wind energy conversion systems as power filters: a control

Wind Energy Conversion Systems as Power Filters: A

Control Methodology

by

Barry Rawn

A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

Copyright c© 2004 by Barry Rawn

Abstract

Wind Energy Conversion Systems as Power Filters: A Control Methodology

Barry Rawn

Masters of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

2004

In recent years, wind energy conversion systems have been deployed in large numbers in

electricity grids throughout the world. Their impact on the operation of power systems

is a growing area of research. This thesis presents a control methodology for wind tur-

bine systems that are interfaced to the grid through power-electronic converters. It is

shown that fast power fluctuations excited in the conversion system by the wind can be

contained rather than transferred to the grid. Also, the use of the turbine hub’s energy

storage capacity for filtering slow power fluctuations is explored. It is shown that a sta-

bility limit exists on the maximum filtering cut-off frequency, and that this limit can be

conservatively estimated. Application of the control methodology results in a simplified

model of the wind energy conversion system as a first-order filter of the incoming wind

power.

ii

Dedication

In their co-supervision of my MASc research, Dr. Peter Lehn and Dr. Manfredi Maggiore

have invested their financial resources and much of their valuable time with a wonderful

enthusiasm. Through their engagement, patience, encouragement, and rigour, I have

been given a significant educational gift for which I can only be grateful.

The documentation and communication of the potential contributions of our work is,

on the other hand, a debt that is within my capacity to repay. Therefore, this document

and the thesis contained within is dedicated to them.

iii

Acknowledgements

I would like to thank my parents for their support throughout my education, and during

this research. By providing every fundamental condition as given, they have co-authored

all of my life’s accomplishments.

The development of my thesis has been greatly aided by many discussions with my

colleagues in the Energy Systems Group. A special contribution has also been provided by

Dr. Torbjorn Thiringer of the Chalmers University of Technology in Goteborg, Sweden.

With a remarkable generosity, he has provided me with not only valuable measured and

technical data concerning the Alsvik wind farm, but also friendly explanations via e-

mail. Dr. Ted Davison of the System Control Group also provided a helpful consultation

concerning certain aspects of control design.

I would especially like to acknowledge the essential contribution made by my thesis

committee, which is formed by Dr. Francis Dawson, Dr. Reza Iravani, and Dr. Lacra

Pavel. I sincerely appreciate their efforts in reviewing my work, and conducting the oral

examination. Finally, I would be extremely remiss if I did not express here my gratitude

to the University of Toronto for its financial support during my time here as a graduate

student.

iv

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Objectives and Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . 5

2 System Modeling 7

2.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Wind Characterization and Modeling . . . . . . . . . . . . . . . . . . . . 8

2.3 Spinning Blades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Aerodynamic Conversion Efficiency . . . . . . . . . . . . . . . . . 13

2.3.2 Rotational Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Generator and Back to Back Converter . . . . . . . . . . . . . . . . . . . 20

2.6 Converter Grid Interface and Power System . . . . . . . . . . . . . . . . 22

2.7 Summary of Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Control Methodology, Design and Performance 26


3.2 System Structure and Function . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Control Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

v

3.5 Sgrid: Design of Cgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Sint: Separation of Fast and Slow Dynamics . . . . . . . . . . . . . . . . 35

3.7 Sfast: Design of Cinternal . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.8 Sslow: Design of Cexternal . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.8.1 Stability of the Turbine Hub . . . . . . . . . . . . . . . . . . . . . 45

3.8.2 Stability Case Study . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8.3 Supervisory System and Recovery Controller . . . . . . . . . . . . 58

3.8.4 Design of Filtering Pgrid Reference . . . . . . . . . . . . . . . . . . 62

3.8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Results 73


4.2 Safety Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 Performance Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Conclusions 88

5.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A Technical Specifications 92

A.1 Wind Turbine Parameters: . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A.2 Drive Train Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A.3 Generator Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

A.4 Converter Parameters: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Bibliography 94

vi

Chapter 1

Introduction

1.1 Background

In recent years, concern over the long-term health and environmental effects of conven-

tional electricity generation has been growing. This has been part of a larger movement

toward an energy infrastructure that is more sustainable. Considerable research effort

has been directed at technologies that extract electrical energy from renewable sources.

Of these technologies, horizontal axis wind turbines are one of the most economically

viable, and have been deployed in huge numbers in Germany, Denmark, Spain, and the

United States.

No longer experimental showcases, ‘farms’ of wind turbines are now serving up to

14% of electricity needs in Denmark [1], and 4% in Germany [2]. As the penetration

of wind energy into power systems increases, the effects of wind farms on power system

operation will become an increasingly important factor in their profitability and man-

agement. Standards already exist for power quality issues like harmonic injection and

‘flicker’, a phenomenon that is discussed later in the literature overview. In the near

future, ancillary services like generation/load regulation, load following, voltage control,

and frequency responding spinning reserve will also become relevant for wind farms. Tur-

1

Chapter 1. Introduction 2

bine and farm operators will be asked to specify which types of ancillary services they

need or can provide [3]. In a restructured electricty environment, these services may also

be sold or traded [4].

There are a number of wind turbine technologies, and they have different capabilities

and effects with respect to these power systems issues. In light of such issues, wind

turbine configurations and farm models are being more carefully examined to determine

their potential and limitations [3], [5]. The first wind turbines were typically constant

speed turbines with induction machines and gearboxes connected directly to the grid.

This configuration is still common in Denmark; it is the least flexible configuration,

and has the greatest negative impact, sometimes necessitating the installation of com-

pensating devices. The majority of large turbines being installed today are much more

sophisticated variable-pitch, variable speed turbines with doubly-fed induction machine

generators (DFIGs). Such systems achieve variable speed operation (which is desirable to

increase energy capture and reduce blade loading) at minimum cost [6]. They employ a

back-to-back power electronic converter to energize the rotor windings of the doubly-fed

machine through a connection to the grid. Because of this, they also offer control of

reactive power at the grid interface.

The most flexible variable speed wind turbine configuration is one in which a syn-

chronous generator with many poles or an induction generator with a gearbox is connected

to the grid through back-to-back power electronic converters. One converter interfaces

with the turbine generator, and the other interfaces with the grid. In such an arrange-

ment, the converter system conveys the full power, rather than just a portion as in the

case of a DFIG. For this reason, such systems are sometimes called ‘full load’ converters

[5]. While this increases the necessary rating of the switches, it also offers the possibil-

ity of substantially reducing the interaction between the turbine and the power system.

As the price of high-power semiconductor switches drops, the ‘full load’ configuration

becomes more attractive because of its increased flexibility, speed range and simplified


construction. It has been suggested that such configurations will be more common in the

future [7].

Commercial wind turbine technology has so far been designed to maximize energy

production. In the future, operators may need to trade off efficiency of operation with

flexibility of operation. It has become important to understand and quantify the capa-

bilities of wind energy conversion systems to optimize their interaction with the grid, as

opposed to optimizing energy efficiency[4].

1.2 Literature Overview

The modeling of wind turbines has been extensively studied [8],[9],[5],[10]. More de-

tailed studies of effects particular to wind turbines [11],[10],[5] and measurement studies

[12],[13], [14],[15] help clarify what level of detail is appropriate for a given line of inquiry.

The control of variable speed wind turbines has also been widely addressed, with a

focus on optimizing energy capture and minimizing torque variations (e.g. [16],[17]). It

is generally recognized that two important advantages to variable speed operation are

the reduction of fatigue in the blades and drive train and an increase in energy capture,

and that these factors are of importance to the profitability of a wind turbine system.

These factors are included in standard control objectives, as discussed by Leithead [18],

[19] and Novak [20]. Leithead also conducts a review and classification of variable speed

wind turbine control strategies [19]. He provides a survey of work in the field, from

which he observes that some studies are ‘feasibility’ studies that examine a strategy

conceptually, while later ones are ‘design’ studies, in the sense that they cover a complete

implementation. The latter type of study addresses operation of the turbine in both above

and below rated wind speed conditions, where different control strategies are required,

while a feasibility study typically focuses on the operating range within rated values to

demonstrate a method’s basic concept.


It is commonly mentioned that variable speed systems equipped with converters re-

duce grid integration problems. This is partly because variable speed operation allows

many fluctuations to be absorbed into the turbine hub as speed changes. In addition to

this partial buffering of power fluctuations, systems with power electronic converters can

also regulate reactive power. By comparison, torque fluctuations applied by the wind to

fixed speed systems are transmitted as both real and reactive power fluctuations.

Many grid-integration studies can be found that focus on voltage variations caused

by wind energy conversion systems within a specially defined voltage-frequency envelope

known to be irritating to the human eye. Such variations are referred to as ‘flicker’,

and are a result of the fluctuations in power due to the wind, and the dynamics of the

conversion system. In general, it is found that variable speed systems with ‘full load’

converters reduce flicker substantially through both their buffering of power fluctuations,

and their control of reactive current to maintain voltage. However, such adjustment

does not eliminate the transfer of power fluctuations to the grid; it merely changes their

form from voltage magnitude fluctuations to voltage angle fluctuations. Also, limitations

exist on the size of the fluctuations that can be compensated in this manner, depending

on the rating of the converter and the strength of the network. No studies were found

that examined in detail the specific limits of voltage regulation by wind turbine systems

equipped with converters. Some studies of the ability of such systems to ‘ride-through’

voltage dips do exist [21], which reflects recent changes to the regulatory environment.

The frequency range of concern for the phenomenon of flicker is between 1 and 25 Hz

[22]. The power variation of wind energy conversion systems over minutes and hours can

lead to variations of system frequency in some grids [23], in cases where the penetration

of wind power is high. This makes it necessary to significantly tap the spinning reserve

of conventional generation units. There is little literature on the subject of controlling

wind energy conversion systems with ancillary services in mind. It has been suggested

in [4] that the inertia of the turbine hub might be employed to provide a small amount


of spinning reserve over short time periods in response to frequency variations. Droop

characteristics, dynamic voltage regulation, and survival of faults were properties listed as

possible future requirements for wind farms to meet in [3]. A simple control scheme for a

DFIG turbine has been proposed in [24] that allows it to provide a useful power response

to frequency deviations along these lines. Pitch control is used to give the turbine a

droop characteristic that helps stabilize frequency variations, and control of the converter

is altered such that the turbine’s output power responds to changes in frequency with a

small apparent inertia. As of June 2004 the authors of [24] were unaware of any practical

implementations. Such a use of wind farm power variations for system stabilization was

proposed and studied for a multi-bus power system in [23]. It was concluded that wind

farms could partially mitigate their potentially adverse effect on system frequency with

such an approach.

1.3 Objectives and Scope of the Thesis

In this work, a variable speed wind energy conversion system will be studied from the

perspective of improving its interaction with the power system. Instead of following the

paradigm of controlling the dynamics of the turbine to extract an optimal power from the

wind, the concept of extracting a reasonable but grid-specified power from the system

will be explored. Toward this end, a control methodology will be developed that:

• achieves the regulation of conversion system dynamics without involving the con-

trols of the grid interface converter, in order to free them for power systems control

objectives.

• maximizes the use of the turbine hub’s inertia to filter incoming wind power fluc-

tuations.

These two aspects of the methodology will address grid integration issues related to

fast and slow variations, respectively. The first aspect will be shown to eliminate the


variations associated with flicker. The second aspect will be used to demonstrate the

extent to which the conversion system can be used to smooth the desired wind power.

The thesis will be a ‘feasibility’ study, in the sense that it will focus on presenting a

new method, rather than guaranteeing a control design for a particular variable speed

system. The control methodology will be modular, with a structure that decomposes

the system into different control tasks. The focus of the thesis will be on those control

tasks associated with the dynamics of the wind turbine rather than on those of the grid

interface, which are non-trivial but more standard. In particular, a full treatment of grid

voltage regulation is outside the scope of this thesis. Relatively simple grid interface

controls will be presented, leaving open the possibility for more sophisticated controls to

be developed separately.

Chapter 2

System Modeling

2.1 Chapter Overview

There are several aspects of wind energy conversion systems that set them apart from

more traditional forms of power generation. In this chapter, a discussion is made of

how these aspects may be appropriately represented. The modeling of mechanical and

electrical elements is also examined. Finally, the range of validity of the model is outlined.

The chapter begins with a description of the nature of the wind, and how it may be more

Figure 2.1: Diagram of model used, showing a division into chapter sections.

simply represented in terms of its harmonic content for control studies (Section 2.2). The

turbine portion of the model is based on a particular three bladed machine from which

7

Chapter 2. System Modeling 8

measured data were obtained, and physical specifications available [10]. The aerodynamic

nonlinearity of the blades is examined, and the effects experienced by the rotating blades

of a wind turbine as they pass through the wind field are evaluated (Section 2.3). A two-

mass torsional model is developed to describe the mechanical drive-train of the turbine

(Section 2.4).

