directly to the grid without the intervention of any power conditioning device. Wind

Wind

Rotation

Figure 2.1: Top view of VAWT (left) and HAWT (right) showing motion of blades in relation to wind. 2.3 AERODYNAMIC CONSIDERATIONS In the first stage of energy conversion, wind is converted to mechanical energy through the turbine. Figure 2.1 shows two wind turbine configurations. For horizontal axis wind turbines (HAWTs) the plane of rotation is perpendicular to the oncoming wind, and hence these types receive uniform application of power. For vertical axis wind turbines (VAWTs) the aerodynamics are more complex [76] and blades only receive power at certain points of rotation. The applied power is characterized by a torque ripple [77,78]. Fatigue issues due to this mode of operation along with the inability to mount the turbine up high where strong winds can be experienced have discouraged the use of 10

the VAWT in large wind turbines. The HAWT dominates the market and is the focus of most development efforts [69] due to its simplicity, lower cost, and greater efficiency in capturing wind.

Figure 2.2: Turbine rotor blade pitch angle. Modern wind turbines primarily employ lift forces that act on the blades to produce motion [79]. The blades can also play a critical role in limiting the power in Region 3 to prevent overheating of the generator. In such instances, the lift force is reduced and drag is increased to impede the motion. In active control schemes, this is achieved by increasing the pitch angle, as illustrated in Fig. 2.2, using electric or hydraulic actuators [20]. A blade designed to stall in higher wind speed is also used to reduce the complexity, cost and maintenance of the system [12,13,22].

A yaw

mechanism is also implemented in HAWTs to control the rotor plane position with respect to the oncoming wind [80]. This can be achieved using either an actively controlled motor or a simple tail [79]. The larger wind turbines generally rely on the 11

former approach while the latter may be more feasible for some small wind turbines [18]. However, many small wind turbines may actively use this to control power in Region 3 by turning the turbine away from the wind [23]. This method reduces cost by eliminating the need for pitch control, although there is a trade off as yaw control does not respond as readily. To maintain an acceptable speed ratio, and hence power coefficient, the speed at the outer tip of the rotor blade operates within a general range for all HAWTs. For wind turbines operating onshore the upper limit is constrained to around 70 m/s to avoid excessive noise levels [27]. Since the tip speed has a dependency on rotor diameter the smallest wind turbines have the highest speeds, around 500 RPM [26] or more, while those in the megawatt class can move as slow as 10 RPM [17,27]. For types A, B, and C listed in Table 2.1, power from the slow moving turbine shaft must be transformed through the gearbox to match the high working speed of the generator [27,81,82]. Consequently, large wind turbines require a much greater speed increase. Wind turbine systems in the megawatt class can have gear ratios upwards to 1:100 [83,84]. This has necessitated gearboxes with multiple stages [85]. 2.4 ELECTRICAL POWER GENERATION Electrical production generally takes place through either a synchronous or induction generator, which differs in the way that the rotor is magnetically energized [74,86]. To produce electrical power, mechanical torque is applied to the internal rotor to overcome the reaction torque of the surrounding stator field. As shown in Fig. 2.3, this field is continually rotating through poles 1, 2, and 3, at what is known as the synchronous speed [75]. For the generator to work, the rotor speed must move in relation to the synchronous speed which is dictated by the load-side of the generator. The ability 12

to deviate from this ‘fixed’ synchronous speed is what establishes the degree to which the rotor speed may vary. Although DC alternators coupled with an inverter could facilitate this process more readily, the cost and weight renders these machine types impractical with exception to very small turbines [17].

Figure 2.3: Sequence for synchronous electromagnetic field. In the induction machine, the stator current induces an electromagnetic field between itself and the enclosed rotor [87] to make the reactive power. To create this power, which is ultimately output with the produced power, it must first be pulled from an external source such as the grid or a capacitor bank [88]. For the common machine type that is connected to a 60 Hz grid a synchronous speed of 1800 RPM is typical [87]. For generator action (conversion from mechanical to electrical) the rotor must move at a slightly higher speed than the stator field. This difference in speed, known as ‘slip’ allows variation in the rotor speed, which increases approximately linearly with the amount of torque applied [74]. Type A wind turbines incorporate the squirrel cage induction generator (SCIG), which can slip 1 to 2% [40,41] above the synchronous speed. The SCIG is a mature product that is reliable and highly efficient. It has the lowest cost among the four types listed in Table 2.1 [39,49]. The wound rotor induction generator 13

(WRIG) found in type B turbines enables slip up to 10% by actively varying rotor resistance [41]. These fixed speed generators produce power that is in phase with the load-side and therefore may be directly connected. For the synchronous machine, the rotor has fixed north and south poles that must move at precisely the same speed as the rotating electromagnetic stator field [89]. This lack of compliance would otherwise be problematic for wind energy conversion if these machines were not decoupled from the load by power conversion equipment [44,58,90]. Type D wind turbines incorporate these electronics with synchronous generators to achieve variable speed operation [89,91]. The polarity of the rotor is developed by passing current through a wound rotor or by permanent magnets [92,93]. The latter approach is characterized by high efficiency [81], greater strength, and is the recent trend due to a decline in the cost of the magnetic materials [7]. External excitement is not required for these machines which facilitate grid independence [92]. A class of multipole low speed synchronous generators has been developed for use in low speed wind turbines [69,86] to eliminate the need for the gearbox. However, this customized solution carries significant cost, and can weigh as much as the gearboxes that have been removed. Recent developments in microprocessor technology have improved reliability and lowered the cost for integrated devices used in frequency and voltage conversion [90]. As shown in Fig. 2.4, a common approach for frequency conversion is to rectify the AC power from the generator to DC that can be fed, in a controlled manner, to a subsequent inverter that outputs AC electricity at the load frequency and voltage [17]. In this configuration, the rotor speed is also varied by controlling the torque at the generator [90]. This technique is already being used in large wind turbines. However, in case of small wind turbines, the power conversion equipment still lacks technical maturity, is costly to justify, and has a record of unreliable performance [14–16]. The doubly-fed 14

induction generator (DFIG) found in type C implements two coil windings, one of which is connected to the grid and the other through the power converter [21,94].

This

approach reduces the cost of the converter and allows the rotor speed to slip from 40% below to +30% above the synchronous speed [17]. DC Generator Side AC

Grid Side AC

Figure 2.4: AC to DC to AC power smoothing by electronic conversion. 2.5 GEARBOX ISSUES There are a number of difficulties with large gearboxes used in the megawatt class of wind turbines. Installation requires special equipment for mounting the unit at the top of the tower [85,95] and gearbox reliability has been a leading cause of failure among wind turbines in the megawatt class [65,96]. An NREL study [97] traces the majority of these breakdowns to worn bearings that deposit particles on or allow poor alignment of gear teeth. Only a limited amount of research pertaining to gearbox tribology has been conducted to resolve these problems [98]. Fixed-speed types are especially vulnerable as wind gusts translate to cyclical loading of the drive train [17,19]. Such events take the unit out of commission while the onerous tasks of removal, repair, and reinstallation are performed [99]. These problems have justified replacement of the gearbox with directly driven low speed synchronous generators that require costly power electronics [59]. Although gearbox failure ranks high among the large class of wind turbines, this scenario improves among the small turbine class where this component is responsible for only 5% of the repair, comparable to the reliability of the power generator. [100]. 15

Variable speed wind turbines that can adapt the rotor speed to wind changes are able to mitigate peak loads in the gearbox [20] and this has led to improved reliability. It has also been suggested that poor gearbox performance may be the result of insufficient design [95]. There are alternative solutions that have improved performance such as splitting the torque between the main generator and a controlled energy storage device that operates as either a generator or motor to extract or supplement power in response to the load variation [101]. By sharing the output torque among four high speed shafts, Clipper Windpower has reduced the load by a factor of 16 and reduced the weight of the gearbox 20% to 50% when compared to other 2.5 MW models [85]. Similar measures have been suggested for helicopter gearboxes [102], which carry loads and have gear reductions of similar magnitude to wind turbines, as a method to reduce weight and volume. The idea of implementing a continuously variable transmission (CVT) has appeared in recent literature [15,16] where it is used for fine adjustment of the gear ratio while a gearbox performs the coarse speed increase. In this arrangement, an induction generator is used to make a direct connection to the grid preventing the need for power conversion equipment. While the generator runs at a fixed speed, the CVT allows the rotor to operate at a variable speed to maximize the aerodynamic efficiency. Moreover, variable speed operation can reduce the mechanical fatigue in the drive train that has plagued fixed speed turbines. The CVT concept is not new and has been proposed periodically in the past [29–34]. However, it has failed to get traction mostly due to limited durability, which is a concern for this device [28]. Experience in the automotive industry suggests the CVT is best suited for low torque applications, such as the Honda Insight, which produces 167 N-m of torque at an engine speed of 1000 to 1700 rpm [35]. The low-torque-carrying characteristic of the CVT is due to the CVT transferring power 16

through friction force rather than using a more robust gear mesh. The wind turbine gearbox could be exposed to a torque greater than 700,000 N-m with a speed of 18 RPM [103]. Therefore, the current CVT technology is not suitable for mid-size to large wind turbine systems. However, with technology development, the CVT could play a major role in the power transfer from the rotor to the generator in the future. 2.6 VARIABLE RATIO GEARBOXES The power coefficient of the wind turbine can be modulated through the blade pitch angle or the speed ratio. The state of the art approach for adjusting the speed coefficient in Region 2 is limited to varying the generator operating torque. This approach may be too expensive for distributed wind turbines. However, changing the gear ratio of the drive train might be a low cost alternative. Even though a CVT is proposed to achieve this goal [15,16], it cannot be widely applied to wind turbine applications due to the material limitations as previously discussed. There is a need to design a device that can withstand the torque level imposed by the wind turbine and still have the capability to vary the drive train gear ratio. A variable-ratio gearbox (VRG), based on the automotive manual transmission design, is proposed in this research to satisfy this requirement [104,105]. These types of gearboxes are a mature technology with proven reliability in the automotive industry [24]. The VRG is an automatically controlled gearbox with meshed gears and discrete gear ratios. The power transfer inside the VRG is mainly through the gear mesh rather than through friction as in CVTs. A VRG with a set of ‘n’ gears can provide the controller a set of ‘n’ power coefficients, one of which can be selected and applied based on the desired performance for a given scenario. This approach is promising for the fixed-speed class of wind turbines because it allows the rotor speed to adjust 17

independently of the generator speed. Through this mode of operation, the VRG enables this class of wind turbines the flexibility to track wind speed and adjust the rotor speed as necessary to increase power production. 2.7 PROPOSED WIND TURBINE SYSTEM Based on the start of art knowledge for wind turbine systems, this research proposes to develop a variable ratio gearbox that increases the turbine wind energy capture and improves the system economic viability. In order to minimize the system cost and maximize its reliability, a horizontal axis wind turbine with a squirrel cage induction generator is used. The wind turbine system consists of a rotor, main gearbox, the VRG, a mechanical brake and a squirrel cage induction generator. The results from the analysis will demonstrate the feasibility and the benefit of employing the VRG. 2.8 METHODS OF CONTROL Control strategies are crucial to the energy conversion process and exist at multiple levels [19,106]. At the top level there is a supervisory control responsible for performing startup and shutdown of the turbine in relation to the wind. The next level is the turbine control which includes the functions used to control the power level [99]. This is also most commonly referred to in literature and includes tasks such as varying generator torque, pitching rotor blades, and adjusting the turbine yaw angle [19]. The latter technique is only used on horizontal axis turbines, and refers to adjusting the rotor plane orientation in relation to the oncoming winds. The lowest level of control relates to the internal control of the devices such as individual hydraulics, servo motors and electronic components that are used to support the function at the turbine level [19,106].

18

0.6 0 degrees 1 degrees 2 degrees 5 degrees 10 degrees 15 degrees

Power Coefficient, Cp

0.5 0.4 0.3 0.2 0.1 0 -0.1

0

2

4

6 8 10 12 14 rotor tip to wind speed ratio, 

16

18

20

Figure 2.5: Power coefficient as a function of speed ratio and blade pitch angle. 2.8.1 Control Mechanisms One of the control goals is to achieve high aerodynamic efficiency, which refers to the ability of the turbine to extract power from the wind and is represented by the power coefficient [79]. As shown in Fig. 2.5, this parameter is affected by the rotor blade pitch angle, the speed at the tip of the rotor, and the wind speed [20,107,108]. The work of Betz [109] has shown that this value is limited to 0.59 since the wind must also be moving to exit the system, thus retaining a portion of the kinetic energy. Through the application of turbine control, the power coefficient,

, is maximized or limited,

respectively, in the partial and full load regions [107]. Figure 2.1 illustrates the power coefficient versus the speed ratio at different pitch angles. Only low pitch angles are useful in extracting power from low winds [101,110], which correspond to higher values of the speed ratio, , as shown in Fig. 2.5. The capability of pitch control is limited in Region 2. On the other hand, the rotor speed can be lowered in relation to these lower wind speeds [42] to shift the speed ratio from the right side toward the middle of the 19

curve in Fig. 2.5 where the power coefficient has a higher value. Turbine types A and B have very limited ability to vary speed and therefore do not extract as much power from Region 2 as types C and D of the variable speed class do [13,19]. In Region 3 the controller seeks to reduce the power coefficient to prevent overheating of the power generator [27,59]. In this full load region, the generator runs at maximum capacity and the rotor speed is held constant [27,79]. To achieve the goal of maintaining constant a constant torque so that the generator is kept at its rated power, the controller increases the pitch angle, which corresponds to the lower power coefficient values at the left side of Fig. 2.5. For systems that employ passive stall [12,13,22], these higher winds also shift the speed ratio to the left with a similar effect, although with slightly less efficiency such that the generator actually operates just below the rated power or full power. For small wind systems, furling is often used to rotate the rotor away from the wind, thus limiting the power [23]. 2.8.2 Control Algorithms Control objectives of wind turbine systems often aim at maximizing power production and work through the power coefficient [15,111]. In the partial load region, the power coefficient is maximized by adjusting the rotor speed [111]. To limit power in the full load region, the rotor blades are rotated and used as brakes [19]. Traditional feedback control and adaptive control have been developed for wind turbine systems with satisfied stability and robustness. However, an optimal control is desired in order to maximize the wind turbine efficiency, which is critically needed for distributed wind turbines. Nonlinear optimization strategies, such as maximum power point tracking [112– 114] and variable structure control [115–117] are commonly used to design the wind turbine controller, based primarily on electrical power production [118]. However, they 20

are highly time consuming and take long time to converge to the desired operating point. More control algorithm exploration is required for optimal operation of the wind turbine system. The control goal for the VRG-enabled wind turbine is also to maximize power production. When operating in Region 2, it is desirable to apply the gear ratio that provides the greatest amount of torque, and thus maximum power. In Region 3, the system will try to achieve full power using the brake, or simply maintain the power production above 90% using a different gear ratio when it is not practical to use the brake. Since the history of the gear ratio selection and level of brake use influence energy capture, brake pad wear and gear teeth fatigue, dynamic programming is used to optimize the process over a known time-horizon.

This approach has demonstrated

abilities to manage power in hybrid vehicles [119–122], perform motion planning for robotic manipulators [123–125], and establish operating parameters that prolong the life of manufacturing tools [126–128].

Here, the dynamic programming (DP) process

enables the development of a control algorithm derived for time-based disturbance data. The DP analysis will choose the optimal VRG ratios and defines how each ratio is used with the brake in various operating conditions. The dynamic programming technique provides global optimization through an exhaustive search, which renders the technique to be computational expensive. Efforts that significantly reduce the computation time have been part of this body of research.