A simple power-balance model of a back to back voltage source converter system is

then introduced. It is assumed that field-oriented controls are applied such that one

converter controls the torque and flux of the generator, and the other converter produces

direct and quadrature voltages at the grid frequency (Section 2.5). A simple model of the

grid interface and transmission system completes the description of electrical dynamics

(Section 2.6).

As a complete energy conversion system, the model is applicable over a certain range

of situations. Assumptions underlying the model are summarized in Section 2.7. Within

this range, a control methodology is formulated and elaborated in the next chapter.

2.2 Wind Characterization and Modeling

The atmosphere of the earth is continually forced by solar radiation, and possesses its

own cycles of heat exchange with the oceans and the land. The wind arises to dissipate

the resulting thermal gradients. Seasonal and daily variations of the wind can mostly be

attributed to its role in energy transport within the climate system [25]. The character of

faster variations depends on these complex and turbulent processes, but is also affected

by local geographical features and ground cover. As a result, the wind field is non-uniform

both temporally and spatially.

Time variations on the scale of tens of minutes to seconds are of particular relevance

to the control of a wind energy conversion system. Such variations may either be tracked

closely or smoothed. Slower variations are transferred to the system, and become a power-


planning issue. A ‘spectral gap’ in the spectrum of the wind from about 10 min/cycle to

2 hours/cycle, within which the energy content is low, is generally assumed [26],[25],[27]

to separate the faster turbulent variations from the slower variations. Fig 2.2 shows the

shape of the overall wind spectrum. This motivates a standard practice [26],[28],[27] of

Figure 2.2: Spectral content of the wind, in cycles/hour. The ‘spectral gap’ in the fre-quency range around 1 hour is used to separate the wind into a slowly varying componentand a fast turbulent component.

representing the wind as a linear combination of a slowly varying mean value vw, and a

turbulent component composed of a finite number of sinusoids over the frequency range

of the fast wind variations. These sinusoids are assumed to have a particular amplitude

Ai at each a discrete frequency ωi, and a random phases ψi:

vw(t) = vw +N∑

i=0

Aicos(ωit+ ψi) (2.1)

The amplitudes Ai are based on a spectral density function S(ωi) that has been empir-

ically fit to the wind turbulence. One such function that is commonly used is the von


Figure 2.3: Von Karman power spectral density function S(ωi) (top) and correspondingamplitude spectrum A(ωi) (bottom). Values used to generate the displayed curves werevw=10 m/s, L=180, and σ=2.

Karman distribution [27]

S(ωi) =0.475σ2 L

vw

[

1 +(

ωiLvw

)2] 5

6

(2.2)

This distribution depends on the mean speed vw, on the roughness of the surrounding

landscape (as described by a characteristic length scale L), and on the standard deviation

of the wind speed σ. All of these parameters can be obtained from site data in order to

calibrate the density function S(ωi). The amplitudes Ai can then be determined for each

discrete frequency ωi from the area under S(ωi) over a certain range:

Ai(ωi) =2

π

√

1

2[Svv(ωi) + Svv(ωi+1)] [ωi+1 − ωi] (2.3)

Both S(ωi) and A(ωi) are shown in Fig 2.3. Distributions such as those of Weibull [26]

or van der Hoven [27] can be employed to model the variations of the mean speed vw,

using site specific parameters from wind data.

The artificial model (2.1) is sometimes useful when a simplified, clean description of


the wind’s turbulence is required. In this work, the spectral density (2.2) is used along

with (2.3) in the design of a control reference signal (Section 3.8.4). It is also used to

generate a artificial wind time series that is employed to demonstrate the effect of the

controlled system in the frequency domain.

The spectrum of an actual wind measurement is more complicated than may be

expected from the preceding discussion. In Fig 2.4, the spectrum of an artificial wind

time series is compared with that of a measured one. The actual measured wind speed

has non-stationary statistics and a broad frequency content that are not captured by the

artificial model used. Changes in direction also contribute discontinuities to the time

series, which alters the spectrum.

Figure 2.4: Comparison of spectra belonging to measured and artificial wind time series.Artificial spectrum based on vw=10 m/s, L=180, and σ=2.

For realistic testing of control designs, actual wind data are preferable. A large wind

time series, measured at the Alsvik wind farm on the island of Gotland, Sweden, is

shown in Fig 2.5. One can observe the slow daily variation in the first time series, and


1 2 3 4 5

5

10

15

Time (hours)

30 40 50 60

4

6

8

10

Time (min)

v w

(m/s

)

170 175 180 185 190 195 200

4

6

8

10

Time (s)

Figure 2.5: Measured wind time series from Gotland Island, over five hours, half an hour,and half a minute


the variation over minutes and seconds in the other two series. Segments of this time

series are used to generate most performance results.

2.3 Spinning Blades

The blades of a wind turbine transfer a portion of the wind’s kinetic energy to the

rotational energy of the turbine hub via forces and torques. The fact that the turbine

hub rotates has two important consequences, which will be discussed in subsections 2.3.1

and 2.3.2. First, the efficiency with which the blades convert energy depends both on

the speed of the turbine hub, ωh, and on the wind speed vw(t). Second, the rotational

motion of the blades through the wind field results in an interaction that can introduce

harmonics of the speed ωh into the power extracted.

2.3.1 Aerodynamic Conversion Efficiency

The efficiency with which energy is extracted from the air depends on the shape of the

blades, and the effective angle of attack of the cross-section of the blade, which is defined

in Fig 2.6. As the blade moves, the wind velocity veff it experiences depends partly on

its own motion through the air, and partly on the incoming wind vw(t). At a distance

r along the blade, veff is determined by the vector sum of the wind speed vw and the

tangential speed rωh, as indicated by the dashed lines in Fig 2.6.

The blade is inclined at some angle β to the horizontal. The angle of attack is defined

as the angle between the central chord of the airfoil (Fig 2.6, dotted line) and the incoming

wind velocity. When ωh = 0, the angle of attack is simply β. As ωh increases, however,

the effective angle of attack is

φ = β − atan

(rωh

vw(t)

)

(2.4)


Figure 2.6: Wind velocity encountered by blade. The effective wind velocity veff isdetermined by the vector sum veff = rωh +vw(t). The angle φ made between the centralchord of the airfoil (dotted line) and the apparent wind direction veff (dashed vectors)is the effective angle of attack, and is in general different from the pitch of the blade β.

Even for a fixed pitch blade, this angle changes with variations in vw(t) and ωh. For

properly designed blades, the effective angle of attack is the same along the length R

of the blade. The dimensionless ratio of the two speeds Rωh and vw is defined as the

tip-speed ratio λ and is used to represent this changing aspect of the turbine blades.

λ(t) =Rωh

vw(t)(2.5)

where R is the radius of the rotor.

The tip-speed ratio λ determines the efficiency of aerodynamic power conversion. This

efficiency is given the symbol Cp, and is a function of λ. Fig 2.7 shows Cp(λ) for the

turbine being modeled.

When λ is small, the angle of attack is large, and the air flow tends to separate from

the blade surface and transfer more energy to vorticity, resulting in the effect known as

aerodynamic stall and low efficiency of conversion. When ωh is high enough to produce


Figure 2.7: Cp(λ) curve. An efficiency of energy power conversion Cp corresponds to eachoperating tip-speed ratio λ.

a specific ratio λopt, the angle attack is at an intermediate, optimal value that maximizes

conversion efficiency. This λopt corresponds to the peak of the Cp curve in Fig. 2.7. The

turbine hub speed ωh necessary to achieve this optimal efficiency is specified as

ωopt(t) =λopt

Rvw(t) (2.6)

For speeds higher than ωopt(t), the conversion efficiency drops off again.

The aerodynamic power Paero (ωh, vw(t)) extracted by the turbine is equal to the

available wind power Pwind(t) multiplied by the conversion efficiency Cp(λ). The wind

power is defined as the kinetic energy of the air flowing through the cross-sectional area

of the turbine.

Pwind(t) =1

2ρπR2vw(t)3 (2.7)

where ρ is the density of air. The extracted power Paero (ωh, vw(t)) is therefore

Paero (ωh, vw(t)) = Pwind(t)Cp(λ(t)) (2.8)


The Cp(λ) curve of a wind turbine shown in Fig 2.7 was derived from a figure in

[10]. Such curves can be either experimentally determined or estimated through iterative

calculations such as the blade element method [10],[29]. The Cp(λ) curve is used to

determine power-speed curves or torque speed curves, and give the aerodynamic torque

Taero (ωh, vw(t)) as a function of λ(t) and the wind speed vw(t):

Taero (ωh, vw(t)) =Paero (ωh, vw(t))

ωh

=1

2

ρπR2vw(t)3Cp(λ(t))

ωh

(2.9)

The torque Taero (ωh, vw(t)) is the input for the mechanical model, which is discussed in

Section 2.4.

2.3.2 Rotational Effects

Fluctuations introduced into the actual aerodynamic torque due to the turbine hub’s

rotation can be the result of both deterministic and stochastic phenomena.

The wind field encountered by one of the blades varies as a result of the increase

of average wind speed with height (referred to as wind shear), and the alteration of the

airflow by the presence of the turbine tower (referred to as tower shadow). Such variations

are deterministic and tend to occur, due to symmetry, at multiples of the turbine speed

ωh. A particularly important multiple of the turbine speed is nωh, where n is the number

of blades. As each of the blades passes through a given part of the wind field, fluctuations

at this frequency (called the blade-passing frequency) are created in the shaft torque. An

example of the type of variations that can occur is shown in Fig 2.8(a).

Other variations in the shaft torque are stochastic in nature. It was observed in Sec-

tion 2.2 that the wind speed has a certain spectral content. The wind speed experienced

by a rotating blade is different, because it moves through a spatially varying wind field.

The spectral content of the torque transferred by the n blades to the turbine hub is


(a) Deterministic effects: Non-uniform wind-field effects onshaft torque

(b) Stochastic effects: Alteration of wind spectrum due torotational sampling.

Figure 2.8: Nature of deterministic and stochastic rotational effects. In (a), periodiccomponents in the shaft torque of a wind turbine demonstrate the existence of determin-istic variations that are a function of turbine hub angle. Such variations can be assumedto be a result of wind shear and tower shadow. Mean value of torque is 20 kNm. In(b), a schematic depiction is shown of the rotational sampling effect on the spectrum ofstochastic wind turbulence.


therefore also different. The total energy of the variations is the same, but energy is

shifted to multiples of the blade passing frequency [11], as depicted schematically by the

peaks in Fig 2.8(b).

It is thought that these two types of effects contribute equally to the periodic pulsa-

tions of the aerodynamic torque [12], and with some effort can be extracted separately

from measured signals [13], [15], as was done to produce Fig 2.8(a). While the standard

deviation of the combined effects can be as high as 20% of their average value [14], their

relatively high frequencies provoke only small changes in the speed of the turbine hub,

due to its large inertia. Therefore, while the effects may well be cruicial for studies con-

cerning blade design or detailed turbine simulators, they were excluded from the model

used for this work after a brief evaluation.

2.4 Mechanical System

Multiple-mass models of wind turbines that account for several blade and tower modes

have been developed in the past [8]. Detailed studies of how these modes can be discerned

in the power output of fixed-speed turbines have been undertaken [14],[30]. However, it is

widely acknowledged [8],[31],[10],[32] that a simplified two-mass model, as shown in Fig

2.9, is sufficient for studies concerned with generator control and power systems stability.

A two-mass model is based on two lumped inertias Jh and Jg that loosely correspond

to the turbine hub and blades, and to the generator mass. The inertias Jh and Jg may be

determined more accurately through parameter identification methods than by physical

specifications [20]. The drive system connecting the two inertias is commonly modeled

as a torsional spring with negligible damping and a stiffness Ks. Compared to the shaft

stiffnesses of other kinds of turbine generators, Ks is relatively low, especially in the

case of drive-trains that employ a gearbox. In this case, there is also a gearbox ratio


Figure 2.9: Mechanical model of drive-train and rotational inertias. Two lumped inertiasJh and Jg are joined by a torsional spring having stiffness Ks, and subject to the twotorque inputs Taero and Tgen.

N that relates the speed of the low-speed shaft to the speed of the high-speed shaft.

For configurations where a synchronous machine with a large number of poles is used, a

two-mass model may also be necessary if the number of poles is sufficiently large [5]. The

resulting low frequency torsional mode is significant for both direct-connected stability

studies [33] and variable speed control schemes [19],[17]. The mechanical model is driven

by two torques, one coming from the aerodynamics of the turbine blades (Taero), the other

from the electromagnetic torque Tgen exerted by the interacting fields of the generator.

The differential equations modeling the rotational motion of two inertias connected

by a gearbox and referred to the high-speed side are as follows:

dθh

dt= ωh

dωh

dt=

1

Jh

(Taero(ωh, vw)

N−Ks (θh − θg)

)

dθg

dt= ωg

dωg

dt=

1

Jg(Ks (θh − θg) − Tgen)

(2.10)

where θh and θg are the absolute angles of the two inertias, and ωh and ωg are their


angular velocities. A convenient change of variable is

θsum = θh + θg

θdiff = θh − θg

(2.11)

The sum θsum = θh + θg relates to the absolute angle of the center of inertia of Jh and Jg.

In this model, this ever-advancing angle is not of interest. The difference θdiff = θh − θg

determines the shaft torque linking the two masses. Dropping θsum, one obtains the

following model for the mechanical system:

dωh

dt=

1

Jh

(Taero(ωh, vw)

N−Ksθdiff

)

dθdiff

dt= ωh − ωg

dωg

dt=

1

Jg

(Ksθdiff − Tgen)

(2.12)

2.5 Generator and Back to Back Converter

The inclusion of a back to back converter in the system imparts it with considerable

control freedom. Each converter can be assumed to allow control of the magnitude and

phase angle of a three-phase voltage, provided that the capacitor maintaining the dc-

link between them is maintained at a sufficiently high and constant dc voltage . The

validity of such a fundamental frequency model hinges on how the bandwidth demanded

of such controlled voltages compares to the switching frequency of the converter. Provided

that subsequent control designs have a sufficiently small bandwidth, the assumption of

controlled three phase voltages can be used to simplify the system model.

The largest simplification comes in the use of the converter interfaced to the generator.

Through the application of field-oriented techniques [34],[35], the flux of the machine can

be regulated to the rated value, and the torque controlled to a set-point with a bandwidth

sufficient for most mechanical transients [35]. In this work, it is assumed that if such


controls are applied, the exact nature of the machine being controlled can be neglected,

and it can be represented only as a set of operating constraints for a commanded torque

Tgen. This controlled torque becomes a new degree of control freedom.

The other converter clearly must produce a three-phase voltage at the frequency of

the grid. Field-oriented techniques can be also applied in this case to simplify the control

of the grid; this is discussed in detail in Section 2.6. The model of the generator and

converter system is thus reduced to the dynamics of the dc-link capacitor Cdc. On the

dc-link, the ac dynamics of the two converters can be represented by two currents idc1

and idc2 obtained by a power-balance across each converter. Such a model is depicted in

Fig 2.10 The differential equation governing the capacitor voltage is as follows:

Figure 2.10: Fundamental frequency model of the dc-link of the back to back converter.By a power balance, the ac dynamics of each converter are represented as a dc currentsource.

Cdcdvdc

dt= idc1 − idc2

Neglecting generator and converter losses, the real power absorbed from the generator

by its associated converter is simply the product of the controlled torque Tgen and the


speed of the generator ωg:

idc1 =Pmech

vdc

=Tgenωg

vdc

while the real power delivered to the grid depends on a product of the states of the

network and the controlled 3-phase voltages of the converter. For now, it can simply be

named Pgrid.

idc2 =Pgrid

vdc

The model of the dc-link dynamics that interfaces between the mechanical model defined

in the previous section and the ac-electrical model developed in the next section becomes:

dvdc

dt=Tgenωg − Pgrid

Cdcvdc(2.13)

2.6 Converter Grid Interface and Power System

The connection of the converter to the grid occurs through an interface inductor and

possibly a transformer. This connection can be modeled using an inductance LV SC and a

resistance RV SC that corresponds to conduction losses. The power systems network local

to a wind farm or single wind turbine is commonly the low voltage distribution system,

and can be relatively ‘weak’ (i.e. low XR

ratio). The grid parameters influence the voltage

at the point of common coupling where the converter interfaces with the power system,

as do various disturbances.

It is not the objective of this work to develop optimized controls for the grid interface,

or to extensively explore their performance and limitations. Therefore, in this work the

network will be simply modeled as an ideal voltage source. In this case, the model is the

simple circuit depicted in Fig 2.11.


Figure 2.11: Single-phase equivalent circuit of three-phase converter interface and net-work

vtabcand vsysabc

are the sinusoidal three-phase converter and system voltages, respec-

tively, and iabc is a vector of the three-phase currents of the system. KVL equations for

the three-phase system of Fig 2.11 can be arranged to form the differential equations

Lvscdiabc

dt= −Rvsciabc + vtabc

− vsysabc(2.14)

which describes the dynamics of the current iabc. Such a description is inconvenient

because of its time-varying nature and the definitions of quantities like power. Through

the standard time-varying transformation described in [34], (2.14) can be transformed to

a frame of reference synchronized to one of the voltages of the system, at the frequency

of the grid ωsys. If the voltage at the point of common coupling is relatively stiff (as

is assumed in Fig 2.11), then it can be assumed that the frequency ωsys is constant, or

slowly varying. It can also be assumed that the dynamics of the synchronizing phase-

locked loop may be neglected. Given these assumptions, a change to direct, quadrature

and zero-sequence state variables from the regular abc phase variables is made, resulting


in a time-invariant model of the grid interface:

Lvsc +

didq0

dt+

0 −ωsys 0

ωsys 0 0

0 0 0

idq0

= − (Rvsc +Rsys) idq0 + vtdq0− vsysdq0

(2.15)

For a balanced 3-phase system, only two independent equations exist; zero-sequence state

variables are zero. The direct and quadrature components are retained.

Lvscdiddt

= −Rvscid + Lvscωsysiq + vtd − vsysd

Lvscdiqdt

= −Rvsciq − Lvscωsysid + vtq

(2.16)

where the d-q axis has been aligned with the system voltage and subsequently vsysq= 0.

With the use of the time-varying transformation, simple, decoupled expressions for the

real and reactive powers flowing into the point of common coupling are obtained:

Ppcc =3

2vpccd

id

Qpcc =3

2vpccd

iq

(2.17)

The real power Pgrid flowing out of the converter in terms of these dq-frame variables is

Pgrid =3

2

(vtdid + vtq iq

)(2.18)

2.7 Summary of Assumptions

The main modeling assumptions made can be summarized as follows:

1. Variations in the wind speed encountered by the blades due to wind shear, tower

shadow and rotational sampling may be neglected due to their high frequency relative to

the natural frequency of the turbine hub.


2. The generator torque can be controlled by the converter with a bandwidth suf-

ficiently high that it can be regarded as a system input. It is also assumed that the

reactive power required by the generator is either adequately managed by the converter

controls, or that it does not impact the achievement of the commanded torque.

3. Losses in the converter and generator may be neglected.

4. The converter voltages can be controlled with a bandwidth high enough for all

control purposes, and the effects of switching harmonics on both the generator and the

electrical grid may be neglected for the purposes of this work.

5. The torsional resonance between the turbine and generator inertias is a dominant

oscillatory mode that can be adequately modeled by a lumped two mass linear system.

6. For analysis, the wind speed vw can be approximated as a finite sum of sinusoids

having a von Karman distribution, with random phases, varying about a mean value.

The model has been chosen to allow a focus on the special characteristics of wind

energy conversion systems, rather than the details of power electronic and power system

components (which, while non-trivial, are already widely studied). Such a model is also

more general by its lack of assumption about the type of machine used. It has been

deemed appropriate for more carefully exploring possible control philosophies that can

be applied to variable speed wind turbine conversion systems. However, it consequently

does not give a guarantee of performance for a specific system.

Chapter 3

Control Methodology, Design and

Performance


In Chapter 1, the increasing importance of minimizing the transmission of power distur-

bances to the grid by wind energy conversion systems was discussed. In this chapter,

a control methodology is introduced that gives priority to this goal. It is based on the

modeling elements of the back-to-back converter-based wind energy conversion system

that were introduced in Chapter 2. First, the structure and function of the system is

studied in terms of its components, and control objectives are defined. Then, the nature

of both a typical control methodology and the proposed methodology are discussed using

block diagrams. Finally, control designs associated with the proposed methodology are

presented.

26

Chapter 3. Control Methodology, Design and Performance 27

3.2 System Structure and Function

Putting together the modeling elements of Chapter 2, we have the state space model

(3.1). The model can be viewed as a number of interconnected subsystems. Two ways of

subdividing the system are denoted by the braces at the left and right sides of (3.1). The

division on the left underlies the standard approach to controlling the system. That on

the right is used to inform a new control approach. Both yield insight into the structure

of the system.

Smech

Sdc−link

dωh

dt= 1

Jh

(Taero(ωh/N,vw(t))

N−Ksθdiff

)

dθdiff

dt= ωh − ωg

dωg

dt= 1

Jg(Ksθdiff − Tgen)

dvdc

dt=

Tgenωg−1.5(vtdid+vtq iq)

Cdcvdc

Sslow

Sfast

Sint

Sgrid

diddt

= −Rvsc

Lvscid + ωsysiq +

vtd−vsysd

Lvsc

diqdt

= −Rvsc

Lvsciq − ωsysid +

vtq

Lvsc

Sgrid

(3.1)

The usual way of viewing the model (3.1) is as an interconnection of a mechanical subsys-

tem (Smech) with a 3-phase electrical subsystem (Sgrid) via the DC-link (Sdc−link). Such

a division is simply based on the engineered components that make up the conversion

system. Taken individually, in the absence of external and control inputs, the subsystems

Smech and Sgrid possess characteristic frequencies, which are summarized in Table 3.1. A

block diagram of this view is provided in Fig 3.1.

Table 3.1: Summary: Characteristic Frequencies of Engineered SubsystemsSubsystem Smech Sdc−link Sgrid

Characteristic Frequencies 7 Hz torsional 60 Hz dq frame coupling


Figure 3.1: Subsystems based on engineered components.

Another way to view the system can be based on the its storage elements. In Table

3.2, the energy storage capacity of each element associated with the conversion system is

listed. The open-loop dynamics of the coupled controlled system (3.1) could be quantified

by the eigenvalues of its linearization, but the nature of the control has not yet been

determined. Some insight about dynamics can instead be gained by considering the

storage capacities of the system, based on typical operating points. A per-unit inertia

H can be calculated for every element using the formula common to power systems

engineering, where the storage capacity is divided by the rated power of the system.

H =storage capacity (J)

rated power (J/s)(3.2)

Such a quantity characterizes the response time of an element to energy fluctuations, and

thus reveals the potential dynamics inherent to the system. Three distinct time scales

are apparent, which suggests another way of subdividing (3.1).

The interface inductance (which is contained within the subsystem Sgrid) at the right

side of Table 3.2 has its own timescale and is unavoidably coupled to the grid. It is

reasonable for it to remain as Sgrid, as in the usual way (Fig 3.1) of viewing the system.

The remaining elements, however, can be defined as ‘internal elements’, in the sense that

their dynamics should ideally be decoupled from the grid. In general, they can be grouped

together under this definition as a subsystem Sint, as suggested by the bracket on the


Table 3.2: Summary: Energy Storage Elements of Wind Energy Conversion SystemStorage Turbine Shaft Generator DC Link Interface

Element Hub Strain Rotor Capacitance InductanceState ωh θdiff ωg vdc id, iq

Stored Energy 12Jhωh

2 12Ksθdiff

2 12Jgωg

2 12Cdcvdc

2 12Lvscid

2

Storage Capacity 0.6 MJ 1.1 kJ 24 kJ 1.8 kJ 10 JH 3 s 6 ms 137 ms 10 ms 50 µs

Time Scale Slow Intermediate Fast

right hand side of (3.1). However, the relatively large storage capacity of the turbine hub

suggests that these internal elements should be further divided into two subsystems Sslow

and Sfast. Such a subdivided system is depicted in the block diagram of Fig 3.2, where

Sfast is actually composed of the elements on the ‘intermediate’ time scale of Table 3.2.

Figure 3.2: Subsystems based on energy storage capacity.

3.3 Control Objectives

However one chooses to subdivide it, the system’s primary function is to convey a time-

varying mechanical power to the grid in the form of three phase voltages and currents.

To operate safely while carrying out its function, the system must be designed such that

variables remain within their operating limits. Beyond this, performance objectives can

be defined that improve the quality of the grid interface, optimize energy extraction, or

maximize the containment of energy fluctuations by the conversion system. This results


in a number of control objectives.

Safety Objectives:

S1. The capacitor voltage vdc must be regulated within a certain range of its nominal

value.