21

Chapter 3: Wind Turbine Drivetrain Model The Variable Ratio Gearbox (VRG) works in conjunction with the main gearbox, which produces most of the speed increase from the low-speed rotor shaft to the highspeed generator shaft. Figure 3.1 shows the proposed configuration of the VRG-enabled drive train, which consists of a turbine rotor, brake, main gearbox, VRG, and Squirrel Cage Induction Generator (SCIG). In this research, the mechanical brake is placed on the low speed shaft. In this location, the brake can be used to mitigate dynamic wind loads

Rotor

Main Gearbox Generator

+ Brake

VRG

Control System

Figure 3.1: VRG-enabled wind turbine configuration.

Significant portions of this chapter have previously appeared in publications.

22

before they reach the drivetrain. A simple rule-based controller determines when to shift gears or apply the mechanical brake. During operation the brake is used briefly to stabilize the turbine rotor during gear shifting. These events are relatively short in duration (lasting less than 3 seconds), low in frequency, and are negligible in computing heat generation and wear due to prolonged brake usage as these occur in the simulations in this study. To demonstrate the proposed methodology, a low-order dynamic model of a 100 kW, stall-regulated, fixed speed, wind turbine drivetrain has been developed in a MATLAB/SIMULINK environment based on [104,101,48]. Fig. 3.2 identifies the model components and demonstrates how wind data is used to simulate power production.

Wind data

vw

Rule-based control

Tt

General profile

wt

Tip-speed ratio calculation

wt

vw

VRG Ratio

Disk Brake

pdi

qin

F

GR

Brake model `

 Power coefficient calculation

Tm

cp

GR

Tb

vw Turbine torque calculation

Tt

wt

wt

wg

Mechanical Model

Tg

Induction generator

Tg

Pg

Figure 3.2: Wind turbine model. 3.1 AERODYNAMIC MODELING The process starts as wind, flowing at a speed,

, passes through the turbine.

Power is produced by the turbine rotor according to [10,74,75], (3.1)

23

where, = turbine rotor diameter = density of air The power coefficient,

, shown in Fig. 3.3, varies according to the blade pitch angle, ,

and the wind speed ratio,

[129]. –





(3.2)

where, (3.3) Constants,

through

, are assigned [129–131] to match the behavior of a

particular wind turbine system.

Power Coefficient, Cp

0.5

0

-0.5 0 0

5 5

10 10 15

15 20

Speed ratio, 

20 Pitch angle,

Figure 3.3: The effect of speed ratio and pitch angle on the power coefficient. 24

The extracted power is realized as torque,

, that is applied at a rate of

,

through the low speed shaft. (3.4)

qconv Vane

View A

dh, l View B-B

Tt, wt

Disk rotor

y

B

qrad

B

Tb qin View A Vane detail

Pad

Fb

pdi, f Figure 3.4: Components of the disc brake. 3.2 DISC BRAKE MODELING An automotive disk brake is selected for small wind turbine applications. The disk brake presented here is similar to that found in a pickup truck or large passenger vehicle. To facilitate heat dissipation, the rotor is vented, and the pads feature a durable ceramic-based lining. As shown in Fig. 3.4, the system works by applying a force,

, to

the pads, which press and slide against the rotor disk to produce braking torque,

,

according to [132], (3.5) 25

where, = inner rotor diameter = outer rotor diameter The amount of torque that can be removed with the disk brake is constrained to avoid scenarios where excessive heat or pad pressure occur and lead to system failure. 3.2.1 Limitations Due to Heat The heat generated by the brake sliding surfaces is primarily rejected through the rotor disk [133]. When the temperature of the disk surface reaches

, around 315C,

fade starts to occur. This means that above this point the heat reduces the frictional capability of the liner, and leads to lower and unreliable braking capability. The situation could happen during full load operation, when the brake could potentially be applied hours at a time. To identify such operating scenarios, each braking instance is considered to occur for an infinite amount of time, with the temperature of the brake system initialized to

. As long as braking produces heat at a rate that is less than the heat

rejection rate, the temperature stays below the critical value and the level of torque is acceptable. This relationship is represented by the follow inequality (3.6) where, = rate of generated heat = the heat rejection rate 3.2.1.1 Heat Produced at the Brake It is assumed that the power removed by braking converts to heat in the rotor disk based on the number of pads,

, brake torque, 26

, and turbine rotor speed, wt,

w

(3.7)

3.2.1.2 Heat Rejected by the Brake This research assumes all of the heat is rejected through the disk, which has a uniform temperature distribution, through convection,

, and radiation,

[133]. (3.8)

As shown in Fig. 3.4, convection takes place through rotor vanes, while radiation takes place from the outer exposed rotor surfaces. Heat removed by convection occurs according to [134]. (3.9) where, = total surface area of all the vanes in which convection takes place = temperature of the brake = ambient temperature cooling air The convection coefficient,

, is based on the vane geometry (Fig. 3.4) and nature of

the airflow [133]. (3.10) where, = hydraulic diameter of the vane = length of the vane The Reynolds and Prandlt numbers,

and

, are based on an air velocity, , through

the vane according to [133], w

(3.11) 27

where, = area at vane inlet = area at vane outlet The amount of heat removed from the brake by radiation is found by [134]. (3.12) where, , = exposed area of the disk = surface emissivity = surrounding temperature 3.2.2 Limitations Due to Pad Pressure In the case of uniform wear analysis [132], failure will occur along the inner diameter of pad, where maximum pressure is applied. To avoid failure at this point,

,

should not exceed the maximum recommended pad pressure, (3.13) where the pressure is related to the braking torque,

, by, (3.14)

where, = friction coefficient between pad and rotor = angle of rotor section that contacts pad

28

T 1, q 1

Tt, qt

bt

GR

kt

Tb

Jt

Tg

bg T 2, q 2

kg

Jg

qg

Figure 3.5: Diagram of drivetrain dynamics model. 3.3 GEARS AND SHAFTS MODELING As shown in Fig. 3.5, torque conveys through the two-mass mechanical system, overcoming the generator electromagnetic torque to produce electrical power.

Two

independent variables, q and q , characterize the dynamics of this event through the following equations, q



q where

– –

is the turbine rotor inertia,

speed shaft,

q –

q

q –

q

(3.15)

q

q

(3.16)

q

is the frictional loss,

is the inertia of the generator rotor,

stiffness of the high-speed shaft. The expressions of the gear ratio,

q –

, according to, 29

is the stiffness of the low-

is the bearing loss, and through

is the

give consideration to

(3.17)

(3.18)

(3.19)

(3.20)

3.4 ELECTRIC GENERATOR MODELING The squirrel cage induction generator is modeled to simulate the transformation of mechanical torque into electrical power.

Three-phase, alternating current, creates a

rotating synchronous field that surrounds the rotor. The rotor is induced by, and moves relative to, this rotating field, and is described in the d-q coordinate frame [135,136]. The amount of flux voltage is affected by the resistances and rotating velocities of the stator and rotor, denoted by , in each reference axis. w y

y

(3.21)

w y

y

(3.22)

w

w y

y

(3.23)

w

w y

y

(3.24)

The flux voltage terms are dependent upon the rotor speed, w , and stator speed, w as well as the respective resistances,

and

. These voltage components are related

through the flux-linkage current terms, , which are expressed in each reference axis for the stator and rotor as, 30

y y y y where y represents the flux terms, respectively, and

and

(3.25)

are the rotor and stator inductances,

is the magnetic inductance. The amount of generator torque applied

against the turbine rotor input during electricity production is determined by, y

y

(3.26)

3.5 MODEL VERIFICATION The model is verified by simulating the performance of the 100 kW fixed speed system.

The gear ratio is set to 21.59, which is typical for this wind turbine

configuration, and the generated power is recorded over a discrete range of normal operating speeds that are used as input. An example of the generated power versus wind speed for a 100 kW wind turbine system is shown in Fig. 3.6, where the dotted red line represents the typical power a fixed-speed turbine can produce [137]. The solid blue line is our simulation result. The two lines both have a maximum power around 17.8 m/s, which is the speed where the wind turbine works most efficiently. These two curves are in good agreement thus indicating the approach used in this study is justifiable.

31

4

12

x 10

10

Model Wu and Chen, 2008

power [W]

8 6 4 2 0 0

5

10 15 wind speed [m/s]

20

25

Figure 3.6: Verification of model to 100 kW power curve. 3.6 WIND DATA CONSTRUCTION The National Renewable Energy Laboratory (NREL) website provides wind data for several wind turbine sites in the United States [138]. Each profile consists of three years of wind speed data recorded at a height of 100 m. Since the VRG targets small to medium-size wind turbines, the wind profiles are adapted here to a reduced height of 50 m. This is achieved using the power law [139] as given by (3.27) where, = wind speed at the height, = the recorded wind speed, = height at which wind speed is being predicted = height at which wind speed is recorded

32

The exponent, , is computed as a function of height and each individual wind speed. It is given by (3.28) Several wind profiles were converted to the 50 m height and 20 were selected to represent the full range of wind categories given in Table 3.1. Wind Mean wind speed Power density class* [m/s] [w/m2] 1 0 – 200 0.0 < Vw  5.6 2 200 – 300 5.6 < Vw  6.4 3 300 – 400 6.4 < Vw  7.0 4 400 – 500 7.0 < Vw  7.5 5 500 – 600 7.5 < Vw  8.0 6 600 – 800 8.0 < Vw  8.8 7 800 – 2000 8.8< Vw  11.9 *as referenced at 50m hub height Table 3.1:

Classes of wind power density at 50 m [140].

The intention in selecting the wind profiles, used as inputs to the VRG, was to get a reasonably even-spaced distribution, in terms of power density, across the seven wind classes. Among the many profiles reviewed, it was found that the power density and wind speed are not always in agreement when it comes to classifying the wind profiles. Hence, the power density criteria was determined to be the most appropriate for this task with attempts made to select profiles that had mean wind speeds fitting into, or at least being close, to the range of wind speeds for the designated class. As shown in Table 3.2, each class generally contains three wind profiles, situated near the bottom, middle, and top, of the range with respect to power density. However, only one profile was included for wind class 1, since sites that fall into this category, are 33

rarely used for wind turbines. Conversely, four profiles were selected for class 7 because it has such a wide range of power density. Wind class 1 2

3

4

5

6

7

Table 3.2:

Set No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Site 12951 3412 11719 28570 6474 6684 22814 7302 31266 2002 174 3134 13731 21291 22818 16577 7858 19520 12796 11721

Power density [W/m2] 192.9 244.0 259.9 289.4 329.9 363.0 391.8 430.8 468.6 487.4 523.0 555.2 580.3 649.9 690.0 747.6 918.2 1004.3 1082.3 1242.2

[m/s] 4.97 5.76 5.71 5.47 6.84 6.91 6.95 7.34 7.30 6.29 7.13 7.87 7.34 7.75 8.26 8.59 9.21 9.38 10.16 10.05

National Renewable Energy Laboratories (NREL) wind turbine sites used in study.

3.7 CONTROL FRAMEWORK ESTABLISHMENT The 100 kW wind turbine system modeled in this research is suitable for supplying power to roughly 10 to 20 homes depending on wind conditions [104]. In this arrangement it connects to the local grid, both directly and through a battery that reconciles the variations between the electricity production and the demand. Two levels 34

of management, supervisory and servo, are used to control the wind turbine. This study focuses on a framework that will be implemented at the supervisory level or the turbine level, although, the servo level is briefly discussed in terms of how it interacts with the supervisory level. In this research, the mode control first sets the operating mode to one of the four conditions shown in Table 3.3. When the operating mode is 'On', wind speeds are suitable for power production, the system operates, and the mode controller then defines when the gear ratio needs to change, what gear to apply, and when to apply the mechanical brake during operation. Mode Off

Description The system is off when the wind speed is outside of the operating range, which is bound by the cut-in and cut-off speeds. During this time the rotor is locked down and the generator is disconnected. Start Up This mode occurs after the wind turbine has been off, and the wind speeds have shifted back within the operating range. During this time the turbine rotor starts and is accelerated to its operating speed. Once this occurs, the generator is reconnected to the grid. On In this mode the wind speed is within the operating range. The rotor is running at one of six discrete speeds and the generator is connected to the load. This is also the only mode where turbine control is active. Shut Down This mode occurs when the wind speed shifts outside of the operating speed, and it is necessary to shut down the turbine. The generator is disconnected from the grid, and the rotor is brought to a stop. Table 3.3:

Operating modes.

During the “On” mode, the supervisory controller determines the optimal gear ratio selection and when to apply the mechanical brake to maximize the energy capture with prolonged system life. This control development will be the focus of this research. At the component level, control is applied through servos that perform gear shifting and braking. These servo controllers receive the set points from the supervisory controller 35

during normal operation. During 'Start Up' the brake actively stabilizes the rotor speed as the VRG is engaged. During 'Shut Down' the brake is used to stop the rotor. 3.8 CONTROL PROBLEM FORMULATION The control structure, illustrated in Fig. 3.7, performs decision-making tasks for changing the gear ratio and applying the mechanical brake. The wind turbine model presented in the previous section is a nonlinear system.

The dynamics have state

equations of the form, (3.29) The state variables,

are taken from the mechanical and generator dynamic

models, q q q q

(3.30)

Control is applied through the variable, , and is affected by a disturbance, w, or wind speed. (3.31) (3.32) Output from the system includes measurements, , and performance metrics, , (3.33) (3.34)

36

 

w  Vw 

GR u   Fb 

z  Pg VRG-enabled drivetrain

qt  y  Tt 

Figure 3.7: Control problem formulation. During normal operation, the control system monitors the wind speed and estimates key parameters. A set of rules is used to evaluate the parameters with respect to a set of operating boundaries. Moreover, they control the system by changing the gear ratio or applying the brake. The parameters relating to the brake system and rotor speed are compared to constant values. The operating range for wind and torque varies and is summarized through a look-up table. The dynamic wind turbine model is developed in the MATLAB/SIMULINK environment, based on the above techniques.

37

Chapter 4: Rule Based Control Design Results for this research are obtained using a simulation model that produces power curves, for both the variable ratio gearbox (VRG) and original configurations, shown in Fig. 4.1. The plots illustrate that the VRG enables the wind turbine to peak at the maximum power level several times, as indicated by the dotted green line, while the fixed-speed configuration only reaches the maximum power level at one point, as indicated by the blue line. The additional area found under these peaks represents gain in produced power for an even distribution of winds in the range of 0 to 25 m/s. The power boundaries shown in Fig. 4.1 are indicative of a design requirement that all the gear ratios enable the system to operate between 90% and 100% of full production for all wind speeds greater than

.

4

x 10 10

power [W]

8

Fixed speed VRG-enabled Vr, FS Vr, VRG

6 4

PFL, max PFL, min

Region 2

2 0 0

5

10 15 wind speed [m/s]

Region 3

20

25

Figure 4.1: Comparison of VRG-enabled production to original production, with traditional operating regions shown.

Significant portions of this chapter have previously appeared in publications.

38

The shift in the rated speed, also redefines the partial and full load operating modes, which are shown as Regions 2' and 3' in Fig. 2. The optimal control design will further increase the power capture capability of the VRG-enabled drivetrain with the use of a mechanical brake. In that approach the brake and gear ratio are coordinated to enable full-load operation for a wider range of the wind speed. Without the brake, maximum power can only be held at the peaks of the 'VRG-enabled' curve as shown in Fig. 4.2. Hence, full load operation is only achieved momentarily, since a change in wind 4

x 10

electric power [W]

10 8

VRG-enabled VRG w/brake PFL, max PFL, min

6 Region 2'

Region 3'

4 2 0 0

5

10 15 wind speed [m/s]

20

25

Figure 4.2: Trend for VRG-enabled when used with mechanical brake, shown with new operating regions. speed will either reduce power, or produce more torque than the generator can accept. This requires the VRG to shift up to the next gear that produces less torque, indicated by the drop next to each peak, and results in lower electrical power. On the other hand, using the mechanical brake dissipates excess turbine torque, enables the same gear to be used across the drop, and allows full load operation for a broader range of wind speed. The resulting gain in wind capture is realized in the flat sections at the top of the 'VRG 39

with brake' curve. Applying the mechanical brake to limit power is generally acceptable in the case of small wind turbines, with a rotor diameter of 60 m or less, which produce low rotor torque [74]. Furthermore, this torque level also permits the brake to be used at the low speed shaft. In Chapter 5, measures will be taken to avoid excessive heat and pressure that could result in brake system failure. 4.1 GEAR RATIO APPLICATION Most of the gear ratios are used in at least two, and in some cases, three different operating segments of the control map shown in Fig. 4.3. Here, each segment establishes the range of wind speed through which a particular gear ratio may continuously operate. Figure 4.3 graphically defines the general operating range for each gear ratio with respect to the wind speed. The data shown here is obtained by simulating the performance of each gear ratio across the typical wind speed range (0 to 25 m/s) and choosing the one that produces the greatest amount of electrical power for the VRG-enabled system.

GR = GR = GR = GR = GR = GR = 0

5

14.71 17.1 19.15 20.54 21.03 21.59 10 15 wind speed [m/s]

20

25

Figure 4.3: General wind speed operating ranges for gears in 100 kW VRG-enabled model. 40

For those segments in Region 2' the objective is to select the gear ratio that produces the greatest torque. Sections in Region 3' must select the gear ratio that produces the greatest amount of torque and does not exceed the maximum amount allowed by the generator. Use of the brake will limit mechanical torque, and allow the gear ratio to maintain a full power. Steps taken at the design stage insure that a gear ratio is always available in this region to maintain a production level of at least 90% of full capacity. Each operating segment, , is defined in terms of wind limits of and torque limits of

and

and

, shown in Table 4.1. These limits are used to control

the gear ratio. The control structure is reconfigurable so that the look-up table can be replaced, and rules can be modified to achieve different results for a variety of wind turbines sizes, wind conditions, and performance goals. This approach facilitates the optimization study presented in Chapter 5.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

21.59 21.03 20.54 19.15 17.10 14.71 17.10 19.15 20.54 21.03 21.59 21.59 21.03 20.54

Table 4.1:

4.9 8.4 8.7 9.1 10.0 11.4 12.2 12.5 13.2 14.4 15.1 17.8 21.9 24.1

8.4 8.7 9.1 10.0 11.4 12.2 12.5 13.2 14.4 15.1 17.8 21.9 24.1 25.0

16 192 207 238 312 461 557 527 527 558 551 527 527 555

191 206 237 311 460 583 583 583 584 584 584 584 584 584

Data used in look-up table for 100 kW VRG-enabled wind turbine. 41

Turbine control

Check vw,

Tm,

Mode == ‘On’

qin, pdi, Mode

No End

Yes Get i(vw)

(vmin(i)- vc) ≤ vw ≤ (vmax(i)+ vc)

No Apply Brake

Yes

Yes No

qin ≤ qoutmax &&

No

pri ≤Nopmax

Tg ≤ Tg max vw < vr, VRG

No

Yes Decrease ratio

i-Yes

Tmax(i) ≥ Tm

Tmin(i) ≤ Tm No

Yes

No

Increase ratio

Yes

Tmax(i) ≥ Tm

i++ No Change gears

Tmin(i) ≤ Tm

No

Yes

Get GR(i) Tmin(i), Tmax(i), vmin(i), vmax(i)

Yes

Figure 4.4: Decision making structure implemented during normal operation.

42

4.1.1 Normal Operation The measured variables are continuously evaluated in the decision-making process illustrated in Fig. 4.4. The first step in the process is to verify that the wind speed,

, is within the operating range for the current operating segment, , that is bound

by

and

. Doing this avoids a situation in which the wind speed changes faster

than the rotor speed can adjust, and coincidentally, settles at a new wind speed where the gear ratio can produce torque within the operating range defined by the previous gear sector. A confidence term,

, exaggerates the operating range so that wind speed only

initiates a gear change for the described situation. The actual design intention is for gear shifting to occur in response to the mechanical torque. Control through this parameter enables the generator to run at full capacity without damaging the generator. When the system operates in Region 2' the gear ratio does not change as long as the wind speed and mechanical torque stay within the normal ranges defined by the current segment, . In Region 3' the same rules apply, with an exception made when the mechanical torque exceeds the upper sector limit, of

. In this case the mechanical brake is applied,

and used as long as the braking does not produce excessive heat or pressure. 4.1.2 Gear-Changing Events When gear shifting is required, the sequence of events shown in Fig. 4.5 is carried out. The wind speed is detected, and the control data for the associated segment, , is referenced from the look-up table. The rotor speed is detected to determine if the speed must be increased or reduced to accommodate the new gear ratio. When changing from a higher to a lower gear ratio, the electromagnetic torque is first removed and the gear is then disengaged in the VRG. This causes the turbine rotor to speed up, and the new gear is engaged when the rotor speed is sufficient to turn the generator at its required speed. 43

When switching to a higher gear ratio the brake is used to slow the rotor down to match the required speed. Change gears

Disengage rotor

Check

Yes

No

qt 

qg

qt

qg nom  qt  GR (i )

Apply brake

No

qt 

min

GR (i ) Yes

qg

max

No

GR ( i ) Yes

Engage rotor

End

Figure 4.5: Decision making structure for changing gears. 4.2 MODEL ANALYSIS This section describes how the aforementioned model and control structure are used in this study. Simulations were conducted to characterize the performance of each VRG gear ratio in conjunction with the brake. The subsequent chapter explains how the VRG gears are determined in the design process. The upper plot of Fig. 4.6 shows the 44

amount of mechanical torque that each gear ratio can potentially deliver to the generator. Flat portions of the curves indicate ranges of wind speed, where the turbine torque and the brake can work collaboratively to maintain full load production. The range for these segments is limited by the model constraints, and is discussed below. The 'Selected profile' is assembled from segments from the individual gear ratio curves. The lower plot shows how the selected gear ratios are used to construct the power curve.

mechanical torque [N-m]

1000 Selected prof ile GR = 14.71

800

GR = 17.1 GR = 19.15

600

GR = 20.54 GR = 21.03

400

GR = 21.59

200 0

0

5

10

15

20

25

0

5

10 15 wind speed [m/s]

20

25

power coefficient, Cp [-]

0.4 0.3 0.2 0.1 0

Figure 4.6: Simulated performance of each VRG ratio across operating spectrum.

45

The lower plot in Fig. 4.6 also demonstrates how variations in the power coefficient factor into the gear application process. Each of the curves represents a VRG ratio. All the curves have the same basic profile that shifts to the right as the gear ratio decreases. The power coefficient plot implies the highest gear ratio is most effective in capturing wind energy at low speeds. At around 8.4 m/s, the second highest gear ratio produces the highest coefficient, and this trend continues with each lower gear becoming more effective at capturing wind. Hence, these lower gears are applied in Region 2', to extract the maximum amount of power from the wind. At 10.7 m/s, the lowest gear is able to extract the greatest amount of wind power, with the capability of this dropping off as the curve approaches cut-off speed. maximize power in region 2'.

The highest coefficient value is applied to

For wind speeds above

, the greatest power

coefficient is not used. Instead, the coefficient that produces the most power without exceeding the maximum generator torque input is used. This causes the gear ratio to shift towards higher gears that are less effective at capturing wind, through around 17.8 m/s. Past this point wind speeds have increased significantly to produce greater power, but the power coefficient has dropped off more rapidly, and hence the system begins to shift back to the lower, more effective power coefficients. Figure 4.7 characterizes the response of the constrained parameters in the disk brake model. The plots confirm that the selected gear ratios can be used at the designated wind speeds and remain within the operating boundaries. The peaks in the top plot are indicative of wind speeds where the brake is applied to maintain maximum allowable torque, and thus maximum power production. For peaks below the original rated speed,

46

pad force [N]

6000 4000 2000 0

0

5

10

15

20

25

10

15

20

25

10 15 wind speed [m/s]

20

25

5

heat rate [W]

pad pressure [Pa]

x 10 10

pmax 5

0

pdi

0

5

6000 qomax

4000

qin

2000 0

0

5

Figure 4.7: Brake characteristics during operation. , an increase in wind speed corresponds to an increase in power, and requires more brake force to maintain the same mechanical torque. Above the wind speed of , the situation is reversed, and an increase in wind speed means a decrease in power. The drop corresponds to the power coefficient dropping off in high winds. For this configuration of gear ratios, the brake pad force is generally limited by the maximum pad pressure, as shown in the middle plot. Heat is only a limiting factor in one of the peaks 47

as shown in the bottom illustration. Since the brake acts on the low speed shaft, the rotating speed of the turbine rotor plays a significant factor in determining the constraining parameter. A decrease in gear ratio corresponds to a higher turbine rotor speed. It will produce more heat than a higher ratio that moves slowly with the same amount of pressure. In the case of this wind turbine model, our analysis shows that only the highest gear is limited by the pressure, while the lower gear ratios move the rotor fast enough that the heat is the limiting factor. Figure 4.8 shows gains in electrical capacity in the power curve (top) correspond to the use of lower gears (bottom) in coordination with the mechanical brake to produce higher torque. The area between the power curves at the top plot depicts the gain in energy that is achieved by applying the brake to the VRG-enabled wind turbine. The new curve is much flatter than the previous one, and achieves full power operation for a broader range of wind speeds. Adding brake capability to the VRG-enabled system produces 2% more area under the power curve then the previous technique. This brings the total captured energy increase over the orginal fixed speed system to about 9%. The flattened sections at the top of the power curve indicate full power can be achieved across 43% of the wind speed in Region 3', where the electrical power is output at an average of 98.8 kW. The lower plot demonstrates how the gear ratio changes with respect to wind. The proposed technique involves the use of the mechanical brake, and allows lower gear ratios to be used across greater sections of wind speed. The extended range shown in the curve for the VRG with the brake corresponds to the flattened section in the power curve, and indicates wind speeds where the mechanical brake is used to maintain the power level. The result will be translated into the maps used in the rule-based structure during normal production. Even though the above result was obtained based on a uniform distribution of wind speed from 0 to 25 m/s, the framework established by this research 48

could be employed for control design based on wind site data. This will be demonstrated in Chapter 5, where an optimization strategy is used to maximize the power generation based on unique operating conditions. 4

12

x 10

power [W]

10 8 6 4 2 0

0

5

10

15

20

25

15

20

25

22

Gear ratio [-]

20 VRG-enabled VRG w/brake

18

16

14

0

5

10 wind speed [m/s]

Figure 4.8: Electricity production gains (top) and the corresponding gear ratio vs. wind speed (bottom).

49

Chapter 5: System Performance This portion of the research provides an initial look at the performance of the variable ratio gearbox (VRG) in the wind turbine. The purpose of the study is to determine if the VRG is a practical concept for wind energy conversion. At this point there is no coordination between the brake and VRG, and only the latter is controlled. This approach facilitates the investigation and is more conservative since using the brake further increases the level of production. The dynamic wind turbine model demonstrates the effect the gear ratio has on various parameters in wind energy conversion. By varying the gear ratios, one can determine the effect this has on the power curve. As shown in Fig. 5.1, lower gear ratios correspond to an increase in the amplitude of the power curve. Moreover, with the exception of the low speed region less than 10 m/s, each lower gear ratio curve envelops those that have higher ratios. For example, the gear ratio of 14.71 makes a ‘shell’ that surrounds all the higher gear ratios. As shown in Fig. 5.1 the lowest gear ratio of 14.71 intersects the maximum power line at a lower wind speed than any other curves, which means it will harness the most energy from the wind at this point. Beyond this point of intersection the power transferred through this gear ratio is too great for the generator. Hence, this gear ratio is not practical for wind speeds above that point. In the conventional single speed turbine, only the gear ratio curve that is entirely below the maximum power line provides the continuous operation across the wind spectrum. In the case of stall-regulated turbines, those gear ratios end up being only ideal for a single rated wind speed. On the other hand, if a mechanism is put in place to

Significant portions of this chapter have previously appeared in publications.

50

allow these outer shells to be used below the maximum power line, greater wind capture can occur in these low and high wind regions. 5

3.5

x 10

3

power [W]

2.5 2 1.5

Maximum power GR = 14.71 GR = 17.1 GR = 19.15 GR = 20.54 GR = 21.03 GR = 21.59

1 0.5 0 0

5

10 15 wind speed [m/s]

20

25

Figure 5.1: Power curve changes in response to varied gear ratio. To understand how this ‘shell’ effect occurs requires an explanation of the constitutive parameter behavior across the wind speed continuum. As shown in Fig. 5.2 the lower gear ratios are associated with higher tip-speed ratios. This effect shifts the power coefficient curve to the right for the lower gear ratios. At very low wind speeds around cut-in, the higher speed ratios have a greater power coefficient. At this point there is also very little variation in the torque, and hence the higher power coefficient allows these higher gears to be more effective near the cut-in speed. In the range of 8 to 12 m/s, the power coefficient peaks. Beyond this point, the power coefficient increases as the gear ratio decreases. Moreover, above this region the mechanical torque delivered to the generator becomes more pronounced for the lower gear ratios. In the high-wind region, the power curve normally drops off for fixed speed, stall regulated, wind turbines. 51

However, the increased power coefficient and torque of the VRG’s lower gears allows the power curve to stay close to the maximum level until the cut-off speed is reached.

20

1800

18 16

GR = GR = GR = GR = GR = GR =

0.5

0.4

Power coefficient [-]

Speed ratio [-]

14

14.71 17.1 19.15 20.54 21.03 21.59

12 10 8

14.71 17.1 19.15 20.54 21.03 21.59

GR = GR = GR = GR = GR = GR =

1600 1400

Mechanical torque [N-m]

GR = GR = GR = GR = GR = GR =

0.3

0.2

14.71 17.1 19.15 20.54 21.03 21.59

1200 1000 800 600

6 400

4

0.1 200

2 0

0

5

10 15 20 wind speed [m/s]

25

0

0

5

10 15 20 wind speed [m/s]

25

0

0

5

10 15 20 wind speed [m/s]

25

Figure 5.2: Effect of gear ratio on speed ratio (left), power coefficient (middle), and torque (right). In examining a set of power curves, such as those in Fig. 5.1, the design objective is to look at a point along the wind speed axis and choose the curve associated with the greatest power that does not exceed the generator rating. It is clear that the lower gears will be most productive in the low and high wind regions. Toward the middle of the spectrum, higher gear ratios yield lower efficiencies and provide suitable torque without overheating the electrical generator. For a conventional stall-regulated turbine, the best gear ratio to use is that associated with the power curve that has a peak value consistent with the rating of the electrical generator. For our analysis, this curve serves as the baseline that is compared 52

to the power generated by using multiple gear ratios. This baseline ratio is also the highest gear ratio used in the VRG. Without this curve, a discontinuity occurs across the wind spectrum. Additionally, curves generated with higher gear ratios fall below the baseline curve ‘shell’, are less efficient, and do not make any contribution that could otherwise be made by those with lower ratios and therefore are not used. On the other hand, there is a low limit the gear ratio can reach. Therefore the lowest useful ratio must be identified. It was pointed out in Fig. 5.1 that lower gear ratios intersect the maximum power line at lower wind speeds than the higher gear ratios. Beyond the lowest useful ratio, this dynamic changes and the intersection begins to shift to the right. The simulation shows that the gear ratio selection is always constrained by this low limit. The lowest gear ratio used is associated with the power curve that intersects the maximum power line at the lowest wind speed. 5.1 GEAR RATIO SELECTION METHODOLOGY The VRG enables the fixed-speed wind turbine to realize greater energy production at lower wind speeds. Above the rated wind speed, lower gears are used again to maintain a level of power that would otherwise drop-off. For Region 3' it is important to select gear ratios that allow the gear-changing transition to take place without exceeding the maximum power line. Establishing this threshold prevents large changes in torque that would jolt the drivetrain. It also limits the amount of spacing between the individual ratios in the VRG. This means the design must contain enough gears to make these transitions smoothly while at the same time minimizing the number of gear ratios to avoid complexity. Keeping these constraints in mind, the VRG design can be implemented through an algorithm as described in Fig. 5.3. The sequence starts by considering the baseline 53

curve as previously explained.