S2. Torsional oscillations in Smech due to its characteristic frequency must be damped.

S3. The turbine hub speed ωh must be bounded above zero, and below its maximum

safe speed.

Performance Objectives:

P1. The reactive power injected at the point of common coupling must be regulated.

P2. The safety objectives S1 and S2 must be achieved by employing, if possible, the

energy stored in the turbine hub, rather than energy from the grid.

P3. The turbine hub speed ωh must be controlled to track changes in the wind speed

vw(t) in order to maintain a desired conversion efficiency; e.g. near-optimal efficiency.

P4. The turbine hub speed ωh must be managed such the energy storage of the

turbine hub acts as a filter, absorbing wind energy fluctuations above a certain cut-off

frequency.

The safety objectives can be met by a number of existing control structures. However,

the proposed performance objectives are better facilitated by a control structure inspired

by Fig 3.2. In the next section, a methodology based on such a structure is presented

and contrasted with a methodology that is common in the literature.

3.4 Control Methodology

In a typical approach to control the system (3.1), a turbine controller ensures the extrac-

tion of the optimal wind power, and a grid controller ensures the delivery of that power.

The specifics of the approach are illustrated by Fig 3.3, and described by the following

steps.


Figure 3.3: Block diagram of typical control structure. Grey areas denote parts of thesystem addressed by a particular controller.

1. Tgen is chosen to regulate ωh to a certain speed specified by the optimal tip-speed

ratio of the turbine blades, to achieve the objective P3. This is accomplished either by

speed feedback, or torque feedback. Torsional oscillations are damped as a result of this

control. Thus, the objectives S2 and S3 are also achieved.

2. The reference i∗td for the current id is used to regulate the capacitor voltage vdc in

Sdc−link and achieve the objective S1 by compensating for the variations of Tgen and ωg.

3. The reference i∗tq for the current iq can be applied to regulate the reactive power,

addressing the objective P1.

This standard approach achieves all of the safety objectives outlined in the previous

section, as well as the performance objectives P1 and P3. However, it transfers the

influence of the ‘internal’ states of the conversion system to the grid. This is because

each step of the method is designed to compensate for the fluctuations introduced by the

previous step.

A control paradigm of minimizing of grid disturbance requires an internal containment

of fluctuations of this kind by the conversion system. Therefore, a control methodology

is proposed based on a different control structure that is depicted in Fig 3.4. In this

approach, an electrical power Pgrid is requested by an external controller, and an internal


controller extracts it. This method is illustrated by Fig 3.4 and described by the following

steps, which begin at the grid interface rather the turbine.

Figure 3.4: Block diagram of proposed control structure. Grey areas denote parts of thesystem addressed by a particular controller.

1. The controller Cgrid generates references for the control inputs vtd and vtq to

regulate the currents id and iq to desired references. These references are selected to

deliver an electrical power Pgrid and regulate the reactive power injected at the point of

common coupling. This achieves the objective P1.

2. An internal controller Cinternal generates a reference for Tgen to regulate the fast

subsystem Sfast. Specifically, the capacitor voltage vdc is maintained at a nominal value,

and torsional oscillations are damped. This achieves the objectives S1 and S2.

3. An external controller generates a reference for Pgrid based on the wind speed

vw(t). The reference is designed with the stability limits of Sslow in mind. However, it

is not based on the turbine hub speed ωh, except in the event that the objective S3 is

endangered. In that case, a stabilizing reference is momentarily applied, guaranteeing


that S3 is met.

In such an approach, the influence of the ‘internal’ states on the grid is eliminated.

The states of Sfast are never directly coupled to Sgrid. An indirect connection through

Cexternal is avoided through careful design of the reference Pgrid∗. In the event that hub-

speed feedback is needed to satisfy the objective S3, filtering can mitigate the effect of

the coupling.

The reference Pgrid∗ can be chosen to deliver optimal power through hub-speed track-

ing and achieve the performance objective P3. Such an approach is widely dealt with

in the literature [17],[16]. This work will examine the objective P4 of employing the

turbine hub’s energy storage capacity to absorb wind power fluctuations. Such a objec-

tive is at partly at odds with the objective P3, but the proposed control structure and

methodology allows a trade off between the two to be selected.

In the following sections, control designs are outlined for each step of the methodology.

First, the design of the controller Cgrid is discussed. Then the decomposition of Sint into

the fast and slow systems Sfast and Sslow is formalized. This allows the independent

design of Cinternal and Cexternal.

3.5 Sgrid: Design of Cgrid

It is desired to regulate the point of common coupling of the conversion system to the

grid in order to achieve two objectives. First, the specified active power Pgrid must be

delivered. Second, the reactive power injected at the point of common coupling must be

kept constant. Given a more detailed model of the grid, a controller could be developed

that addresses additional performance objectives. Here, the simplest possible controller

for the model developed in Chapter 2 is presented.


The model for the converter interface with the point of common coupling is as follows:

diddt

= −Rvsc

Lvscid + ωsysiq +

vtd − vpccd

Lvsc

diqdt

= −Rvsc

Lvsciq − ωsysid +

vtq

Lvsc

(3.3)

In order to regulate the grid interface to achieve the stated control objectives, it

is prudent to use a nested control structure in which an inner controller regulates the

currents id and iq. This helps protect the converter from over-currents. References for

the controlled currents can then be developed to achieve the control objectives.

A standard approach that yields decoupled control of id and iq is outlined in [34] and

introduces a cancellation of the cross coupling ωsys terms using a feed-forward term. The

converter voltages vtd and vtq are chosen as

vtd∗ = Lvsc (xd + vpccd

− ωsysiq)

vtq∗ = Lvsc (xq + vpccd

+ ωsysid)

(3.4)

where xd and xq are proportional-plus-integral feedback signals of the current reference

errors:

xd = −KPied −KIi

∫

(id − id∗)dt

xq = −KPieq −KIi

∫

(iq − iq∗)dt

The resulting uncoupled first order systems

diddt

= −Rvsc

Lvscid − xd

diqdt

= −Rvsc

Lvsciq − xq

can be tuned to a level of performance that gives the fastest possible tracking of the

current references id∗ and iq

∗ that can still be reasonably achieved by a practical converter


system.

The real power delivered to the grid is proportional to the current id. The reference

for id is chosen to be

id∗ = Pgrid

∗/vpccd(3.5)

to achieve the first control objective, using the definitions (2.17). The reactive power

injected at the point of common coupling is proportional to the current iq. It is simplest

to assume that unity power factor is desired; thus, the reference for iq is chosen as

iq∗ = 0 (3.6)

It would also be possible to supply a specific reactive power required by the grid. The

range of possible reactive powers would be determined by the current rating of the con-

verter.

3.6 Sint: Separation of Fast and Slow Dynamics

The subsystem Sint of (3.1) may now be addressed, considering the electrical power Pgrid

as an input, along with the input Tgen:

dωh

dt= 1

Jh

(Taero(ωh/N,vw(t))

N−Ksθdiff

)

dθdiff

dt= ωh − ωg

dωg

dt= 1


dvdc

dt=

Tgenωg−Pgrid

Cdcvdc

Sslow

Sfast

(3.7)

It is desirable to decompose Sint into two separate subsystems Sfast and Sslow, each with

their own control input, so that the design of the controllers Cinternal and Cexternal may

proceed. An analytical approach to separating a system like Sint into independent fast


and slow subsystems has been outlined in [36], and is based on a singular perturbation

model of the system, which has the form

dx

dt= f(t, x, z, ǫ) (3.8)

ǫdz

dt= g(t, x, z, ǫ) (3.9)

where some of the state equations (3.9) have relatively fast dynamics because of the

presence of a small parameter ǫ. This can be appreciated by dividing the state equations

of this fast system by ǫ; it becomes evident that a small ǫ corresponds to a large time

derivative, and therefore fast dynamics.

The approach to separating the two subsystems of a singular perturbation model

is based on the idea that their dynamics occur on different time scales. It is assumed

that the states of the fastest subsystem settle to steady-state values that may depend

on the states of the slow subsystem. This allows a simplified representation of how

the subsystems interact. Within the slow subsystem, the influence of the fast states is

replaced by their steady state values. Within the fast subsystem, the influence of the

slow states is assumed to be constant. In the following development, this concept will

be formalized and then applied to decompose Sint into fast and slow parts. The stability

properties of such a decomposed system will also be examined.

First, the slow subsystem (3.8) is isolated. One begins by observing that in the case

that ǫ = 0, the state equations in the variable z are no longer dynamical states, but

equations for the steady states of the fast system, which are denoted z, and are given by

some function h(t, x):

0 = g(t, z)

z = h(t, x)

These steady states are used to represent the influence of the settled fast states on the


slow states. In (3.8), the slow state equations are a function of the fast states z. These

equations can be approximated by a reduced, ‘steady state’ system with state variable x

and solution x(t) that is obtained by assuming that z is equal to z, and substituting the

function h(t, x) where z occurs:

reduced system: → ˙x = f(t, x, h(t, x)) (3.10)

This reduced system no longer depends on z. To similarly isolate a fast subsystem

based on (3.9), the dependence of (3.9) on the slow system must be given a simplified

representation. For this purpose, a constant ξ0 is introduced that is equal to the state of

the slow system at a time t0.

ξ0 = x(t0)

The isolation of the fast system (3.9) can then proceed, beginning with a change of

variable

y = z − z = z − h(t, x) (3.11)

such that the fast dynamics are centered on the steady states z. The dynamical equations

have the form:

ǫy = ǫz − ǫ∂h∂t

− ǫ∂h∂x

∂x∂t

= g(t, x, y + h(t, x)) − ǫ∂h∂t

− ǫ∂h∂xf(t, x, y + h(t, x))

(3.12)

We then define a new time variable τ as a new time scale for the fast system that is

based on the arbitrary time t0

τ =(t− t0)

ǫ(3.13)

In this time scale, one can appreciate that as ǫ→ 0, τ → ∞ for a finite time t > t0. This

idealizes the relatively rapid settling of the fast system to its steady state. Also, solving


for the original time variable

t = t0 + ǫτ

makes it clear that on the time scale defined by τ , the time t can be approximated by t0

in the limit as ǫ→ 0. In this limit,

t→ t0

x(t) → ξ0

from the perspective of the fast system.

After making a transformation to the new time variable τ , the fast or boundary-layer

system is obtained by setting ǫ = 0 and substituting t0 and ξ0 to eliminate the original

time variable t:

boundary-layer system: →dy

dτ= g(t0, ξ0, y + h(t0, ξ0)) (3.14)

Such a boundary-layer system is studied over the expected operating range of ξ0 in order

to account for the slow variations of the reduced system.

To apply the preceding formalism to the system Sint and derive a singular perturbation

model for (3.7), it is necessary to first identify the fast and slow states. The relatively

large value Jh of the turbine hub inertia was recognized in Section 3.2 for giving the state

ωh its own slow timescale. A time variable scaled by Jh will be temporarily introduced

to emphasize this fact:

thub = tJh

→ dxdt

= dxdthub

dthub

dt= 1

Jh

dxdthub

(3.15)

Substitution into (3.7) puts it into a form suggestive of a singular perturbation model,


where the parameter ǫ is identified as 1/Jh:

dωh

dthub=(

Taero(ωh/N,vw(t))N

−Ksθdiff

)

(1Jh

)dθdiff

dthub= ωh − ωg

(1Jh

)dωg

dthub= 1


(1Jh

)dvdc

dthub=

Tgenωg−Pgrid

Cdcvdc

x = f(t, x, z, ǫ)

ǫz = g(t, x, z, ǫ)

(3.16)

Before separating Sint into two subsystems, a notation for the variables of the fast and

slow subsystems and for the steady-state values must be defined. The reduced ‘steady-

state’ system will be denoted by a ωh. The steady-state values of the states in (3.16) are

denoted by a bar over their variable names:

z =[θdiff , ωg, vdc

].

The transformation (3.11) can then be applied to the fast states [θdiff , ωg, vdc] to obtain

the states [y1, y2, y3] centered on the steady-state values z.

The control input Tgen will be defined as having a slow, steady-state component Tgen,

and a fast component Tfast that is zero at steady-state (Tfast will be specified in later

sections):

Tgen = Tgen + Tfast

Separation can then be carried out determining the steady-state values of the boundary-

layer system

z = h(t, x)

By solving the last three lines of (3.16) for the case where the left hand side is zero, and


Tgen = Tgen, one obtains:

ωh = ωg

Tgen = Ksθdiff

ωgTgen = Pgrid

(3.17)

These solutions dictate the steady-state values z for the fast states θdiff , ωg and the

steady-state control input component Tgen:

θdiff = 1KS

Pgrid

ωh

ωg = ωh

Tgen =Pgrid

ωh

(3.18)

The desired steady-state value for vdc is not actually dependent on the state ωh; it is

assignable. In practice it would be the nominal dc-link voltage dictated by the converter

specifications. Here, it is simply left as vdc.

In order to generate two separate systems for analysis that are not coupled by state

variables, the following substitutions are made. First, the state θdiff in the first equation

of 3.16 is replaced by its steady state value defined in 3.18. Then, the first equation of

3.16 is returned to the original timescale t from thub. This forms the reduced system:

reduced system: dωh

dt= 1

Jh

(

Taero(ωh/N, vw(t))N −

Pgrid

ωh

)

(3.19)

which has Pgrid as a control input, and describes the hub dynamics under the assumption

that the states [θdiff , ωg, vdc] identified as fast have settled to their steady states. The

use of the state variable name ωh reflects this underlying assumption.

The remaining equations of 3.16 are transformed to the variables [y1, y2, y3], using the

steady-state values in 3.18 and vdc. ωh is replaced by ξ0, its value at a given time t0. This


yields the boundary-layer system:

boundary layer system: dy1

dτ= −y2

dy2

dτ= 1

Jg(Ksy1 − Tfast)

dy3

dτ=

ξ0 + y2

Cdc(vdc + y3)Tfast

(3.20)

The variables y evolve on the singular perturbation time scale τ . At this time scale, it

is assumed that the state ωh changes so slowly that it can be designated as the constant

ξ0.

The decomposition of Sint into these fast (3.20) and slow (3.19) subsystems simplifies

control design, as well as the consideration of overall system stability. It has been proven

[36] that if the dynamics of the boundary system in a singularly perturbed model are

sufficiently fast (i.e. ǫ is sufficiently small), the exponential stability of the origin of the

reduced (3.10) and boundary (3.14) systems on their own, is sufficient to guarantee the

stability of the origin of the interconnected system.

In the next two sections, the designs of the controllers Cinternal and Cexternal are based

on the models (3.20) and (3.19). Thus, the state variables under consideration become

ωh and

y1

y2

y3

In Section 3.7 a controller is designed that satisfies the first two safety control objec-

tives S1 and S2, and makes (3.20) locally exponentially stable. This reduces the issue of

overall system stability to the stability of (3.19), which is studied in Section 3.8. There,

it is shown that (3.19) has a natural stable region, and can be made exponentially stable

through appropriate control when necessary to guarantee stability.


3.7 Sfast: Design of Cinternal

The control objectives assigned to the internal controller are the regulation of the capac-

itor voltage vdc to a constant nominal value, and the damping of torsional oscillations.

The change of variables applied to Sfast in the previous section make these objectives

equivalent to the problem of regulating the states y1, y2 and y3 to zero, independent of

the state ξ0 of the reduced system.

dy1

dτ= −y2

dy2

dτ= 1

Jg(Ksy1 − Tfast)

dy3

dτ=

ξ0 + y2

Cdc(vdc + y3)Tfast

(3.21)

Practical methods exist to measure all of the required quantities, and thus determine the

states above using the steady state values[θdiff , ωg, vdc

]defined in the previous section

as (3.18). However, determining the actual deviation

y1 = θdiff − θdiff

= θdiff −1

KS

Pgrid

ωh

(3.22)

could be inaccurate due to the approximate knowledge of Ks, especially in the case of a

drive train with a gearbox. Therefore, a controller will be developed based on knowledge

of the deviation y2 (which could be obtained using optical encoders at either end of the

drive-train to determine the angular velocities ωh and ωg) and the dc voltage deviation

y3.

y2 = ωg − ωh

y3 = vdc − vdc

(3.23)


A partial state feedback control law multiplied by a nonlinear scaling

Tfast =vdc + y3

ξ0(−K1y3 −K2y2) (3.24)

where K1 and K2 are positive gains, can be substituted in (3.21). After some algebraic

manipulation, one obtains:

y1

dτ

y2

dτ

y3

dτ

=

0 −1 0

Ks

Jg

1Jg

vdc

ξ0K2

1Jg

vdc

ξ0K1

0 − 1CdcK2 − 1

CdcK1

y1

y2

y3

+

0

+ 1Jgξ0

K2y3y2 + 1Jgξ0

K1y23

− 1Cdcξ0

K2y22 − 1

Cdcξ0K1y3y2

(3.25)

which has the form of a linear system x = Ax with a perturbation g(x)

x = Ax+ g(x) (3.26)

that vanishes at the origin of the transformed system, which in this case corresponds to

[θdiff , ωg, vdc] =[θdiff , ωg, vdc

]. This perturbation has a higher order, so it does not affect

the linearization, and is overpowered by the linearization terms for small y. Hence, if

the linear part of the system is stable, then the origin of the boundary system is locally

exponentially stable.

It was found that while any pair of positive gains K1, K2 result in a stable system, the

eigenvalues of the system were poorly damped for many gains. The location of eigenvalues

was not very sensitive to the changes in the operating point ξ0. Therefore, a value for ξ0


in the middle of the operating range was chosen, and an optimization procedure was used

to find gains K1 and K2 that placed eigenvalues as far to the left as possible. Viewing

K1 and K2 as parameters of the eigenvalues of A, this can be specifically stated

[K1, K2] = argmin max ℜ (λ (A))

i.e Find K1 and K2 that minimize the real part of the largest eigenvalue of A.

3.8 Sslow: Design of Cexternal

It is desired to design a controller Cexternal to generate a reference for Pgrid that achieves

the safety and performance objectives set for the turbine hub. The decomposition per-

formed in Section 3.6 resulted in the single differential equation (3.27) that describes the

slow dynamics of the turbine hub. The use of the state variable ωh indicates an under-

lying assumption that the fast states of the system settle and convey the steady state

torque Pgrid/ωh:

dωh

dt=

1

Jh

(Taero(ωh/N, vw(t))

N−Pgrid

ωh

)

(3.27)

The reference Pgrid∗ will be chosen to deliver a filtered version of the available wind

power. This means that power fluctuations occurring at a frequency higher than the cut-

off of this filtered power will be absorbed by the turbine hub, and consequently become

speed fluctuations.

When a power Pgrid is imposed on the turbine hub, the hub’s dynamics can become

unstable. This constrains what power references can be achieved, which has implications

for both the objectives S3 and P4. In subsection 3.8.1, the nature of the equilibria of

(3.27) and their attractive regions is studied in general, and the potential for instability

is shown. Two specific cases where Pgrid∗ is a filtered version of the wind power are


examined in subsection 3.8.2. From this study, guidelines are provided for anticipating

and averting the instability.

In subsection 3.8.3, a design is outlined for a supervisory control structure that allows

a general reference for Pgrid to be used, but intervenes if the instability is encountered to

ensure that safety objective S3 is met.

Subsection 3.8.4 outlines the design of the filter used to determine a particular Pgrid

reference that achieves P4. The design attempts to maximize the absorption of higher

frequency wind fluctuations by the turbine hub without causing instability. This design

can reduce coupling of hub-speed dynamics to the grid, to the extent that the design

reduces the occasions where the supervisory control is required to intervene to guarantee

the safety objective S3.

The entire section is summarized and discussed in subsection 3.8.5.

3.8.1 Stability of the Turbine Hub

A study of the stability of (3.27) must begin with an analysis of its equilibria by con-

sidering the condition dωh/dt = 0. Aerodynamic quantities defined earlier (2.9) are

substituted:

0 =Taero(ωh/N, vw(t))

N −Pgridωh

Pgrid =Taero(ωh/N, vw(t))

N ωh

= Paero(vw(t), ωh) ((2.9), line 1)

(3.28)

Thus, equilibrium occurs when the three-phase electrical power Pgrid is equal to the

power Paero(vw(t), ωh) extracted by the turbine. Because both Paero(vw(t), ωh) and Pgrid

can vary in time, the location of the equilibria are time-varying. Fig 3.5 shows that there

can be up to 2 equilibria, or none.

The location of equilibria depends on both the wind speed vw(t) and Pgrid, as shown

in Fig 3.6. Given that equilibria exist, the sign and size of their associated eigenvalues


(a) Pgrid held constant (b) vw held constant

Figure 3.5: Equilibria of (3.27). Equilibria are determined by the intersection of thetwo curves Paero(ωh, vw(t)) and Pgrid, both of which may vary in time. Two, one, or noequilibria may exist. The possible cases are demonstrated in (a) for different wind speedsand in (b) for different grid powers. The equilibria are marked as dots.

can be determined from the linearization of (3.27), taking the equilibrium location as

the operating point.libria depends on both the wind speed vw(t) and Pgrid, as shown

in Fig 3.6. Given that equilibria exist, the sign and size of their associated eigenvalues

can be determined from the linearization of (3.27), taking the equilibrium location as

the operating point. Their stability and location influence the dynamics of (3.27). Its

linearization at a particular operating point ωhop,vw

op, neglecting wind changes, is

d∆ωh

dthub=

1

Jh

(

1

N

∂Taero

∂ωh

∣∣∣∣ωh

op,vwop

−Pgrid

ωh∗2

)

︸︷︷︸

, ∂f∂ωh

∆ωh (3.29)

The top curve of Fig 3.6 shows how the location of equilibria is determined by the

relationship between the maximum available wind power Pmax(t) and the power removed

and delivered to the grid, Pgrid. The bottom curve of Fig 3.6 shows the eigenvalue

∂f/∂ωh of the linearization at each equilibrium. Arrows indicate the effect of variations.

It is evident that when two equilibria exist, they occur in a stable/unstable pair, which

can be designated ωs(t) and ωu(t). The arrows show that as Pgrid exceeds Pmax(t), the

two equilibria approach and then annihilate one another in a saddle node bifurcation,


such that no equilibria exist. Because the relationship between Pmax(t) and Pgrid can in

general vary with time, however, this condition may be temporary.

The lower curve of Fig 3.6 also shows that the unstable equilibrium ωu(t) can have

a relatively large eigenvalue, which indicates the local dynamics can be fast. A sense of

global dynamics can be obtained by fixing the two equillibria and then plotting the value

of dωh/dt as a function of ωh to obtain the flow field for the state equation (3.27). This

is shown in Fig 3.7. Because the flow field can change in time with the equilibria, two

cases are shown.

The left hand case shows that for speeds slower than the unstable equilibrium ωu(t),

dωh/dt is negative and its magnitude increases sharply as ωh decreases. Thus, the possi-

bility of a fast instability exists in which the hub speed ωh quickly falls toward the speed

ωh = 0. This phenomenon will be referred to elsewhere in this work as hub-speed col-

lapse. For speeds greater than ωu(t), the equilibrium ωs(t) is attractive; i.e. the flow field

points toward ωs(t). Thus, ωu(t) defines the boundary between a time-varying region of

attraction, or R.O.A, for ωs(t), and a R.O.A for ωh = 0.

The right hand case shows a situation where no equillibria exist. While such a condi-

tion persists, the flow field is negative everywhere, pointing to the point at zero speed. It

is notable, however, that the flow field has a minimum magnitude in the vicinity of where

ωu(t) and ωs(t) met before disappearing; this is in fact the speed ωopt corresponding to

Pmax(t).

It is important to recognize that the disappearance of equillibria is certainly not a

sufficient condition for hub-speed collapse, as long as the condition is not prolonged. It

is conceivable that an alternation between the two situations depicted in Fig 3.7 could

result in an oscillation of the hub speed ωh that is bounded, and centered on the value

ωopt where maximum power is obtained. However, it is clear that the nonlinearities and

time-varying nature of the system make definitive statements on stability elusive.


Figure 3.6: Stable/unstable pair of equilibria ωs(t) and ωu(t) of turbine hub speed, andtheir dependence on the relation between Pgrid and the maximum of the power-speedcurve, Pmax(t). The wind speed vw(t) is held constant. The intersection of the twopower curves determines the number and location of the equilibria. The linearization∂f∂ωh

characterizes their stability and associated eigenvalue. Arrows indicate how these

aspects change as Pgrid approaches Pmax(t). For a given wind speed, Pmax(t) is obtainedat a speed ωopt(t). As Pgrid approaches Pmax, the two equilibria approach one another.For Pgrid = Pmax(t), a single equilibrium exists at ωopt(t), and for Pgrid > Pmax(t), noequillibria exist.


Figure 3.7: Flow fields of (3.27) for two cases. The time derivative dωh/dt is shown as afunction of ωh, and arrows indicate the direction and magnitude of the flow field. In (a),two equilibria exist, and there is a region of attraction (R.O.A) within which the flowfield points toward the stable equilibrium ωs(t). There is also an unstable region withinwhich the flow field points toward ωh = 0. In (b), the flow field is everywhere negative,leaving only the point ωh = 0 attractive.