This curve peaks at the maximum power line, and

eventually drops below the minimum power line that is established by our threshold to avoid large torque changes. The wind speeds at the two points where this curve intersects the minimum power lines is then determined.

Each of these wind speeds is then

considered separately to construct a power curve that intersects the maximum power line by varying gear ratios. The purpose of this exercise is to determine the gear ratio at the point that provides continuity while the change is made from the higher gear that intersects the minimum power line, to the lower gear that intersects the maximum power line. Ultimately the curve that has the highest gear ratio is kept. This approach captures maximum power from the wind, while also maintaining continuity.

Begin with the power curve for the highest gear ratio

Record the gear ratio for the high wind speed and generate the associated power curve

Record the two wind speed points where this curve intersects minimum power line

Record the gear ratio for the low wind speed and generate the associated power curve

Find gear ratios for both power curves that intersect maximum power line at these wind speeds No Is the high wind speed less than the cut out speed?

No

Yes

No

Does the low wind speed have the highest gear ratio?

Figure 5.3: Design algorithm used to develop VRG. 54

Is the low wind speed ratio less than the lowest useful ratio? Yes

Yes End

In this approach, the side that is not used does not drop down to the minimum power band, and thus experiences less change in torque. Our simulation indicates the right side prevails in determining the power band. In the event the wind speed on the right side is greater than the desired cut-out speed, this side no longer needs to be considered. At this point the left side dictates the gear ratio. The process continues until the lowest useful gear is reached, which then becomes the final gear in the VRG. As an option, the number of gears may be limited and only the higher gears are used. The VRG power curve is ultimately determined, and can then be used to calculate the area under the curve that represents the total energy captured. This area is compared to that of the baseline. This metric can further be split into Regions 2 and 3, which are respectively separated below and above the rated speed of 17.8 m/s. 5.2 ENERGY PRODUCTION RESULTS To quantitatively understand the effect of using multiple gear ratios, the VRG is compared to the conventional fixed-speed gearbox. The 100 kW wind turbine model simulated the energy production of normal operating wind speeds to generate power profiles.. As pointed out in the previous section, the VRG requires a particular number of gears to transition in a continuous manner across the wind speed spectrum, and also to reach the lowest gear possible that yields the greatest wind capture in the low wind region. Figure 5.4 shows the full set of gear ratios with the addition of the lower power boundary that is used by the design algorithm to limit changes in torque during full-load operation. It is possible to operate the VRG without using the full range of ratios, although the chosen set must include the maximum gear ratio and each of the lower consecutive curves up to the desired number of speeds. Three scenarios are presented

55

here – the first two are limited to four and five speeds, while the last case is the maximum number of gears required to reach the lowest useful gear possible. 4

12

x 10

10

power [W]

8 6 4

Maximum power Minimum power GR = 21.59 GR = 21.03 GR = 20.54 GR = 19.15 GR = 17.1 GR = 14.71

2 0 0

5

10 15 wind speed [m/s]

20

25

Figure 5.4: Gear ratios used in design of VRG. The VRG power curve shown in Fig. 5.5 illustrates the effect of the four-speed VRG in boosting aerodynamic efficiency in low and high wind regions. An amount of 10% power variation is allowed in this study. The VRG curve produces a steeper slope on the left hand side when compared to the baseline curve. On the right hand side the power is also greater than the baseline. The highest operating gear is always the baseline curve, while transitions to lower gears are marked by the peaks on the left and right hand sides. The same gears used to boost power on the low wind region are also used in the high-speed region to minimize power. The changes in gear ratio as a function of wind speed for this case study are discussed later in the chapter. With the exception of those gear changes that occur just after cut-in, all transitions take place within the power band

56

that is defined by the maximum and minimum power lines. The changes that occur in the high wind region, span the entire band and limit the step size between the gears. 4

12

x 10

10

power [W]

8

Maximum power Minimum power Single speed 4 speed VRG

6 4 2 0 0

5

10 15 wind speed [m/s]

20

25

Figure 5.5: Power curve for a four-speed VRG. The power curve shown in Fig. 5.6 is for a five-speed VRG and illustrates how more gear ratios increase area under the power curve. In this case, maximum power is first reached around 12.5 m/s versus 13.2 m/s for the four-speed case. There is no change on the right hand side since only three gears, denoted by three ‘peaks’ are needed to take the power curve out to the cut-out speed. The last scenario presented in Fig. 5.7 implements six gears, and is the full set required to reach the lowest useful gear. Figure 5.7 shows that maximum power is first achieved at 12.2 m/s versus 12.5 m/s for the five-speed case. The addition of this final gear does not push the full power envelope out much further than what occurred in the previous step of going from four to five gears. As shown in Fig. 5.7, peaks on the left side of the curve generally get closer together at the outside as shifting to lower gears 57

occurs. This suggests that the greatest gain in area under the curve occurs with the addition of the first few gears, beyond that, adding lower gears makes a less significant contribution to wind energy capture. 4

12

x 10

Maximum power Minimum power Single speed 5 speed VRG

10

power [W]

8 6 4 2 0 0

5

10 15 wind speed [m/s]

20

25

20

25

Figure 5.6: Power curve for a five-speed VRG. 4

12

x 10

10

power [W]

8

Maximum power Minimum power Single speed 6 speed VRG

6 4 2 0 0

5

10 15 wind speed [m/s]

Figure 5.7: Power curve for a six-speed VRG. 58

The results can be further analyzed by quantitatively measuring the power curve area, in terms of the fixed-speed ranges for Regions 2 and 3. The single-speed, stallregulated case has an area of 140.9 Wm/s, and is used as a baseline to determine the amount of increase for each VRG scenario. The results in Table 5.1 suggest that a fourspeed VRG increases overall efficiency by 6.3%, and also provides complete coverage in region 3 through the cut-off speed. As expected, the case with six speeds offers the greatest versatility in varying rotor speed and captures the greatest amount of wind. It should be noted that the percent increase in Region 2 jumps the most between the lower gears, while there is very little change in the higher gears. This is especially true in going from five to six gears, where the increase only expands from 7.8 to 8.0%. The effect per gear added tapers off. It is also observable that there is no increase in the power curve area in Region 3 beyond three speeds. Still, the lower gears are needed to realize the maximum benefit in Region 2. The six speed VRG increases overall energy production the most, and this number of gears can readily be packaged into the VRG design. Total area (x104 Wm/s)

Gears No. 2 3 4 5 6

Low 21.0 20.5 19.1 17.1 14.7

Table 5.1:

High 21.6

Single 140.9

VRG 145.7 147.8 149.6 150.8 151.0

% inc 3.4% 4.9% 6.2% 7.0% 7.1%

Region 2 area (x104 Wm/s) Single VRG % inc 75.6 77.4 2.3% 78.6 3.9% 80.4 6.3% 81.5 7.8% 81.7 8.0%

Region 3 area (x104 Wm/s) Single VRG % inc 65.3 68.4 4.7% 69.3 6.1% 69.3 6.1% 69.3 6.1% 69.3 6.1%

Comparison of VRG to single speed turbine.

The six ratios used in the VRG for the case study are provided in Table 5.2. In this form Gear 1 and 6 are the lowest and highest ratios, respectively. As shown in Fig. 5.8, the selected VRG ratios do increase energy production for the general range of wind 59

conditions. The lower gears shift the point when full power is reached from the original rated speed,

, to the lowest wind speed possible as indicated by

. Moreover,

the combination of ratios maintains a level of power in Region 3' that is within 90 to 100% of full capacity. Gear 1 2 3 4 5 6

Ratio 14.71 17.10 19.15 20.54 21.03 21.59

Table 5.2:

VRG Gear ratios.

Through the application of the appropriate gears, the direct connect system can perform at a mean level of 96.7% of full capacity in Region 3'. The previous study suggests the gear ratio, 21.59, used in the original fixed speed system should remain in the VRG (used as gear 6). This gear also happens to be the highest ratio and the only gear that provides continuous coverage across the entire power curve. As shown in Fig. 5.8, the original fixed-speed gear ratio produces the greatest amount of energy just above cut-in and around the rated speed. Otherwise, there are two sections, one below and one above the rated speed, when the VRG works to enhance power. Figure 5.9 shows that the first section of wind speed to experience improved efficiency occurs from 8.4 to 15.1 m/s. When starting at the low end of this interval and moving to slightly higher wind speeds, the gear ratio decreases, the power coefficient holds steady at about 0.35 rather than dropping off as the fixed speed ratio does. At 9.7 m/s the VRG increases the production by 5% and at 11.4 m/s, the gain in efficiency becomes slightly greater than 20%. Just above these speeds, the lowest gear ratios 60

provide relatively high torque to the generator, boost power by more than 35% over that of the original fixed-speed system, and enable full power to be reached at 12.2 m/s. As wind speeds approach the rated wind speed of the original system, higher gear ratios are applied to reduce the power coefficient and prevent the generator from overheating. Here, the gain in efficiency drops to 10%, around 14.4 m/s, and then approaches zero at 15.1 m/s, where the original fixed-speed gear ratio maintains full power. 4

x 10 10

power [W]

8 6 4

Fixed speed VRG Vr,FS

2

Vr,VRG Full-load boundary

0 0

5

10 15 wind speed [m/s]

20

25

Figure 5.8: Additional area added to power curve by using a VRG. As pointed out in Chapter 2, control of the pitch angle provides little benefit in the low wind region. Recall in Fig. 5.2, that the low pitch angles correspond to the higher speed ratios, which are indicative of the low speed region. Unlike pitch angle control, the VRG is effective in the low wind speed region and actively adjusts gear ratios to boost wind capture. As shown in Fig. 5.9, the gear ratio initially starts out in the highest gear, and provides the greatest torque at this point necessary to initiate partial-load power generation. This is explained by Fig. 5.2, which shows that there is little variation in 61

torque near cut-in, however an increase in efficiency can be found by using the highest gear. When the wind speed increases to higher values, in this case from roughly 8.4 to 11.3 m/s, the lower gear ratios provide a considerable gain in the torque. The gears then shift to these lower ratios in this range. The lowest gear possible is followed until around 12.2 m/s, where the power curve reaches full-load operation, and at this point the switch is made to higher gear ratios that have lower torque and a slightly lower power coefficient to maintain the power at or below the maximum generator operating point. The gear ratio climbs at this point to utilize the portions of the power curve (recall from Fig. 5.4) that are within the maximum and minimum power boundaries. This trend continues, with the highest gear ratio passing through the rated speed of 17.8 m/s. As the wind speed increases through region 3, the lower gear ratios come back at around 22 m/s. As the cut-off speed is approached, the power coefficient reduces the torque such that the lower gears will not overheat the generator and can once again provide the greatest amount of power. Through this high wind region, the VRG continues to shift gears out to lower speeds so that the system can follow the portion of the power curve that is within the maximum and minimum power band. Throughout Region 3, the overall torque stays within 10% of the generator rated power.

62

4

power [W]

15

x 10

10 5 0

0

Gear 1 Gear 2 Gear 3 Gear 4 Gear 5 Gear 6 5

10

15

20

25

power coefficient [-]

0.4 Fixed speed VRG 0.2

0

0

5

10

15

20

25

0

5

10 15 wind speed [m/s]

20

25

gear ratio [-]

25 20 15 10

Figure 5.9: Amount of power (top), power coefficient (middle), and gear applied in (bottom) relation to wind speed for a VRG-enabled wind turbine. 5.3 ENERGY PRODUCTION GAIN IN VARIED WIND CONDITIONS The ability of the VRG to increase the production for a particular turbine location depends on the distribution of wind speed into the VRG-enhanced sections. Histograms of four sites are presented in Fig. 5.10 to demonstrate the frequency of such occurrences in these sections. Here, the bins are allocated into 0.25 m/s intervals, and the quantities 63

represent the number of ten-minute recordings. The four profiles were selected to show the diverse range of wind behavior that occurs at various sites and presented in the order of increasing power density. The general trend here is that the distribution curve is flattened and stretched into the high wind speed area as power density increases. Only the wind that occurs within the operational limits is capable of generating power. For site 3412 (set no. 2), from the NREL website, nearly half of the wind speed is below cut-in and hence, a turbine at this site will operate about 54% of the time. Also, the low wind speed means that a significant portion of the wind occurs before the 8.4 m/s mark. Still, about 18% of the time, wind does occur in the first VRG enhanced section. For site 22814 (set no. 7), which is located near the high end of class 3, the level of utilization is 67% as a higher volume of wind occurs above the cut-in. This shift also means that 29% of the wind is between 8.4 and 15.1 m/s, and the VRG may provide a greater benefit to this profile than the previous one. For the third site, 13731 (set. no. 13), the level of utilization is 72% as the wind speed continues to shift away from the area below the cut-in. In comparison to the previous profiles, this curve is stretched further to the right with 40% of the occurrences in the first region where the VRG can provide additional efficiency. The distribution for site 12796 (set no. 19) is much broader than all of the previous, with wind speed extending up to 80 m/s (not shown). Since a significant portion of the wind exceeds cut-off speed, the level of utilization drops to 60% for this class. The broad distribution means that 23% of the winds occur in the first VRG enhanced region with 3.3% occurring in the second. With a larger amount of wind occurring around the rated speed of the original system, much of the energy will be produced using the highest gear ratio.

64

occurences [-]

5000 Set no. 2 Operating limits Increased efficiency

4000 3000 2000 1000 0

0

5

10

15

20

25

30

occurences [-]

5000 Set no. 7 Operating limits Increased efficiency

4000 3000 2000 1000 0

0

5

10

15

20

25

30

occurences [-]

5000 Set no. 12 Operating limits Increased efficiency

4000 3000 2000 1000 0

0

5

10

15

20

25

30

occurences [-]

5000 Set no. 19 Operating limits Increased efficiency

4000 3000 2000 1000 0

0

5

10

15 wind speed [m/s]

20

25

30

Figure 5.10: Histograms showing distribution of wind of select wind profiles.

65

5.4 REAL TIME EFFICIENCY GAIN BY USING VRG Previous sections reveal the potential efficiency increase by using the VRG. In this section, the real-time efficiency gain will be discussed. A model of a 100 kW wind turbine was used to simulate the performance of the VRG-enabled system on a variety of wind profiles obtained from NREL. The VRG enhances the efficiency of the wind turbine by applying more efficient gear ratios at wind speeds where the original fixedspeed system lacks efficiency. During partial load operation the segment of wind that experiences increased efficiency are those in the range of 8.4 m/s up to 15.1 m/s, and during full load operation the boost is realized above 21.9 m/s. Simulation results for a 24 hour period, selected from site 31266 (set no. 9) are used to illustrate the gain in performance for a typical wind profile as shown in Figure 5.11. At the start of the period the system experiences partial load operation and the gain in efficiency fluctuates between 20 and 25%. A drop in power occurs at about 2.5 hours, where the wind speed drops close to 8.4 m/s, and there is no appreciable gain. At about 5 hours the wind speed increases from 10.5 m/s to a maximum value of 24 m/s before it drops off at about the 11 hour point. During this time the original fixed speed production level drops off when the wind speeds go well above the rated speed, but the VRG maintains a power at 90% or more of capacity, with gains in efficiency reaching 25% at some points. The VRG makes less of a contribution between 11.5 to 13.6 hours as the wind speed drops down near cut-in where the fixed speed gear ratio is sufficient. At 14.2 hours the system is able to maintain a production level between 90% of full capacity or better for the remainder of the 24 hour period. Here, the VRG fills in the low spots on the power profile, with exception to a gap from the 18th hour to the 23rd hour, where the wind speed stays near the rated speed of the fixed speed system.