3.8.2 Stability Case Study

Two main difficulties are apparent from the preceding discussion. First, it is not the

absolute value of the turbine hub speed, but its relation to the moving equillibria ωu(t)

that is important. Second, the location of ωu(t) depends on both vw(t) and Pgrid. In this

subsection, a consideration of stability will be simplified by

• making the power Pgrid a function of the wind speed vw(t), thus making the relation

between Pgrid and Pmax(t) more specific.

• basing the visualization and discussion of dynamics on the tip-speed λ(t), which

characterizes the operating point in a way that is impossible using ωh or vw(t)

alone.

In order to draw more useful conclusions about stability, a specific case of the power Pgrid

will be examined. First, some important concepts will be established.


The tip-speed ratio λ(t) was introduced in Chapter 2.

λ(t) =Rωh

vw(t)

The main nonlinearity of the system, the Cp curve, is a function of λ, not ωh or vw(t) alone.

Thus, λ will be used to determine the proximity of the operating point to the maxima

of the power and torque curves, which are depicted in Fig 3.8, and are a consequence of

the Cp curve’s shape.

The tip-speed ratio at which the maximum power Pmax(t) is extracted is λopt. When

a power is being extracted at a lower-than optimal efficiency, there are two tip speed

ratios at which this can occur. In Fig 3.8(a) they are labeled as λs and λu, because they

correspond to the stable and unstable equilibria.

The torque curves of a wind turbine also have a peak that occurs at some λstall, λstall <

λopt, as shown in Fig 3.8(b). Below λstall, the aerodynamic torque drops off sharply as

the speed ωh decreases. This peak in torque contributes to making hub-speed collapse

such a fast instability.

It is useful to introduce the vw − ωh plane, depicted in Fig 3.9 to gain a better

understanding of how the equilibria ωs(t) and ωu(t) move due to changes in both Pmax(t)

and the choice of Pgrid. The operating conditions vw(t) and ωh specify a point on the

plane, as shown in Fig 3.9. The locus of points on the vw − ωh plane corresponding to

a constant tip-speed ratio λ is a line passing through the origin with slope λ/R. Several

are marked in Fig 3.9 as examples.


(a)

(b)

Figure 3.8: Aerodynamic features and their associated tip-speed ratio λ. a) A powerless than Pmax(t) can be achieved at two λ, λs, and λu. b) Both the power extractedand the torque exerted on the turbine hub have maxima at λopt and λstall respectively.λs > λopt > λu > λstall.


Figure 3.9: The vw − ωh plane. An operating point and three lines of constant tip-speedratio: λs, λopt, and λu are shown. The tip-speed ratio λ(t) is related by the radius R tothe angle α of the point with the vw axis.


The vw −ωh plane facilitates the study of the special case where the wind input vw(t)

and the power Pgrid are as defined below:

vw(t) = vavg + Vwcos (2πfwindt) (3.30)

dvfilt

dt=

2π

τ(vw(t) − vfilt) (3.31)

Pgrid = Pwind(vfilt)Cp(λ∗), λ∗ < λopt (3.32)

Pwind =1

2ρπR2vfilt

3 (3.33)

The wind speed vw(t) is simplified as a single sinusoid with amplitude Vw, varying at

a frequency fwind around the average wind speed vavg . A filter with time constant τ is

applied to the wind-speed (3.31), and used to calculate a filtered version of the wind

power Pwind(vfilt). A certain fraction of this power, where the fraction is determined

by the efficiency Cp(λ∗) at some desired tip-speed ratio λ∗, is chosen as the power Pgrid

demanded from the turbine (3.33).

The desired tip-speed ratio λ∗ should be chosen such that λ∗ ≥ λopt. For some

λ∗ > λopt, two extreme values for the filter constant τ will be analyzed:

τ ≃ 0 → vfilt(t) = vw(t)

τ >> 0 → vfilt = vavg

The first example corresponds to a situation where almost no filtering occurs, and the

second case represents a situation where the wind variation at frequency fwind around

vavg must be entirely absorbed by the turbine hub. For the first case, the positions of

the equilibria can be directly calculated. At equilibrium, ωh = 0 implies that

Pgrid = Paero(vw(t), ωh)

Pwind(vw(t))Cp(λs) = Pwind(vw(t))Cp(λ(t))

Cp(λ∗) = Cp(λ(t))


for which two solutions exist, due to the fact that the Cp(λ) curve possesses a maximum

ωs(t) =λs

rvw(t)

ωu(t) =λu

rvw(t)

(3.34)

In Fig 3.10(a), the movements of the two equilibria along lines of constant λ are

shown.

For the case where τ is large, the equilibrium condition gives

Pgrid = Paero(vw(t), ωh)

const = Pwind(vw(t))Cp(λ(t))

Rather than moving along lines of constant λ, the equilibria move along a curve of

constant power, which is marked in Fig 3.10(b). For the wind variation shown, they can

disappear and reappear.

In Fig 3.11, a time series is shown to illustrate dynamics for the case where τ is large.

The position of both equilibria is shown. It can be seen how they influence ωh. During

the time that no equilibrium exists, ωh must decrease, as occurs in Fig 3.11 at t = 44s.

In the event that ωu(t) reappears below ωh, ωh will remain within the region of

attraction of ωs(t). If ωu(t) reappears but remains greater than ωh, the hub speed must

eventually collapse to zero. This is the outcome in Fig 3.11. At t = 95s, the unstable

equilibrium ωu(t) reappears, but ωh has dropped below it. The time derivative of ωh

increases sharply, and ωh falls quickly. Of special note is the coincidence of a peak in the

aerodynamic torque with this decrease. This peak is the one occurring at λstall. From

this example, two signs of impending hub-speed collapse can be observed:

λ(t) < λstall

dωh

dt< 0

(3.35)


(a) τ ≃ 0 (b) τ >> 0

Figure 3.10: Movement of equilibria for two extremes of filter time constant τ . Bracketsmark the range of equilibrium movement (y-axis) corresponding to the sinusoidal windspeed variation about the mean value vavg (x-axis). In 3.10(a), the range of equilibriummotions for a small τ is relatively narrow and predictable by 3.34. In 3.10(b), where alarge τ is used in the filter, the range of movement is wider. At some low wind speed,the equilibria meet and disappear in a saddle-node bifurcation (open circle), reappearinglater as this threshold speed is crossed during the wind’s increase in speed.


The first condition is not sufficient to guarantee collapse. An increase in wind can cause

λ(t) to drop below λstall, but also raise Pmax(t) well above Pgrid, ensuring the existence

of ωu(t) at some low speed. The situation of concern is when the stall region λ(t) < λstall

has been entered due to a slow decrease in speed, as shown in Fig 3.11.

Figure 3.11: Hub speed and torque time series demonstrating a hub speed collapse. Thestable and unstable equilibria ωs(t) and ωu(t) are shown as dashed and dotted lines,respectively. At t = 45 s, both equilibria disappear, and reappear at t = 95 s such thatωh is within the unstable region. The hub speed then decreases rapidly. In the torquetime series, the two terms of (3.27) are shown. The peak of the aerodynamic torque isapparent as collapse occurs. One also perceives a sharp increase in the controlled torquePgrid

ωh.

The preceding example has demonstrated how choosing a Pgrid based on a filtered

version of the wind power can result in hub-speed collapse, if the wind filter time constant

τ is made large enough. As discussed in subsection 3.8.1, this instability is associated


with either of the prolonged conditions

ωh < ωu(t)

∄ωu(t)

(3.36)

and proceeds especially quickly once

λ(t) < λstall. (3.37)

However, the comparison of equilibrium motion offered by Fig 3.10(a) and 3.10(b)

shows another important effect of increasing τ from zero. The motion of equilibria are

wider, encouraging larger hub speed variations. While it is not evident from Fig 3.10(a),

a τ of zero prevents the turbine hub from tracking changes in the wind speed, which

it must do to maintain its efficiency of conversion. To cause the turbine hub to track

an increase, for example, in wind, and subsequently produce more power, the extracted

power must lag the available power in such a way that the turbine hub is accelerated.

Thus, it is important to select a τ large enough that useful hub-speed variations for

tracking or filtering can occur, but not so large that instability becomes an issue. In the

next section, a supervisory system and controller are designed that ensure the identified

instability will not cause a violation of the safety objective S3 (i.e. operation with the hub

speed bounded above zero). Then, section 3.8.4 shows how the problem of determining

a range of τ for safe operation can be solved using model data and the wind turbulence

models introduced in Chapter 2.


3.8.3 Supervisory System and Recovery Controller

To remove the danger of a hub speed collapse, a supervisory system is incorporated in

the Cexternal controller. This system generates two references

Pperf∗ : a performance control reference

Psafe∗ : a safety control reference

(3.38)

A block diagram of Cexternal is provided in Fig 3.12, showing the reference P ∗

perf based

on a filtered version of the wind, and a recovery controller used to generate P ∗

safe. The

logic used to switch between the references Pperf∗ and Psafe

∗ is depicted in Fig 3.13.

During normal operation, the turbine hub speed ωh is permitted to take on any value

necessary to achieve the requested power. Thus, the nominal performance reference

Pperf∗ is not a function of ωh. However, if the supervisory system detects the onset of a

hub speed collapse, it becomes necessary to stabilize the speed. The reference for Pgrid is

switched by the supervisory system to Psafe∗, which is generated by a recovery controller

based on hub speed feedback. This controller regulates the speed of the turbine hub ωh

to a reference speed

ωh∗ =

λ∗

Rvfilt (3.39)

which is derived from the filtered wind speed and the desired operating tip-speed ratio

λ∗. It is the speed at which the requested power would be obtained, if the wind speed

vw(t) were in fact equal to the filtered value vfilt. During normal operation, the actual

hub speed tends to vary around ωh∗, because the wind contains significant variations at

above the cut-off frequency of the wind filter.

The hub speed feedback controller ensures that ωh approaches ωh∗. Because the actual

wind speed vw(t) varies around vfilt, the power required to reach the speed ωh∗ can be


different from the filtered power Pperf∗. However, at some point in time tresume

vfilt = vw(t)

Pperf∗ = Psafe

∗

(3.40)

and the supervisory system resumes the nominal performance control reference.

Figure 3.12: Block diagram of Cexternal controller.

An example of the system’s action during a recovery is shown in Fig 3.14. At a time

tonset, the system detects the onset of a hub speed collapse, and switches the reference

for Pgrid to Psafe∗. The speed ωh increases until it closely approaches ωh

∗. The power

Psafe∗ is at some level determined by the actual wind speed vw(t) at this point, but a

short time later, at tresume, Psafe∗ matches Pperf

∗, and the supervisory logic resumes the

nominal reference.

The detection of collapse onset is performed using measurements of the turbine hub

speed ωh and the wind speed vw(t) to evaluate the conditions listed in Fig 3.13. The


Figure 3.13: Supervisory logic within Cexternal

recovery controller is a simple proportional-integral feedback controller.

Psafe∗ = KP (ωh − ωh

∗) +KIη

η = ωh − ωh∗

(3.41)

At the instant tonset when a switch of control reference is made, it is important to ensure

the continuity of Pgrid∗, i.e.

Psafe∗(tonset

+) = Pgrid∗(tonset

−)

= Pperf∗(tonset

−)

(3.42)

This is achieved by initializing the integral state η appropriately using the values of

ωh, ωh∗, and Pperf

∗ at tonset. Substituting (3.42) in the first line of (3.41) and solving for

η, one obtains

η(tonset+) =

Pperf∗(tonset

− −KP (ωh(tonset−) − ωh

∗(tonset−))

KI(3.43)


Figure 3.14: Action of recovery control: Time series of Pgrid reference, and variablesassociated with the turbine hub speed. Before tonset, the hub speed ωh falls during aperiod where ∄ ωu(t). After activation of the recovery controller at tonset, the equilibriumωu(t) reappears at t = 2235 s and drops below the hub speed ωh(t), which then increasesand approaches the reference speed ωh

∗. A short time later,Psafe∗ matches Pperf

∗, andthe supervisory logic resumes the nominal reference.


3.8.4 Design of Filtering Pgrid Reference

The reference to extract a filtered power from the turbine hub was first introduced as a

stability case study in subsection 3.8.2:

dvfilt

dt=

2π

τ(vw(t) − vfilt)

Pperf∗ = Pwind(vfilt)Cp(λ

∗), λ∗ ≥ λopt

Designing the reference P ∗

perf above is a matter of choosing the filter time constant

τ and the desired tip-speed ratio λ∗. It was noted in section 3.8.2 that a minimum τ is

necessary for tracking wind variations. It was also established that the phenomenon of

hub speed collapse poses a stability limit on how large τ can be. When filtering of power

by the turbine hub is desired, τ should be made as large as possible, within this stability

limit. Thus, there is an operating envelope defined by a τmin and τmax for each tip-speed

ratio λ∗.