66

A gain in efficiency greater than 35% is realized when using the lowest gear ratio of 14.71, instead of that used in the original configuration. An instance of this occurs in the section of the power curve at the point where full power is first achieved. Wind turbine sites that frequently have wind near this speed have the greatest benefit.

wind speed [m/s]

25 20 15 10 5 0

0

5

10

15

20

4

x 10

power [W]

10

5 Fixed speed VRG 0 0

5

10

15

20

time [hrs]

Figure 5.11: Plot showing wind speed (top), and corresponding production, (bottom), occurring in 24-hour period for site 31266 (set no. 9). Figure 5.12 shows a 24-hour period from site 31266 that often runs near full capacity by using the VRG. In this example 2070 kWh of energy is produced, versus the 1642 kWh by the standard fixed speed, and an overall gain of 26% in production is incurred. Hence, it would be beneficial to implement the VRG where there is a high penetration of winds around this speed. Table 5.3 summarizes the efficiency gain by using VRG for the typical wind site for all seven classes. Although the wind in the 67

overall profile frequently falls below this speed, The VRG provides a significant gain in efficiency of 9.50% over the three year period.

wind speed [m/s]

25 20 15 10 5 0

0

5

10

15

20

4

x 10

power [W]

10

5 Fixed speed VRG 0 0

5

10

15

20

time [hrs]

Figure 5.12: Plot showing wind speed (top), and corresponding production, (bottom), occurring in 24-hour period for site 31266 (set no. 9). In addition to the amount of energy produced, the overall power coefficient, provided for each profile in Table 5.3

, is

This parameter represents the aerodynamic

efficiency of the turbine, and is given for both the original and VRG-enabled systems. In both cases, the coefficient remains fairly consistent through the first three wind classes (for both the VRG-enabled and fixed speed systems) but begins to decrease in the fourth class around set no. 9. This is because the power coefficient is being used to boost efficiency in these lower wind classes, while limiting it as wind speed increases in the upper wind class. In general, the VRG boosts the power coefficient at an average value 68

of .013, and this is evident in the increased power production. The average increase in efficiency for all of the wind profiles in the group was 8.55%. In examining this through all seven classes of wind, it appears as though the wind profiles just above the middle from a power density of 400 to 700 W/m2, are generally well above this average. It is in this range that the greatest increases in efficiency are experienced at both profile 12 and 15, with 10.73% and 10.13% respectively. The lowest levels appear to occur at the ends of the spectrum, with set no. 2 and set no. 19 only having an increase of 7.47% and 7.18% respectively.

Nevertheless, the maximum and minimum gains do not stray

significantly from the average of 8.55%. Wind class 1 2

3

4

5

6

7

Set No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Table 5.3:

Site 12951 3412 11719 28570 6474 6684 22814 7302 31266 2002 174 3134 13731 21291 22818 16577 7858 19520 12796 11721

[kWh] 327,112 396,278 427,283 441,287 574,259 587,504 609,297 690,383 697,580 613,499 657,448 839,085 716,076 751,114 900,408 945,453 789,764 1,022,070 872,956 1,029,166

[kWh] 352,817 425,897 465,004 481,536 623,517 637,173 665,602 754,640 763,867 665,265 711,623 929,118 775,911 814,355 991,639 1,024,867 848,904 1,101,983 935,641 1,106,576

Increase [%] 7.86 7.47 8.83 9.12 8.58 8.45 9.24 9.31 9.50 8.44 8.24 10.73 8.36 8.42 10.13 8.40 7.49 7.82 7.18 7.52

[-] 0.294 0.289 0.289 0.287 0.299 0.293 0.289 0.293 0.284 0.275 0.274 0.282 0.269 0.270 0.275 0.268 0.243 0.253 0.233 0.231

[-] 0.304 0.297 0.300 0.300 0.310 0.304 0.302 0.307 0.298 0.290 0.286 0.300 0.283 0.283 0.293 0.283 0.256 0.266 0.245 0.245

Performance results for cross section of class 1 - 7 wind profiles. 69

12

percent increase [%]

10 8 Region 2 Region 3 Overall

6 4 2 0

0

200

400

600

800

1000

1200

1400

2

power density [W/m ]

Figure 5.13: Trends in the increase in efficiency for a VRG-enabled wind turbine. Figure 5.13 illustrates how the VRG increases production in relationship to each profile's power density. The second-order, fitted curve suggests that the greatest gain in overall production will occur for turbines operating in class 4 through 6 conditions, with the amount of increase being smallest at the extremes. This trend is predictable when considering the histograms (Figure 3), which demonstrated how winds in the middle classes would have the highest level of penetration into the VRG enabled regions, while those profiles nearest the end would have fewer opportunities for the VRG to operate. The 'Region 2' curve shows how the VRG improved the production for all of the wind turbines through all the classes of wind. This curve also has the greatest impact on the overall increase considering that all of the wind turbines in the study operate at partial load during the majority of the time. Winds above the rated speed occur much less frequently, but can realize greater efficiency when they fall into the second VRG enabled

70

region (Figure 3).

The increase through this mode of production, indicated by the

'Region 3' curve, provides the greatest benefit to wind profiles in class 6 and 7. The results in this Chapter indicate that a VRG can increase the energy production of a wind turbine. Moreover, the design methodology provides good ratios to simulate the VRG enabled system. The VRG allows the rotor speed to vary in relation to the fixed-speed of the power generator. The ability to vary the rotor speed at discrete gear ratios coupled with control allows for the application of a power coefficient that can boost efficiency in low wind regions. The data suggests the VRG provides a substantial benefit in the low speed region where variable speed control is currently the only available option. A reliable VRG can be developed similar to the manual automotive transmission, which is a reliable low cost device. For small wind turbines this will be a practical alternative to the costly power electronics required for variable speed operation. The VRG also limits power in the high wind region, thus avoiding the need for pitch control.

71

Chapter 6: Optimal Control Design A variable ratio gearbox (VRG) enables discrete variable speed operation for a distributed wind turbine with a squirrel cage induction generator (SCIG) and fixed rotor blades. Through this capability the rotor speed can now be adjusted in relation to the wind speed to extract greater energy from the wind. This results in higher energy production as shown in the previous chapter. However, greater power can be extracted from the wind by applying dual control to the VRG and mechanical brake. In Region 3' of Fig. 6.1, the VRG gear ratio can be set so the turbine produces torque slightly above that required by the generator, while the mechanical brake dissipates some of this torque so that the generator only receives the amount needed for full-load production. However, it is necessary to recognize the finite life of the brake pad. A decision-making algorithm is presented in this Chapter to find the optimal trade-off between using the brake to maximize the wind energy capture, and maintaining lining until scheduled pad replacement occurs. This is achieved through an optimal control design that determines the VRG gear ratios, when gear-shifting occurs, and periods of brake use. To account for different wind conditions, which affect the level of brake utilization and power production, the National Renewable Energy Laboratories (NREL) wind profiles are used in this study [138]. Control objectives of wind turbine systems often aim at maximizing power production and work through the power coefficient [15,111]. In the partial load region, the power coefficient is maximized by adjusting the rotor speed [111]. To limit power in the full load region, the rotor blades are rotated and used as brakes [19]. Nonlinear

Significant portions of this chapter have previously appeared in publications.

72

optimization strategies, such as maximum power point tracking [112–114] and variable structure control [115–117] are commonly used to design the wind turbine controller, based primarily on electrical power production [118]. The control goal for the VRGenabled wind turbine is also to maximize power production. When operating in Region 2', it is desirable to apply the gear ratio that provides the greatest amount of torque, and thus maximum power. In Region 3', the system will try to achieve full power using the brake, or simply maintain the power production above 90% using a different gear ratio when it is not practical to use the brake. 4

x 10

electric power [W]

10

Fixed speed VRG w/control

8 6 Region 2'

Region 3'

4 2 0 0

5

10 15 wind speed [m/s]

20

25

Figure 6.1: Power curve for VRG when used with brake to limit torque in Region 3'. Since the history of the gear ratio selection and level of brake use influence energy capture and brake pad wear, dynamic programming (DP) is used to optimize the process over a known time-horizon. The dynamic programming process enables the development of a control algorithm derived for time-based disturbance data. The DP

73

algorithm chooses the VRG ratios and defines how each ratio is used with the brake in various operating conditions. Simulate GR(j) thru vw(k)

Compute Pg,

.

Wpad,

Tm, , qin, pdi

No

Tm ≤ Tgmax

Apply Brake

Yes

Tgmin ≤ Tm

No

.Pg(j) = 0

Wpad(j) = ∞

No

qin ≤ qoutmax &&

pdi ≤ pmax

Yes

.

Pg(j), Wpad(j) to DMM

Yes

End

Figure 6.2: Modified rules used with program optimization. 6.1 MODEL RECONFIGURATION FOR OPTIMIZATION The dynamic wind turbine model simulates performance data for this portion of the research. For the optimization problem, the rule-based controller is replaced with a simple set of rules, shown in Fig. 6.2, so that gear ratios may be evaluated individually at any wind speed. In this process, the decision-making module communicates with the simulation model to determine the amount of power produced and brake pad consumed, for each gear ratio over a section of the wind profile. For simulation purposes, the wind input is discretized into 0.01 m/s increments to match the resolution of the NREL wind 74

data. A gear ratio can be used if it produces mechanical torque within the acceptable operating range for the electrical generator. The mechanical brake may be applied to limit the torque, as long as this does not lead to excessive heat or pad pressure. For gear ratios that do not work, the power and wear parameters are zero and infinity, respectively. The brake model in this problem also considers the finite life of the brake pad. Hence, the liner's rate of wear must also be considered in the model to estimate the pad life. At the end of each simulation period, the decision-making module uses a cost function to quantify the performance of each operating scenario. 6.2 BRAKE PAD LIFE Overuse of the friction brake will necessitate frequent pad replacement and can even cause system breakdowns. To avoid this, dynamic programming is used to establish the operating technique based on the unique wind conditions at a particular site. For example, a wind turbine operating in wind class 7, which has the highest average wind speed, will experience more braking time in Region 3' than a system operating in class 1. The brake use has to be optimized for the anticipated wind speed over its desired service life. The life of the pad depends on its thickness, which is determined to be around 12.7 mm, and the rate of wear, which is assumed to be uniform [132]. Using the inner edge as a reference point, the liner thickness is removed at a rate,

, which is a function of the

sliding speed, applied load, and material characteristics [141]. w

The total wear,

(6.1)

, is calculated in terms of the total wind profile time, N,

using (6.2) 75

6.3 OPTIMIZATION PROBLEM FORMULATION The goal of the optimization problem is to find the combination of gears that provides the most electrical energy generation without excessive brake maintenance for each of the wind site profiles.

Dynamic programming is applied to obtain the

optimization algorithm. In general terms, the model of the wind turbine system can be expressed in the discrete-time format as (6.3) In this arrangement

are taken from the dynamic models described in Chapter

3, while control is applied through the variable,

, that uses the gear ratio and brake

force as inputs. q q q q

Dynamic programming seeks to find the appropriate control inputs for will minimize the cost function, , over a finite time period of

that

. (6.4)

The instantaneous cost function, , can further be replaced by the performance metrics for generated power,

, and the rate of brake pad wear, (6.5)

Here generator, and

is the maximum amount of power that can be produced by the is the greatest possible rate of wear and occurs when the maximum

allowable brake pressure is applied. The weight function, , can be varied between 0 to 1 to favor brake pad conservation on the low end and increased energy production at the 76

high end. The instantaneous cost, , is also between 0 and 1, and decreases with a rise in power or a reduction in the rate of wear. Operating constraints are also used in conjunction with the selection process to eliminate gear combinations that are not practical. As a first requirement, a combination set must have ratios that produce enough torque to operate the generator when the wind speed is above the cut-in speed. (6.6) In Region 3' it is desirable to maintain a minimum power of at least 90% of the full load power,

. (6.7)

In this region the gear ratio can be used in conjunction with the brake to limit mechanical torque in cases when full power is reached.

During normal operation,

between cut-in and cut-off speeds, the VRG combination must contain at least one ratio that allows the generator to operate without producing more torque than the generator can accept. (6.8) The amount of brake torque is limited by the heat produced at the sliding surface and pressure applied to the brake pad. To avoid reduced braking capability due to fade [133], the system must be able to dissipate heat more readily than it is being added, such that (6.9)

77

To avoid failure to the pad [132], the pressure,

, at the point on the inner radius

(where the pad applies the greatest amount of pressure) is constrained by, (6.10) In addition, the brake liner must not be fully consumed before the scheduled maintenance, which occurs every three years in this study. The reduction in the liner thickness,

, is determined by Eq. (6.2) and must be less than the original thickness,

, for the combination to be valid. (6.11) 6.4 CONTROL DESIGN There are three decision-making modules, shown in Fig. 6.3, that interact with the 100 kW wind turbine model to select the gear ratios and define the control technique for a given wind site. The first step, or module, explores the design space to build a set of gear ratios that will be evaluated. subsequent steps.

The optimization problem is then performed in the

Preliminary programming eliminates gear ratios that violate the

operating constraints.

Finally, dynamic programming selects the VRG ratio that

produces the most power without consuming excessive brake pad for a specific wind profile.

To reduce computational expense, the latter two modules eliminate a

combination from any further analysis once it is deemed invalid. The steps presented here allowed the consideration of 131 million possible gear combinations and take roughly twelve hours to be processed* in MATLAB.

*Performed on a MS Windows-based 64-bit system with an Intel Core i7-950 processor and 12.0 GB of RAM.

78

Control optimization problem

All possible combinations

Gear design space

vw,gen, GR

Pg

Feasible combinations

Preliminary programming

vw,gen, GR

Pg

Dynamic programming

vw,site, GR

Optimal combination Weight coefficient Total energy Total pad life

P . g, Wpad

Interface to simulation model

Figure 6.3: Wind turbine decision-making model. 6.4.1 Gear Ratio Design Space The design space includes a range of useful gear ratios bound at each end by a lowest and highest ratio. To define this range of useful gears, the model uses an iterative process evaluating the performance of individual gear ratios in terms of the power curve. All of these ratios can achieve full power production at a wind speed that is 25 m/s or less, where normal operation occurs. However, at some wind speeds these ratios produce more torque than the generator can accept. Only the highest gear ratio is continuous between cut-in to cut-off speed. This is also the same ratio used in the fixed speed system and has the same power curve [104]. A decrease from the highest ratio shifts the power curve peak to the left (towards lower speeds) and up above the maximum power level, where it is non-continuous. The lowest useful gear ratio is the one that can achieve full load operation at the lowest wind speed. Gear ratios lower than this shift the power curve back to the right. They have lower efficiency than the ratios already included, and therefore, are excluded from the useful set. On the other hand, ratios greater than the highest useful one do not achieve full load production. Moreover, these high ratios always produce less power than the highest useful ratio, and provide no benefit.

79

Using the lowest and highest ratios as endpoints, the useful range is discretized into ratios that can be evaluated in the subsequent modules. We chose to have 70 discrete ratios in the set,

, and this creates a level of resolution similar to that obtained when

selecting the actual gear teeth at the design stage. Previous experience demonstrated that the VRG requires six gear ratios to maintain a power level of at least 90% in Region 3' [104]. This arrangement produced more than 131 million possibilities as indicated by Table 6.1. Preliminary trial runs, demonstrated that the highest gear ratio is always part of the optimal solution. This became a mandatory ratio in each set, which reduced the number of possible combinations, in Combination 1 2

131,115,985 Table 6.1:

, to just over 12 million.

Gear Gear 1 2 14.71 14.81 14.71 14.81 14.71 14.81

Gear 3 14.91 14.91 14.91

Gear Gear Gear 4 5 6 15.01 15.11 15.21 15.01 15.11 15.31 15.01 15.11 15.41

21.09 21.19

21.29

21.39 21.49 21.59

Gear combinations generated.

6.4.2 Preliminary Programming Combinations that are retained following preliminary programming, have at least one gear ratio that can provide the proper amount of mechanical torque at each of the wind speed points that is greater than the cut-in speed. For any given combination, , the cut-in speed is the lowest possible cut-in speed among the six ratios in that set. As illustrated in Fig. 6.4, for the wind speed on interval, performance of all 70 ratios in

, the module evaluates the

. If the wind speed is below the cut-in speed, no

action is taken. Otherwise, the performance model assigns zero power to any ratio, that violates one of the constraints in equations (6.6) through (6.10). The program then 80

searches through these results and finds the gear ratio that produces the maximum amount of power for that combination. Combinations that have zero power for all of the gear ratios will also return a maximum power of zero. The code discards any invalid combinations before moving to the combinations in

interval, to reduce,

, the number of

for subsequent steps. Eliminate invalid combinations, k

Get

Pg(GRset,vw(k)) n > ncomb

No

vw(k) ≥ vwcut-in(n)

No

Yes Yes end

Compute

max(Pg(GRcomb(n) Ç GRset))

Eliminate

n++

No

Gcomb(n)

Pg(n) > 0 Yes

Figure 6.4: Decision-making structure for preliminary programming. A power curve can be constructed for each combination from the results of the above programming steps. The VRG curve shown at the top of Fig. 6.5 is an example of a valid combination, while the bottom curve violates the requirement that power must be maintained above 90% of full power when operating in Region 3'. Without preliminary programming, the same invalid combinations would be eliminated every time the 81

dynamic optimization is executed on one of the wind profiles, and create unnecessary computational expense in the process. For the 100 kW system, the technique reduces the number of combinations from 12 million to 575,413 for the subsequent dynamic programming process. 4

x 10 10

power [W]

8 6 4 2

VRG profile Power constraint

0 0

5 4

x 10

10 15 wind speed [m/s]

20

25

10

power [W]

8 6 4 2 0 0

VRG profile Power constraint 5

10 15 wind speed [m/s]

20

25

Figure 6.5: Example of a valid (top) and an invalid (bottom) VRG combination.

82

6.4.3 Dynamic Programming The final decision-making module verifies the total wear constraint described by Eq. (6.11), finds the optimal gear combination, and defines how the VRG and brake will be controlled at each unique turbine site. 6.4.3.1 NREL Wind Data Sets This dynamic programming analysis was performed on wind data obtained from the NREL website for 20 turbine sites [138]. Each wind profile consists of 157,680 points with a 10-minute resolution that spans the years 2004 through 2006. Wind speed between these points is predicted using a power spectral density function [139]. For each site, the wind speed was recorded at a height of 100 m. The power law [139,142] was used here to predict the wind speed at 50 m, which is considered to be a realistic hub height for the 100 kW turbine in the case study. 6.4.3.2 Programming Technique The dynamic programming module works through the wind profile one step at a time, accessing the simulation model to determine the performance of each ratio in As shown in Fig. 6.6, the power and brake wear metrics are then used in equation (6.4), to find the minimum cost,

of VRG combination , for each of

weights. Using the

best gear ratio in each case, the cost and performance metrics are then added to the cumulative variables for cost, , total energy production,

, and total wear

, for

each scenario. Through experimentation, it was determined that weight needed to vary between 0.9 and 1, and this range was discretized into .005 intervals. Values below 0.9, underutilized the brake such that the liner was never fully consumed for any of the wind profiles. This was because many operating intervals, , were in Region 2', or in 3' at a point where the brake was not even used. In such cases, the braking term in the cost 83

function was always zero, giving it great influence in the cost function. If the time horizon is longer, or if a brake pad with a thinner liner is used, then the value of need to be reduced. After all weights,

would

, are considered, the program will try to find at

least one of these weights for the VRG combination that still has liner material remaining. If the above attempt is not successful, combination

is eliminated and the

program then moves on to evaluate the next one. It is unnecessary to check the other constraints as preliminary programming insures that each of these combinations will always have at least one valid gear. Compute cost function, k

Get

Pg(GRset,vw(k)) Wpad(GRset,vw(k)) n > ncomb

No

vw(k) ≥ vwcut-in(n)

No

Yes Yes end Find

Yes

No

m>m

min(L(GRcomb(n), (m)))

Compute

J(n, m) = J + L E(n, m) = E + Pg×.Dt Wpad(n, m) = Wpad + Wpad× Dt

Eliminate

n++

m++

Yes

Gcomb(n)

Wpad > Woriq for all m

No

Figure 6.6: Dynamic programming algorithm for computing the cost at step . 84

energy produced [kWh]

5

x 10

Optimal point

8.24 8.22 8.2 2.11

2.115

2.12

2.125

2.13

2.135 5

amount of wear [mm]

x 10

12 10 8 2.11

2.115

2.12

2.125

2.13

2.135 5

x 10 4

cost function [-]

x 10 8

7.98

7.96 2.11

2.115

2.12 2.125 gear combination [-]

2.13

2.135 5

x 10

Figure 6.7: Plot showing total energy produced (top), brake wear (middle), and cost (bottom) for site 16577 (set no. 16) for weight, = 0.960. Once the program has searched the entire wind profile, the metrics for the total cost, energy, and wear are used to choose the optimal gear combination, and associated weight value that prescribes the level of brake utilization. The relationship among these variables is described in Fig. 6.7. As illustrated, the lowest cost corresponds to the combination that produced a high level of energy, while the brake pad is utilized to the point it is almost completely consumed. As shown in Fig. 6.8, the selection process first 85

finds the best VRG combination corresponding to each

weight and then finds the

combination that produces the most energy. There is no need to check the brake wear at this point, as combinations associated with excessive wear have already been removed. It is impossible to simply look at the combinations in

, and choose the one

with the lowest cost as this does not necessarily correspond to the case that produces the maximum power, as demonstrated in Fig. 6.9. This is because the cost function is only useful in finding the best solution within a given instance,

, of weight. The curvature of

the surface is attributed to the order in which the combinations were systematically created and placed in Table 6.1. The uneven edge in both plots, near the higher

values,

is due to voids where gear combinations are discarded because of excessive use of the brake.

Find optimal combination for profile

No

m++

Yes

m>m

Find

max(Pg(GR

))

Keep

Find

min(J(ncomb, (m)))

GRp,

p

end

Figure 6.8: Technique for finding optimal VRG combination, and weight variable, , that will be used to operate the wind turbine.

86

Figure 6.9: Surface plot showing total cost (top) and total energy (bottom) as a function of for each of the 575,413 gear combination for site 16577 (set no. 16). 87

This research establishes a framework to apply dynamic programming techniques to maximize the electrical power production and minimize the brake wear for a wind turbine equipped with a VRG and mechanical brake. The six gear ratios used in the VRG have been carefully selected using dynamic programming, which relied on performance data generated by a 100 kW fixed speed model based on a wind speed profile. A cross section of wind profiles, collected at various turbines sites, was obtained from the National Renewable Energy Laboratory, and used as the inputs into the wind turbine. 6.5 CASE STUDY BASED ON A REAL WIND SITE The profile from site 21291 (set no. 14) is used as an example to demonstrate the optimization process. The results of the analysis are shown in Fig. 6.10. The average wind speed, 7.75 m/s, for this profile designates the site into wind class 6; however, the wide variation across the profile is indicative of what happens in all classes of winds. The lower plot shows how the additional power steadily increases over that produced by the fixed-speed model. The percent of increase incurred through the VRG is 9.82%, which is close to the mean increase for all of the wind profiles. In the lower plot, the brake wear is superimposed over the power and illustrates the trade-off between energy production and brake wear. The dynamic programming identified the appropriate gear ratios to produce the maximum energy without wearing the brake pads out before the end of the desired three-year period. The weight of 0.980 produced the most energy of all the feasible solutions and used the brake pad to a level of 99.8%. In this instance the weight being less than 1.0, meant that a gear ratio with higher torque could have been used at more wind speeds in this profile. However, an alternative gear that produced less power was used instead to prevent wearing down the pad before it could be replaced at the end of its desired three-year life. 88

wind speed [m/s]

40 30 20 10 0

0

20

40

60

80

100

120

140

160

140

14 12 10 8 6 4 2 0 160

8 6 4 2 0

0

20

40

60

80 100 time [weeks]

120

pad thickness [mm]

added energy [kWh]

4

x 10

Figure 6.10: Plot showing wind speed (top), additional energy produced by the VRG (bottom, left axis) and cumulative brake pad used (bottom, right axis) over a three year period, for site 21291 (set no. 14), programmed with a weight coefficient, = 0.980. The top plot of Fig. 6.11 shows the power produced by the wind turbine. The area underneath the selected profile is much larger than that under the highest gear ratio (GR = 21.59) curve, which is also the power curve for the original fixed speed system. The additional area is indicative of how the VRG provides better wind capture across the wind spectrum. A higher weightabld 7,2 t value, , would have produced a power curve that is flatter at the top, but the system would require more frequent braking. To acquire more energy, the brake can be used more often if multiple disk brakes are implemented, thicker

89

4

electric power [W]

x 10 10

Selected prof ile GR = 14.71 GR = 14.91 GR = 15.11

5

GR = 15.31 GR = 15.51 GR = 15.71

0 0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10 15 wind speed [m/s]

20

25

gear ratio [-]

22 20 18 16

8000 6000 4000 2000 0

turbine speed [rad/s]

power dissipated [W]

14

15 10 5 0

Figure 6.11: Results for site 21291 (set no. 14) with weight, = 0.980, showing VRG power curve (top), gear data used for control (upper-middle), power dissipated (lower-middle), and turbine speed (bottom). 90

pads are used, or if the amount of time between brake-pad maintenance is decreased. The second plot of Fig. 6.11 is based on the dynamic programming results corresponding to the overall optimal weight. The data here establishes the optimal gear ratio for a given wind speed, and can be used in the rule-based controller, presented in Chapter 3, to operate the VRG-enabled turbine. The amount of power that is dissipated by the braking process is shown in the third plot. Intervals where the brake is used, and full power is maintained, align with the flat portions in the top curve. The bottom plot demonstrates how the turbine rotor speed varies, to produce the optimal amount of torque delivery. The turbine speed is set in relation to wind speed, and is used to control the power coefficient, and hence, ultimately the generated power. 6.6 WIND ENERGY INCREASE ACROSS SEVERAL WIND CLASSES Ultimately, 20 wind profiles were analyzed to determine the performance of the VRG across seven wind classifications. As shown in Table 6.2, an increase in wind class corresponds to greater power density and energy production. The increase in energy produced, setup,

, by the VRG enabled turbine over that produced with the original turbine , has a mean value of 9.80% for all the sets. This is significant considering that

5-7% of energy variation is equivalent to 1 billion dollars of revenue if 20% of total energy supply comes from the wind [2]. There is no apparent trend in the percent of increase among sites within one wind class. Site 3412 (set no. 2) experiences the lowest gain of 8.27%, and site 3134 (set no. 12) has the greatest increase at 12.21%. Thus, the VRG can likely provide a consistent gain in energy production irrespective of location. The weight coefficient, , is at the maximum value of 1.000 for classes 1 through 4. This is because these wind classes generally consist of lower wind speeds that promote operation in the partial load region where braking is not required. The column 91

on the far right shows that these pads will last beyond the three-year requirement. In the case of classes 1 and 2 winds, the brake can last up to 10 years and more without maintenance. The weight coefficient value approaches 0.945 towards the high end of class 7 indicating the controller has sacrificed power to prolong brake life. In such cases, the pad life extends slightly past the three years. This high level of pad utilization indicates the strength of the dynamic programming strategy to push maximum power production and leave just enough brake pad to avoid unscheduled maintenance. Wind class [50m] 1 2

3

4

5

6

7

Set No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Table 6.2:

Site

12951 3412 11719 28570 6474 6684 22814 7302 31266 2002 174 3134 13731 21291 22818 16577 7858 19520 12796 11721

Power density [W/m2] [m/s] 192.9 4.97 244.0 5.76 259.9 5.71 289.4 5.47 329.9 6.84 363.0 6.91 391.8 6.95 430.8 7.34 468.6 7.30 487.4 6.29 523.0 7.13 555.2 7.87 580.3 7.34 649.9 7.75 690.0 8.26 747.6 8.59 918.2 9.21 1004.3 9.38 1082.3 10.16 1242.2 10.05

[kWh] 327,112 396,278 427,283 441,287 574,259 587,504 609,297 690,383 697,580 613,499 657,448 839,085 716,076 751,114 900,408 945,453 789,764 1,022,070 872,956 1,029,166

[kWh] 355,408 429,035 469,436 486,397 628,048 642,803 672,639 761,966 772,913 673,735 720,217 941,496 786,260 824,890 1,004,355 1,037,586 861,207 1,116,221 949,994 1,122,559

Increase

Weight

[%] 8.65 8.27 9.87 10.22 9.37 9.41 10.40 10.37 10.80 9.82 9.55 12.21 9.80 9.82 11.54 9.74 9.05 9.21 8.83 9.07

[-] 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.965 0.980 0.980 0.960 0.960 0.965 0.950 0.960 0.945

Simulation results for cross section of class 1 - 7 wind profiles. 92

Pad life [years] 13.630 10.689 7.558 6.692 7.894 6.035 4.840 4.641 3.592 3.465 3.437 3.028 3.000 3.034 3.007 3.009 3.009 3.011 3.001 3.004

Figure 6.12 shows the analysis results for each site variant of the VRG-enabled system. In wind classes 1 through 6, only small differences exist among the ratios selected for the VRG. In class 7, the two profiles with the highest power density indicate an increase in the ratios for gears 1 through 3. This rise reflects how the optimization technique chooses gear ratios with a bias towards stronger winds, which occur more frequently in these cases. Irrespective of wind class, the difference between the higher gear ratios is always less. This is because the sections of the VRG power curve that are constructed from gears 4 and 5 have a relatively low slope in Region 3'. Hence, these can be used over longer ranges of wind speed, and only require small changes in the gear ratio to still maintain the power level required in Region 3'.

22 21

gear ratio [-]

20

Gear 1 Gear 2 Gear 3 Gear 4 Gear 5 Gear 6

19 18 17 16 15

1 2 3 4 5 6 7 8 9 10 11 12 13 1415 16 17 18 19 20 set no. [-]

Figure 6.12: Optimal VRG ratios for each set of wind data. The similarity of gear ratios suggests that a standard VRG design may be used with wind turbines that are operated in classes 1 through 6. Class 7 wind profiles require deviations from this design, mostly in the cases of gears 1, 2, and 3. One explanation for 93

the similarity in the ratios is how these selections always maximize the power capture during partial load operation and are capable of pushing the power curve up across a broad range of wind speed in the full load region as shown in Fig. 6.12.