This design problem will be more precisely defined and solved through the introduc-

tion of a simplified model of the controlled system and a simplified model of the wind

input vw(t). The form of the model for the controlled system immediately yields a par-

ticular τ that facilitates the tracking of speed variations. In order to obtain an estimate

of the stability limit on large τ , the two models are combined with a geometric criteria

defined in the vw − ωh plane.

The simplified model of the controlled system is obtained by substituting the reference

Pperf∗ into the turbine hub speed state equation

dvfilt

dt=

2π

τ(vw(t) − vfilt)

dωh

dt=

1

Jh

(Taero(ωh/N, vw(t))

N−Cp(λ

∗) · Pwind(t)

ωh

)

and linearizing at an operating point ωhop, vw

op. Through a Laplace transform, this yields


a single-input, single-output linear system with a transfer function G(s) from the wind

speed to the turbine hub speed:

∆ωh = G(s)∆vw

=∂Taero

∂vw

Jh

(s+ ωzero)

(s+ ωpole) (s+ ωfilt)∆vw

(3.44)

where

ωzero =2π

τ

∂Taero

∂vw−

1

ωhop

∂Pwind

∂vw

∂Taero∂vw

ωpole =

∂Taero

∂ωh

−Pwind(vw

op)

(ωhop)2

Jh

ωfilt =2π

τ

(3.45)

and all partial derivatives are evaluated at ωhop, vw

op. The dc-gain of (3.44) is

λ∗

R(3.46)

A distinctive value for τ is that which causes the zero ωzero and the pole ωpole to be

equal, defined as τmatch. Such a choice compensates the natural time constant of the hub,

leaving the system (3.44) with a first-order response to wind changes (Fig 3.15, dashed

line). The relatively flat magnitude and small phase of the response indicates that the

hub-speed will tend to follow the variations of the wind up to the corner frequency at

ωfilt. This implies, by (3.46)

∆ωh∼= |G(0)|∆vw

∼=λ∗

R∆vw

(3.47)

This implies that the tip-speed ratio λ∗ is preserved, to the extent that the above relation


holds. Such a result is more clearly shown by expressing λ(t) in terms of the linearization

variations and operating point:

∆λ =r

vwop

∆ωh −rωh

op

(vwop)2 ∆vw

Setting the condition ∆λ = 0 recovers the condition shown in (3.47):

r

vwop

∆ωh =rωh

op

(vwop)2 ∆vw

∆ωh =ωh

op

vwop

∆vw

=λ∗

R∆vw

In the case where τ > τmatch, the simplified model of the controlled system develops a

resonant peak located between the two poles (Fig 3.15, solid line). This peak corresponds

to the large variations in hub speed that occur when the turbine hub is absorbing wind

power fluctuations. These speed variations are larger than those necessary to track a

desired tip-speed ratio, and so significant variations in λ(t) can also be expected.

The large speed variations due to the resonant peak are clearly of importance to the

stability of the hub. Their allowable size has much to do with their phase relationship

with the wind, which, from Fig 3.15, can evidently be a leading or lagging one around

the frequency of the resonant peak. Both the gain and phase of the system response are

significant for stability. In order to combine them, the variation of the tip-speed ratio

λ(t) will be studied. Visualizations will be made in the vw − ωh plane.

For simplicity, the analysis of the system (3.44) will be based on single representative

frequency. The frequency most relevant to the stability of the hub is the resonant peak

ωpeak, which occurs at the logarithmic mean of the two poles of (3.44):

ωpeak = 10log10(ωpole+ωfilt

2)


Figure 3.15: Bode plots of the transfer function G(jω) for two τ . λ = 8.7, Solid line:τ = 100, Dashed line: τmatch = 48.85


For the purposes of this design problem, the wind vw(t) must therefore be approxi-

mated as a single representative sinusoid with a certain frequency ωpeak and amplitude

Vw:

vrepr(t) = vavg + Vwcos (ωpeakt) (3.48)

The wind input vw(t) is composed of many frequencies. As discussed in Chapter 2,

it can be usefully decomposed into a slowly varying mean wind speed vw and turbulence

component that is represented by summation of sinusoids with a particular amplitude

distribution

vw(t) = vavg +

N∑

i=1

Aicos(2πfit+ ψi)

where the amplitudes Ai are determined from a spectral density function.

It is necessary to determine an amplitude Vw for the representative sinusoid (3.48).

Rather than being a representation of the wind’s energy at a particular frequency, or an

approximation of the entire wind spectrum, Vw must capture the effects of those wind

fluctuations in wind energy that are absorbed into the hub as speed variations. It will

be assumed that only the frequency components above the cut-off frequency of the wind

filter are likely to result in such fluctuations. Thus, a representation of the relevant wind

frequency components is simply the sum of all components having frequencies greater

than ωfilt, and the amplitude Vw is defined as:

Vw =N∑

i=k

Ai

where k is such that ωk ≥ ωfilt, as depicted in Fig 3.16.

The simplified models (3.44) and (3.48) are useful because they can be used to describe

the operating point’s trajectory in the vw−ωh plane as a parametric curve. An operating


Figure 3.16: Discrete harmonics used to artificially represent turbulence (line arrows),and the sinusoid at used to represent components of the wind that are absorbed in hubspeed fluctuations (solid arrow). Components of the wind at frequencies lower than thecut-off frequency ωfilt of the wind filter (shaded area) are not included in Vw.


point for the transfer function G(s) can be defined as

vwop = vavg

ωhop =

λ∗

rvavg

The response of the simplified model to the representative wind input can then be deter-

mined. The trajectory of the operating point is given by the two components

vw(t) = vwop + Vwcos (ωpeakt)

ωh(t) = ωhop + |G(jωpeak)| · Vwcos (ωpeakt+ ∠G(jωpeak))

(3.49)

Definitions concerning the curve and examples of the effect of the parameter τ are shown

in Fig 3.17. In particular, a minimum angle αmin, which corresponds to the lowest

operating λ, is defined graphically. From Fig 3.17(b) it can be seen that increasing τ

results in a curve that deviates more from the line of constant tip-speed ratio λ∗, such

that the angle αmin decreases. In the previous subsection on stability, it was established

that hub speed collapse occurred when the hub speed ωh was lower than the unstable

equilibrium ωu(t), whose is roughly given by λunstable. Thus, the following stability limit

condition on αmin is proposed:

αmin ≥ αunstable

Having defined a geometric way to estimate the stability limit on the design parameter

τ , it becomes possible to make a more precise formulation and solution of the filter

reference design problem:

Given the mean wind speed vw, and a desired tip-speed ratio of operation λ∗ ≥ λopt,

define an operating point ωhop, vw(t)op for the transfer function G(s) ( (3.44),(3.45)) as


(a) (b)

Figure 3.17: Operating point trajectories of linearized model. 3.17(a): The magnitudeand phase response of (3.44) to a representative wind sinusoid with amplitude Vw deter-mine a closed curve in the vw − ωh plane, which has a minimum angle αmin with respectto the vw axis. 3.17(b): As τ increases, the minimum angle αmin decreases.


follows:

vw(t)op = vw

ωhop =

λ∗

rvw

Define the representative wind sinusoid vrepr as in (3.48), and through (3.49) obtain a

curve in the vw −ωh plane, for which αmin and αunstable are defined as in Fig 3.17(a) and

Fig 3.9.

The filter reference design problem is to determine, for every desired tip-speed ratio

λ∗, the operating envelope for the filter time constant τ defined by

τmin < τ < τmax

where τmin is given by

τmatch = 2πJh

∂Taero

∂vw

(∂Taero

∂vw−

1

ωhop

∂Pwind

∂vw

)

∂Taero∂vw

(∂Taero

∂ωh−Pwind(vw

op)

(ωhop)2

) (3.50)

and τmax is that τ for which

αmin = min

(

atan

(ωh

op + |G(jωpeak)| · Vwcos (ωpeakt+ ∠G(jωpeak))

vwop + Vwcos (ωpeakt)

))

t ∈ [0, 2π]

= αunstable

The problem associated with finding τmax can be solved using a number of approaches.

For each λ∗ over a range starting from λopt, a τ was calculated in order to construct an

estimate of the stability limit as a function of the operating tip-speed ratio λ∗. Simu-

lation trials were conducted over the same range of λ∗ for a range of τ . During these


trials, interventions by the supervisory control system to avert a hub speed collapse were

counted. It was found that for each λ∗, there was a threshold τ greater than which the

number of interventions increased dramatically.

In Fig 3.18, this threshold is plotted against λ∗ as an indicator of the actual stability

limit on the size of τ . The calculated stability limit τmax obtained from the simplified

models is plotted as a comparison, and appears to be a conservative but reasonable

estimate.

Figure 3.18: τ -λ operating envolope. Comparison of the stability limit on the filteringtime constant τ indicated by simulations and the estimate τmax obtained from simplifiedmodels. The apparent threshold implied by the simulation study is 30% larger. A basictradeoff exists between a wider operating envelope for τ , and the operating conversionefficiency, which decreases with λ. The bottom range of operation is demarcated byτmin = τmatch, as defined by (3.50).


3.8.5 Summary

In this section, a controller Cexternal was designed to generate a reference for the power

delivered to the grid. The structure of the controller is variable. For the majority of the

time, it generates a reference that is independent of the turbine hub speed ωh and based

on a filtered wind speed measurement.

dvfilt

dt=

2π

τ(vw(t) − vfilt)

Pgrid∗ = Pwind(vfilt)Cp(λ

∗)

However, the imposition of an arbitrary power on the turbine hub by such a reference

creates a potential instability. The nature of this instability was studied for the case

of the power reference above in Section 3.7.1. A conservative estimate of how the filter

time constant τ determines a stability limit for a given operating tip-speed ratio λ∗ was

developed in Section 3.8.4. This provides an operating envelope, depicted in Fig 3.18,

within which the reference can be designed to deliver the filtered power reference while

avoiding the instability.

Such an operating envelope is approximate, and heuristically derived. Wind mea-

surements are also likely to be error prone. Therefore, the controller Cexternal is given

the ability to detect the onset of the instability, and apply a speed-feedback controller

that ensures the stability of the turbine hub speed. In the event of an instability, the

controller switches to a reference Psafe∗ based on hub speed feedback.

The result of applying such a controller is that in nominal operation, the model of

the wind energy conversion system reduces to a simple linear filter of the available wind

power that is independent of conversion system states. During a recovery transient, a

dip in power and a coupling of turbine hub dynamics to the grid would occur.

Chapter 4

Results


In this chapter, the proposed control methodology and controller designs are evaluated.

Through simulation studies, the efficacy of the control system in achieving the control

objectives outlined in Section 3.2 is examined. First, the behaviour of the controlled

system is studied with respect to the safety objectives. Then, performance trade-offs and

limitations are explored.

4.2 Safety Objectives

The controller Cinternal was assigned the task of satisfying safety objectives S1 and S2:

the regulation of capacitor voltage and the damping of torsional vibrations. A sample of

the performance of the controlled system is shown in Figs 4.1 and 4.2, which show the

variations of the states of Sfast around their steady state values.

In Fig 4.1, two extreme cases for the power reference are considered; nominal opera-

tion using the filtering reference Pperf∗ with a large time constant, and a speed feedback

controller that regulates the hub speed to the optimal speed reference. The feedback

controller causes a power flow with a higher frequency content, and results in larger fluc-

73

Chapter 4. Results 74

tuations of the states controlled by Cinternal. Thus, it can be concluded that the controller

can achieve the safety objective for range of cases, but not with uniform performance.

Fig 4.2 shows the variations of Sfast over a short time period, and demonstrates

that variations in the capacitor voltage are slow compared to typical converter switching

frequencies.

Time series have already been presented in Section 3.8.2 of the behaviour of the

recovery controller that ensures that the turbine hub speed is bounded above zero. It is

also of interest, however, to examine the variations in hub speed that occur nominally.

Typical variations of the hub speed as a function of operating point are summarized

in Table 4.1; all operating points were simulated with τ = τmax in order to study the

limit on how large speed variations can become. Average speeds are higher for operation

at larger tip-speed ratios, and the speed range also increases as a result of the larger τ

that become possible. From the table, it seems likely that physical speed limits would

likely play a role in determining the range of possible operation.