94

Chapter 7: Gear Train Design and Analysis The control design presented in the previous chapter quantified the ratios for the variable ratio gearbox (VRG). The process focused on optimal control using energy production and brake wear as performance metrics. In this chapter, the goal is to find a combination of commercially available gears that emulate the values and performance of the gear ratios selected by the optimal control design. Accordingly, this chapter presents a methodology for selecting the gears to minimize (1) overall gear mass and (2) the amount of energy lost by deviating from the control design ratios. The process in this chapter aids the mechanical design. It is also an initial step towards a future goal of the VRG research to develop design optimization software. The gear selection technique is developed in the MATLAB environment, and demonstrated through the 100 kW simulation model. The technique will be used to find gear designs that meet the demands of the diverse wind conditions of all of the sites in the case study. The VRG collaborates with the main gearbox to reduce the rotor torque and increase the speed, as required by the induction generator. For a small wind turbine, a parallel shaft gearbox is used [74,143] to produce the majority of this increase. The controller works through the VRG to vary the gear ratio and thus rotor speed. Figure 7.1 illustrates the design concept for a six-speed VRG, which is similar to the manual automotive transmission. [144–146]. The gear train consists of two gearsets, each consisting of one pinion and gear, where energy transfers from the former to the latter through meshed teeth [144]. During normal operation, power that is applied to the VRG input shaft conveys to the second shaft through gearset 1. A servomotor moves the shifting mechanism [144] to engage the collar into gearset 2. The selected set of gears delivers power to the third shaft that drives the generator. 95

To shift mechanism

Dogteeth collar From main gearbox

To generator

Gearset 1 Pinion Gear

Gearset 2 with interchangeable ratios

Figure 7.1: Gear configuration for six-speed VRG. The VRG design in the 100 kW case study has 14 spur gears, or seven gearsets. One of these is the first set in the train, while the remaining six interchange to set the ratio. In this case, both the pinion and gear may have anywhere from 17 to 289 teeth. The range is based on the gears that are actually commercially available [147] and suitable for this application. This assortment creates roughly 75,000 possibilities for each of the seven gearsets, and hence, 1.27 x 1034 possible VRG combinations. The task for the design methodology is to find the best combination of gears to satisfy the design goals set forth. 96

Gearboxes modify speed and torque in vehicles, aircraft, watercraft, and industrial equipment. Many times, the design optimization techniques for these applications seek to minimize noise [148–150], energy loss [150–152],overall weight [152–155], or a combination of criteria [156,157]. The procedures are bound by physical limitations such as the mechanical durability of the gears [148,155,158]. The design strategies are similar to those for the gearboxes that are used in the wind turbine [10,74,143]. Gear selection analyses involve massive sets of possible design configurations [159–161]. Multiple design parameters are optimized, and only a limited number of gears sizes are considered in light of limited processing capabilities [162]. To minimize cost it is desirable to use commercially available gears in the VRG. Unfortunately, most all of the existing research approaches optimization through custom gear geometry without any provision for the designer to select commercially available gears.

There is also a lack of research

pertaining to the design of variable ratio

gearboxes, which have more variables and require greater computational expense [163,164]. In many instances the gear train optimization is broke down into smaller problems where classic optimization techniques can be applied. However, the technique is complicated by discontinuities in the VRG performance data. Furthermore, these strategies do not consider all of the interrelationships among the involved parameters, and cannot always find the actual solution that is optimal. Hence, a technique is needed to find the optimal combination of gears that can be used in the VRG. 7.1 METHODOLOGY A decision-making algorithm is developed to select the gearset sizes for a VRG design that can be used at multiple turbine sites. These sites should have reasonably similar wind conditions to maximize the capability of the design. For this particular 97

study, there is a design for both normal and high wind density sites. The former is dedicated to the National Renewable Energy Laboratories (NREL) study sites 1 through 18, and the latter is for sites 19 and 20. To acquire the greatest energy production for a given wind site, it is necessary to match the control design ratios as closely as possible. However, the ability to match these ratios increases with gearsets that have more teeth, but also greater mass. Hence, there is a trade-off between maximizing energy production and minimizing the total weight of the VRG gears. To explain the analysis, it is helpful to describe the basic aspects of the gearsets that can be assembled from the commercially available components. A gear module of 3 is appropriate for transmission design [165] and is used for the VRG application as well. For a variable speed gearbox, it is necessary to mount the interchangeable gearsets on common axes. Hence, all six of the gearsets used for those ratios, which are referred to here as

, must have the same mounting distance, as measured from the center of the

pinion to the center of the gear. The top-right plot in Fig. 7.2 shows contour lines for gearsets that have a common mounting distance. The length of the line is indicative of the number of gearsets that are in each group. It is desirable to use the gearsets that are nearest the origin. They have the lowest mass as indicated by the plot at the top right corner. However, the gearsets in this region provide very limited design flexibility. There are a limited number of gearsets to work with, as already pointed out. The plot in the lower left corner shows lines of constant gear ratio. These lines converge near the origin, which means that there are fewer gearsets with available ratios. The last plot in the lower right corner also supports this notion. The isolines here are indicative of the change in ratio from one gearset to the next. The area near the top of this plot represents gearsets that have the greatest granularity in ratio, and provide the most flexibility in matching the control design ratios.

The trends in this plot illustrate to the design 98

problem. On the one hand, it is desirable to use gearsets near the origin that will provide low mass. However, it is necessary to move away from this low-mass area to find ratios that match the performance of those in the control problem. There is a trade-off between the gearset weight and the VRG performance. Ratio [-]

Mass [Kg]

1

200

0.40.27 7

100

0. 7

3 1 1.4

0

3.73

250

20 0 17 5

75 200

50

150

25

100

10 50

75

50

2.14

12 5 10 0

0 15

3 1.4

150

0

number of teeth, gear [-]

0.09

250

0. 7

300

5 12 0 0 1

number of teeth, gear [-]

300

11 .43

100 200 number of teeth, pinion [-]

0

300

0

100 200 300 number of teeth, pinion [-]

Centerline Distance [mm]

Granularity [-]

250

40 0

80 0

50 0

70 0

200

20 0

100

40 0

50 0

60 0

30 0

150

0

number of teeth, gear [-]

number of teeth, gear [-]

300

50 0

100 200 number of teeth, pinion [-]

300

250 200

05 0.0 6 0 0.0 07 5 0.0

150

1 0.0

100

5 0.01 5 0.02 0.05

50

0.01 0.015 0.025 0.05 0.25

50 100 150 200 250 number of teeth, pinion [-]

Figure 7.2: Typical trends for module 3 gearsets, showing the ratio (top left), total mass (top right), total teeth (bottom left), and ratio granularity (bottom right).

99

7.1.1 Procedure The optimal combination of gearsets is found through two algorithms. The first one builds a set of all the combinations that are applicable for all of the sites considered by the design. In the second algorithm, these combinations are evaluated using an objective function to identify the optimal solution.

The process uses the dynamic

programming results from the control design previously discussed in Chapter 5. This site-specific data includes the gear ratios and associated performance results for roughly four-hundred thousand VRG combinations that produce energy within 1% of the ideal level. Prior to running the algorithm, the gearset data is computed for the spur gears based on the desired gear module. This includes the geometry, mass, and allowable tooth loading associated with each possible gearset. The first algorithm develops a collection of gearset combinations for the VRG. Combinations identified through this process must meet the performance requirements for all sites, denoted as , considered for the design. The initial step is to group all of the gearsets together that have a common mounting distance, or the same number of total teeth adding both pinion and gear. These are the gearsets that will be considered for .

The process is computational expensive.

Each unique group of gearsets is

examined by the algorithm one at a time. The technique works by first trying to find the gearsets for six ratios in

, and then finding a ratio,

, for the first gearset. The

process begins with the gearset 2 group that has the lowest number of teeth. Each subsequent examination, indicating by an increment in , indicates a new collection of gearsets having one more tooth than those in the set previously examined. Table 7.1 characterizes the

gearsets that are in the first group ( = 1)

examined. Each gearset has a total of 39 teeth, thus insuring a common mounting distance. This is the first group to be examined because it is the one with the least 100

amount of teeth that provides at least six combinations. The subsequent examination ( = 2) considers gearsets that have a tooth sum of 40, and provide seven ratios. This trend continues until a tooth sum of 305 ( = 267) is reached, where there are 272 gearsets to be examined. Moving forward, the number of gearsets is reduced by one until a maximum number of teeth of 573 is reach ( = 535). However, it is unlikely the algorithm will advance to this size of gearsets. = 1, 39 teeth Gearset No. 1 2 3 4 5 6 Table 7.1:

Number of teeth Pinion Gear 17 22 18 21 19 20 20 19 21 18 22 17

Ratio 0.773 0.857 0.950 1.053 1.167 1.294

Characteristics for first group having at least six gearsets.

Processing the gearsets for

in this order allows the algorithm to find the

optimal combination without having to check every possibility that exists. In the first step, shown in Fig. 7.3, the algorithm builds a broad set of combinations, each having seven unique gearsets. The goal is to capture all of the combinations that are practical for use in the VRG. This step of the process is conservative and therefore includes some combinations that are not valid. In the successive step, these combinations are discarded when no corresponding performance data is found in the lookup table. The energy production performance is then quantified in the next step for all of the remaining combinations. The fatigue analysis evaluates the effect of the dynamic loading that will occur at each site over a twenty-year period. It is most efficient to perform this analysis near the end of the cycle, when the group of gearset combinations has been reduced by 101

previous steps. Combinations that last twenty years meet the performance requirement criteria. The mass is then computed in the following step, and stored with the associated energy production and gearset data for each remaining combination.

Select n

Gearset data Site-specific data

sites

Build VRG combinations based on

i++

Ng2(i)

Range of feasible ratios for n sites Eliminate combinations excluded from lookup table

Replace M2 as necessary Commercial gearset data, Ng2 groups

Compute energy production for each combination Energy lookup table For n sites

Yes Is 99.9% energy level achieved at all sites?

No

Perform fatigue analysis Applied loads at n sites

Replace M1 as necessary

Eliminate combinations that fail Allowable gearset loads Compute weight of remaining VRG combinations

Mmin(i) > M2 Store remaining gearset combinations for set i

Combinations remaining No

No

Yes

End

Yes

Yes

No

Figure 7.3: Procedure for finding valid gearset combinations for a given set of wind sites.

102

After storing the data for group

the algorithm searches for the minimum mass

values required to use the objective function. The mass,

, corresponds to the gearset

combination that has the lowest mass among those that simply meet the performance criteria. After each examination of group , it is replaced if a new combination of lower mass is found. After a couple of cycles the minimum has always been found since the gearsets generally increase in size with each new group, . The expression 'generally' is used here, since it is still possible to find a combination of lower mass, although becoming less likely, as increases. However, as gearsets with more teeth are considered the granularity of the ratios increase. This affords greater opportunity for the algorithm to find gearsets that match the control design ratios more closely. Therefore, the level of production generally increases with each examination. The mass,

, represents the

minimum mass that can be found for combinations that produce at least 99.9% of the total maximum energy of all n sites. This parameter determines when the algorithm can stop searching. This occurs after continuing to repeat the cycle to the extent that the mass of the gears in group, , is so great, that no optimal solution is possible in any subsequent groups,

, and so on. Once the domain of practical gearset combination is established, the second

algorithm applies an objective function [166,167]. The function assigns a value to each combination based on the energy it produced at

sites and the mass of the gearets. (7.1)

where, (7.2) (7.3) 103

The first term,

, is indicative of how the total gearset mass compares to the minimum

mass found in the set of valid combinations. The second term,

, represents the amount

of energy produced for each combination, , in group . It reflects an average based on the resulting production at each site,

, that is considered.

The objective function

searches for a minimum value that is indicative of the optimal solution.

Figure 7.4: Objective function data, Pareto curve, and optimal solution of second VRG design designated for sites 19 and 20. The performance data for each combination is evaluated in the objective function by varying the weight, , from zero to one at an increment of .01. The combination with the lowest value at each of these weights creates the Pareto curve shown in Fig. 7.4. For this analysis, the point that is nearest the origin corresponds to the gearset combination that is most optimal. It is determined using, (7.4)

where, = vector of

values for Pareto curve combinations

= vector of

values for Pareto curve combinations

104

7.1.1.1 Building combinations Building gearset combinations is the first step for each new set of gears that pass through the algorithm. A combination is defined as all six of the gear ratios that are used in the system such that, (7.5) The process uses a database of dynamic programming results (in Chapter 5) to find gearsets combinations that are practical for the VRG design. These results include gear combinations that produced at least 99% of the maximum level for all of the NREL sites considered by the design. They are used to envelop the gearset ratios that should be considered by the algorithm. As combinations are created, all six of the ratios in

are

checked to insure they fall within the respective ranges that are given in Table 7.2. Any combination with one or more ratios outside of the respective range is eliminated in real time. The process is conservative and captures all of the valid combinations in addition to some that are not valid. The subsequent step will only be able to quantify the energy production for combinations that are valid, and will discard invalid ones. Experience with the algorithm proved it is the most efficient method of developing valid gearset combinations. Gear Min Max

1 14.71 18.29

Table 7.2:

2 14.81 20.19

3 17.80 20.79

4 20.19 21.39

5 20.89 21.49

6 21.58 21.60

Allowable range of VRG ratios for combination building.

The gear ratio,

represents the total amount of speed increase from the rotor to

the generator. The amount of gear increase is the product of the main gearbox and VRG ratios, given respectively as

, and

. 105

(7.6) Recall from Fig. 7.1, that the VRG ratio is the product of the first and second gearset. (7.7) There are seven gearsets in the VRG with the first being used for remaining six used, interchangeably, in

, and the

, to vary the ratio. (7.8)

where the variable

represents the ratio in gearset 2 that corresponds to gear .

Substituting equation (7.7) into (7.6), and rearranging it gives, (7.9) At this point, the goal is to find one gearset for the ratios in

and six for

such that all six of

meet the requirements in Table 7.2. Due to the mounting distance

constraint, it is not always possible to find gearsets for

that satisfy Equation (7.4).

In the instances where all six are found, it is almost always possible to find a gearset for that produces the desired output. Reducing the ratios to relative ratios facilitates the search to find gearsets that work in

. Relative ratios are found by dividing all of the gearset ratios by the

highest ratio in the set. These are compared to the control design ratios, which are also put in relative terms by dividing the values in Table 7.2 by the highest gear ratio. The range for the relative ratios used to find gearsets for algorithm first tries to find gearsets for

are given in Table 7.3. The

, and then finds one for

range in Table 7.2.

106

to satisfy the

Gear Min Max

1 0.681 0.843

Table 7.3:

2 0.686 0.935

3 0.824 0.963

4 0.935 0.990

5 0.967 0.995

6 0.999 1.000

Allowable range of relative ratios for combination building.

7 Gearset ratio Gear 6, jth set Set j

6

ratio [-]

5 4 3 2 1 0 0

10

20

30

40 50 gearset j

60

70

80

90

Figure 7.5: Sorted ratios used to build subgroups of combination sets. Each set , that is examined through the algorithm analyzes a unique collection of gearsets considered for number of teeth

that have the same mounting distance,

, or also the same

. The collection of gearsets is sorted in terms of the ratio produced,

as shown in Fig. 7.5. Each point in the plot represents a gearset with the corresponding ratio given on the ordinate. The program begins searching the gearsets at the right side, and moves to the left one gear at a time. At each step, , there is a new highest gear that is used as a divisor to generate the relative ratios for all of the points in the boxed area. The right side of the box ends with the smallest ratio that is able to satisfy the lowest range, corresponding to gear 1. To economize the search, the program only assembles 107

Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Pinion teeth Gear teeth Ratio Relative ratio Table 7.4:

Gearsets considered for 1 2 3 29 30 35 59 58 53 0.600 0.622 0.739 0.686 0.710 0.845 30 31 36 58 57 52 0.622 0.644 0.765 0.710 0.736 0.874 31 32 37 57 56 51 0.644 0.667 0.791 0.736 0.762 0.904 32 33 38 56 55 50 0.667 0.690 0.818 0.762 0.789 0.935 33 34 55 54 0.690 0.714 0.789 0.816 34 35 54 53 0.714 0.739 0.816 0.845 36 52 0.765 0.874 37 51 0.791 0.904

, 4

5

38 50 0.818 0.935 39 49 0.846 0.967

39 49 0.846 0.967

Example of available gearsets for set , subset . 108

teeth 6 40 48 0.875 1.000

combinations in which each consecutive gear ratio increases in size. For example, it would be inefficient to consider a combination in which gear 4 has a lower ratio than that of gear 3. Table 7.4 is indicative of the sample data points found in the boxed-in area. There are six possible ratios for gear 1, seven for gear 2, four for gear 3, two for gear 4, and so on. At this point the program computes all of the possible combinations from subset . It then checks to see if each one fits within Table 7.2. If so, it finds a solution for gearset 1 that satisfies Table 7.3, and stores the combination for further consideration. After all of the combinations are determined for set , the program advances to the next step where more criteria is applied to eliminate invalid combinations. 7.1.1.2 Energy production The optimal control design in Chapter 5 found the VRG gear ratios that produced the most energy at each of the NREL sites. At that time, performance data was also recorded for the VRG combinations that produced energy at 99% of the maximum level at each site. All of the combinations stored in this database have adequate brake lining for the sites being considered and provide continuous operation through the normal range of wind speed. In this chapter, the data is reformatted into a five-dimension lookup table that predicts energy production as a function of the gearset ratios being evaluated. )

(7.10)

Figure 7.6 illustrates the relationship among these gear ratios in the production of energy. Recall that the ratios are discretized at roughly .01 increments. Gear ratios 1 and 2, (

and

), have the greatest amount of variation, and are shown on the abscissa

and ordinate in all of the plots. Moving from the top to the bottom row shows the effect

109

16

15 15

16

0.2

0.4

17 18 Energy loss [kW-h]

0.6

0.8

1

1.2

19

1.4

20

1.6

1.8

2 4

x 10

19

18

18

18

GR 2

19

17

17

17

16

16

16

18

16

18

16

18

GR 1

GR 1

GR 1

GR 3 = 19.49, GR 4 = 20.59, GR 5 = 21.19, GR 6 = 21.59

GR 3 = 19.49, GR 4 = 20.69, GR 5 = 21.19, GR 6 = 21.59

GR 3 = 19.49, GR 4 = 20.79, GR 5 = 21.19, GR 6 = 21.59

19

19

18

18

18

GR 2

19

GR 2

17

17

17

16

16

16

16

18

16

18

16

18

GR 1

GR 1

GR 1

GR 3 = 19.79, GR 4 = 20.59, GR 5 = 21.19, GR 6 = 21.59

GR 3 = 19.79, GR 4 = 20.69, GR 5 = 21.19, GR 6 = 21.59

GR 3 = 19.79, GR 4 = 20.79, GR 5 = 21.19, GR 6 = 21.59

19

19

18

18

18

GR 2

19

GR 2

GR 2

GR 3 = 19.19, GR 4 = 20.79, GR 5 = 21.19, GR 6 = 21.59

19

16

GR 2

GR 3 = 19.19, GR 4 = 20.69, GR 5 = 21.19, GR 6 = 21.59

GR 2

GR 2

GR 3 = 19.19, GR 4 = 20.59, GR 5 = 21.19, GR 6 = 21.59

17

17

17

16

16

16

16

18 GR 1

16

18 GR 1

16

18 GR 1

Figure 7.6: Illustration of energy data used to predict combination performance in VRG.

110

of

, while moving from the left to the right column shows the effect of

range of

The

is narrow in comparison to those shown here, and is not represented. The

contoured surface shown here provides an accurate tool for interpolating gearset ratios. The table data is used to both build and evaluate the performance of valid gearset combinations for all

sites. The endpoints, or minimum and maximum ratios for each

gear in the lookup table, are extracted and used in Table 7.2. As pointed out in that section, this data defines the rough set of potential gear ratios in the initial step of the algorithm. The purpose is to insure that the process captures all of the valid gearset combinations, including the optimal one. It also acquires some invalid combinations, which are eliminated in the subsequent step if just one of the ratios is outside of the lookup table boundaries. For example, in the first step the program will accept a ratio of 16 for

, since it is within the range of that gear. In the second step, the program

searches this ratio in terms of the lookup table. The ratio is valid if top row of Fig. 7.6, but it is invalid for larger values of

is at 19.19, in the

. The energy production is

determined if all of the gearset ratios are found in the table, otherwise the combination is discarded. This phenomenon (explained in Chapter 5) has to do with the way the gears work together to provide continuous production through the normal operating range of wind speed (0 to 25 m/s). 7.1.1.3 VRG Gearset Mass Calculation The mass of the gearsets is based on the geometry of hubless spur gears that are commercially available [147]. The total mass of the VRG gearsets is the sum of the mass from the first gearset and that of the other six that are used interchangeably: (7.11) where, 111

= Mass of the first gearset = Mass of the second gearset for gear 7.1.1.4 Fatigue Failure The spur gears used in the VRG are susceptible to tooth failure, which can occur through bending stress or pitting of the contacting surface [168–170]. The American Gear Manufacturer's Association (AGMA) has developed formulas for predicting failure, and the approach is most commonly employed by designers as part of the gear selection process [171,172]. The purpose of the AGMA formulas and factors is to provide a uniform approach to rating gears in various applications. The amount of bending stress,

, that a tooth experiences is computed using

[168], (7.12) where, = tangential load applied = application factor = dynamic factor = metric module number = face width = size factor = load distribution factor = geometry factor for bending On the right hand side of the equation, the first set of terms represent the loading characteristics, the second set relates to gear geometry, and the last group pertains to 112

tooth form [170]. Similarly, the amount of contact stress,

, that occurs is given by

[168], (7.13) where, = elastic coefficient = application coefficient = dynamic coefficient = size coefficient = pinion pitch diameter = load-distribution coefficient = surface-condition coefficient = geometry factor for pitting The AGMA factors and coefficients that are denoted by influence the load capacity. The factors

and

and

, respectively,

pertain to the geometry of the gear teeth

and are computed for both the pinion and gear. Values for these parameters are found in reference tables and selected on the basis of the type of application [171,173]. There are also formulas available to determine the amount of allowable stress that the gear tooth can with stand for both modes of failure. For bending the allowable stress,

, is, (7.14)

where, = bending strength of material = life factor 113

= temperature factor = reliability factor Similarly, the allowable contact stress,

, is, (7.15)

where, = contact strength of material = life coefficient = hardness-ratio coefficient = temperature coefficient = reliability coefficient In this form, equations are used to analyze gear teeth subjected to a constant load each cycle. However, the loading that occurs in the wind turbine drivertrain is variable. Accordingly, the site data includes a profile of the loads as imparted by the varied wind speed over time, (7.16) The AGMA formulas are adapted to this input by setting the allowable stress equal to the actual stress, and solving for the applied load. (7.17) (7.18) The allowable tooth loads, coefficient, and

and

, for this analysis have a life factor,

, and life

, set to unity. This further associates the loads with a finite set of cycles,

, that can be used to create an endurance curve [174]. 114

The input load is a function of the rotor torque and gear reduction at the point of application. The tooth loads for the first and second gearset are given by, (7.19) (7.20) Relating the applied loads to the allowable loads, enables the use of the endurance curve shown in Fig. 7.7, to predict the number of cycles that can occur at a given load [174]. Through Miner's Rule [34], the life, , of the wind turbine at a particular profile is given as, (7.21)

where, = Duration of the profile = Number of actual cycles at load ratio, = Number of allowable cycles at load ratio, .

4.0

load ratio

W app/W b 3.0

W app/W c

1.0 0.7 0.5 2 10

4

10

6

10 allowable cycles, Ni, [-]

8

10

Figure 7.7: Endurance limit based on ratio of applied to allowable loading. 115

10

10

The algorithm determines the life of both the pinion and gear with respect to bending and pitting failure. The combination is kept if the life exceeds 20 years in all four cases and for all seven gearsets. 7.2 RESULTS Based on the above methodology, algorithms are developed to find the optimal gearset combination that maximizes energy production and minimizes mass.

The

selection is customized to the wind conditions at one or more sites. For the case study of the 100 kW wind turbine there are two different VRG designs. These are based on the results of the control design and similarity of the ratios in each group. The first is for typical wind speeds and is recommended for use at sites 1 though 18. The second VRG configuration is suitable for the high wind power density found at sites 19 and 20. Figure 7.8 illustrates the combination data considered by the objective function. When the parameters are expressed through the objective function, the optimal selection is the point on the Pareto curve that is nearest to the origin.

116

Figure 7.8: Mean energy loss and gearset mass data for combinations considered in objective function for sites 1 through 18 (top) and sites 19 and 20 (bottom). The five Pareto points nearest the origin are given in Tables 7.5 and 7.6, respectively, for the normal and high power density designs. algorithm found the combination with the lowest mass,

For each search, the

, that can maintain a minimum

life of twenty years and produce at least 99% of energy as the ratios found in the control design. Recall, that the combination associated with

is the one with the lowest mass

that satisfies the minimum performance requirements. It has a mounting distance of 207 mm, a mass of 279 Kg, and is used in the objective function as a baseline. It happens to be the minimal solution found in both searches for the normal and high power density design. The optimal solutions indicate that the algorithm had to increase the mounting distance by around 10 to 25 mm to find ratios that could match the performance of the 117

control design without failure. If this baseline configuration were used outright, the energy loss would be around 6,900 and 7500 kWh, respectively, for each normal and high power density site. However, the combinations found through the objective function reduces this to an average of 321 and 1082 kWh for sites in each design. The trade-off is a gain in gearset mass of roughly 16 kg for both the normal and high power density design. This increase in mass is a reasonable trade off since the optimal solution reduces the energy loss to 0.13% or less for all of the sites. Rank 1 2 3 4 5

[-] 0.147 0.153 0.185 0.227 0.254

Table 7.5:

Rank 1 2 3 4 5

[kWh] 321 423 257 788 255

[Kg] 141.6 139.5 148.3 136.0 157.6

Characteristics of the top five Pareto combinations for normal wind power density VRG design used at sites 1 though 18.

[-] 0.248 0.280 0.304 0.333 0.334

Table 7.6:

[mm] 225.0 219.0 235.5 216.0 238.5

[mm] 231.0 246.0 249.0 223.5 252.0

[kWh] 1082 785 708 1647 674

[Kg] 143.7 156.5 161.0 139.2 165.7

Characteristics of the top five Pareto combinations for high wind power density VRG design used at sites 19 and 20.

The search algorithm searched through a mounting distance of 264 and 285 mm, respectively, for both the normal and high power design combination. These distances are well above any of the points along the Pareto curve and suggest the optimal solution was well within this search envelop. For the normal power VRG design, the gearsets in 118

each have 150 teeth, which correspond to a mounting distance of 223.5 mm. In the high power design, these gearsets have 154 teeth, with a distance of 229.5 mm. Wind turbines in these sites experience high loads more frequently, which necessitates slightly larger gearsets that have greater load allowance. The gearsets selected for the normal and high power designs appear in Tables 7.7 and 7.8, respectively. In each design, the first gearset is actually an increaser while the six interchangeable gearsets are reducers. The reducer gears provided the algorithm with the greatest flexibility (see the granularity plot in Fig. 7.2) to match the control design gears from each group, , of gearsets.

Pinion teeth Gear teeth

Table 7.7:

Gear 1 49 101 15.72

Gear 2 53 97 17.70

Gearset 2 Gear 3 Gear 4 56 58 94 92 19.30 20.43

Gear 5 59 91 21.01

Gear 6 60 90 21.60

Optimal gearsets selected through multiobjective function for normal power density design at sites 1 through 18.

Pinion teeth Gear teeth

Table 7.8:

Gearset 1 135 50 Ratio

Gearset 1 132 58 Ratio

Gear 1 57 97 16.05

Gear 2 61 93 17.91

Gearset 2 Gear 3 Gear 4 64 66 90 88 19.42 20.48

Gear 5 67 87 21.03

Gear 6 68 86 21.59

Optimal gearsets selected through multiobjective function for high power density design at sites 19 and 20.

119

Wind Set class No. [50m] 1 1 2 2 3 4 3 5 6 7 4 8 9 10 5 11 12 13 6 14 15 16 7 17 18 19 20 Table 7.9:

Site

12951 3412 11719 28570 6474 6684 22814 7302 31266 2002 174 3134 13731 21291 22818 16577 7858 19520 12796 11721

[kWh] 327,112 396,278 427,283 441,287 574,259 587,504 609,297 690,383 697,580 613,499 657,448 839,085 716,076 751,114 900,408 945,453 789,764 1,022,070 872,956 1,029,166

[kWh] 355,408 429,035 469,436 486,397 628,048 642,803 672,639 761,966 772,913 673,735 720,217 941,496 786,260 824,890 1,004,355 1,037,586 861,207 1,116,221 949,994 1,122,559

[kWh] 355,319 428,940 469,298 486,278 627,867 642,621 672,454 761,807 772,685 673,329 719,897 941,210 785,662 824,462 1,004,045 1,037,079 860,515 1,115,368 948,875 1,121,515

[%] 8.65% 8.27% 9.87% 10.22% 9.37% 9.41% 10.40% 10.37% 10.80% 9.82% 9.55% 12.21% 9.80% 9.82% 11.54% 9.74% 9.05% 9.21% 8.83% 9.07%

[%] 8.62% 8.24% 9.83% 10.20% 9.34% 9.38% 10.37% 10.35% 10.77% 9.75% 9.50% 12.17% 9.72% 9.77% 11.51% 9.69% 8.96% 9.13% 8.70% 8.97%

[%] 0.03% 0.02% 0.03% 0.03% 0.03% 0.03% 0.03% 0.02% 0.03% 0.07% 0.05% 0.03% 0.08% 0.06% 0.03% 0.05% 0.09% 0.08% 0.13% 0.10%

Performance results for VRG wind turbine following gearset selection.

The gearset selection in Table 7.7 is for the normal VRG design. The ratios found here are relatively close to those found in the previous chapter. The amount of loss due to these deviations is error is less than 0.1%, for all of sites 1 through 18 as shown in Table 7.9. Moreover, many of the low wind sites having an error as low as .02 to .03%. The deviations among the control design ratios, and those in Table 7.8, for the high power VRG design are slightly greater than those for the normal deign. This is explained due to both designs having the same minimal mass baseline, and the need for the higher speed design to find gearsets characterized by slightly greater strength. To obtain a 120

design of reasonably low mass, the objective function compromised some energy production. Hence, the amount of energy decrease is slightly higher for these last two sites. Nonetheless, the loss is within 0.13% and practically negligible inasmuch as matching the performance is concerned. The ability to find VRG gearsets with relatively low mass, and the low amount of energy loss demonstrate the capability of this design approach.

121

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Vita

John Hall was born in Macon, Missouri, and lived there until graduating from the from Macon High School. John graduated from the University of Missouri-Rolla in 1992, with a Bachelor of Science in Mechanical Engineering, Mechanical. As a student he was awarded the prestigious General Motors Scholarship and recognized as an International Gas Turbine Institute Scholar in his final year. Following graduation, he entered the Westinghouse Electrical Manufacturing Management Program, and spent five years working primarily in the power generation division. Following this John worked to develop material handling robot for the semiconductor industry. He has also been a consultant to numerous clients in industry. After 12 years of industrial experience, John entered the Graduate School of Engineering at The University of Texas in Austin, where he graduated with a Master's Degree in 2005.

Permanent address: [email protected] This dissertation was typed by the author.

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