Table 4.1: Summary: Speed variations in Filtering modeOperating λ∗ Average Min Max Range τmax

Point (rad/s) (rad/s) (rad/s) (s)

1 6.93 (opt) 91 55 150 1:2.7 603 8.12 105 56 160 1:2.9 965 9.82 122 55 187 1:3.4 2107 10.4 142 44 225 1:5.1 590

Figs 4.3 - 4.5 show typical variations in ωh and Pgrid during nominal operation. Figs

4.3 and 4.4 compare two extremes using the filtering Pperf reference: operation around

the optimal tip-speed ratio with a minimum filter time-constant, and operation at a high

tip-speed ratio with the maximum filter time-constant.

The tip-speed ratio chosen for comparison in Fig 4.4 results in especially smooth

power output, but involves speed variations that may be unrealistic. The variations

of speed Fig 4.3 are more moderate, but would not necessarily be allowable at higher


Figure 4.1: Performance of Cinternal. Plots of the states of Sfast demonstrate the degreeof regulation possible with the proposed control structure. The case of both the nominal,filtered Pgrid reference (black) and optimal power tracking using speed feedback (gray)are shown. The steady state value ωg varies between 100-300 rad/s, θdiff varies between0.05-0.3 radians, and vdc is 600V. It is evident from the plots that while the regulation isacceptable, its quality is determined by the nature of the power flow.


Figure 4.2: Performance of Cinternal, short time-scale. The steady state value ωg variesbetween 100-300 rad/s, θdiff varies between 0.05-0.3 radians, and vdc is 600V. The fluctu-ation of the capacitor voltage is relatively slow compared to typical converter switchingfrequencies.


wind speeds. In both cases, the filtered power (gray line) is less on average than the

power extracted through operation around the optimal tip-speed ratio. Thus Fig 4.3 and

4.4 both illustrate how either smooth power delivery or optimal power delivery can be

selected through the choice of λ∗ and τ . This trade-off is quantitatively demonstrated in

the Performance Objectives subsection.

Fig 4.5 demonstrates the range of operation at a given tip-speed ratio made possible

by changing τ from τmin to τmax.

Figure 4.3: Comparison of operation around the optimal tip-speed ratio of 6.93 (black)with a wind filter time constant of τmin = 32s, and operation around the tip-speed ratioλ = 9.82 using a wind filter time constant close to τmax at τmax = 200s.


Figure 4.4: Comparison of operation around the optimal tip-speed ratio of 6.93 (black)with a wind filter time constant of τmin = 32s, and operation around the tip-speed ratioλ = 10.4 using a wind filter time constant close to τmax at τmax = 600s. Considerablesmoothing is evident, but very large variations of hub speed and tip-speed ratio areoccurring.


Figure 4.5: Two extremes of operational envelope at λ = 8.12.

4.3 Performance Objectives

The performance objectives defined for the controller Cexternal in Chapter 3 were con-

cerned with the regulation of the grid interface and the nature of power variations. Only

the latter objectives P2 − P4 will be examined. First, the trade-off between achieving

the goal P3, optimal power tracking, and P4, filtering of wind fluctuation by the tur-

bine hub, is examined. Then, the spectra of power flows in various parts of the system

will be studied to evaluate the efficacy of the proposed control structure for containing

fluctuations. This will help to evaluate the performance goals P2 and P4.

Higher λ∗ values allow the use of longer wind filter time constants. This comes at

the price of poorer conversion efficiency due to a sub-optimal λ∗. In Fig 4.6, the amount

of energy by the conversion system over 100 minutes is plotted. As may be expected,

it decreases as λ∗ increases away from λopt. Also, a large wind filter time constant τ

further reduces the average efficiency of conversion. This is because speed changes cause


variations in the tip-speed ratio λ(t). cause the conversion efficiency to drop below that

achieved at λ∗ much of the time.

Figure 4.6: Energy captured over a 123

hour time period over a range of operating tip-speed ratios λ∗. It can be seen that operation at higher tip-speed ratios has a lowerefficiency of conversion, and that this is worsened when τ is τmax.

The benefit of operating at a more inefficient tip-speed ratio would be the increase

in wind filter time constant τ . As shown in the previous section during the discussion

of hub-speed and power time series, larger filter time constants result in smoother power

delivery. This is more clearly shown by examining the spectra of power variations.

Figure 4.7 schematically depicts how power flows through the wind energy conver-

sion system. A portion of the wind power Pwind(t) is extracted by the turbine blades,

depending on the power conversion efficiency Cp(λ(t)). Some of the converted power,

Pshaft(t), is removed through the shaft to Sfast. The rest flows into the turbine hub,

causing changes in speed. This power is marked as Phub(t). A portion of the shaft power

Pshaft(t) is removed from the conversion system as Pgrid(t). The remaining power flow

not transferred to the grid is called Pfast(t).

Phub(t) and Pfast(t) are the power fluctuations that are contained within the blocks


Figure 4.7: Power flow through the wind energy conversion system.

Sslow and Sfast. An examination of the spectra of the three power flows Pwind(t), Pshaft(t)

and Pgrid(t) should show that as power passes through the blocks Sslow and Sfast, its

content at higher frequencies is lowered. The nature of the fluctuations being removed

should be evident from the spectra of Phub(t) and Pfast(t).

In order to demonstrate these effects clearly, an artificial wind input is applied to the

system. The spectrum of such a signal is much cleaner than a measured wind time series,

as was first observed in Section 2.2, Fig 2.4. This makes an evaluation of the spectral

response of the system easier. However, it should be pointed out that the spectrum of

the wind power is less clean due to the cubic nonlinearity involved, as is evident from

Fig 4.8(a) and 4.8(b).

Two extreme cases are compared: the use of the filtering reference Pperf with τ = τmax

and a high λ∗, and the use of hub speed feedback to track the optimal speed ωopt(t). The

first case is shown for the artificial wind input in Fig 4.9. In general, the observations

that follow are also apparent in the spectra for a measured wind input, Fig 4.10.

From Fig 4.9(a) it is clear that Pgrid is a low-pass filtered version of the wind power

Pwind(t), as one would expect from the filtered reference Pperf . A comparison of Pshaft


(a) Spectrum of artificial wind series.

(b) Spectrum of artificial wind power.

Figure 4.8: Comparison of wind spectrum and wind power spectrum. Harmonics presentin the wind power are a result of its cubic dependence on windspeed.


with Pgrid shows that it contains slightly more energy at high frequencies.

Fig 4.9(b) shows how the spectra of Phub contains the peaks present in the wind power

above a certain cut-off frequency (approximately 5 mHz) where the power Pshaft does not.

Below this frequency, the opposite is true; the spectrum of Pshaft has the low-frequency

spectrum of the wind power, but Phub does not. This demonstrates the achievement of

the performance objective P4.

In 4.9(c), a similar effect is observed in Pfast. In general, the magnitude of the

power fluctuations in Sfast is an order of magnitude lower than those shown by Sslow in

Fig 4.9(b). This is reasonable, given the relative energy storage capacities of the two

subsystems. Above 0.1 Hz, Pfast begins to possess more high frequency content than

Pgrid, and a bump can be observed in the spectrum in the 1-2 s range. This is the

frequency range of the internal modes of the controlled system, which are excited by

wind power fluctuations but not transfered to Pgrid. Thus, the objective P2 is also being

met.

Fig 4.11 shows the case of optimal power tracking using speed feedback. The spec-

trum of Pgrid is close to that of the wind. In both Fig 4.11(c) and Fig 4.12(c), a small

bulge in the spectrum Pfast not transferred to Pgrid is evident. Thus, even when tight

speed regulation is employed using Cexternal, the control structure still results in some

containment of fluctuations.


(a) The three power flows Pwind, Pshaft, and Pgrid.

(b) Power variations contained within Sslow

(c) Power variations contained within Sfast

Figure 4.9: Artificial wind input: Power flow spectra for Pperf .





Figure 4.10: Measured wind input: Power flow spectra for filtering mode





Figure 4.11: Artificial wind input: Power flow spectra for optimal power tracking.





Figure 4.12: Measured wind input: Power flow spectra for optimal power tracking.

Chapter 5

Conclusions

5.1 Contributions

The objective of this thesis was to explore the concept of extracting a well-regulated

power from a wind energy conversion system. The two main contributions of the thesis

are as follows:

A) a control structure that eliminates the interaction of grid dynamics with fast

conversion system dynamics, and allows the specification of real and reactive power at

the grid interface.

B) a control design procedure that allows the use of the turbine hub as a power filter

by conservatively estimating its stability limit.

These contributions are complementary parts of a control strategy. The resulting

controlled conversion system can be modeled simply as a first-order filter of the available

wind power.

However, the contributions can also be evaluated separately. The first main contri-

bution is independent of the power requested from the system by the controller Cexternal.

The fast dynamics of the turbine are never directly introduced into the grid. This comes

with a trade-off discussed in the next section. The second main contribution is a general

88

Chapter 5. Conclusions 89

procedure that could be applied to other types of wind turbine systems, and may have

applications to other problems concerning energy storage in the hub.

5.2 Limitations

It is important to enumerate the inherent limitations of each thesis contribution. The

implications of these limitations can be partially discussed, but also extend into the

domain of future work.

LA1. It has been suggested that the use of energy from the grid to achieve the regula-

tion of states ‘internal’ to the conversion system should be avoided, and that the energy

of the turbine hub may be made employed instead. However, the available control is lim-

ited in its effects. As a result, the performance of capacitor regulation is unfortunately a

function of wind turbulence and conversion system parameters. In particular, the dc-link

capacitor must be sized large enough to ensure that fluctuations are within acceptable

limits. The bandwidth of Cinternal is also limited, due to the underlying limitations of

the converter and the machine controller.

LB1. Operation that employs the wind turbine as a filter involves larger speed vari-

ations and operation at sub-optimal efficiency (i.e λ∗ > λopt ), which implies a higher

average turbine hub speed. For some operating points, physical limitations on hub speed

and generator speed would define the operating envelope, rather than stability limits.

The only speed limits discussed in the literature assume that rated power is flowing at

rated speed, which would not the case in the technique proposed. Further research is

required to more carefully assess this limitation.

LB2. Assuming a measurement of the wind was a useful simplifying assumption

for a preliminary study of a new control method. However, windspeed measurement is

typically avoided in practical implementations. While windspeed and aerodynamic torque

can be estimated to some degree of accuracy with knowledge of the turbine characteristic


[19], [37], [38], approaches that employ torque and hub speed measurements are more

practical. While use of the windspeed has simplified analysis and design, it would be

necessary to evaluate and possibly reduce the dependence of the method on windspeed

measurement before attempting to implement it.

5.3 Future Work

Possible future work includes both immediate extensions and verification, as well as

studies involving more background research.

• An important extension of the work in this thesis would be a detailed investiga-

tion of the proposed Cexternal controller’s sensitivity to wind measurement, and the

mitigation of this sensitivity. Some form of self- adjustment in the control system

could also correct for a changing Cp curve, and other factors that cause the actual

power extracted by the turbine to be different from the estimate.

• Further evaluation of the practicality of the proposed method would require a closer

examination of the torsional oscillations during nominal operation and a comparison

with those attributed to other methods. The physical origin of the safety limits

on generator and hub speed should also be investigated, and used to evaluate the

nominal speed ranges found in simulation.

• An examination of the specific nature of power controls useful for ancillary services

should be studied for the case of a specific variable speed system in a known power

grid. The extent to which small power variations may be commanded from a wind

turbine should then be evaluated. A realistic assessment of the impact of wind

energy conversion systems on ancillary services would also consider the practical

operational limits (e.g. start-up, shut down).

• A grid interface controller to regulate the voltage at the point of common coupling


should be designed based on a more detailed model of the grid. Its performance

in regulating voltage for the case of the proposed turbine controller could then be

compared with that of a more conventional method for the same wind power input.

This would provide another evaluation of the method.

Appendix A

Technical Specifications

A.1 Wind Turbine Parameters:

Rated Power: 180kW

Rotor Diameter: 23.2m

Rotor Speed: 42rpm

Blade Profile: NACA− 63200

Gearbox Ratio: 23.75

A.2 Drive Train Parameters

(specified on high-speed side of shaft)

Turbine Inertia: 102.8 kg ·m2

Generator Inertia: 4.5 kg ·m2

Shaft Stiffness: 2700 Nm/rad

92

Appendix A. Technical Specifications 93

A.3 Generator Parameters

Nominal Voltage: 400V

Pole Pairs:3

A.4 Converter Parameters:

DC-link Capacitor: 10mF

DC-link Voltage: 600V

Interface Inductance: 1mH

Interface Resistance: 5mΩ

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wind energy conversion systems as power filters: a control

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